Nonlinear analysis and synthesis techniques for aircraft control

Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari 365 Declan Bates, Martin Hagström (Eds...

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Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari

365

Declan Bates, Martin Hagström (Eds.)

Nonlinear Analysis and Synthesis Techniques for Aircraft Control

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Editors Declan Bates Control and Instrumentation Research Group Department of Engineering University of Leicester U.K. Email: [email protected]

Martin Hagström Dept. of Autonomous Systems Swedish Defence Research Agency 164 90 Stockholm Sweden Email: [email protected]

Library of Congress Control Number: 2007931119 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-73718-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-73718-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and SPS using a Springer LATEX macro package Printed on acid-free paper

SPIN: 11946915

89/SPS

543210

Preface

Despite many signiﬁcant advances in the theory of nonlinear control in recent years, the majority of control laws implemented in the European aerospace industry are still designed and analysed using predominantly linear techniques applied to linearised models of the aircrafts’ dynamics. Given the continuous increase in the complexity of aircraft control laws, and the corresponding increase in the demands on their performance and reliability, industrial control law designers are highly motivated to explore the applicability of new and more powerful methods for design and analysis. The successful application of fully nonlinear control techniques to aircraft control problems oﬀers the prospect of improvements in several diﬀerent areas. Firstly, there is the possibility of improving design and analysis criteria to more fully reﬂect the nonlinear nature of the dynamics of the aircraft. Secondly, the time and eﬀort required on the part of designers to meet demanding speciﬁcations on aircraft performance and handling could be reduced. Thirdly, nonlinear analysis techniques could potentially reduce the time and resources required to clear ﬂight control laws, and help to bridge the gap between design, analysis and ﬁnal ﬂight clearance. The above considerations motivated the research presented in this book, which is the result of a three-year research eﬀort organised by the Group for Aeronautical Research and Technology in Europe (GARTEUR). In September 2004, GARTEUR Flight Mechanics Action Group 17 (FM-AG17) was established to conduct research on ”New Analysis and Synthesis Techniques for Aircraft Control”. The group comprised representatives from the European aerospace industry (EADS Military Aircraft, Airbus and Saab), research establishments (ONERA France, FOI Sweden, DLR Germany, NLR Netherlands) and universities (Bristol, DeMontfort, Liverpool and Leicester). FM-AG17 was initially chaired by Dr. Markus H¨ ogberg of FOI Sweden, and subsequently by Dr. Martin Hagstr¨om, also of FOI. The overall objective of the Action Group was to explore new nonlinear design and analysis methods that have the potential to reduce the time and cost involved with control law development for new aerospace vehicles, while simultaneously increasing the performance, reliability and safety of the resulting controller. This

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Preface

objective was to be achieved by investigating the full potential of nonlinear design and analysis methods on demanding benchmarks developed within the project, in order to focus the research eﬀort on the issues of most relevance to industry. Since nonlinear methods generally make more demands on the designer in terms of theoretical background and understanding, a secondary objective of the group was to present the results obtained in such a way as to clarify the beneﬁts, limitations and eﬀort required to implement the various techniques in an industrial context. Over the course of the Action Group, two workshops have been organised to present the results obtained within the project: the ﬁrst by FOI in Sweden (2006) and the second by ONERA in Toulouse (2007). Two industrial benchmarks were developed within the Action Group to provide challenging and industrially relevant applications for the various nonlinear control law design and analysis techniques to be investigated in the project. The ADMIRE (Aerodynamic Model in a Research Environment) benchmark provides a realistic platform for the evaluation of ﬂight control laws for highly manoeuvrable aircraft and includes a complete description of the closed-loop dynamics of a delta-canard aircraft over a wide ﬂight envelope. The Airbus On-Ground transport aircraft benchmark provides a highly detailed simulation model of the complex dynamics of a large transport aircraft during rolling on the runway. For each of these benchmarks, design and analysis challenges were formulated by the industrial members of the Action Group, comprising a detailed list of control problems and speciﬁcations which were to be addressed by the various nonlinear techniques explored in the project. Complete details of the two industrial benchmarks, together with their associated research challenges are contained in Part I of this book. Parts II and III of the book describe the application of advanced nonlinear control techniques to the Airbus and ADMIRE benchmarks, respectively. Finally, Part IV of the book contains an industrial evaluation of the results of the project, and provides some concluding remarks. This book is the result of a huge amount of eﬀort on the part of all of the participants in FM-AG17. The editors are extremely grateful to the academics who worked through gaps in research grant funding, to the members of national research laboratories who worked through increasingly stringent budgetary limitations and to the industrial participants who worked through their weekends to ensure the timely completion of this book. All of the participants in the Action Group would also like to express their thanks to the industrial and academic evaluators from outside the group who have contributed to this work through their constructive comments and reviews. May 2007

Declan G. Bates Martin Hagstr¨ om

Contents

Part I: Benchmarks and Design and Analysis Challenges 1 The AIRBUS On-Ground Transport Aircraft Benchmark Matthieu Jeanneau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 On-Ground Transport Aircraft Nonlinear Control Design and Analysis Challenges Matthieu Jeanneau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 The ADMIRE Benchmark Aircraft Model Martin Hagstr¨om . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4 Nonlinear Flight Control Design and Analysis Challenge Fredrik Karlsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Part II: Applications to the Airbus Benchmark 5 Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design Andres Marcos, Declan G. Bates, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . .

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6 Nonlinear LFT Modelling for On-Ground Transport Aircraft Jean-Marc Biannic, Andres Marcos, Declan G. Bates, and Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 On-Ground Aircraft Control Design Using an LPV Anti-windup Approach Clement Roos, Jean-Marc Biannic, Sophie Tarbouriech, and Christophe Prieur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8 Rapid Prototyping Using Inversion-Based Control and Object-Oriented Modelling Gertjan Looye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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Contents

9 Robustness Analysis Versus Mixed LTI/LTV Uncertainties for On-Ground Aircraft Clement Roos, Jean-Marc Biannic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Part III: Applications to the ADMIRE Benchmark 10 An LPV Control Law Design and Evaluation for the ADMIRE Model Maria E. Sidoryuk, Mikhail G. Goman, Stephen Kendrick, Daniel J. Walker, and Philip Perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 11 Block Backstepping for Nonlinear Flight Control Law Design John W.C. Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 12 Optimisation-Based Flight Control Law Clearance Prathyush P. Menon, Declan G. Bates, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . 259 13 Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods Mikhail G. Goman, Andrew V. Khramtsovsky, Evgeny N. Kolesnikov . . . . . . . . . . 301 Part IV: Industrial Evaluation and Concluding Remarks 14 Industrial Evaluation Matthieu Jeanneau, Fredrik Karlsson, Udo Korte . . . . . . . . . . . . . . . . . . . . . . . . . 327 15 Concluding Remarks Declan G. Bates, Martin Hagstr¨om . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Part I

Benchmarks and Design and Analysis Challenges

1 The AIRBUS On-Ground Transport Aircraft Benchmark Matthieu Jeanneau Airbus-France, Department of Stability and Control, Toulouse, France [email protected]

1.1 Introduction This chapter describes the behaviour of a transport aircraft and its systems during rolling. Notations and conventions are given ﬁrst, followed by the main equations of motion. Loads affecting aircraft motion are then described and their modelling given. Finally a short aircraft behaviour analysis is provided. This chapter aims at offering the reader a clear understanding of the control application and its requirements.

1.2 Notations and Conventions Before discussing the physics governing the dynamics of the aircraft on the ground, we will give the notations, the conventions and the main coordinate systems used in this chapter. D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 3–24, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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1.2.1

M. Jeanneau

Body-Fixed Coordinate System

This coordinate system, also called the ”aircraft coordinate system” , is a mobile coordinate system (c.g.; XAC , YAC , ZAC ), see Fig. 1.1. Its origin is the centre of gravity of the aircraft and its three longitudinal, lateral and vertical axes correspond respectively to the three longitudinal, lateral and vertical axes associated with the aircraft symmetry characteristics. The translations and rotations of this coordinate system are therefore directly linked to the motion of the aircraft.

Fig. 1.1. Aircraft coordinate system

1.2.2

Earth’s Coordinate System

This coordinate system (O; XE , YE , ZE ) is a Galilean coordinate system where the origin is a ﬁxed random reference point in space. In general, this point is taken as being equal to the initial position of the centre of gravity. It would also be possible to choose a reference point as an airport or the intersection of the Greenwich meridian and the equator. Such modiﬁcations only impact the initial value of the centre of gravity coordinates. The three axes (XE , YE , ZE ) of this coordinate system are oriented respectively towards the north, the east and downward. The transformation matrix to go from the aircraft’s to the earth’s coordinate system - see Fig. 1.2 and 1.3 - is: ⎛ ⎞⎛ ⎞⎛ ⎞ cos Ψ − sin Ψ 0 cos θ 0 sin θ 1 0 0 (1.1) MAC→E = ⎝ sin Ψ cos Ψ 0⎠ ⎝ 0 1 0 ⎠ ⎝0 cos ϕ − sin ϕ ⎠ 0 0 1 − sin θ 0 cos θ 0 sin ϕ cos ϕ

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Fig. 1.2. Aircraft θ and Ψ rotations

1.2.3

Aerodynamic Coordinate System

The aerodynamic coordinate system is a mobile coordinate system (c.g.; Xaero , Yaero , Zaero ) associated with the orientation of the aircraft velocity vector in relation to the air mass (Vair ) - see Fig. 1.4 Its origin is the centre of gravity. The longitudinal axis Xaero is oriented in the direction of this “air” velocity vector. In relation to the aircraft coordinate system, the coordinate system associated with the aerodynamic coordinate system is obtained by a rotation of angle α (angle of attack) around the YAC axis and of angle βaero (aerodynamic sideslip) around the ZAC -axis. The transformation matrix is: ⎞⎛ ⎞ ⎛ cos α 0 − sin α cos βaero sin βaero 0 0 ⎠ MAEROR→AC = ⎝ − sin βaero cos βaero 0⎠ ⎝ 0 1 0 0 1 sin α 0 cos α ⎛ ⎞ (1.2) cos α · cos βaero sin βaero 0 = ⎝ − sin βaero · cos α cosβaero sin α · sin βaero ⎠ sin α 0 cos α 1.2.4

Wheels Coordinate Systems

These coordinate systems comprise a set of mobile coordinate systems associated with each wheel. For a given wheel (indexed i ), the wheel incoordinate system (Oi ; Xwheeli , Ywheeli , Zwheel i ) is positioned at the contact point between the wheel and the ground

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M. Jeanneau

Fig. 1.3. Aircraft ϕ-rotation

(mean force application point). The plane (Xwheeli , Ywheeli ) is tangential to the surface of the ground at this point (the Zwheeli axis is orthogonal to it) and the Xwheeli axis belongs to the wheel plane. For a perfectly horizontal pavement, such a coordinate system can be deduced from the earth’s coordinate system by a rotation around the vertical axis so as to orientate it in the wheel plane. For the main landing gear wheels, this orientation angle can be taken as equivalent to the aircraft heading. Because the nose landing gear is movable, its angle of deﬂection must be taken into account for the nose wheels. Finally there are two “wheel” coordinate systems: one NW (Nose Wheel) coordinate system and one MLG (Main Landing Gear) coordinate system.

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Fig. 1.4. Schematic representation of aerodynamic coordinate system

Fig. 1.5. Angles linking aerodynamic and aircraft coordinate

1.2.5

Geometry

The main distances impacting the aircraft’s equations of motion are shown in Fig. 1.7.

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M. Jeanneau

Fig. 1.6. Wheel coordinate systems

1.3 Aircraft Dynamics The core of the model is based on differential equations derived from the fundamental principle of dynamics and the Euler angle formalism. 1.3.1

Equations

Applying Newton’s second law of motion gives the following equations: ∂V F=m +Ω∧V ∂t ∂(I · Ω) M= + Ω ∧ (U · Ω) ∂t With

(1.3) (1.4)

⎡

⎡ ⎤ ⎤ Vx p = roll rate Ω = ⎣ q = pitch rate⎦ and V = ⎣Vy ⎦ Vz r = yaw rate

the centre of gravity displacement velocity projected into the aircraft coordinate system. F represents the sum of the external forces applied to the system and M the sum of the moments. Variables m and I represent the mass and the inertia matrix of the aircraft.

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Fig. 1.7. Main geometrical elements used in the model

The projection onto aircraft axes gives: Fx − (qVz − rVy ) V˙x = m Fy V˙y = − (rVx − pVz ) m Fz V˙z = − (pVy − qVx ) m

(1.5)

Taking into account aircraft symmetry properties, from 1.4 we obtain: p˙ = iIxx · M p + iIxz · Mr + I p pq · pq + I pqr · qr q˙ = iIyy · Mq + Iq pr · pr + Iq pp · p2 + Iqrr · r2

(1.6)

r˙ = iIzz · Mr + iIxz · M p + Iq pq · pq + Iqqr · qr (5-6) give the 6 degrees of freedom representation. Variables are the longitudinal, lateral and vertical velocities and the roll, pitch and yaw rates. These equations allow

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M. Jeanneau

the behaviour of the aircraft to be described from the sum of the exerted forces and moments: longitudinal and lateral tyre/ground contacts, engine thrusts, aerodynamic loads, gravity, etc. 1.3.2

Aircraft Attitude and Position

The derivatives of the three Euler angles relevant to the attitude of the aircraft are given by equations: ˙ = q sin ϕ + r cosϕ Ψ cos θ θ˙ = q cos ϕ − r sin ϕ

(1.7)

˙ ϕ˙ = p + (q sin ϕ + r cosϕ) tan θ = p + Ψsin θ The position of the aircraft can be determined by integration of the projection onto the earth’s coordinate system of its velocity previously determined in the aircraft coordinate system. This change of coordinate system can be done on the basis of the matrix 1.1. This leads to: x˙ = {VX cos θ + (Vy sin ϕ + Vz cos ϕ) sin θ} cos Ψ + (−Vy cos ϕ + Vz sin ϕ) sin Ψ y˙ = {VX cos θ + (Vy sin ϕ + Vz cos ϕ) sin θ} sin Ψ + (−Vy cos ϕ + Vz sin ϕ) cos Ψ

(1.8)

z˙ = − VX sin θ + (Vy sin ϕ + Vz cos ϕ) cos θ The model corresponding to this representation is a 12th order nonlinear model, whose state variables are: • • • •

The centre of gravity velocity expressed in the aircraft axis. The three angular velocities of the aircraft in pitch, roll and yaw. The three coordinates giving the position of the centre of gravity. The three angles (ϕ, θ, ψ) giving the attitude of the aircraft.

The corresponding state vector is later increased to take into account the dynamics speciﬁc to certain sub-assemblies of the system such as the wheels, the braking system, the nose wheel steering systems, and the engines.

1.4 Forces and Moments 1.4.1

Aerodynamic Loads

These loads are represented macroscopically by a set of moments and forces applied to the centre of gravity and projected either onto the aircraft coordinate system or onto the aerodynamic coordinate system. These aerodynamic loads and moments depend on the status of the system (velocities, attitude, altitude), external conditions (velocity and direction of the wind, etc.), the

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11

conﬁguration of the aircraft (slats, ﬂaps, spoilers, etc.) and the position of the aerodynamic control surfaces (ailerons, rudder, elevators, etc.). Their determination is based on the identiﬁcation of aerodynamic coefﬁcients. In the model these loads are expressed in the aircraft coordinate system as illustrated in Fig. 1.8.

Fig. 1.8. Aerodynamic loads and moments expressed in aircraft-axis

The aerodynamic loads and moments are usually represented in the form of functions proportional to the dynamic pressure ρ, to the reference surface of the aircraft S, to the square of the air velocity Vair and to aerodynamic coefﬁcients Cx , Cy , Cz , Cl , Cm , Cn 1 2 Fxaero = − ρSVair Cx 2 1 2 ρSVair Fyaero = Cy 2 1 2 Fzaero = − ρSVair Cz 2 (1.9) 1 2 ρS eaeroVair M paero = Cl 2 1 2 Mqaero = Cm ρS caeroVair 2 1 2 ρS eaeroVair Mraero = Cn 2 with eaero the wing span and caero the aerodynamic chord. The aerodynamic coefﬁcients are determined from the angles of attack, sideslip, pitch, roll and yaw rates, deﬂection of the rudder, ailerons, elevators and the conﬁguration of the aircraft, etc. Commonly these coefﬁcients are identiﬁed using polynomials

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M. Jeanneau

and interpolation tables. For the purpose of the GARTEUR FM AG17 study, the aerodynamic coefﬁcients were obtained from neural networks developed by Airbus within the scope of a thesis [129]. The neural networks are identiﬁed from a certiﬁed Airbus reference model. The learning data are obtained by means of a high number of simulations where the parameters are deﬁned by random sampling in the operating range to be covered. The modelling tool used to ﬁne-tune the neural models also allows an accurate correlation evaluation to be done between the reference model and the model obtained. This analysis is based on a statistical evaluation performed using other sets of simulation data randomly sampled. The main reason for using this type of aerodynamic coefﬁcient modelling method is to obtain a model which is easily usable for simulation and control, and which does not require signiﬁcant computation capabilities but is guaranteed to be representative of the actual aircraft. 1.4.2

Gravity

The weight of the aircraft is considered to be applied at its centre of gravity along the vertical axis. When the attitude of the aircraft is not null, it induces longitudinal and lateral forces due to the projection onto aircraft-axis: FxG = − sin θ · m · g FyG = cos θ sin ϕ · m · g FzG = cos θ cos ϕ · m · g

(1.10)

Fig. 1.9. Projection of gravitational force onto aircraft coordinate system

1.4.3

Engines

The thrusts generated by the engines allow control of the longitudinal velocity of the aircraft. In addition, by applying differential thrusts to the left and right engines, a yaw torque is generated that impacts the lateral motion. There is a two-step modelling procedure for the engines. The ﬁrst considers the quasi-steady case. The aim is to develop a function to determine the balanced thrust associated with an N1 target at given conditions: Mach number, temperature, pressure altitude, etc. As for the aerodynamic coefﬁcients, this function is based on a neural network identiﬁed from an Airbus reference model.

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Fig. 1.10. Engine thrust forces

Secondly the dynamics of this system are represented using a second order nonlinear representation. This representation is based on a linear model augmented with saturations in amplitude and in velocity. The frequency, the damping and the saturation thresholds vary according to N1 and the ﬂight conditions. The resulting model can be summarised by the following equations: ¨ = − 2 ·zN1 (N1, . . .) · ωN1 (N1, . . .) · N1 ˙ + ωN1 (N1, . . .)2 · (N1c − N1) N1 ˙ min (N1, . . .) ≤ N1 ˙ ≤ N1 ˙ max (N1, . . .) N1 and N1min (. . .) ≤ N1 ≤ N1max (. . .)

(1.11)

The limit points correspond to the taking into account of the utilisation conditions: Mach number, temperature, pressure altitude, etc. 1.4.4

Shock Absorbers

The aim of the shock absorbers is to dampen the forces transmitted by the wheels and to dissipate the energy due to the touchdown of the aircraft during landing. Seen from the aircraft, the shock absorbers ﬁlter the vertical forces from the wheels. Seen from the wheels, they ﬁlter the vertical load variations due to the pitch and roll moments. The forces generated by these types of shock absorbers can be represented simply using stiffness and damping elements dependent respectively on the compression and the compression rate of the landing gear leg. To do this, it is possible to directly use graphs obtained from tests or functions such as polynomial functions identiﬁed on the basis of these graphs. The force then obtained is oriented in the landing gear axis. If the landing gear concerned has a non-negligible inclination (with regard to the aircraft coordinate system) this force must then be projected into the aircraft coordinate system. As the inﬂuence of a possible inclination of a landing gear on the longitudinal and lateral behaviour of the aircraft is very indirect and extremely small, the shock absorbers are considered, throughout the study, as oriented along the aircraft coordinate system “vertical” axis.

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1.4.5

M. Jeanneau

Forces Related to the Rolling on the Ground

The forces called the “rolling” forces include all forces induced by the contact between the tyres and the ground. These forces are applied to the contact surface and are in general expressed in coordinate systems associated with the wheels. These coordinate systems are such that the x-axis is oriented in the plane of the wheel considered, and the z-axis is orthogonal to the surface of the ground at tyre/ground contact point. For a wheel, the z-axis force component is associated with the strength of the ground. For a rigid pavement, this resultant opposes the projection of the force obtained from the compression of the shock absorbers. The longitudinal and lateral components (along the x- and y-axes) are due to frictions between the tyres and the ground. These two forces are brieﬂy described hereafter: Lateral Component The lateral force appears when the velocity of the wheel is no longer oriented in the plane of the wheel: the wheel slips on the ground. This sideslip can be compared to a lateral velocity component generating deformations in the tyre and friction forces. This force is generally presented as a nonlinear function of the sideslip angle. To simplify the modelling, the two wheels of a given bogie are superimposed at the centre of the landing gear so as to represent the mean behaviour. The local sideslip angles at the nose wheels and right and left main landing gears are then written: VyNW − θNW βNW = arctan VxNW Vy + LxNW × r = arctan − θNW Vx VyMLG R βMLG R = arctan Vx R MLG (1.12) Vy − LxMLG × r = arctan Vx − LyMLG × r VyMLG L βMLG L = arctan Vx L MLG Vy − LxMLG × r = arctan Vx + LyMLG × r The forces generated by the interactions between the tyres and the ground are mainly due to two effects: the lateral deformation of the tread (and therefore of the tyre) and frictions between the tyre and the ground. These two actions are based on complex physical phenomena associating: • Frictions between a more or less rough pavement with poorly known properties and a sculpted tyre surface, which can heat up, deform and wear. • Deformations of the tyre as a whole, which associates a complex geometry with a heterogeneous matrix formed by the tyre reinforced with metal. • The compression and the heating of the gas inside the tyre.

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Due to the complexity, macroscopic models obtained from a mathematical identiﬁcation of test data is used. An example of such results is given in Fig. 1.11. From these curves, it is possible to develop a mathematical function taking into account the main characteristics in this study, which are: • The derivative at the origin of the force versus the local sideslip angle (this derivative is also called “cornering gain”: Gy ). • The maximum force obtainable (Fymax ). • The corresponding sideslip angle, which will be called the optimum sideslip angle (βOPT ). These elements depend in a complex manner on many parameters such as the load applied to the wheels, the grip on the ground (dry, wet, snowbound, icy runway, etc.) and the velocity of the aircraft. These elements are taken into account according to the models used. The main equation of the model is based on the LAAS modelling [38]: FyW = 2Fyw max ×

βw OPT × βw β2w OPT + β2w

(1.13)

This model characterises the behaviour of the tyre by two characteristic parameters. In (1.13), these parameters are the maximum force obtainable and the corresponding sideslip angle. The cornering gain (efﬁciency of the tyre for low sideslips) can then be deﬁned by: ∂FyW 2FyW max GyW = = (1.14) ∂βW 0 βW OPT Thus, for example, it is possible to rewrite the expression (1.13) and to characterise the behaviour of the tyres either from the cornering gain and the optimum sideslip angle, or from the cornering gain and the maximum force: FyW = GyW × βW ×

β2W OPT β2W OPT + β2W

or

βW

FyW = GyW × 1+

G2yW 4×Fy2W max

× β2W

(1.15)

The forces generated by the tyres depend to a great extent on the vertical load. This dependency can be taken into account in the previous representation by identifying the characteristic parameters of the model. For example, this identiﬁcation can be based on a polynomial representation such as: FyWmax = (a1 · FzW + a2 ) FzW ∂FyW Gy1W = A1 FzW + A2 Fz2W GyW = = ∂βW βR ≈0 Gy2W = B0 + B1 FzW

if FzW < SFz if FzW > SFz

(1.16)

Longitudinal Component The longitudinal force associates the rolling drag and the braking forces. The rolling drag is mainly due to energy losses associated with the compression of the tyre (periodic

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Fig. 1.11. Example of test results furnished by Michelin for an Airbus-aircraft’s nose wheel tyres and for different vertical loads

deformation of the tread due to the ﬂexibility of the tyre and the rotation of the wheel). A force which is constant or which changes slightly with the speed is usually used models to model this phenomenon. Braking forces appear when the tread velocity is different from the wheel travel speed (difference due to the application of a braking torque on the wheel). This induces “slip” and therefore friction of the tyre on the ground. The slip is normally characterised by a slip velocity deﬁned as: VLSW =

VxW − rW · ωW VxW

(1.17)

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where VxW is the local longitudinal velocity expressed in the coordinate system of the wheel considered, rW is the radius of the tyre and ωW is the rotational velocity of the wheel. The slip velocity is therefore directly related to the dynamics of the wheel via its rotational velocity. The dynamics of the wheel can be represented by the following differential equation: ˙W = ω

1 (CBRK (PBRK ) − rW · FxW (VLSW )) IW

where CBRK :braking torque applied to the wheel, PBRK :braking pressure in the brakes, FxW :longitudinal force from slip of the tyres, IW : wheel moment of inertia. (1.18) However, since the moment of inertia of the wheel is neglected in the benchmark, the above dynamics (1.18) may also be neglected. The longitudinal behaviour of the tyre is very similar to its lateral behaviour. Thus only the macroscopic modelling is considered. This modelling is based on the same mathematical assumptions as those used in the lateral case. They depend on several characteristic elements such as: • The derivative at the origin of the force obtained according to slip velocity. • The maximum force obtainable (Fxmax ). • The corresponding slip velocity, called VLS OPT (Optimal Longitudinal Slip Velocity). These elements are identiﬁed from test data comparable to those shown in Fig. 1.12. The modelling is thus based on the following equation: FxW = 2FxW max ×

VLSW OPT × VLSW 2 2 VLS + VLS W OPT W

(1.19)

The torque generated by the brakes is immediately translated into a longitudinal force applied to the tyre: FxW =

CBRK (PBRK ) rW

(1.20)

Coupling Between Lateral and Longitudinal Components The formulas given previously for the lateral and longitudinal models apply only to tyres producing uncoupled lateral and longitudinal forces. Thus a tyre delivering a lateral force cannot at the same time deliver a maximum longitudinal force (and vice versa).

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M. Jeanneau

Fig. 1.12. Example of longitudinal characteristics of a tyre

The most commonly used model of this coupling is based on the traction circle concept. This approach consists of modifying the characteristic parameters of the model to remain constantly within a given circle, called the traction circle. This type of modelling is based on the following criteria:

• FW = (FyW )2 + (Fx W )2 must have its maximum around the traction circle. Hence:

FyW FxW max

2

+

FxW FxW max

2

= 1.

• If the slip velocity is null or if the wheel does not slip, the expressions of the lateral and longitudinal forces must remain unchanged. • For a positive ﬁxed slip velocity value, if β increases, the longitudinal force determined from the equivalent coupled model must be reduced. In the same way, for a ﬁxed positive value of β, if the slip velocity increases, the lateral force determined from the coupled model must also be reduced. • If the slip velocity and the sideslip angle remain small, the impact of the longitudinal or lateral coupling must not cause substantial modiﬁcations to the forces obtained. This coupling is taken into account in the simulation model. The control laws studied must therefore be robust to these coupling effects. Note, however, that since the modelling of the tyres is to a great extent uncertain by nature (uncertainty concerning type of ground and its condition), the taking into account of the coupling is just one component of all uncertainties affecting the tyres, and probably not even the most severe.

The AIRBUS On-Ground Transport Aircraft Benchmark

19

Conclusion on Computation of Forces Related to Rolling The various forces related to rolling were modelled “wheel by wheel”. To take these forces into account in the aircraft mechanics equations, they must be brought into the aircraft coordinate system by making suitable changes. Therefore the moments associated with these forces take into account the wheels lever arms (distances of the force application points from the centre of gravity). It is therefore possible to break down the computation of these forces into ﬁve steps: • Determination of the local velocities and the associated sideslip angles. • Determination of the slip from the dynamics of the wheel. • Computation of the longitudinal, lateral and vertical forces in the coordinate systems related to the wheels. • Projection of these forces into the aircraft coordinate system. • Determination of the associated moments. 1.4.6

Braking System

The braking system can be divided into ﬁve subsets [80] as illustrated by Fig. 1.13, of which the main components are: • The brake discs: quasi-static modelling of brake gain. • The hydraulic system: servovalves and pistons. • The system control logic: slave control in pressure and anti-skid ﬁlter. The reference model used for this study includes these elements to form a third order set (wheel dynamics, hydraulic system and its control). The mathematical representation of the behaviour of the system described above is based on an intermediary modelling level allowing the main elements comprising its dynamics to be reconstructed. The behaviour of the brake discs are modelled by a gain (GBRK ) the value of which can vary to a great extent from one brake to another. The generation of the braking torque (CBRK ) from a braking pressure applied in the pistons (PBRK) is relatively fast. For better representativeness, it is however in general represented by a ﬁrst order transfer function: CBRK (s) =

GBRK · PBRK (s) 1 + τBRK · s

(1.21)

The braking pressure is generated by the hydraulic system from a ﬂow rate (QBRK ) sent to the pistons: t

PBRK = 0

β QBRK dt VBRK

(1.22)

Variable β’ represents a compressibility coefﬁcient; which allows variations in the volumes of the pistons (and the nonlinear stiffness coefﬁcient associated with it) to be compared to the compressibility of the hydraulic ﬂuid. The value of βBRK can vary according to the pressure applied in the pistons.

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M. Jeanneau

Fig. 1.13. Schematic representation of the braking system

The ﬂow rate at piston inlet is generated by the difference between the supply pressure Pa and the return pressure Pr and is related to the position of the servovalve slide valve (xBRK ). This√ﬂow rate is written: QBRK = ηS (xBRK ) ∆P where: ⎧ ⎪ ⎨∆P = 0 ∆P = Pa − PBRK ⎪ ⎩ ∆P = PBRK − Pa

if xBRK = 0, if xBRK >0 if xBRK <0

η is the ﬂow coefﬁcient (1.23) S( · ) represents the section for the passage of the hydraulic ﬂuid through the servovalve. It is deﬁned by: (1.24) S(xBRK ) = K.xBRK The position of the servovalve slide valve can be deduced from the following equation: K2 S2 − K1 S1 S2 − S1 xBRK = (1.25) · (a SV BRK − ISVBRK) + x0BRK + · PBRK r r ISV BRK is the servovalve control current. S1 and S2 make reference to the slide valve sections in the left and right chambers of the servovalve. r is the stiffness of the spring

The AIRBUS On-Ground Transport Aircraft Benchmark

21

re-centring the slide valve. Constants K1 , K2 and a SV BRK characterise the behaviour of the ﬁrst stage of the servovalve. The slave control of the braking pressure can be approximated by a ﬁrst order corrector. Finally, the anti-skid ﬁlter is a complex ﬁlter. Its aim is to prevent the wheels from skidding. It limits the braking pressure target so as to maintain the slip velocity close to its optimum value. To do so, a detection algorithm compares the wheel velocities with the velocity of the aircraft. 1.4.7

Nose Wheel Steering System

The nose wheel steering system can be divided into four subassemblies including the system itself and its control system [3] as illustrated in Fig. 1.14. • The BSCU (Braking and Steering Control Unit). This subassembly deﬁnes the servovalve control current (ISVNW ) from the pilot’s target and the deﬂection angle obtained. • The hydraulic part determines the characteristic linking the wheel deﬂection rate to the ISVNW deﬁned previously. This characteristic is nonlinear and depends on the difference in pressure between the actuator chambers.

Fig. 1.14. Nose wheel steering system

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M. Jeanneau

• The mechanical part allows this pressure difference to be determined from forces exerted on the landing gear leg: dry frictions, viscous frictions and torque generated by the wheels. • Wheels. The aim of this part of the model is to compute the forces exerted on the aircraft and to determine the torque value generated on the landing gear leg. In the BSCU, the slave control of the nose wheel steering system is: ISVNW = k × (θNWc − θNW )

(1.26)

Amplitude and velocity control saturations are: θ NW min < θNW < θNW max θ˙ NW min < θ˙ NW < θ˙ NW max

(1.27)

The derivative of θNW is then a nonlinear function of ISV . The nonlinearities of this function reﬂect ﬂow rate losses due to the diaphragms in the hydraulic unit.

1.5 General Architecture of the Model The following diagram summarizes the aircraft-on-ground model developed for the GARTEUR FM AG17 project:

Fig. 1.15. Architecture of the aircraft-on-ground model

The AIRBUS On-Ground Transport Aircraft Benchmark

23

1.6 Short Analysis of Aircraft-on-Ground Behaviour This section provides a global and qualitative understanding of the behaviour of the aircraft during rolling. The analyses hereafter do not take into account the aerodynamic forces, which may be neglected at low rolling speed. At higher rolling speeds, the aircraft will display similar behaviour, which will be made more stable by the aerodynamic forces tending to make its trajectory straight (in wind axis). 1.6.1

Stability of Aircraft When Rolling

When rolling, the stability of the aircraft can be directly associated with a ratio between the moment generated by the main landing gears and the moment due to the nose wheels. Indeed, as the nose wheels are located forward of the centre of gravity, their action on the dynamics of the aircraft is destabilising. The main landing gears, on the contrary, due to their rearward position, generate a stabilising moment. For example, let us consider the case of a straight trajectory. Let us assume that we have a disturbance (for example a wind gust) inducing a slight sideslip of the aircraft to the right (Vy > 0). This wheel sideslip angle then induces a negative force on each of the landing gears. For the nose landing gear, this force is translated into a negative yaw moment which increases the rotation of the aircraft and therefore its sideslip. The force from the main landing gears generates on the contrary a positive moment, which tends to bring the aircraft onto a straight trajectory. If this moment is lower than the moment due to the nose landing gear, the aircraft is then unstable (and will start to spin). 1.6.2

Turn Initiation Case

When a turn is initiated, the deﬂection of the nose landing gear wheels increases the sideslip angle. The lateral force thus produced generates a yaw moment, which triggers the turn. During the turn, sideslip also appears at the main landing gears. This sideslip generates a force opposing the centrifugal force and the lateral force from the nose wheels. This force thus produces a moment opposing the rotation of the aircraft. To keep the rotation stable, it is then necessary to conserve a sufﬁcient nose wheel deﬂection to generate a moment opposing the one produced by the main landing gears. The lateral dynamics of the aircraft when turning can therefore be explained on the basis of these observations: • The aim of the lateral velocity dynamics is to generate a sufﬁcient sideslip at the main landing gears to compensate for the centrifugal force and the force due to the nose wheels. • The yaw dynamics directly result from the balance between the moment due to the deﬂection of the wheels and the stabilising effect of the main landing gears. 1.6.3

Behaviour at Grip Limit

We now study the case of sideslip angle reaching or exceeding the optimum value (β OPT ). The force generated by the wheels concerned is then at its maximum.

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M. Jeanneau

Effect on Nose Wheels If the force on the nose landing gear reaches its maximum value, the resulting yaw moment is at its maximum. This limitation is thus translated by a saturation of the turn initiation dynamics. This limitation is in general (dry runway case) not encountered. Indeed, the nose wheel steering deﬂection rate limitation (hydraulic ﬂow rate limitation at piston input) limits the sideslip angle to a value lower than its optimum value. Effect on Main Landing Gears When the main landing gears reach their grip limit, the increase in the yaw rate is then reﬂected by some sideslip. The force generated by the tyres no longer counteracts the centrifugal forces and the aircraft continues its turn by sliding. When the turn made is such that the sideslip at the main landing gears reaches its optimum value (β OPT ), the turning radius obtained then corresponds to the minimum radius that can be attained by the aircraft under such conditions.

2 On-Ground Transport Aircraft Nonlinear Control Design and Analysis Challenges Matthieu Jeanneau Airbus-France, Department of Stability and Control, Toulouse, France [email protected]

2.1 Introduction This chapter provides the main guidelines for the control design and analysis of a “rolling on the ground” control law. The rolling of aircraft is a very challenging task in term of piloting. The overall design of transport aircraft is clearly not optimized for rolling on ground but for ﬂight. Its natural rolling qualities are very poor, both in term of stability and performance. Besides the coupling of aerodynamic loads, engine-thrusts, gravity and friction loads at the contact point of each tyre produce highly nonlinear and time-varying dynamics. These dynamics are strongly inﬂuenced by many parameters such as the velocity, the runway state, aircraft conﬁguration (mass and inertia) and external disturbances like wind turbulence or gusts. The control objectives may also vary depending on the rolling phase: taxiing, runway acceleration prior to take-off, runway deceleration after landing, and runway deceleration after a rejected take-off. The following sections provide speciﬁcations for the design of a rolling on-ground control law for the Airbus on-ground transport aircraft benchmark described in chapter 1.

2.2 Control Architecture and Objectives 2.2.1

Aircraft Controls

The available controls while aircraft are on ground are the following: Rudder Deﬂection The rudder deﬂection directly effects the lateral aerodynamic loads. The higher the velocity, the higher the loads and the efﬁciency of the rudder. Nose-Wheel Deﬂection The nose-wheel deﬂection creates sideslip on the nose-wheel tyres that creates a lateral force. The efﬁciency of the lateral friction force induced depends on the vertical payload, which decreases with the velocity of the aircraft. Therefore, the higher the velocity, the higher the lift, the lower the lateral loads induced by a nose-wheel deﬂection. D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 25–33, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

26

M. Jeanneau

N1R and N1L : Right and Left Engine Speed Each engine thrust is controlled by the FADEC through a commanded rotation speed called N1. The N1 orders sent to the FADEC (Full Authority Digital Engine Computer) are classically used for aircraft speed control. The higher the N1, the higher the thrust and the higher the acceleration. Classically, symmetric thrust orders are sent to the left and right engines in order to prevent any disturbance on the lateral dynamics. However differential thrust orders can be deliberately sent in order to generate a yaw moment and help the lateral ground control. PBRK R and PBRK L : Commanded Right and Left Braking Pressure The braking torque applied either to the left or the right bogie is controlled through a commanded braking pressure PBRK . The classical use consists in applying symmetrical pressure to the left and right brake discs in order to avoid lateral disturbance during decelaration. However differential thrust orders can be deliberately sent in order to generate a yaw moment and help the lateral ground control. 2.2.2

Measures Available

The following measures are available for the longitudinal and lateral control of the aircraft on-ground motion: • p, q, r, the roll, pitch and yaw angular rates • p, ˙ q, ˙ r˙: roll, pitch and yaw angular accelerations • Nx , Ny , Nz : load factors measured at IRS and also available at centre of gravity, which are accelerations expressed in g • PBRK , the braking pressure applied on each bogie • N1R and N1L , the right and left engine speed • θNW and θ˙ NW , the nose-wheel angular deﬂection and angular velocity • ISVNW , the current for the control of the nose-wheel steering system. 2.2.3

Pilots Controls

The following controls are classically available for the pilot to control the plane on ground: pedals, tiller (a small lateral wheel located close to the lateral side-stick), throttles and pedal brakes. Other controls are available on board transport aircraft such as airbrakes, but the design of a control function for the on-ground motion should focus on these four controls. 2.2.4

Piloting Objectives and Philosophy

Currently, the pilot commands are directly transferred to the control surfaces, i.e. the pilot performs both the longitudinal and the lateral control of transport aircraft on ground manually.

On-Ground Transport Aircraft Nonlinear Control Design

27

“Piloting by objectives”, which has become standard on-board modern transport aircraft since the introduction of the ﬂy-by-wire concept on board the A320, is today limited to the ﬂight phase. However the possibilities offered by this concept should also make it a common tool for on-ground control in a very near future. In the context of this GARTEUR project, the control design shall focus on two different ways of piloting. The ﬁrst solution consists of converting pilot commands into commanded accelerations. The commands will be expressed as longitudinal and lateral targets in the form of ( Nxcockpit ; Nycockpit ), where Nxcockpit and Nycockpit are the longitudinal and lateral horizontal accelerations felt by the pilots in the cockpit. Due to the proximity of the pilots to the IRS (Inertial Reference System) that measures the accelerations, NxIRS and NyIRS can approximate Nxcockpit and Nycockpit . The second solution consists of converting pilot orders into commanded longitudinal acceleration at the centre-of-gravity and yaw rate. The pilot orders are then homogeneous to ( Nxcg ; r ), i.e. the derivative of the aircraft velocity and the yaw rate. This solution has already been investigated and studied at Airbus.

2.3 Design and Analysis Criteria 2.3.1

Main Time-Domain Constraints

Here are the main constraints on the aircraft time-responses to commanded step inputs: • No overshoot on the pilot commands • No steady error −3 x • | dN dt | < 1.2 ms

dNx / dt

Nx

Overshoot

Time−response

Time Response−rate delay

Fig. 2.1. Illustration of main criteria on the Nx response to a commanded step input

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M. Jeanneau

2.3.2

Criteria to Optimise

The following time-domain criteria have to be optimized by the control laws: • Minimise response time to reach steady state • Minimise the initial delay • Maximise the initial dynamic response: dNy x – either dN dt and dt – 2.3.3

or

dNxcg dt

and

dr dt

(depending on the piloting objectives chosen)

Robustness Constraints

The main components affected by uncertainties are listed hereafter. Braking System The braking efﬁciency is highly variable from one brake to another: up to ±50% due to manufacturing non-homogeneity in the materials. G BRK = k G BRKnominal with k ∈ [0.5, 1.5]

(2.1)

Note that different values kleft and kright must be considered for right and left brakes. Tyre Characteristics The grip of tyres is strongly affected by the runway characteristics (icy, wet or dry). In the SimulinkT M benchmark described in Chapter 1, the default data correspond to a dry runway. To cover all possible conditions, a multiplication gain k is added to the main tyre characteristics: Fymax for the lateral friction model and Fxmax for the longitudinal friction model. This gain k varies from 0.2 in icy conditions to 1 in dry conditions. Of course this gain k is the same for the three gears (nose-wheel, left main landing gear and right main landing gear). Aerodynamic Model One should consider uncertainties in the aerodynamic coefﬁcients of up to ±25%. For instance the roll aerodynamic coefﬁcient becomes: Cl = k ·Clnominal with k ∈ [0.75, 1.25]

(2.2)

Nose-Wheel Angle Measurement The measured value of the nose-wheel deﬂection can be affected by a bias, possibly reaching a maximum value of 1◦ . θNW

measured

= θNW ± 1◦

(2.3)

On-Ground Transport Aircraft Nonlinear Control Design

29

Uncertainties on the Thrust Model Due to many uncertainties in the atmospheric conditions, a ±10% multiplicative uncertainty k should be considered on the thrust forces computed by the engines models: thrustR and thrustL . An additive uncertainty Fu on these forces may also affect the modelling. thrustR = k ·thrustRnominal + Fu thrustL = k ·thrustLnominal + Fu with k ∈ [0.9, 1.1] Fu ∈ [−5000 N, 5000 N]. (2.4) These uncertainties are symmetric, i.e. the same uncertainty must be considered on both engines. Wind Wind is deﬁned in the earth coordinate system. Considering an aircraft lined up with the runway, the wind to be considered is the following: • Lateral wind along Yrunway up to 30 kts (15m/s) • Longitudinal wind along Xrunway – Up to 30 kts (15 m/s) for front wind – Up to 10 kts (5 m/s) for rear wind These winds must be tested at high speed only, i.e. for velocities above 70 kts, because the aircraft’s aerodynamic modelling of the wind effects is not appropriate at lower speeds (sideslip β is too high).

2.4 Typical Manoeuvres Hereafter are presented reference manoeuvres for the assessment of aircraft on-ground control laws. Some are textbook manoeuvres which do not reﬂect operational usage. Others are more representative of operational manoeuvres encountered during aircraft acceleration, deceleration, runway exit and taxiing phases. 2.4.1

Low Speed Lateral Manoeuvres

Doublet in Ny (or r) In a doublet manoeuvre, a command to change attitude in a given direction is followed by a command to change attitude in the opposite direction by the same amount. This is of interest for aircraft having the right heading on a runway, but not aligned exactly in the middle of the runway.

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M. Jeanneau

Fig. 2.2. Doublet lateral manoeuvre at low speed

90◦ Runway Exit

Fig. 2.3. 90◦ runway exit

U-turn The U-turn manoeuvre has to be accomplished at 7 knots. Aircraft must be able to perform this manoeuvre on a 45 m wide runway. The U-turn manoeuvre is made of the followings: • Straight towards the border with an angle of 30◦ • Reach the border with the nose-wheel • Apply full right input to turn

2.4.2

High Speed Lateral Manoeuvre

Step in Ny (or r) In this manoeuvre, illustrated by Fig. 2.5, the assessment will test different values for u. On top of the usual time-domain criteria (time-response, overshoot, initial delay, . . . ) the maximum achievable umax before the nose-wheel slips should be monitored.

On-Ground Transport Aircraft Nonlinear Control Design

31

Fig. 2.4. U-turn

Fig. 2.5. High-speed lateral manoeuvre

2.4.3

High Speed Longitudinal Manoeuvres (Runway Manoeuvres)

This is a typical acceleration/decelaration manoeuvre, representative of the longitudinal motion of aircraft on the runway, either after landing, or prior to take-off. The two associated manoeuvres are: • Full acceleration from 5 kts to 150 kts • Full deceleration from 150 kts to 5 kts

Fig. 2.6. High-speed longitudinal manoeuvre

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M. Jeanneau

2.4.4

Low Speed Longitudinal Manoeuvre (Taxiway Manoeuvres)

Multistep Inputs to Reach Different Commanded Speeds This manoeuvres aims at assessing the ability to reach and follow different speed targets during taxiing operations. The two following sequences will be considered: • 10 kts → 20 kts → 30 kts → 20 kts → 10 kts • 40 kts → 50 kts → 60 kts → 50 kts → 40 kts 2.4.5

Coupled High-Speed/Low Speed, Lateral/Longitudinal Manoeuvre

This manoeuvre concatenates previous manoeuvres to illustrate operational needs for a transport aircraft on-ground. The sequence is: • • • • • • • •

Starting from 150 kts along the X axis Decelerate to 30 kts Turn to take a 30◦ exit while decelerating to 20 kts Decelerate to 10 kts Make a 60◦ turn Accelerate to 20 kts Decelerate to 5 kts Perform a U-turn

Fig. 2.7. Example of an operational sequence on-ground

On-Ground Transport Aircraft Nonlinear Control Design

33

For the assessment of control law performance, the following criteria will be evaluated: • Total length necessary for this manoeuvre • Total duration • General aircraft behaviour

2.5 Conclusions The on-ground benchmark proposed in this GARTEUR project is highly representative in terms of the behaviour and dynamics of transport aircraft. The challenges in term of control and analysis described in this chapter shall be considered as guidelines to propose innovative and novel solutions for: the control design methods, the non-linear analysis methods, and in terms of overall control strategies. For instance, regarding the longitudinal motion, available actuators are engines and brakes. It is of great interest in a research project such as this to investigate solutions that mix these two actuators. Regarding the lateral motion, the use of differential braking, or differential engine thrusts for enhancing the lateral piloting would be an innovative strategy.

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3 The ADMIRE Benchmark Aircraft Model Martin Hagstr¨om Department of Autonomous Systems, Swedish Defence Research Institute, SE-164 90, Stockholm, Sweden [email protected]

Summary. In this chapter we describe the simulation model ADMIRE. ADMIRE is an advanced generic simulation model of a modern delta-canard ﬁghter aircraft. The model is based on the Generic Aerodata Model (GAM), developed by Saab AB. Keywords: aircraft simulation model, generic aerodata model, GAM, ADMIRE.

3.1 Introduction In the mid 1990’s the need for a complex, realistic and non-classiﬁed aircraft model for academic use was discussed at the aircraft manufacturer Saab in Sweden. Within the academic community new methods for control law design had been developed but these were usually applied to simple models which lacked the challenges of coupled and nonlinear behaviour that modern ﬁghter aircraft posed for control law designers. The discussions resulted in a national research project where Saab in cooperation with the Royal Institute of Technology (KTH) produced the Generic Aerodata Model (GAM). Although GAM sufﬁciently deﬁnes the dynamics of an aircraft for an aerodynamisyst’s analysis, it requires a substantial effort for the control engineer to utilise it for control law design purposes. This led the former Aeronautical Research Institute of Sweden (FFA) to develop a complete simulation model based on the GAM-data. The result of this effort was the AeroData Model In Research Environment, ADMIRE. ADMIRE integrates the GAM-data with models of engine, actuators, atmosphere and sensors.

3.2 Description of ADMIRE GAM GAM is an unclassiﬁed aerodynamic model of a delta canard ﬁghter aircraft. It includes a complete description of the dynamics over a large envelope, as well as effects of landing gear and airbrakes. It is open-loop unstable and includes several realistic coupled aerodynamical effects which pose a challenge for the control engineer. There are 11 control surfaces; nosewing, four leading edge ﬂaps, four elevons (combinations of ﬂaps, elevators and ailerons) and a rudder. Engine airﬂow mass ratio, D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 35–54, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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M. Hagstr¨om

landing gear in or out, airbrakes in or out, nosewheel door open or closed are further inputs to the model. Outputs from GAM are non dimensional force, moment and hinge moment coefﬁcients in the aerodata reference point, see ﬁgures 3.5 and 3.6. The outputs are interpolated from tables which, except for the above mentioned inputs, depend on angle of attack (α), sideslip angle, (β), Mach number and altitude. The actual forces and moments need to be calculated. The aerodata tables, interpolation routines and other calculations were originally written in Fortran. GAM is described in detail in [10]. 3.2.1

ADMIRE

ADMIRE is an implementation of GAM in a simulation environment where the aerodynamic data from GAM is complemented with models of the engine, actuators for the surfaces, and sensors. In the simulation model the resulting forces and moments are calculated. The earth relative position, velocity and attitude of the aircraft and winds are computed in realtime. ADMIRE comes with a basic control system for stabilisation and basic handling qualities and an executable implementation, currently in SIMULINKT M with the GAM-data tables and interpolation functions rewritten in C. The aerodata tables are also extended in ADMIRE compared to the original GAM-data and allow simulation up to 90◦ in angle of attack (α, AoA) and up to 30◦ in sideslip angle (β) for all Mach numbers lower than 0.5. ADMIRE contains twelve states representing the dynamics of the aircraft plus additional states due to the presence of actuators, sensors and the Flight Control System (FCS). Available control effectors are left and right canard, leading edge ﬂaps, four elevons, rudder and throttle settings. The model is ADMIRE 1 2 3

u0fcs(1)

4

Fes1

5

Total Computer Delay

u0fcs(2)

6 7

Vt

Transport delay version

u0fcs(3) Fas1

8

Saturators Ratelimiters and Actuators

9 10 11 12

u0fcs(4) 13

Frp

14 15

0

Control System

dle

Aircraft Response

16 17 18

0 19

ldg

20 21

0

22

dty

23 24

0 25

dtz

26

dist

DisturbParam

27 28 29 30 31

Fig. 3.1. An overview of the simulink implementation of ADMIRE

The ADMIRE Benchmark Aircraft Model

37

equipped with air brakes and a choice to have the landing gear up or down. The model is prepared for the use of atmospheric turbulence as external disturbance and thrust vectoring capability. The FCS contains a longitudinal and a lateral part. The longitudinal controller provides pitch rate control for low Mach numbers and load factor control for larger Mach numbers. There is an α-limiter functionality active during pitch rate mode. The longitudinal controller also contains a rudimentary speed controller. The lateral controller enables the pilot to perform initial roll control around the velocity vector of the aircraft and angle of sideslip control. Sensor models used by the FCS are incorporated in the model, together with a 20 ms computer delay on the actuator inputs. There is the possibility to vary some parameters and uncertainties within given tolerances, in order to facilitate robustness analysis. The available uncertainties consist of inertia, aerodynamic, actuator and sensor, all presented in section 3.2.8. Although the ADMIRE model is now quite mature and has been extensively used for several years, further improvements to the model are still possible. The bundled FCS does not utilise the full envelope of the GAM-data. Effects of engine airﬂow are not fully implemented and the fuel consumption does not affect the mass and inertial data which remain constant over time. Some coordinate transformations are still deﬁned in Euler angles which introduces a singularity in the computations of the dynamics. There is an implementation of thrust vectoring which is not yet validated. This book contains examples of how to design more advanced FCS for the ADMIRE. 3.2.2

Model Data and Envelope

Figure 3.2 illustrates the control surface conﬁguration. The aircraft conﬁguration data used in ADMIRE is described in Table 3.1. The data gives an idea about the size of the aircraft. In [10], data for mass and inertia depending on the amount of fuel are tabulated for three levels of fuel; 100%, 60% and 30%. In ADMIRE the mass and inertia are constant but could be modelled as functions of fuel consumption which is part of the engine model. The mass and inertia are chosen for a nominal case with a mass representing 60% of fuel loaded. The calculations of the resulting forces and moments with effects from gravity and aerodynamics transformation are done in different frames. The relative distance between the aerodynamic reference point, deﬁned in the aerodynamic frame SU , (see ﬁgure 3.5) and the centre of gravity (c.g.), deﬁned in the the body ﬁxed frame, SB , (see ﬁgure 3.4) are used to transfer the moments and forces from one frame to another. The FCS is scheduled in altitude and Mach number and designed for the nominal model described in Table 3.1. A change in the centre of gravity will change the control performance. The ﬂight envelope for the GAM-data extends to Mach 2.5 and an altitude of 20 km. The envelope for the engine model is valid up to Mach 2. With the bundled FCS the ADMIRE ﬂight envelope is restricted to Mach numbers less than 1.2 and altitudes below

38

M. Hagstr¨om

Fig. 3.2. Principal layout of the control surface conﬁguration

Table 3.1. Nominal conﬁguration data Component: Wing area Wing span Wing chord(mean) Mass Ix Iy Iz Ixz

Value: 45.00 10.00 5.20 9100 21000 81000 101000 2500

Unit: m2 m m kg kgm2 kgm2 kgm2 kgm2

6 km. Within the ﬂight envelope there are additional constraints due to the aerodynamics. Angle of attack, angle of sideslip and the control surface deﬂections are limited as shown in Figure 3.3. Due to (assumed) structural reasons and concern for the well-being of a hypothetical pilot, the normal load factor is constrained to −3g ≤ nz ≤ +9g over the whole envelope. 3.2.3

Aircraft Dynamic Model

The aircraft dynamics are modelled as a set of twelve ﬁrst order nonlinear differential equations of the form x˙ = f (x, u, p) (3.1) y = g(x, p)

The ADMIRE Benchmark Aircraft Model AoA

39

Beta

30 80

20 60 10 40 0 20 −10 0 −20

−20 −30 0

0.5

1

1.5

2

2.5

0

(a) Angle of attack.

0.5

1

1.5

2

2.5

2

2.5

(b) Angle of sideslip.

Canard

Leading edge flap

60

15

50

10

5

40

0 30 −5 20 −10 10 −15 0 −20 −10

−25

−20

−30

−30

0

0.5

1

1.5

2

2.5

(c) Canard deﬂection angle.

−35

0

0.5

1

1.5

(d) Leading edge ﬂap deﬂection angle.

Elevon

Rudder

30

30

20

20

10

10

0

0

−10

−10

−20

−20

−30

−30 0

0.5

1

1.5

2

(e) Eleven deﬂection angle.

2.5

0

0.5

1

1.5

2

2.5

(f) Rudder deﬂection angle.

Fig. 3.3. Envelope of the ADMIRE aerodata model. The original GAM data is expanded for higher angle of attack.

where x is the state vector, u is the input vector, y is the output vector and p is the uncertainty parameter space vector. The state equations used are listed below Total velocity V˙T ˙ Angle of attack α Angle of sideslip β˙ Roll rate p˙b

= = = =

(ub · u˙b + vb · v˙b + wb · w˙ b )/VT (ub · w˙ b − wb · u˙b )/(u2b + w2b ) (v˙b ·VT − vb · V˙T )/(VT2 · cos β) (C1 · rb + C2 · pb ) · qb + C3 · Mx + C4 · Mz

40

M. Hagstr¨om

= C5 · pb · rb − C6 (p2b − rb2 ) + C7 · My = (C8 · pb − C2 · rb )qb + C4 · Mx + C9 · Mz = (qb · sin φ + rb · cosφ)/ cos(θ) = qb · cosφ − rb · sin φ = pb + tanθ · (qb · sin φ + rb · cos φ) = cosθ · cos ψ · ub + (sin φ · sin θ · cos ψ − cosφ · sin ψ) · vb + (cos φ · sin θ · cosψ + sin φ · sin ψ) · wb Position SB -frame y˙v = cosθ · sin ψ · ub + (sin φ · sin θ · sin ψ + cosφ · cosψ) · vb + (cos φ · sin θ · sin ψ − sinφ · cos ψ) · wb Position SB -frame z˙v = − sin θ · ub + sin φ · cos θ · vb + cosφ · cos θ · wb

Pitch rate Yaw rate Heading angle Pitch angle Bank angle Position SB -frame

q˙b r˙b ˙ ψ θ˙ φ˙ x˙v

where u˙b v˙b w˙ b C1 C2 C3 C4 C5 C6 C7 C8 C9 Γ

= rb · vb − qb · wb − g0 · sin θ + Fx /m = −rb · ub + pb · wb + g0 · sin φ · cos θ + Fy /m = qb · ub − pb · vb + g0 · cos φ · cos θ + Fz/m = ((Iy − Iz )Iz − Ixz Ixz )/Γ = ((Ix − Iy + Iz )Ixz )/Γ = Iz /Γ = Ixz /Γ = (Iz − Ix )/(Iy ) = Ixz /(Iy ) = 1/(Iy ) = (Ix (Ix − Iy ) − IxzIxz )/Γ = (Ix )/Γ = Ix Iz − Ixz Ixz

(3.2)

The output vector consists of the state variables plus additional variables deﬁned by the equations: ub vb wb uv vv wv nz ny M γ CD CL

= VT · cos α · cos β = VT · sin β = VT · sin α · cos β = cos θ · cos ψ · ub + (sin φ · sin θ · cosψ − cosφ · sin ψ) · vb + (cos φ · sin θ · cos ψ + sin φ · sin ψ) · wb = cos θ · sin ψ · ub + (sin φ · sin θ · sin ψ + cosφ · cos ψ) · vb + (cos φ · sin θ · sin ψ − sin φ · cos ψ) · wb = − sin θ · ub + sin φ · cos θ · vb + cosφ · cos θ · wb = −FZaero /(m · g0 ) = −FYaero /(m · g0 ) = VT /a(h) = arcsin(cos α · cos β · sin θ− (sin φ · sin β + cosφ · sin α · cos β) · cos θ) = CN · sin α + CT · cos α = CN · cos α − CT · sin α

and CC ,Cl ,Cm ,Cn , Fx , Fy , My

(3.3)

The ADMIRE Benchmark Aircraft Model

41

There is a singularity in the numerics at θ = 90◦ . Note also that the bank angle (φ) is not limited to ±180◦. These limitations will be removed in future versions. The equations are described in more detail in [219]. 3.2.4

Aerodata Model

The aircraft aerodata modelling consists of aerodata tables, interpolation routines and aerodata algorithms. This is a standard way of performing aerodynamic modelling. In these aerodata tables an interpolation is made and the six resulting aerodynamic coefﬁcients, (CT , CN , CC , Cl , Cm , Cn ), are calculated. These coefﬁcients are calculated with respect to a reference point as depicted in ﬁgure 3.5. The aerodynamic reference point coincides with the nominal c.g. of the aircraft. The ﬁgures 3.4 and 3.5 are reproduced with the kind permission of Saab AB. ADMIRE is implemented in the MATLAB/SIMULINK environment as S-functions based on C-code. As mentioned earlier the original aerodata model, GAM, is valid for Mach numbers up to 2.5, altitudes up to 20 km, angles of attack up to 30◦ and sideslip angles up to 20◦ .The model has been extended on two occasions. First the envelope was extended for angles of attack up to 90◦ at Mach numbers less than 0.5 for the longitudinal part only. That is, at high angles of attack the lateral motion would be governed by aerodynamic coefﬁcients only valid at 30◦ angle of attack. A complete report on this work can be found in [235]. This led to the second extension where the lateral part of the aerodynamic data was extended, accordingly, up to 90◦ angle of attack. The latest work on the lateral part is quite rough and based only on assumptions about what lateral dynamics this type of aircraft might have. There is no real effect of separation modelled. For instance, the aircraft is assumed to suffer from yaw instabilities at high α, when the airﬂow around the ﬁn is strongly turbulent and disturbed. Further, the control efﬁciency of the surfaces is assumed to deteriorate.

YB

OB

XB

ZB

Fig. 3.4. Body ﬁxed frame, SB -frame

42

M. Hagstr¨om

CN CT

YU Cm OU

CC Cn

Cl

XU

ZU

Fig. 3.5. Deﬁnition of aerodynamic coefﬁcients, SU -frame

However large the ﬂight envelope of GAM, ADMIRE’s is smaller since it is constrained by the bundled FCS. It is scheduled for altitudes up to 6 km and Mach numbers up to 1.2. With a different FCS the data is valid up to 20 km and Mach 2.0. In Figure 3.5, the deﬁnition of the direction of the forces and moments from the aerodata is shown. The aerodynamic forces are given in the form of body ﬁxed normal, tangential and side forces. The aerodynamic reference point (OU ) and the centre of gravity (OB ) are given in Figure 3.6. The reference point is ﬁxed but the location of the c.g. can change. In the nominal case these two points coincide. Deviation in c.g. from the aerodynamic reference point will give additional effects in the moment equations. The aerodynamic model is built up in a conventional way, by interpolating (unstructured) data tables to obtain the different contributions. Different aerodata tables are used at different Mach numbers. A transition is made between Mach numbers 0.4 and 0.5 and at Mach number 1.4. The aerodata contains static aeroelastic effects and coupling between lateral and longitudinal dynamics of the aircraft, i.e. CNβ . The total aerodynamic forces and moments acting on the aircraft are calculated in the following way: Fx Fy Fz Mx My Mz

= = = = = =

−q¯ · Sref ·CTtot −q¯ · Sref ·CCtot −q¯ · Sref ·CNtot q¯ · Sref · bref ·Cltot − zcg · Fy + ycg · Fz q¯ · Sref · cref ·Cmtot − xcg · Fz + zcg · Fx q¯ · Sref · bref ·Cntot + xcg · Fy − ycg · Fx

(3.4)

Note that xcg , ycg , zcg is not a ﬁxed coordinate but the relative distance between the centre of gravity and the aerodynamic reference point.

The ADMIRE Benchmark Aircraft Model

43

YB

OU

OB

XB

ZB

Fig. 3.6. Deﬁnition of reference frames Thrust without AB at Tss 0.8, and with AB maximum Tss respectively 4

x 10 14

12

10

Thrust

8

6

4

2

0 2

2

1.5

1.5

1

1

0.5

4

x 10

0.5 0

Altitude

0 Mach number

Fig. 3.7. Engine thrust data

3.2.5

Engine Model

The engine model contains data in two 2-dimensional tables describing the engine thrust. The two tables contain the available thrust from the engine, one with activated afterburner and the other without. The thrust is a function of the altitude and the Mach number, see Figure 3.7.

44

M. Hagstr¨om

The input to the engine is the Throttle Stick Setting (Tss ), which takes values between 0 and 1. When Tss is greater than or equal to 0.8 the afterburner is active. In the present version of the model there is a blending between the two tables smoothing the transition. The blending is done to make the model more smooth which simpliﬁes the trimming. ⎧ ⎪ if Tss < 0.78 ⎨Table1 Thrust = 25(Tss − 0.78)(TableH − TableL ) + TableL if 0.78 ≤ Tss < 0.82 ⎪ ⎩ if Tss > 0.82 Table2 (3.5) where TableH = Table2 at M = 0.82 TableL = Table1 at M = 0.78 To make the ratio between the static thrust and the maximum take-off weight of the aircraft correlate to a value of similar modern aircraft, the tabled thrust is scaled. The scaling factor is a linear function of Tss . Thrust = (0.8 + 0.4 ·Tss ) · Thrusttable

(3.6)

The capture area intake ratio (CAI) is calculated as cai =

m˙ f f ρ∞V∞ Ainlet

(3.7)

where m˙ f is the fuel consumtion, ρ∞ the upstream air density, V∞ upstream airspeed and Ainlet the area of the engine inlet, 0.38 m2 . f is the fuel-air number which is set to be 1/38 without afterburner and 1/25 with afterburner. More on the engine modeling and thrust vector control can be found in [134]. This is not part of the original aerodata model GAM. The implementation of thrust vector control is made in a heuristic fashion and is not validated. Due to the time it takes to accelerate/decelerate the rotating parts of the engine, the dynamic response in the engine is modelled with a simple ﬁrst-order lag ﬁlter. Tss (s) = 3.2.6

0.5 · Tsscom s + 0.5

(3.8)

Actuators

The actuator model used is simply a ﬁrst order transfer function with limited angular deﬂection and maximum angular rate. The time constant for the leading edge ﬂap is chosen to have a different value compared with the other actuators. Although this representation of an actuator is quite standard, it is possible to use more advanced rate and deﬂection limits plus higher order transfer functions, see [62]. The available control actuators in the ADMIRE model are: -

left canard (δlc ) right canard (δrc ) left outer elevon (δloe )

The ADMIRE Benchmark Aircraft Model

-

45

left inner elevon (δlie ) right inner elevon (δrie ) right outer elevon (δroe ) leading edge ﬂap (δle ) rudder (δr ) landing gear (δldg ) air brake (δab ) horizontal thrust vectoring (δth ) vertical thrust vectoring (δtv )

The leading edge ﬂap, landing gear and thrust vectoring are not used in the FCS. The sign of the actuator deﬂections follows the “right-hand-rule”, except for the leading edge ﬂap that has a positive deﬂection down. The “right-hand-rule” means that a positive deﬂection corresponds to a positive rotation assuming that the hinge line is parallel to the respective axis in the body-ﬁxed reference frame SB , see [10] and Figure 3.8. There are four different elevons. Only the outer two are drawn in the Figure 3.8. The inner and outer elevons on each side always move together in the bundled version of the FCS. The maximal allowed deﬂections and suggestions for the angular rate of the control surfaces are given in Table 3.2. The deﬂection limits are deﬁned in the original GAM model and should not be violated. The maximum allowed deﬂections of the actuators depend on the Mach number, see Figure 3.3. Roll-axis

δlc

δrc δle

Yaw-axis, canard- wing

δr Pitch-axis, delta-wing

δloe , δlie δroe

, δrie

Yaw-axis, delta-wing

Fig. 3.8. Deﬁnition of the control surface deﬂections

46

M. Hagstr¨om Table 3.2. Control surface deﬂection limits Control Surface: Min. [◦ ] Max. [◦ ] Angular Rate [◦ /s] Canard −55 25 ±50 Rudder −30 30 ±50 Elevons −25 25 ±50 Leading Edge Flap −10 30 ±20

3.2.7

Sensors

The modelling of the sensors is identical to the models of sensors in the HIRM model [3]. In ADMIRE, models of air data sensors (VT , α, β, h), inertial sensors (pb , qb , rb , nz ) and attitude sensors (θ, φ) are implemented. • Air data sensors:

1 · ξ, 1 + 0.02 ·s

(3.9)

1 + 0.005346 ·s + 0.0001903 ·s2 ·ξ 1 + 0.03082 ·s + 0.0004942 ·s2

(3.10)

1 ·ξ 1 + 0.0323 ·s + 0.00104 ·s2

(3.11)

ξsensed (s) = where ξ = [VT , α, β, h]T . • Inertial sensors: ξsensed (s) = where ξ = [pb , qb , rb , nz ]T • Attitude sensors: ξsensed (s) = where ξ = [θ, φ]T 3.2.8

Model Uncertainties

The parametric uncertainties of the model are; conﬁguration, aerodynamic, sensor and actuator uncertainties. Table 3.3 contains the parameters, their nominal values, upper and lower bounds, units and a description. The nominal values of the parameters are stored within the model, and only values in the range [min;max] should be used. Due to coupling, the values in the table are only valid if one uncertainty is used. If more aerodynamic uncertainties are used simultaneously, the following corrections should be made δk,2 = 0.62 ·δk,1 , δk,3 = 0.46 ·δk,1 , δk,4 = 0.37 ·δk,1 ,

(3.12)

where δk, j is the k’th uncertainty variable in the case with j uncertainties. This means that if for instance δCmα and δCmq are applied simultaneously, the correct values of the uncertainties should be 0.62 ·δCmα and 0.62 ·δCmq . The parametric uncertainties in ADMIRE that are implemented into the code are presented below. The values with parameters are denoted with an asterisk (*), and the nominal values are without asterisk.

The ADMIRE Benchmark Aircraft Model

47

Table 3.3. Measurement errors and parameter uncertainties Longitudinal Uncertainty Range δαerr [-2.0,2.0] deg [-0.08,0.08] δMerr [-0.15,0.15] δxcg δIyy [-0.05,0.05] [-0.1,0.1] δCmα [-0.10,0.10] δCmq δCmδey [-0.01,0.01] δCmδei [-0.03,0.03] δCmδne [-0.02,0.02] δmass [-0.2,0.2]

• Aircraft mass:

m∗ = (1 + δm) · m

(3.13)

x∗cg = xcg + δxcg y∗cg = ycg + δycg z∗cg = zcg + δzcg

(3.14)

• Centre of gravity position:

• Inertial data:

• Roll moment coefﬁcients:

∗ Ixx ∗ Iyy ∗ Izz ∗ Ixz

Ixx · (1 + δIxx ) Iyy · (1 + δIyy ) Izz · (1 + δIzz) Ixz · (1 + δIxz )

Cl∗δ ai Cl∗δr

= Clδ + δClδ · δai

Cm∗ α Cm∗ q Cm∗ δ n Cm∗ δ ey Cm∗ δ

= Cmα + δCmα · α = Cmq + δCmq · qˆ = Cmδn + δCmδn · δn = Cmδey + δCmδey · δey = Cmδ + δCmδ · δei

ei

• Yaw moment coefﬁcients:

= = = =

Cl∗β = Clβ + δClβ · β Cl∗p = Cl p + δCl p · pˆ Cl∗δ = Clδay + δClδay · δay ay

• Pitch moment coefﬁcients:

Lateral Uncertainty Range δαerr [-2.0,2.0] deg δMerr [-0.08,0.08] δβerr [-2.0,2.0] deg δycg [-0.10,0.10] δIxx [-0.20,0.20] δIzz [-0.08,0.08] δClβ [-0.04,0.04] δCl p [-0.10,0.10] δClr [-0.10,0.10] δCnβ [-0.04,0.04] δCnp [-0.10,0.10] δCnr [-0.04,0.04] δCnδna [-0.01,0.01] δCnδr [-0.02,0.02]

ai

(3.15)

(3.16)

ai

= Clδr + δCldr · δr

ei

ei

(3.17)

48

M. Hagstr¨om

• Sensors:

Cn∗basic Cn∗β Cn∗r Cn∗δ na Cn∗δ ay Cn∗δ ai Cn∗δ r

VT∗sensed ∗ Msensed α∗sensed β∗sensed h∗sensed

= Cnbasic + δCn0 = Cnβ + δCnβ · β = Cnr + δCnr · rˆ = Cnδna + δCnδna · δna = Cnδay + δCnδay · δay = Cnδ + δCnδ · δai ai ai = Cnδr + δCnδr · δr

(3.18)

= VT + δVTerr = = = =

VT∗ sensed ∗ acalc (hsensed ,VT∗ ) sensed 1 1+0.02 · s 1 1+0.02 · s 1 1+0.02 · s

+ δMerr

· (α + δαerr ) · (β + δβerr ) · (h + δherr )

(3.19)

• Actuators: (The transfer functions are the same in the model for all actuators.) ξ=

1 · ξcom 1 + (0.05 + δdcbw) · s

(3.20)

where ξ = δrc , δlc , δroe , δrie , δlie , δloe , δr . In the SIMULINK model the actuator transfer functions are preceded by position and rate saturation blocks. 3.2.9

Atmospheric Model

The atmosphere model is the International Standard Atmosphere (ISA), [103]. Only the density and the speed of sound are calculated. The atmosphere is assumed to be dependent on the altitude only. T = T0 + hTgrad ⎧ g ⎨ p T RTgrad h ≤ 11000 m 0 T0 p= ⎩ − g(h−11000) RT0 p0 e h ≥ 11000 m −0.0065 K/m h ≤ 11000 m Tgrad = 0.0 K/m h ≥ 11000 m 288.15 K h ≤ 11000 m T0 = 216.65 K h ≥ 11000 m 101325.0 Pa h ≤ 11000 m p0 = 22632.0 Pa h ≥ 11000 m p ρ= RT Vt M= √ κRT where R = 287, κ = 1.4 and g = 9.81

(3.21)

The ADMIRE Benchmark Aircraft Model

49

Turbulence ADMIRE is prepared for the use of atmospheric turbulence/wind. The available inputs are udist , vdist , wdist and pdist . The ﬁrst three correspond to body referenced wind disturbance in their respective axes and the last is a rotation contribution around the x-axis in SB . Turbulence is not currently implemented in ADMIRE.

3.3 Flight Control System With ADMIRE comes a simple FCS which provides basic stability and sufﬁcient handling qualities within the operational envelope. Although the aerodynamic model envelope is valid up to 20 km, normal ﬂight operation is signiﬁcantly lower. However, in this book several other controllers are presented and applied to ADMIRE and future versions of the simulation model will most likely come with a choice of FCS. As a guide to the design, report [62] was helpful. The FCS contains a longitudinal and a lateral part. The function of the longitudinal controller is pitch rate control (qcom ) below Mach number 0.58 and load factor control (nzcom ) above Mach number 0.62. A blending function is used in the region in between, in order to switch between the two different modes. The longitudinal controller also contains a speed control (VTcom ). The lateral controller enables the pilot to perform roll control around the velocity vector of the aircraft (pwcom ) and to control the sideslip angle (βcom ). The FCS is designed in 29 trim conditions using standard linear design methods. The pilot control inceptors are longitudinal (Fes ) and lateral (Fas ) stick deﬂection, rudder pedal deﬂection (Frp ) and throttle stick setting (Tss ). For simplicity, linear stick gradients are used. The maximum longitudinal stick deﬂection is asymmetric, i.e. it is possible for the pilot to pull a larger command than to push. The control selector (CS) is used to distribute the three control channels, u p , uq and uβ , out to the seven control actuators used by the FCS. For pitch, the CS is calculated using the method proposed in [62]. A scheduling of the CS is done by using the Mach number and the altitude. Since the ﬂight controller is designed in a number of discrete points in the envelope, the gains in the FCS must be adjusted when the aircraft is operating between the initial design points. In ADMIRE all FCS gains and trim conditions are scheduled with the altitude and Mach number. In Figure 3.10 an example of a scheduled gain can be found. In order to model the time delays present in an onboard computer implementation of the control laws, transport delays of 20 ms have been added to the actuators. 3.3.1

Longitudinal Controller

The longitudinal controller has two parts; a pitch and a speed controller. The speed controller is basically a gain and a lead ﬁlter. The controller maintains the commanded speed. The pitch controller has two different modes of functionality. At lower speeds (M < 0.58) the function of the controller is to minimise the tracking error in the commanded pitch rate, which is generated by the pilot. At higher speeds (M > 0.62) the

50

M. Hagstr¨om ADMIRE − Control Law 1 drc

[Altitude]

2 dlc 3 droe

[Mach] 4 drie FCS_cs

5 dlie 6 dloe

3 FCS_Fas 4 FCS_Frp

7 dr

FCS_lat Adding trim values

10 FCS_y

FCS_cs fcsq FCS_lat

FCS_lat_p

5 FCS_dle 6 FCS_ldg

8 dle 9 tss 10 ldg 11 dty 12 dtz

FCS_long

13 u_dist

7 FCS_dty 8 FCS_dtz 9 FCS_dist

14 v_dist 15 w_dist 16 p_dist FCS_ae_tv

1 FCS_Fes 2 FCS_Vt

FCS_long

FCS_Vt_in FCS_Vt_in FCS_Fes_in FCS_Fes_in FCS_Frp_in FCS_Frp_in FCS_Fas_in FCS_Fas_in

Fig. 3.9. An overview of the control law implementation in ADMIRE

controller tracks the commanded load factor. For Mach numbers in between, a blending is performed. The pitch rate controller consists of a stabilising inner loop. The pitch rate and the angle of attack are fed back to the control selector and an outer loop where the tracking error, and the integrated value of it, are fed forward to the control selector. Due to the chosen design method only the deviation values from the trimmed ﬂight condition are fed to the gains. This means that the controller only commands deviation from the trimmed actuator settings. The trimmed values are added in front of the actuators. The purpose of the alpha-limiter is to provide a pitch rate command to the controller when the prescribed limit of angle of attack is exceeded. How this is done is described in [3]. The load factor controller has the same principal structure as the pitch rate controller, described above, except that nz is fed back in the outer loop and it does not contain any alpha-limiter. The longitudinal control laws were synthesised using Pole Placement Methods (PPM). 3.3.2

Lateral Controller

The lateral control system is a so-called lateral-directional control augmentation system (CAS). The body-axis roll rate is fed back to the ailerons to modify the roll-subsidence

The ADMIRE Benchmark Aircraft Model

51

Fig. 3.10. Example of scheduled gain

mode. Closed-loop control of roll rate is used to reduce the variation of roll performance with ﬂight conditions. Figure 3.11 showes the structure of the implementation. The calculation of the ampliﬁcation factors are implemented in a C-function. FCS_lat_fcslat_Fg

FCS_lat_Altitude 1

Level III

FCS_lat_Mach 2 FCS_lat_Fas 3

FCS_lat_fcslat_Fg

Calc of lateral amplfication factors

FCS_lat_p FCS_lat_p FCS_lat_fcslat

Dot Product Fg

FCS lateral stick gradient N to rad/s

FCS_lat_stick

FCS_lat_p 1 Sum4

FCS_lat_stick_p Phase Lag Filter Roll (with initial state)

1

FCS_lat_stck_p

0.25s+1 Dot Product G_roll

3 FCS_lat_pw

U

U(E)

Factor for Aileron−Rudder Interconnect

Sum3

delta p_meas

Dot Product G_ARI FCS_lat_ped_beta

FCS_lat_Frp 4

FCS lateral pedal force N to beta rad

FCS_lat_x0 6 FCS_lat_y 5

FCS_lat_ped_beta 1 Dot Product G_beta

FCS_lat_ped

−K−

2

Gain1

Gain

beta_error U U x Sum5

Sum

Selector [p r Vt alpha phi] U

2 Dot Product G_beta_dot

U(E) Mux

9.81 g

FCS_lat_beta

U(E)

delta beta_meas

U(E)

Sum2

beta ot f(u)

g beta_dot_ estimation

d

Sum1

Gain2

5s 5s+1 Wash−Out Beta_dot (with initial state)

FCS_lat_beta_error FCS_lat_beta_error

Fig. 3.11. The structure of the lateral controller

The inner feedback loop in the rudder channel provides roll damping by feeding back an approximation of the wind-axis yaw rate to the rudder. The wind-axis yaw rate is washed-out so that it operates only transiently and does not contribute to a control

52

M. Hagstr¨om

error when steady yaw rate is present. The yaw-rate feedback is equivalent to β˙ feedback when φ and β are small. When necessary the pilot can command a steady sideslip to the aircraft, because rudder inputs are applied via the rudder pedal gradient to the rudder actuator. The control system will tend to reject this disturbance input, so that the desirable effect of limiting the sideslip will be achieved. The outer feedback loop in the rudder channel provides Dutch-roll damping through sideslip feedback to the rudder. The sideslip contributes to a control error and makes it possible to control the sideslip. The cross-connection is known as the aileron-rudder interconnect (ARI). Its purpose is to provide the component of yaw rate necessary to achieve a wind-axis roll. The lateral control laws were also synthesised using PPM.

Acknowledgements We would like to acknowledge all those who contributed to the development of the model: Hans Backstr¨om (Saab), David Bennet (BAe Systems), Binh Dang-Vu (ONERA), Holger Duda (DLR), Gunnar Duus (DLR), Chris Fielding (BAe Systems), Georg ˚ Hyd´en (FOI), Fredrik Johansson (FOI), Hoﬁnger (EADS), Gunnar Hovmark (FOI), Ake Mangesh Kale (University of Southampton), Harrald Luijerink (TU Delft), Torbj¨orn Nor´en (FOI), Martin N¨asman (FMV), Lars Rundqwist (Saab), Anton Vooren (Royal Norwegian Air Force), David Alan Weaver (FOI), Lars Forssell (FOI), Ulrik Nilsson (FOI) and others who we have forgotten to mention here. This text is based on earlier versions of ADMIRE documentation, [71]. The work of Lars Forssell and Ulrik Nilsson is also acknowledged.

A List of Symbols a bref cref Cl Cl β Cm Cmα Cmq Cn Cnβ Cnr CD CL CT Fas Fes

speed of sound (m/s) Wingspan (m) Mean aerodynamic chord (m) Coefﬁcient of rolling moment Rolling moment coefﬁcient derivative with respect to β Coefﬁcient of pitching moment Pitching moment coefﬁcient derivative with respect to α Pitching moment coefﬁcient derivative with respect to q Coefﬁcient of yawing moment Yawing moment coefﬁcient derivative with respect to β Yawing moment coefﬁcient derivative with respect to r Coefﬁcient of drag Coefﬁcient of lift Coefﬁcient of tangential force Force aileron stick Force elevator stick

The ADMIRE Benchmark Aircraft Model

Frp FXaero FYaero FZaero g h Ix Ixy Ixz Iy Iyz Iz m M nx , ny , nz pb pdem qb qdem rb Sref t Tss u0fcs(1) u0fcs(2) u0fcs(3) u0fcs(4) ub , vb , wb u p ,uq ,uβ VT x, y, z xcg , ycg , zcg xv , yv , zv α β γ δ δlc δldg δle δlie δloe δr δrc δrie

Force rudder pedal Total aerodynamic force in body-ﬁxed x-axis Total aerodynamic force in body-ﬁxed y-axis Total aerodynamic force in body-ﬁxed z-axis Acceleration due to gravity (m/s2 ) Altitude (feet or m) x body moment of inertia (kg ·m2 ) x-y body axis product of inertia (kg ·m2 ) x-z body axis product of inertia (kg · m2 ) y body axis moment of inertia (kg · m2 ) y-z body axis product of inertia (kg · m2 ) z-body moment of inertia (kg ·m2 ) Aircraft total mass (kg) Mach number Load factor along x-, y- and z-axes respectively (g) Body-ﬁxed roll rate (deg/s) Demanded roll rate (deg/s) Body-ﬁxed pitch rate (deg/s) Demanded pitch rate (deg/s) Body-ﬁxed yaw rate (deg/s) Wing surface (m2 ) Time (s) throttle stick setting trim value of pitch stick force commanded speed trim value of roll stick force (usually zero) trim value of pedal force (usually zero) Body-ﬁxed velocities along x-, y- and z-axes respectively Control channel roll, pitch and yaw Total velocity (m/s) Earth axes positions (m) Center of gravity location along x-, y- and z-axes respectively Positions in vehicle carried reference frame (m) Angle of attack (deg) Angle of sideslip (deg) Flight path angle (deg) Vector of system parameters Left canard deﬂection (deg) Landing gear deﬂection Leading edge ﬂap deﬂection (deg) Left inner elevon deﬂection (deg) Left outer elevon deﬂection (deg) Rudder deﬂection (deg) Right canard deﬂection (deg) Right inner elevon deﬂection (deg)

53

54

δroe δth δtv φ θ ρ ψ (˙)

M. Hagstr¨om

Right outer elevon deﬂection (deg) Horizontal thrust vectoring Vertical thrust vectoring Bank angle (deg) Pitch angle (deg) Density of air (kg/m3 ) Heading angle (deg) Derivative with respect to time

4 Nonlinear Flight Control Design and Analysis Challenge Fredrik Karlsson Flight Control System Department, Saab AB, SE-581 88 Linkoping, Sweden [email protected]

Summary. This chapter gives a description of the requirements on the design and analysis of new Flight Control Laws (FCL’s) for the ADMIRE model. The ADMIRE model is currently augmented with “conventional” FCL’s. These are to be replaced with new FCL’s based on new non-linear design methods. The current ADMIRE FCL’s shall be used for the purposes of comparison with the new FCL’s. The objective of the design task is to show the potential of different non-linear methods for aircraft ﬂight control.

4.1 Introduction Previous ﬂight mechanics action groups within the GARTEUR community have looked into different aspects of aircraft control [149] and clearance [68] . The purpose of this group is to show the potential of non-linear methods for design and analysis. The intention is to aim high and reach for the possibly impossible: • the “perfect” general method that can be used generically to design a general FCL, • a single FCL that covers the entire envelope (possibly requiring less gain-scheduling than is common today), • robust FCL’s with high performance, • fast methods for conceptual design, • reduced time and effort for design and analysis compared to current methods, • reduced numbers of simulations needed in the validation and clearance process, and • understandable methods for the average FCL designer. The design teams’ task is to replace the current Flight Control Laws in the ADMIRE model and to demonstrate the differences between the new Flight Control Laws and the old ones.

4.2 Description of ADMIRE This section is included only for information and to give an overview of the ADMIRE model. The model is further described in Chapter 3 and [71]. The model is implemented, together with tools for trimming and linearization in MATLAB/Simulink. The ADMIRE is a generic model of a small single-seat ﬁghter aircraft with a deltacanard conﬁguration. The ADMIRE’s ﬂight operational envelope of [71] is up to Mach 1.2, an altitude up to 6 km, an angle of attack up to ±90 degrees and a sideslip angle D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 55–65, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

56

F. Karlsson

of up to ±20 degrees. The aerodata envelope of ADMIRE is however wider. The ADMIRE contains twelve states (VT , α, β, pb , qb , rb , ψ, θ, φ, xv , yv , zv ) plus additional states due to the presence of actuators, and sensors. Available control effectors are left and right canard (δlc and δrc ), leading edge ﬂap (δle ), four elevons (δloe , δlie , δrie and δroe ), rudder (δr ) and throttle setting (δT ). The model is also equipped with thrust vectoring capability (δth and δtv ) and a choice to have the landing gear up or down (δlg ). The thrust vectoring capability shall not be used in the present design task and the landing gear shall be retracted. The model is prepared for the use of atmospheric turbulence as external disturbances. The ADMIRE is currently augmented with Flight Control Laws in order to provide stability and sufﬁcient handling qualities within the operational envelope (altitude ≤ 6 km, Mach ≤ 1.2). The current FCL contains a longitudinal and a lateral part. The longitudinal controller provides pitch rate control below Mach 0.58. For speeds greater than or equal to Mach 0.62 it provides normal load factor control. The corner speed is by this deﬁnition close to Mach 0.6. In the speed region between Mach 0.58 and 0.62, a blending pitch rate control and normal load factor control is performed. This is not a strict deﬁnition of corner speed, but a satisfactory approximation. The corner speed in fact increases (in terms of Mach number) with altitude. There is an angle of attack limiting functionality active during the pitch rate mode. The longitudinal controller also contains a very rudimentary speed controller. The lateral controller enables the pilot to perform initial roll control around the velocity vector of the aircraft as well as angle of sideslip control. Sensor models used by the ﬂight control system are incorporated in the model, together with a 20 ms computer delay on the actuator inputs, implemented as Pade approximations. The ﬂight control system sampling frequency is 100 Hz. There is also a possibility to vary some parameters and uncertainties within given tolerances, in order to facilitate robustness analyses. The available uncertainties consist of inertia, aerodynamic, actuator and sensor uncertainties. 4.2.1

Model Data and Envelope

There is no conﬁguration description available except for a simple schematic picture, see Fig. 3.1. Aircraft conﬁguration data used in ADMIRE are described in Table 3.1 The data gives an idea about the size of the aircraft. The mass and inertia are a function of the percentage of fuel onboard, and in the nominal case the mass represents 60% of fuel loaded. The ADMIRE’s centre of gravity is located at the aerodynamic reference point in the x-axis and slightly above the reference point in the z-axis. There is of course the possibility of changing the values of the centre of gravity position, mass and mass distribution. If this is done, however, it must be borne in mind that the current FCS is designed for the nominal model. How and within which bounds the conﬁguration parameters can be changed is described in detail in [71].

4.3 Design Objectives The objective of the design is to ﬁnd a new controller for ADMIRE that can make the most of the aircraft within its physical limitations while still ensuring that it is

Nonlinear Flight Control Design and Analysis Challenge

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controllable throughout the entire intended ﬂight envelope. The new controller shall be designed using non-linear methods. The control surfaces shall, in some point of the ﬂight envelope, be used to their maximum in terms of deﬂection and rate of deﬂection. This would require that the ’worst’ manoeuvre in the worst envelope point is found. That can be considered as a separate task, so there is no requirement to prove that such a point has been found. In other parts of the envelope, the deﬂection will certainly be less than the maximum. Some margin to maximum deﬂection and rate saturation should, however, be left for robustness reasons. In general, control surface deﬂection and rate saturation are acceptable as long as there is stability. The maximum control surface deﬂection rate and maximum deﬂection is given in the ADMIRE documentation [71]. The aircraft shall be care free, i.e. it shall not be possible to encounter a departure (loss of aircraft control) regardless of the pilot’s input command. The aircraft shall also have good handling qualities (FQL 1), which means that pilot compensation is not a factor in achieving the desired performance. The aircraft shall be highly manoeuvrable with a maximum roll performance of the order of 300◦/s at high subsonic speed and 1 g, a maximum normal load factor of 9 g, and a maximum angle of attack of 30◦ . There may be difﬁculties with lateral stability at high angle of attack, which could result in a high angle of sideslip. If that is encountered, the maximum angle of attack can be reduced to 26◦ . The minimum normal load factor shall be -3 g and the minimum angle of attack -10◦. The increase rate of normal load factor shall be in the order of 9 g/s at the most. The increase rate of normal load factor shall be at least 4 g/s above corner speed. The FCS shall limit the angle of attack and the normal load factor. A transient 20% overshoot over the limits is acceptable, but it should be less than 5%. The turn performance shall be the highest possible (maximum pitch rate at least in the order of 25◦ /s) at corner speed. Corner speed is deﬁned as the speed where the available instantaneous turn rate is the highest, but also the speed where both maximum angle of attack and maximum normal load factor can be achieved as illustrated in Fig. 4.1. The corner speed, in terms of Mach number, will vary with altitude. Above corner speed, maximum normal load factor will limit the pitch performance and below corner speed, maximum angle of attack will limit the pitch performance. The performance may be reduced by the controller in parts of the envelope to maintain a departure free aircraft. The design envelope is from Mach 0.3 up to 1.4 and altitude from 100 m up to 6000 m. The design shall focus on the altitude 1000 m and 3000 m. Transonic (Mach number in the range 0.9 to 1.1) is an important part of the ﬂight envelope for the Flight Control Laws. Any pitch-up transients due to speed changes shall be minimized. This is of most interest in the transonic region and around corner speed. Unfortunately, the aerodynamic model supplied here does not cover all the complicated aerodynamic phenomena in the transonic region which makes that part of the envelope so challenging from a ﬂight control system point of view. The design shall cover an angle of attack from −10◦ to +30◦ (alternatively reduced to 26◦ as stated on page 61, normal load factor from -3 g to +9 g and angle of sideslip from -10◦ to +30◦ . The angle of sideslip shall be kept small during roll maneuvers. The pitch stick input shall be from -7◦ to +11◦ . The roll stick input shall be from ◦ -8 to +8◦ .

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F. Karlsson

corner speed

AoA,nz

AoA nz

Speed

Fig. 4.1. AoA and normal load factor limits versus speed. Corner speed is marked with a line.

7000

6000

Altitude [m]

5000

4000

3000

2000

1000

0 0.2

0.4

0.6

0.8 Mach number

1

1.2

1.4

Fig. 4.2. Approximate ﬂight envelope for the design. Corner speed and transonic region are illustrated.

The requirements of the U.S. ”Military Speciﬁcation for the Flying Qualities of Piloted Airplanes” (MIL-F-8785C) [169] shall be met. This speciﬁcation gives requirements on e.g. longitudinal manoeuvring characteristics in chapter 3.2.2 and lateraldirectional mode characteristics in chapter 3.3.1. Roll performance requirements are given in 3.3.4.1. Flight phase category A shall be considered for aircraft class IV. The aircraft performance and characteristics shall be veriﬁed with simulations in a nonlinear model of the aircraft. The departure resistance shall be veriﬁed with aggressive pitch and lateral manoeuvring.

Nonlinear Flight Control Design and Analysis Challenge

4.3.1

59

Design Constraints

High frequency gains must be reasonable to reduce the risk for structural coupling in a ﬁghter aircraft. • In this design task, gain from pitch rate to control surfaces shall be less than 9 dB for frequencies above 5 Hz.

4.4 Evaluation The functionality of the proposed controller, both with respect to stability and performance, should be validated by the design team. In order to simplify the clearance process only a limited amount of analysis work is required. The design teams are encouraged to suggest other alternatives to evaluate stability and performance. The descriptions in [68] and [119] give a good understanding of the industrial process for clearance of Flight Control Laws. Only a fraction of the entire clearance process is used here for evaluation. The objective of the evaluation shall also be to identify dangerous ﬂight cases, i.e. cases where the risk of exceeding angle of attack or load factor limit is the highest. Comparison shall also be made with the current ADMIRE controller. Performance shall be demonstrated by performing different manoeuvres using the full nonlinear simulation of ADMIRE: 1. Rapid deceleration turn starting at supersonic speed (M1.2) with roll to the left and full pitch stick command and throttle slammed to idle. The pilot model shall aim for constant altitude. The turn is aborted at subsonic speed (M0.8) at 1000 m. This should demonstrate the problems with gain scheduling in the transonic region in the ﬂight envelope. This should also be demonstrated with errors in Mach number measurement. 2. Rapid deceleration turn starting at M0.9. The maneuver is started with roll to the left and full pitch stick command and throttle slammed to idle. The pilot model shall aim for constant altitude. The pilot model shall aim for maximum AoA/nz (maximum according to section 4.3) during the deceleration turn from M0.9 to minimum speed. This is to demonstrate the capability of the ﬂight envelope protection system, which should result in care free handling of the system. a) The deceleration turn performed at 2g. b) The deceleration turn performed at 4g. 3. Manoeuvre starting at the same condition as manoeuvre 2. Rapid roll at simultaneously high normal load factor and high angle of attack, i.e. at corner speed. This manoeuvre is chosen in order to stress the effects of the dynamic coupling between the roll and pitch axis. The angle of sideslip and the lateral load factor shall be kept low by the ﬂight control laws. 4. Full pitch stick step command both forward and backward respectively, as shown in the upper ﬁgure in Fig. 4.4, to demonstrate care free handling. Fulﬁlment or violation of a time response criterion above corner speed according to Fig. 4.3 shall be demonstrated for this manoeuvre.

60

F. Karlsson

5. Full roll stick step command both left and right respectively to demonstrate care free handling. Maximum achieved angle of sideslip shall be illustrated. Manoeuvres 1, 2 and 3 require pilot models. All design teams should use the same pilot models. The manoeuvres above should also be performed with different, but limited, set of uncertainties. A time response criterion is suggested in Fig. 4.3. Time response criterion

1.2

Normalized normal load factor response

1.1 1 0.9

0.63

0

0

0.3

1.2

3.5 Time [s]

Fig. 4.3. Example of a time response criterion

The steps in the following subsections (4.4.1 to 4.4.4) should be performed as a part of the design process and the result should be documented together with the description of the design methodology. Section 4.4.5 deﬁnes the compulsory maneuvers to be performed and demonstrated in the separate book chapters on ﬂight control law design. 4.4.1

Analysis of Stability Margin and Eigenvalues

In the design process, it is required to identify any ﬂight cases (in terms of M, AoA, dynamic pressure or altitude) where unstable eigenvalues (i.e. those with positive real part) are found. Stability margin also need to be considered in the design. The criteria deﬁned in [120] can be a good support in the design process. The use of nonlinear analysis methods is encouraged in addition. 4.4.2

Deceleration Turn

This kind of manoeuvre is used to check aircraft behaviour in areas where the aerodynamics change rapidly with Mach/AoA (e.g. in the transonic region). This is especially

Nonlinear Flight Control Design and Analysis Challenge

61

important when the controller is scheduled with air data because the adaptation of the gains might then not be correct. Uncertainty in air data measurement should then be considered. 4.4.3

Rapid Roll

Important features to be checked for the nominal and the uncertainty case are maximum roll rates/overshoots, maximum sideslip generated during roll, roll angle overshoot when trying to stop the roll and variation of normal load factor during full stick rapid roll. An aircraft rolling around its velocity vector generates a pitch-up due to gyroscopic effects. As this pitch-up is proportional to the AoA it must be demonstrated that the aircraft will not depart at high AoA and that the available pitch down control power is sufﬁcient for all combination of uncertainties. For ADMIRE it is proposed to additionally include the rapid rolling investigation with respect to maximum generated sideslip and roll rate and maximum generated variation of normal load factor for 1 g (and possibly 3 g) rapid rolling entry conditions (to full roll stick applied in 0.1 s). It should be checked that the variations in normal load factor and the exchange between AoA and AoS are small. The tests should be made at low dynamic pressure and high AoA for the exchange between AoA and AoS. The tests for variation of normal load factor should be made at high dynamic pressure. 4.4.4

Commanded AoA/nz

Identify all ﬂight cases during the pull-up manoeuvres deﬁned below where the positive AoA/nz limits are exceeded. This shall be done for the nominal case and for the uncertainty case. Here the combination of uncertainties which yields the largest exceedance need to be identiﬁed. Two aircraft responses shall be assessed, a full stick rapid pull and a pull in 3 s. The pilot commands are: • A full stick rapid pull on the longitudinal stick that brings the stick from the initial position to the maximum amplitude in the aft direction. (Full stick deﬂection within 0.1 s gives the rate of stick deﬂection.) See top ﬁgure in Fig. 4.4. • A pull in 3 seconds, i.e. a ramp command that brings the stick from the initial position to full aft longitudinal stick in 3 seconds. See bottom ﬁgure in Fig. 4.4. Both commands must be applied from a trimmed condition of straight and level ﬂight, and the simulation should be run for 10 seconds if possible. Otherwise the simulation should be stopped when the pitch attitude angle reaches 90 degrees or the speed is below minimum allowed speed. Note that this means that the SL (Straight and Level) ﬂight routine from ADMIRE3.4e must be used! Push-over manoeuvres are performed in the same manner to test the behavior at the negative AoA and nz limits, as shown in Fig. 4.5. The simulations shall be stopped before the aircraft hits the ground. 4.4.5

Compulsory Manoeuvres for Evaluation

The manoeuvres described in Chapter 4.4 and listed here are to be performed by the design teams with their new ﬂight control system concept. Points in bold shall also be performed with the old ADMIRE ﬂight control system to demonstrate the differences.

F. Karlsson Full stick rapid pull

Stick deflection [deg]

15

10

5

0 −1

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

Full stick pull in 3 seconds

Stick deflection [deg]

15

10

5

0 −1

0

1

2

3

4 5 Time [s]

Fig. 4.4. Pilot commands for testing largest exceedance of AoA and nz limits

Full stick rapid push

Stick deflection [deg]

0

−5

−10

−15 −1

0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

Full stick push in 3 seconds 0 Stick deflection [deg]

62

−5

−10

−15 −1

0

1

2

3

4 5 Time [s]

6

Fig. 4.5. Commands for testing largest exceedance of negative AoA and nz limits

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• • • • • •

Manoeuvre 1, at 3 and 6 km. Manoeuvre 2, at 3 and 6 km. Manoeuvre 2a, at 3 and 6 km. Manoeuvre 2b, at 3 and 6 km. Manoeuvre 3, at 3 and 6 km. Manoeuvre 4, at 1 km at M0.3, M0.5, M0.7, M0.9, M1.1. Manoeuvre 4, at 4 km at M0.5, M0.7, M0.9, M1.1. Manoeuvre 4, at 6 km at M0.5, M0.7, M0.9, M1.1. • Manoeuvre 5, at 1 km at M0.3, M0.5, M0.7, M0.9, M1.1. Manoeuvre 5, at 6 km at M0.5, M0.7, M0.9, M1.1. Some manoeuvres shall also be performed with the same uncertainties as deﬁned in [120].

• Manoeuvre 1, at 3 and 6 km with Mach number measurement uncertainty, δM , -0.04 and +0.04. • Manoeuvre 2, at 3 and 6 km with AoA measurement uncertainty, δα, -2◦ and +2◦ . • Manoeuvre 3, at 3 and 6 km with centre of gravity uncertainty, δxcg , -0.15 m and +0.15 m. • Manoeuvre 4, at 4 km at M0.5, M0.7, M0.9, M1.1 with pitching moment uncertainty, δCmα , -0.1 and +0.1. • Manoeuvre 3, at 6 km at M0.5, M0.7, M0.9, M1.1 with yawing moment uncertainty, δCnβ , -0.04 and +0.04. 4.4.6

Wind Gust Response

Atmospheric disturbances like wind gusts have an important inﬂuence on aircraft motion. The aircraft behaviour when encountering such disturbances must fulﬁll certain requirements. This is an important aspect of control law design. Within this GARTEUR evaluation the aircraft response to a vertical wind gust of 5m/s shall therefore be demonstrated. The ﬂight path angle response should not exceed the boundaries shown in Fig. 4.6 during this manoeuvre. Wind gust modelling For the examination of atmospheric disturbances a wind model is used. The deﬁnition of different wind scenarios is done in an earth ﬁxed coordinate system. Both translatory and rotatory wind components can be conﬁgured generally as: ⎛ ⎞ ⎛ ⎞ uw pw v = ⎝ vw ⎠ , Ω = ⎝ q w ⎠ (4.1) ww g rw g In this evaluation only vertical translatory wind gust is considered, so all terms but ww in (4.1) are zero. For the wind gusts the “1-cosine” shape as proposed in [169] shall be used. A vertical constant “1-cosine” wind gust is given by: Vm ww = 2

w = 0, x < 0 w πx , 0 ≤ x ≤ dm 1 − cos dm ww = Vm , x > dm

(4.2)

64

F. Karlsson Vertical wind gust response

Normalized flight path angle

1

0.5

0

−0.5

0

5

10

15

Time [s]

Fig. 4.6. Vertical wind gust requirement

A vertical short time “1-cosine” gust is described by:

Vm ww = 2

ww = 0, x < 0, x > 2dm πx , 0 ≤ x ≤ 2dm 1 − cos dm

For this evaluation only the short time gust (4.3) shall be used. The shape of both gust types is illustrated in Fig. 4.7 below:

Vertical gust [m/s]

V_m

d_m Distance [m]

Fig. 4.7. Shapes of constant and short time “1-cosine” gust

(4.3)

Nonlinear Flight Control Design and Analysis Challenge

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Gust Amplitude and Length Gust amplitude, Vm , shall be chosen in this investigation to be 5 m/s. The gust length shall be tuned to the closed loop short period frequency, ωSP . Using the relation between distance and speed x = Vt (4.4) the following equation for the gust length is derived dm =

πV ωSP

(4.5)

The short period frequency shall be taken from the eigenvalue calculations for the closed loop longitudinal system (usually the conjugate complex eigenvalue with a frequency between 0.8 to 12). Implementation of Wind Model in the Nonlinear Simulation Aerodynamic forces and moments depend on the aircraft motion relative to the surrounding air which is inﬂuenced by wind. The motion relative to the air is described with the help of translatory and rotatory velocity components in aircraft ﬁxed coordinate system. ⎛ ⎞ ⎛ ⎞ ua pa vab = ⎝ va ⎠ Ωab = ⎝ qa ⎠ (4.6) wa b ra b These velocities are used as inputs for the calculation of aerodynamic forces and moments. They are obtained with the help of the inertial velocities that are basic states of the differential equation system solved in nonlinear simulation and the wind velocities, see (4.7) and (4.8). The wind velocities deﬁned in earth ﬁxed coordinates have to be transformed into the body ﬁxed coordinate system with the transformation matrix Mbg (4.9). ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ua uk uw ⎝ va ⎠ = ⎝ vk ⎠ − Mbg ⎝ vw ⎠ (4.7) wa b wk b ww g ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ pa pw pk ⎝ qa ⎠ = ⎝ qk ⎠ − Mbg ⎝ qw ⎠ (4.8) ra b rk b rw g The transformation matrix from earth in body ﬁxed coordinate system is as follows: Mbg = ⎛

⎞ cos ψ cos θ sin ψ cos θ − sin θ ⎝ cosψ sin θ sin φ − sin ψ cos φ sin ψ sin θ sin φ + cosψ cos φ cos θ sin φ ⎠ (4.9) cosψ sin θ cos φ + sin ψ sin φ sin ψ sin θ cos φ − cosψ sin φ cosθ cos φ

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Part II

Applications to the Airbus Benchmark

5 Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design Andres Marcos1, , Declan G. Bates2 , and Ian Postlethwaite2 1

Deimos-Space S.L., Madrid, Spain [email protected] University of Leicester, United Kingdom dgb3,[email protected]

2

Summary. In this chapter, a general nonlinear symbolic LFT modelling framework and its supporting LFT tools are presented. The modelling approach developed combines the natural modularity and clarity of presentation from LFT modelling with the ease of manipulation from symbolic algebra. It results in an exact nonlinear symbolic LFT that represents an ideal starting point to apply subsequent assumptions and simpliﬁcations to ﬁnally transform the model into an approximated symbolic LFT ready for design and analysis. The development of the framework is supported by a novel algebraic algorithm for symbolic matrix decomposition and two new LFT operations: nested substitution and symbolic differentiation.

5.1 Introduction New control developments in the ﬁeld of on-ground aircraft control are seen as a key technology to reduce the congestion of many airports while increasing safety during onground manoeuvres [52]. These new control developments require mathematical models that consider the aerodynamic forces but also, and more critically, the ground forces affecting the aircraft. The complexity of these ground forces make the modelling task challenging, with highly nonlinear effects and many uncertain parameters inﬂuencing the various phenomena considered. Consequently, it is necessary to develop modelling approaches which are capable of capturing all the complexity of the on-ground aircraft but which also allow simpliﬁcation, in a systematic manner, of the initial high-ﬁdelity model to obtain control-oriented (on-ground aircraft) models. Once the control laws have been synthesized and validated on the simpliﬁed mathematical models, it is required by public certiﬁcation authorities that the resulting control system also be validated in the most complex models. With this certiﬁcation objective in mind, many industries are recently making signiﬁcant efforts to develop and apply nonlinear synthesis and analysis techniques - see for example, the work reported in [68] for recent progress in the aerospace industry. One approach to this certiﬁcation problem, which has been very successful in practice, is to extend traditional linear design and analysis methods to address nonlinear problems. This is the basis of modern synthesis and analysis techniques such as gain

The ﬁrst author was a post-doctoral fellow at the University of Leicester supported by EPSRCUK research grant GR/S61874/01 when this research was performed.

D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 69–92, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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A. Marcos, D.G. Bates, and I. Postlethwaite

scheduling [132], linear parameter varying (LPV) control [19] and integral quadratic constraints [156] among others. It is noted that many of the above techniques work with models based on, or similar to, the linear fractional transformation (LFT) modelling paradigm [176]. Thus, it is desirable to develop a systematic nonlinear modelling framework based on LFT representations, which offers the ﬂexibility and modularity required to obtain the simpliﬁed models used for control design, but also connects these models to the original & more complex models used for certiﬁcation. In this chapter, such a modelling approach is presented. The basic idea of the approach is to rely on symbolic manipulations and LFT representations to provide the desired modelling ﬂexibility and ease of manipulation. In supporting such a modelling framework a novel algebraic matrix decomposition algorithm has been developed together with algorithmic formalizations for two new LFT operations: nested LFT substitution and symbolic LFT differentiation. Chapter 6 presents an application of this approach to an on-ground-aircraft. The layout of the chapter is as follows: Section 5.2 presents the supporting LFT tools and algorithm while Section 5.3 covers the proposed nonlinear symbolic LFT modelling framework.

5.2 LFT and Symbolic Modelling Support Tools In this section, the LFT tools developed in support of the proposed modelling framework are described. The novel symbolic matrix decomposition algorithm, called Logic-Horner-Tree (LHT), is detailed ﬁrst, including its extension to LFT modelling. Subsequently, the formalization of two new LFT operations are given: nested substitution and symbolic differentiation. Their proofs (which also provide their algorithmic implementations) are given in the appendixes. 5.2.1

Logic-Horner-Tree Algorithm

Matrix manipulation is one of the basic cornerstones of many ﬁelds in engineering and mathematics. For example, polynomial/matrix representations and manipulations are ubiquitous in many areas of mathematics and most computer algebra systems such as Mathematica [237] and Maple [95] rely on them. A typical objective (for example, in signal processing and control synthesis) when operating on matrices and polynomials, especially when these are multivariate, is to obtain an equivalent representation of reduced order (in terms of number of parameters and their repetitions). Ideally, the order should be minimal, but minimal representations are in general very difﬁcult to obtain except for some simple cases. In the case of LFT modelling, it is well-known that the problem of ﬁnding a minimal order representation is equivalent to a multidimensional realization [40] which remains an open problem. Indeed, most of the available LFT algorithms search for a minimal representation by exploiting the structure in the LFT models and performing algebraic factorizations on the multivariate matrices: for example, structured-tree decomposition [41], numerical matrix approaches [22], Horner factorizations [233] and symbolic linearizations [234] among others.

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In this section an algorithm is proposed to achieve a lower order representation for multivariate polynomial matrices. This proposed decomposition algorithm is called Logic-Horner-Tree (LHT) to highlight its connection to previous approaches [41, 233, 234] and to emphasize the inclusion of decision logic. Indeed, the algorithm generalizes those approaches, as it follows the layout of the structured-tree decomposition approach but uses the Horner factorization operation which has now been extended for multivariate polynomial matrices. The LHT algorithm is applicable to symbolic matrices whose coefﬁcients are either constants or monomials/polynomials in one or several parameters, i.e. symbolic multivariate polynomial matrices. The symbolic parameters might represent highly complex functions dependent on many other parameters. Furthermore, it can be extended to symbolic rational expressions P(δi ) using the generalized approach from [98] or the factorization result by [21, 147]: ˜ i )−1 N(δ ˜ i) P(δi ) = N(δi )D(δi )−1 = D(δ

(5.1)

˜ i ) and N(δ ˜ i ) are polynomial matrices on the variables δi . where N(δi ),D(δi ),D(δ The algorithm is divided into three main routines: information management (IM), factorization (Horner and afﬁne) and sum decomposition. The iterative combination of the last two routines yields a nested structure for the decomposition: (5.2) M = MB1 + MA1 = MB1 + L1 . . . (MBn + Ln M¯ An Rn ) . . . R1 The matrices Lii , M¯ Aii , Rii are obtained from the factorization of the primary matrix MAii which is obtained together with the secondary matrix MBii from the sum decomposition of M¯ Aii−1 (with MA1 = L1 M¯ A1 R1 ). Each of the algorithm steps is detailed next, starting with the deﬁnition of the metrics used and ending with the extension of the algorithm to LFT modelling. Metrics The main metrics used by the algorithm are the presence degree σ, the factor order f ac, the reduction order red, and the possible reduction order red pos. The above metrics are based on the following standard monomial and polynomial deﬁnitions. Given n symbolic parameters δ1 , δ2 , . . . , δn a monomial m is deﬁned as m = cδα1 1 δα2 2 . . . δαn n where c is a non-zero constant coefﬁcient and αi ∈ ZZ+ represents the corresponding power for the i-th parameter. The extension to negative powers is dii rect noting that δ−α = δˆ αi i where δˆ i = δ−1 i is considered a new symbolic parameter. A i polynomial p is given by a ﬁnite linear combination of k monomials, p = ∑kj=1 m j . The total degree of a monomial is deg(m) = ∑ni=1 αi ; the relative degree of a monomial deﬁned with respect to a parameter is given by deg δi (m) = αi ; the degree of a polynomial is deg(p) = max deg(m j ). The presence degree σ(δi ) is deﬁned as the number of times, including powers, a parameter δi appears in an expression (symbolic monomial, polynomial or matrix). It can be viewed as a polynomial, or matrix, extension of the relative degree of a monomial. The total σ degree is the sum of the σ(δi ) for all the symbolic parameters δi . The

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factor order of a parameter f ac(δi ) is the maximum power to which it can be factored out from an expression. The reduction order for a parameter red(δi ), is the largest reduction in the presence degree of an expression achievable through factorization of that parameter. Assuming there are k factorizable monomials (i.e. each monomial contains the parameter with a minimum order equal to f ac(δi )) in the expression, red(δi ) is given by (k − 1) f ac(δi ). The last metric, red pos (δi ), is similar to red(δi ) but considering there are l non-factorizable monomials in the expression: (k − l − 1) f ac(δi ). Information Management (IM) Routine The IM routine condenses the logic of the algorithm and allows for a completely automated procedure without the need to apply combinatorics to the symbolic parameter vector ordering (a key issue for algebraic symbolic decomposition algorithms). Its main objectives are (these are performed at different stages of the algorithm, see the algorithm pseudo-code in Appendix A): i) to gather the proper information at each step ii) to take a decision regarding operation and parameter ordering iii) to prepare the decomposition matrix for the chosen operation The information gathering pertains mainly to calculating the metrics, called METin the pseudo-code, presented above for all the speciﬁed symbolic parameters and also in identifying the polynomial coefﬁcients. This information is used to decide on the appropriate operation and to prepare the matrix accordingly. For example, even if a high factor order f ac(δk ) is identiﬁed along any row or column, if a higher possible reduction order red pos (δk ) is also obtained the corresponding symbolic parameter can be scheduled for a different operation, see example 1: ⎤ ⎡ δ1 δ23 δ23 δ23 ⎦, we could start using a direct afﬁne facExample 1. Given M = ⎣ 2 δ1 δ3 + δ2δ3 δ33 torization of δ3 , since f ac(δ3 )2row = 2 and f ac(δ3 )3row = 1 which yield red(δ3 ) = (k2row − 1) ∗ f ac(δ3)2row + (k3row − 1) ∗ f ac(δ3)3row = 1 ∗ 2 + 2 ∗ 1 = 4, and which results in M⎡dec1 whose ⎤ ⎡total presence ⎤degree is σ(Mdec1 ) = σ(M) − red(δ3 ) = 11: 1 0 0 δ1 δ23 1 1⎦ Mdec1 = ⎣0 δ23 0 ⎦ ⎣ 0 0 δ3 δ1 + δ2 δ3 δ23 Further manipulations on Mdec1 will yield a total σ = 10. On the other hand, noting that for the initial M the possible reduction order red pos(δ3 ) along the columns is 6, then an afﬁne factorization of δ3 can be performed along the 1st column followed by a sum decomposition ⎤ to ⎡get: ⎤ ⎡ δ3 0 0 δ1 δ3 0 2 ⎣ ⎦ ⎦ ⎣ 0 0 + δ3 δ3 Mdec2 = 0 1 δ1 0 δ2 δ3 δ33 The latter can be further manipulated (columns and row afﬁne factorization of δ3 ) to obtain a ﬁnal total σ = 8. This shows that by postponing afﬁne factorizations along a RICS

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particular direction and using sum decomposition ﬁrst, it might be possible to achieve larger reductions.

The parameter ordering, SIGMA - ORDERING, and the matrix preparation steps depend on the symbolic operation selected by the IM: 1. For Horner factorization, the ordering of the symbolic vector is given by the H ORNER - ORDERING step, and the matrix undergoes a ‘polynomial substitution’ step POLY- SUBSTITUTION. The symbolic vector H ORNER - ORDERING is based on the possible reduction order red pos for each parameter along each matrix dimension (i.e. each parameter results in two values: the sum along the rows and the sum along the columns). This ordering might not give the largest order reduction red for a speciﬁc coefﬁcient but will result in better matrix σ reduction, as will be shown later. In the POLY- SUBSTITUTION step (used before applying afﬁne factorizations), the IM performs a substitution of the sub-polynomials found in the matrix by dummy parameters. The sub-polynomials are the polynomial remainders (or quotients) obtained in the Horner factorization. These substitutions will ease and speed up the task of recognizing which parameters (symbolic and dummies) to afﬁne factorize. After the afﬁne factorization, the IM back-substitutes the sub-polynomials and expands the matrix to prepare it for the sum decomposition step. Furthermore, these last steps allow the IM to appropriately update the metrics for all the symbolic parameters. 2. For the sum decomposition, the IM forms a DECOMPOSITION - LIST that contains information for the sum decomposition routine, which attempts maximizing the reduction in the total σ degree for subsequent afﬁne factorizations. The list is ordered in decreasing possible reduction order red pos and when equal, sub-classiﬁed by σ degree (and if necessary ﬁnally by lexicographic order). Each row in the decomposition list is formed by the parameter number, its position (which row or column), the required metrics, and cells containing the indexes for the factorizable and nonfactorizable coefﬁcients along the speciﬁed row/column – these include summands in polynomial coefﬁcients. This splitting of the matrix coefﬁcients indicates to the sum decomposition routine which coefﬁcients should be assigned to the primary matrix MAii (i.e. factorizable coefﬁcients) or to the secondary matrix MBii (i.e. the non-factorizable). There is also a ’ CONFLICT- ANALYSIS ’ step related to the sum decomposition which is detailed later. After the factorization and the sum decomposition have been performed, the IM evaluates, using the FULLY- DEC and NONFULLY- DEC steps, if the primary matrix MAii can be further decomposed or if the ﬁrst of the non-decomposed MB j j≤ii secondary matrices should be decomposed. The priority is to decompose fully the primary matrix ﬁrst, and subsequently to start with the secondary matrices (there might be more than one, as each time the primary matrix is passed through the decomposition scheme it will generate a secondary matrix). Note that as the secondary matrices are selected for the decomposition they become primary, see Figure 5.1.

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A. Marcos, D.G. Bates, and I. Postlethwaite zz = 1

zz = 2

zz = 3

nested fact ii = 1

nested fact

nested fact ii = 2

ii = 3

nested fact

nested fact ii = 6

ii = 4

nested fact ii = 5

fully decomposed decomposition path main matrix secondary matrix

Fig. 5.1. Graph Logic Horner Tree (LHT) algorithm

Factorization Routine This routine is composed of two main operations: H ORNER - FACTORIZATION and afﬁne factorization A FF - FACTOR. The former is always followed by the later and only occurs if there are polynomial coefﬁcients. The Horner factorization approach is based on the Horner simpliﬁcation for polynomials. For the evaluation of a polynomial of degree k, it requires only k multiplications and k additions which is much less expensive than the number of multiplications for the expanded form [237]. It is also numerically more efﬁcient and accurate. The general univariate case is given by: P (δ) = an + δ · (an−1 + . . . δ · (a1 + δ · ao )) For the multivariate case, an ordering of the parameters must be given. This ordering is not unique and will affect the nesting and the achievable σ reduction. Therefore, all the possible cases should be tested (for n parameters this means n! which is very computationally expensive) unless some logic is used in the ordering. A matrix extension of the polynomial Horner factorization is proposed based on the symbolic vector ordering given in the H ORNER - ORDERING step detailed above. In this manner, see example 2, the emphasis is placed on decreasing the reduction order red of the matrix and not of each individual polynomial coefﬁcient: Example 2. Given M = δ31 + δ31 δ22 δ23 + δ22 δ23 δ22 δ23 , assume it is desired to apply Horner factorization in the {1, 1} coefﬁcient rather than sum decompositions (no afﬁne factorization exists for M). If the maximal order reduction (per element) is used in {1, 1}, the correct element to factorize ﬁrst is δ1 and the resulting matrix is: M1 = δ31 (1 + δ22δ23 ) + δ22δ23 δ22 δ23 . The logical next step is to sum decompose so that δ31 can be afﬁne factorized (i.e. shift the monomial δ22 δ23 in {1, 1} to another matrix, either together with {1, 2} or by itself). This yields a ﬁnal total σ = 11.

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If Horner factorization is performed in M at matrix level (i.e. evaluating impact on {1, 2} as well), the selected element is δ22 or δ23 (actually both but lets assume only one at a time), which yields: M2 = δ22 (δ31 δ23 + δ23 ) + δ31 δ22 δ23 , This can be sum decomposed pushing the monomial δ31 to the other summand-matrix, and using successive afﬁne factorizations of δ22 and δ23 we get a ﬁnal total σ = 10. In the A FF - FACTOR step, MAii = Lii M¯ Aii Rii , parameter ordering is not important since there is no inﬂuence from the factorization of one parameter on the factorization of the others. Nevertheless, for each parameter it is important to identify the ﬁrst direction along which to perform the factorization (i.e. left ≡ rows or right ≡ columns). Direction is important since, by virtue of the nature of factorization, it is mutually exclusive (i.e. once factorization of a parameter along one direction is performed, the possible reduction order red pos for that parameter along the other direction is decreased). This A FFINE - DIRECTION identiﬁcation is based on the following classiﬁcation logic (seven cases) which compares the total order reduction red and the total possible reduction red pos along one direction with those along the other direction: Cases 1-3: Factorize along rows : (1)redrow > redcol + red poscol (2)redrow > redcol & red pos row = red pos col (3)redrow = redcol = 0 & red pos row > red pos col Cases 4-6: Factorize along columns : (4)redcol > redrow + red posrow (5)redrow < redcol & red pos row = red pos col (6)redrow = redcol = 0 & red pos row < red pos col The 7th case occurs when none of the previous cases is satisﬁed. In this case it will be very difﬁcult and computationally expensive to ascertain along which direction to factorize. Hence, a ‘preview’ of the immediate effect the factorization along each direction has on the decomposition is performed (a ‘preview’ of only one step ahead is currently performed but this is a design decision). The classiﬁcation logic is similar to the above but based on the ‘future’ afﬁne factorizations along each direction: M = L f · M fL (left) and M = R f · M fR (right). This approach doubles the number of calculations (as afﬁne factorizations are performed in both directions and then only one is selected) but it maximizes the total order reduction red for the present and subsequent iteration. Sum Decomposition Routine The SUM - DECOMP step decomposes the matrix M¯ Aii into two summand-matrices, M¯ Aii = MAii+1 + MBii+1 , following the DECOMPOSITION - LIST and using a conﬂict logic CONFLICT- ANALYSIS to form the primary and secondary matrices. The CONFLICT- ANALYSIS operates as follows. For the ﬁrst row in the list, the sum decomposition assigns the factorizable coefﬁcients (for that parameter and speciﬁed matrix row or column) to MAii+1 and the non-factorizable coefﬁcients to MBii+1 .

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Subsequently, it moves through the list from top to bottom evaluating if there is any conﬂict, i.e. new factorizable coefﬁcients already placed in MBii+1 or non-factorizable coefﬁcients in MAii+1 . If there is no conﬂict it distributes the new coefﬁcients correspondingly and moves to the next row in the list. Otherwise, the possible reduction order red pos for the parameter / position being evaluated is re-calculated after removing the conﬂicting coefﬁcients. If the new red pos is better or equal than that for the next row in the list, the sum decomposition is performed with the updated coefﬁcients, otherwise the list is re-ordered to account for the new resulting row and the routine proceeds to the next row in the list, see the example: 2 δ1 δ31 δ22 δ22 Example 3. Given the matrix, M = , after analyzing the possible re0 δ1 δ2 + δ3 0 duction order red pos along the rows and columns for the two symbolic parameters, the following decomposition list is obtained: symb. δ1 δ2 δ1 δ2

pos red pos fact indx. non-fact indx. 1st row 2 [ 1,2 ] [3] 1st row 2 [ 2,3 ] [1] 2nd col 1 [ 1,2(1) ] [ 2(2) ] 2nd col 1 [ 1,2(1) ] [ 2(2) ]

After the ﬁrst row in the list is used, the status of the sum decomposition is M = MA 1 + MB 1 : 2 2 3 2 δ1 δ31 δ22 δ22 δ δ δ 0 0 0 δ22 = 1 1 2 + 0 δ1 δ2 + δ3 0 ∗∗ ∗ ∗ ∗ ∗ The second row in the list indicates the subsequent decomposition of the parameter δ2 along the ﬁrst row of M. It is immediately noticed that one of the factorizable indexes conﬂicts with an already assigned coefﬁcient in MB1 . Hence, no sum-decomposition is performed for this parameter and the routine passes to the next row in the table which indicates that the {1, 2} coefﬁcient and the 1st summand of the {2, 2} go to MA 1 : 2 2 3 2 δ1 δ31 δ22 δ22 δ1 δ1 δ2 0 0 0 δ22 = + 0 δ1 δ2 0 0 δ1 δ2 + δ3 0 0 δ3 0 Now, performing an afﬁne factorizations on MA1 yields a ﬁnal total reduction order of 8, which is the best reduction achievable for M.

LFT Modelling Application A paradigm shift in the modelling of dynamic systems occurred in the 1980’s with the introduction of modern robust control theory and its associated modelling framework, the linear fractional transformation [176]. An LFT is generally used to represent an uncertain system using two operators in a linear feedback interconnection:

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FU (M, ∆) = M22 + M21 ∆(I − M11 ∆)−1 M21 where M is the nominal, known part and ∆ represents the system’s uncertainty, see Figure 5.2:

∆ z

w

M y

u

Fig. 5.2. Linear fractional transformation LFT (M, ∆)

In the case of parametric uncertainty (the case of interest in most aircraft applications), the resulting structure is ∆ = diag(δ1 I1 , δ2 I2 , . . . δn In ) where Ii represents an identity matrix of dimension equal to the number of repetitions of the ith parameter. The order of the LFT is then said to be the dimension of ∆, and is an important consideration for the control synthesis and analysis methods currently available in robust control. Many realistic robustness analysis problems easily result in very high order LFTs and therefore it is vital to have efﬁcient (and automated) tools which can compute minimal, or at least close to minimal, representations of these systems. It is noted that the order of an LFT derived from a symbolic expression (where the symbolic parameters are considered uncertain parameters) is equal to the total presence degree σ of the expression [147]. Furthermore, a very important property of LFT systems is that their interconnection results in another LFT. The extension of the LHT decomposition algorithm to LFT modelling is direct using the previous property and, in particular, the symbolic ‘object-oriented’ realization of LFTs as proposed in [147]. The ‘object-oriented’ realization takes the point of view that LFTs are feedback interconnections of matrices and as such are subject to standard matrix operations: addition and multiplication of matrices are equivalent to parallel and series connection of LFTs (see the formulae given in [126,147]). Furthermore, the order of the uncertainty matrix ∆ for the resulting LFT is equal to the sum of the orders of the individual uncertainty matrices. Hence, each of the individual matrices resulting from the proposed algorithm, see equation 5.2, can be transformed into an LFT with a diagonal uncertainty matrix of order equal to the respective matrix total presence degree. The thus obtained LFTs can be manipulated, following the resulting structure of the decomposition (5.2), to obtain a ﬁnal LFT whose uncertainty matrix ∆ is a diagonal matrix of order equal to the sum of the total presence degrees. The applicability of this algorithm to dynamical systems is straight-forward recalling that an LFT is basically a generalization of the notion of state-space where the dynamic system is written as a feedback interconnection of a constant matrix and a diagonal element containing the integrator terms ‘1/s’ and delays [12]. This is valid as well for exact nonlinear modelling, developed in Section 5.3, where the non-linearities are considered as symbolic parameters to be ‘pulled out’ into the ∆ matrix.

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LHT Benchmarking In reference [20] a comparison of other LFT algorithms is given for three benchmark examples: 1. A model of an F16 near a stall bifurcation with 9 states, 5 control inputs and 10 outputs that can be scheduled using a mix of airspeed V and ﬂight path angle γ. 2. The longitudinal motion of a missile [209], a 2 degrees of freedom system with one input and scheduled on angle of attack α and Mach M. 3. A generic physics-based model (no more details are included) given by a polynomial of matrices each multiplied by a monomial formed by three uncertain parameters (x, y, z) of different powers. Speciﬁcally, the compared algorithms are the symtreed [41] implementation found in ONERA’s LFR toolbox [147], NASA’s numerical approximation from [21] and the LFT algorithm available in the new MATLAB robust control toolbox [14]. The benchmarking of the LHT algorithm is given in Table 5.1 for the case of the direct application of the algorithms, i.e. no application of multidimensional reduction techniques (except the ﬂag ‘basic-reduction’ for the MATLAB toolbox –otherwise the comparison will be unfair) 1 . Table 5.1. LFT modelling benchmarking: No reduction cases

NASA ONERA MATLAB LHT

∑ 86 72 235 57

F16 V γ 31 55 24 48 106 129 24 33

Missile Generic ∑ αM ∑ x y 12 4 8 94 9 64 11 6 5 110 9 24 16 8 8 579 268 173 9 45 46 18 15

z 21 77 138 13

From the above comparison it is observed that the LHT algorithm performs much better and actually seems to attempt to minimize the standard deviation between parameters. The latter characteristic is actually a signiﬁcant advantage when there is no information about which parameter is more critical. For example, for the F16 model if it happens that the parameter V can be removed later on, then the effect on the relative order of the LHT algorithm is larger since both parameters have closer values than for the other methods. If 1D or n-D reduction methods [44] are applied (available from ONERA’s toolbox although [14] has its own methods), the results obtained are shown in Table 5.2: Again, it is noted that the LHT algorithm has equivalent or better performance than the other algorithms but with the additional advantage of minimizing at the same time the standard deviation (the LHT algorithm can also emphasize reduction of one parameter over the others if that is preferred). It is remarkable that the effect of the 1

For the LHT algorithm, an initial pre-processing of the system matrices using the variablesplitting operation from [98] has been performed due to the particular structure of the examples (without this pre-processing similar values to ONERA’s toolbox were obtained).

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Table 5.2. LFT modelling benchmarking: best of 1D or n-D reduction methods

NASA ONERA MATLAB LHT

∑ 55 56 53 56

F16 V γ 22 33 23 33 22 31 23 33

Missile ∑ αM 9 45 9 45 14 8 6 8 44

Generic ∑ x y z 45 9 24 12 46 9 24 13 45 9 24 12 45 18 15 12

multidimensional reduction approaches is minimal on the LHT (i.e. the difference before and after their application is negligible in comparison to the other techniques). Finally, it is noted that similarities in the performance of the algorithms is only one aspect of the problem, since the question of which model is better for design or analysis is determined only partially by the total number of parameters (indeed, this is an open problem in general). 5.2.2

Symbolic LFT Differentiation and Nested LFT Substitution Operations

The LFT extension of the LHT algorithm has been shown to rely on standard LFT operations such as concatenation, addition and multiplication to name just a few [126, 147]. Together with these standard operations, the modelling framework proposed in Section 5.3 will require also two special operations: symbolic LFT differentiation and nested LFT substitution. The ﬁrst operation enables us to connect the modelling framework with linear modelling, design and analysis techniques which are based on linear time invariant (LTI) state-space models. The second operation, due to the diagonal structure of the ∆ matrix, allows the substitution of symbolic parameters by approximations or more detailed descriptions (all given also in LFT format) with minor modelling effort. The presented lemmas focus on lower LFT representation but they can be straight-forwardly adapted for upper LFTs. Symbolic LFT Differentiation This ﬁrst LFT operation is based on the LFT differentiation idea found in [147] but adds an additional step which allows for a more complete automatization of the procedure. Indeed, the operation can be viewed as a three-step approach where in the ﬁrst step an LFT is formed for the system under consideration using as inputs the system inputs d and also the symbolic parameters ρ. The second step involves symbolically differentiating with respect to u the part of the LFT containing the ∆ matrix operator. Finally, the third step performs an LFT of the expression from the second step and combines the resulting matrices with those from the ﬁrst step. Lemma 1. Consider a symbolic well-posed lower LFT y = Fl (M, ∆)u where M = [M11 M12 ; M21 M22 ], ∆ = diag(∆1 , ∆2 (ρ)) and u = [ρ d] . Its symbolic lower LFT

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¯ ∆J )σu where σy , σu are deviation linearization, see Figure 5.3, is given by σy = Fl (M, variables with respect to an equilibrium point (yeq , ueq ), e.g. σy = y− yeq ; the coefﬁcient matrix M¯ is given by equation (5.3):

σy _ z

__ __ M M __11 __12 M21 M22

σu J M MJ M11 + M12 M11 12 12 M¯ = J J M21 M22

_ w

∆

J

(5.3)

Fig. 5.3. Symbolic linearized LFT J M J ; M J M J ] and ∆J are respectively the coefﬁcient and unThe matrices M J = [M11 12 21 22 certain components from the lower LFT of L = Fl (M J , ∆J ), and L is given by: ∂ (I − ∆M22)−1 ∆M21 u L= (5.4) ∂u eq

where ∆J is obtained by selecting as symbolic variables the terms ∆1 |eq , ∆2 (ρ)|eq , ueq ∂∆2 (ρ) and the symbolic derivative ∂u . eq

Proof: The proof is given in Appendix B. It represents an algorithmic implementation for the LFT operation. Example 4. The following example based on the chemical reactor model from reference [213], page 287, illustrates the symbolic LFT linearization approach. The component balance model of a chemical reactor (CSTR) is given by: cA f q

cA q − − k1 cA V V cB q + k1 cA − k2 cB c˙B = − V

c˙A =

where q is the feed-ﬂow rate, cA is the concentration of reactant A, cB is the concentration of the intermediate product B and V is the reactor volume. k1 and k2 are constant steady-state values for the feed-ﬂow rate and cA f is the ﬁnal concentration for the reactant A. Introducing deviation variables σcA = cA − cAeq , σcB = cB − cBeq , σq = q − qeq and symbolically linearizing the original nonlinear system yields: cA f eq cAeq qeq )σcA + ( − )σq Veq Veq Veq cBeq qeq =k1 σcA − (k2 + )σc − σq Veq B Veq

σ˙ cA = − (k1 + σ˙ cB

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Using the symbolic LFT approach on the linearized equations, a lower symbolic LFT for the linearized system is obtained, see Figure 5.4 (note that this step is automatic using the LHT algorithm): 11 00 11 00

σc

A 1 0 0 1

σc

B

cA f −k 1 0 ___ V k 1 −k 2 0

−1 __ 0 0 __ −1 V V −1 __ __ −1 0 0 V V

1 0 0 0

0 0 0 0

0 0 1 0

0 1 0 1

0 0 0 0

0 0 0 0

z q

cA

q

cB

0 0 0 0

σc A σc B σq

eq

w

eq

Fig. 5.4. Example: symbolic LFT for linearized CSTR

Alternatively, using the proposed nonlinear symbolic LFT approach, a lower nonlinear symbolic LFT can be obtained for the original nonlinear system, see Figure 5.5: 11 00 00 11

cA f −k 1 0 ___ V k 1 −k 2 0

cA

11 00 00 11

cB

0 0

0 0

1 1

cA cB q

−1 __ 0 V −1 __ 0 V 0 0 0 0

w

z cA

cB

Fig. 5.5. Example: symbolic LFT for nonlinear CSTR

Applying the symbolic linearization LFT operation from Lemma 1 to this nonlinear symbolic lower LFT, the following input-output mapping is obtained: ⎡ ⎤ cA −1 c q c 0 00 w1 cA 0 0 0 1 ⎣ ⎦ cB = A = (I −∆M22)−1 ∆M21 u = I − A w2 0 cB 0 0 0 cB 0 0 1 cB q q Performing a symbolic partial derivative on this input-output mapping with respect to u = [cA , cB , q] yields the term L : ⎡ ⎤ σ q 0 cA ⎣ cA ⎦ σw1 σcB = L σu == σw2 0 q cB eq σq A lower symbolic LFT for L , i.e. Fl (M J , ∆J ), can be obtained assuming cAeq , cBeq , qeq are the symbolic variables to be “pulled out”, see Figure 5.6:

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1

σw

2

__

0 0

0 0

0 0

1 1 0 0

0 0 1 1

1 0 0 0

0 0 1 0

0 1 0 1

0 0 0 0

0 0 0 0

0 0 0 0

z

0 0 0 0

σc A σc B σq __

w q

cA

q

cB

eq

Fig. 5.6. Example: symbolic LFT for CSTR L function

Finally, using the coefﬁcient matrices from Figures 5.5 and Figure 5.6 together with the formulae from Lemma 1, the coefﬁcient matrix M¯ is obtained: cA f 1 cA f −V 0 000 0 0 −k −k J 1 1 ¯ V V M11 =(M11 + M12 M11 )|eq = + = 0 − V1 0 0 0 k1 −k2 0 k1 −k2 0 eq 1 1 1 −V 0 1100 −V −V 0 0 J M¯ 12 =(M12 M12 )|eq = = 0 − V1 0 0 1 1 0 0 − V1 − V1 eq ⎡ ⎤ 100 ⎢0 0 1⎥ J ⎥ M¯ 21 =M21 = ⎢ ⎣0 1 0⎦ 001 J ¯ M22 =M22 = 04×4 The above coefﬁcient matrix together the uncertain matrix ∆J from Figure 5.6 yield a ¯ ∆J ) which is the same as that from Figure 5.4. 2 lower symbolic LFT Fl (M, In order to use linear analysis and design techniques, the numerical form of the linear symbolic LFT must be used. Actually, the latter form is only necessary for the ﬁnal constant matrix M p , and for those terms in ∆ p not required to be symbolic (e.g. some of the parameters might be left symbolic for robust control synthesis or worst-case µ-analysis). As all the symbolic parameters are parameterized by (or independent of) the general equilibrium point (yeq , ueq ), a simple numerical substitution of the chosen equilibrium point sufﬁces to ﬁnd the required linear system model. Nested LFT Substitution This second operation has the aim of facilitating substitution of a (symbolic) parameter in the ∆ matrix operator by another LFT that represents either an approximation or a more detailed expression for that parameter. The term ‘nested’ refers to the generalization of the substitution operation whereby subsequent substitutions can be performed recursively in the newly obtained ∆´s. Lemma 2. Consider a lower LFT, y = Fl (M, ∆(ρ))u, where M = [M11 M12 ; M21 M22 ] and ∆(ρ) = diag(∆1 (ρ), ∆2 (ρ)), as shown in Figure 5.7 (a).

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design

y

w1

u M 11

M 12

M 21

M 22

z1 z2

∆1(ρ) ∆ 2(ρ)

w1 w1 w2

__

z1

z1

∆ 1(ρ)

∆

83

∆

M 111

M 121

M 211

M 221

z1

∆

∆

__

__

w1

∆1

(a)

(b)

Fig. 5.7. Nested LFT: initial lower LFTs

Assume w1 = ∆1 (ρ)z1 can be represented as another LFT, ∆1 (ρ)= Fl (M ∆1 , ∆¯ 1 ) where ∆1 ∆1 ∆1 ∆1 = [M11 M12 ; M21 M22 ], as shown in Figure 5.7 (b). The nested substitution of the ∆1 (ρ) lower LFT into Fl (M, ∆(ρ)) yields another lower ¯ of order equal to the sum of the orders for ∆¯ 1 and ∆2 , deﬁned by: ¯ ∆), LFT Fl (M, ∆¯ 0 ∆¯ = 1 (5.5) 0 ∆2 (ρ) ∆ ∆ ∆ Fl (M, M¯ 11 ) M12 (I − M¯ 11 M22 )−1 M¯ 12 M¯ = ¯ ∆ (5.6) ∆ )−1 M Fu (M¯ ∆ , M22 ) M21 (I − M22 M¯ 11 21 ∆ ∆ 0dim(w)×dim(z) Idim(w)×dim(w¯ 1 +w2 ) M¯ M¯ 12 M¯ ∆ = ¯ 11 = (5.7) ∆ M ∆ ¯ 22 Idim(¯z1 +z2 )×dim(z) 0dim(¯z1 +z2 )×dim(w¯ 1 +w2 ) M21 M ∆1

∆ (ii, ii) = M ∆1 , where M¯ ∆ is composed of zero and identity matrices with elements M¯ 11 11 ∆ ∆ ∆ ∆ ∆ ∆ 1 1 1 M¯ 12 (ii, ii) = M12 , M¯ 21 (ii, ii) = M21 , M¯ 22 (ii, ii) = M22 . The index (ii, ii) is given by the position of ∆1 (ρ) in ∆(ρ).

Proof: The proof, which also represents an algorithmic implementation of the operation, is given in Appendix C. Example 5. Assume that the angle of attack aircraft equation of motion is given in nonlinear symbolic form by (see example 6 for details of the symbols used): α˙ = q −

qS g CL + (sα sθ + cθ cα ) = q − c1ρ8 ρ9 + ρ8 ρ10 m VTAS VTAS

Furthermore, assume that its corresponding symbolic LFT is Fl (M, ∆) where M: M11 = 1 0 M12 = −c1 1 0 0 ⎡ ⎤ ⎡ ⎤ 00 0010 ⎢0 0⎥ ⎢0 0 0 1⎥ ⎥ ⎥ M21 = ⎢ M22 = ⎢ ⎣0 1⎦ ⎣0 0 0 0⎦ 01 0000 and w = ∆z = diag(ρ8 , ρ8 , ρ9 , ρ10 )z with w = [w1 w2 w3 w4 ] , z = [z1 z2 z3 z4 ] .

(5.8)

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Now assume the symbolic parameter in w4 = ρ10 z4 is approximated by ρ10 = c2 (ρ4 ρ6 + ρ5 ρ7 ) which is represented as the symbolic LFT w4 = Fl (M ∆1 , ∆¯ 1 )z4 , see Figure 5.8: w4

_ z

0

c2 c2

0

0

0 0 1 1

0 0 0 0

1 0 0 0

0 1 0 0

ρ

4

0 0 0 0

ρ5

ρ6

z4

_ w

ρ7

Fig. 5.8. Example symbolic substitution: w4 = ρ10 z4 = Fl (M ∆1 , ∆¯ 1 )z4

Following the proof of Lemma 2, the previous LFT Fl (M ∆1 , ∆¯ 1 ) is augmented to contain the input-output mappings w1 = ρ8 z1 , w2 = ρ8 z2 and w3 = ρ9 z3 , yielding another ¯ see Figure 5.9: symbolic lower LFT w = Fl (M¯ ∆ , ∆),

w1 w2 w3 w4

_ z

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 c2 c2 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

z1 z2 z3 z4

_ w

ρ

8ρ 8ρ 9ρ 4ρ 5ρ 6ρ 7

¯ Fig. 5.9. Example symbolic substitution: w = Fl (M¯ ∆ , ∆)z

Finally, using the the coefﬁcient matrices M and M¯ ∆ together with equations (5.245.28), the ﬁnal coefﬁcient matrix M¯ is obtained:

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design

M¯ 11 = 1 0 ⎡ ⎤ 00 ⎢0 0⎥ ⎢ ⎥ ⎢0 1⎥ ⎢ ⎥ ⎥ ¯ M21 = ⎢ ⎢0 0⎥ ⎢0 0⎥ ⎢ ⎥ ⎣0 1⎦ 01

85

M¯ 12 = −c1 1 0 0 0 0 0 ⎡ ⎤ 001 0 0 00 ⎢0 0 0 c2 c2 0 0⎥ ⎢ ⎥ ⎢0 0 0 0 0 0 0⎥ ⎢ ⎥ ⎥ M¯ 22 = ⎢ ⎢0 0 0 0 0 1 0⎥ ⎢0 0 0 0 0 0 1⎥ ⎢ ⎥ ⎣0 0 0 0 0 0 0⎦ 000 0 0 00

¯ ¯ ∆) The uncertain matrix ∆¯ is the same as that shown in Figure 5.9. This ﬁnal LFT Fl (M, can be easily shown to be the lower symbolic LFT corresponding to the angle of attack equation α˙ = q − c1ρ9 ρ8 + c2 ρ8 (ρ4 ρ6 + ρ5 ρ7 ). Note that the same procedure could also be followed in order to substitute ρ9 = ρ1 ρ2 ρ3 into the LFT. Furthermore, due to the LFT property that interconnections of LFTs yield another LFT, the same procedure could be followed for the pitch rate equation and the resulting LFT added to that for the angle of attack to yield the complete aircraft short-period motion model in LFT form. 2

5.3 Nonlinear Symbolic LFT Modelling Approach The basic idea of the proposed modelling approach is to combine symbolic and LFT tools to represent the complex ordinary differential equations (ODEs), which deﬁne the nonlinear system. In this manner, an exact nonlinear symbolic LFT where the structured ∆ matrix contains the nonlinear, time-varying or uncertain terms as symbolic parameters can be obtained. Once this exact nonlinear symbolic LFT is obtained, it is more straightforward and ﬂexible to simplify the model down to a manageable size while retaining sufﬁcient precision for successful control design. An example of the application of the proposed approach to an on-ground-aircraft is given in Chapter 6. Indeed, an advantage of the proposed modelling approach is that it results in a highly structured representation of the nonlinearities, which facilitates their analysis and reduces the likliehood of inappropriate simpliﬁcations and approximations being made during the overall modelling process. This advantage arises due to the LFT nature of the framework together with the diagonal structure of the symbolic nonlinearities in ∆(ρ). Furthermore, once the control system is synthesized using the simpliﬁed LFT model, it can be gradually validated using increasingly complex models up to the full nonlinear model given by the exact nonlinear symbolic LFT. This gradual validation has the advantage of allowing the identiﬁcation of the problematic model terms or controller shortcomings in greater detail, providing speciﬁc feedback if re-design of the controller is necessary (while automatically updating the model if required). Class of Nonlinear Systems Considered It is assumed that the class of nonlinear systems is deﬁned as follows: x˙ = f (x, u) = f1 (x) x + f2 (x) u + f3 (x)

(5.9)

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A. Marcos, D.G. Bates, and I. Postlethwaite

y = g(x, u) = g1 (x) x + g2(x) u + g3(x)

(5.10)

where the nonlinear functions fi (x), gi (x) are given by a polynomial mix of analytic expressions and tabular data. The ﬁrst-order derivative condition for the states is without loss of generality (higher-order derivatives can be substituted by new state variables to transform the system into a ﬁrst-order form). The main structural restriction for this class of systems is the linear dependency of the nonlinear functions on the input vector u, e.g. f (x, u) is a function of f1 (x)x, f2 (x)u and f3 (x). This assumption is indeed quite general and standard for mechanical systems, nevertheless an extension to systems with nonlinear dependency on the inputs can also be considered [152]. The inclusion of the functions f3 (x), g3 (x) which represent those terms (nonlinear, time-varying or constant) that cannot be represented as linear in the states, signiﬁcantly expands the set of nonlinear systems that can be considered. For example, these extra functions often arise in aerospace systems, where their consideration is critical [220] (1992 print). Exact Symbolic LFT Modelling Approach It is highlighted that although seemingly straightforward, the proposed approach has only recently become practical due to the development of specialized symbolic algebra software, such as the LHT algorithm or those in [147, 98], and the related LFT tools and as such it represents a novel and powerful systematic modelling methodology. The basic steps of the modelling approach are: 1. Represent the nonlinear ODEs as a nonlinear state-space 2×2 block matrix using, if needed, ﬁctitious signals u f = 1 ∀ t to include those nonlinear terms not afﬁne on the inputs u or the states x: ⎡ ⎤ x f (x) f2 (x) f3 (x) ⎣ ⎦ x˙ (5.11) = 1 u y g1 (x) g2 (x) g3 (x) uf 2. Declare as symbolic parameters ρk all the uncertain, nonlinear or time-varying terms (including physical parameters that vary with operational condition, e.g. discrete switches). The guiding principle proposed at this stage is to select everything that is not a known constant c j as a symbolic parameter ρk : nk fi (x) = fi (ρn1 1 , . . . , ρ k , c1 , . . . , c j )

(5.12)

where n1, n2, . . ., nk indicate the number of repetitions for each parameter. 3. Transform the resulting nonlinear symbolic 2×2 matrix into a nonlinear symbolic LFT where all the symbolic parameters (and their repetitions) are placed in the ∆(ρ) matrix. This step is automatically carried out using available algebraic LFT modelling software as the proposed LHT algorithm of those found in [98, 147].

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design

87

Remark: Note that the three steps above result in an LFT representation which is identical to the original nonlinear system given by equations (5.9-5.10). 4. Taking advantage of the diagonal structure of ∆(ρ) arising from the previous LFT modelling process it is possible now to carry out [152]: i) simpliﬁcations, ii) model reduction, iii) approximations, and iv) uncertainty characterization, in order to obtain a manageable LFT model for design and analysis. Furthermore, the modularity afforded by the symbolic LFT allows easy updating of the model if any assumption needs to be corrected. Example 6. Assume that the nonlinear angle of attack equation is given by the compact nonlinear equation: α˙ = q −

g qS ¯ CL + (sα sθ + cα cθ ) mVTAS VTAS

The known constants are assumed to be the wing surface c1 = S and the Earth gravity ¯ c2 = g; everything else is treated as a symbolic parameter: dynamic pressure ρ1 = q, lift coefﬁcient ρ2 = CL , aircraft mass ρ3 = m1 , and trigonometric relationships ρ4 = sα , ρ5 = cα , ρ6 = sθ , ρ7 = cθ . The inverse of the true airspeed is also considered as a 1 . Hence, the nonlinear equation is transformed into: symbolic parameter ρ8 = VTAS α˙ = q − c1ρ1 ρ2 ρ3 ρ8 + c2 ρ8 (ρ4 ρ6 + ρ5ρ7 ) Now, for simplicity substitute ρ9 = ρ1 ρ2 ρ3 and ρ10 = c2 (ρ4 ρ6 + ρ5 ρ7 ) in the previous system. Introducing a ﬁctitious input δ f , the following symbolic system is obtained: α˙ = q − c1ρ9 ρ8 δ f + ρ8 ρ10 δ f The corresponding LFT obtained using the LHT algorithm is given in Figure 5.10: 1 0 0 1

α

z

1

0

−c 1 1 0

0

0 0 0 0

0 0 1 1

0 0 0 0

1 0

0 1

0 0

0 0

ρ

8

ρ8

0 0 0 0

ρ9

q δf

w

ρ10

Fig. 5.10. Example: Nonlinear Symbolic LFT for an aircraft angle of attack

The order of the symbolic nonlinear LFT based on the parameters ρ8, ρ9 and ρ10 is 4 (two repeated parameters ρ8 plus one for each of the other two parameters). Note, that if the substituted parameters ρ9 and ρ10 are expanded, the order of the LFT will increase. 2

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General practical considerations that should be considered when applying the proposed modelling approach are: i. Initially, declare each physical parameter as symbolic and only group them as a √ new ρ if the functional expression is complex, e.g. ρ1 = pq but not ρ2 = pq. ii. The symbolic parameters are considered “independent” at this stage (e.g. ρ1 = sθ , ρ2 = cθ and ρ3 = θ). iii. The reciprocal of a parameter is also considered to be a parameter (e.g. ρ1 = m and ρ2 = m−1 ). iv. Select carefully which parameter is extracted from a monomial. v. In general, the selected exact LFT model is that with the lowest LFT order. However, this might not be appropriate if step 4 of the modelling approach is to be used afterwards. vi. The symbolic parameters should be normalized only after all other manipulations have been performed on the symbolic LFT. By performing the normalization last, it is ensured that the physical meaning of the parameters is retained and hence their effects on the system remain easier to understand and study. Furthermore, the diagonal structure of the ∆ matrix means that the order of the LFT remains the same after normalization - see [151] for a ﬂight dynamics example of the dramatic effect normalization of the parameters before the LFT process has on the overall LFT order. This problem has been mentioned before [41, 147], but it is noted that most of the available LFT applications in the literature do not follow this guideline.

5.4 Conclusion In this chapter a modelling framework based on symbolic algebra and LFT representations has been presented. Also, a novel LFT algorithm and the formalization of two additional LFT operations have been developed in support of the framework. While the proposed modelling approach might seem straightforward, its application to realistic problems has only recently become feasible due to the development of specialized symbolic algebra algorithms, such as the LHT and the software implementation of LFT operations such as those in [147]. It results in an exact nonlinear symbolic LFT that represents an ideal starting point to apply subsequent assumptions and simpliﬁcations to ﬁnally transform the model into an approximated symbolic LFT ready for design and analysis. A clear advantage of the proposed modelling approach is that it results in a highly structured representation of the nonlinearities present in the system, which facilitates their analysis and ameliorates the effect that inappropriate simpliﬁcations and approximations might have on the overall modelling process. Furthermore, the approach allows the designer to obtain a connected set of increasingly-simpliﬁed models that can be used to gradually validated the control design from the simplest of the models to the exact nonlinear symbolic LFT. This gradual and systematic approach to validation has the advantage of allowing the identiﬁcation of the problematic model terms or controller shortcomings in greater detail, thus providing speciﬁc feedback if re-design of the controller is necessary. The software implementation of the LHT algorithm, version 2.2, is freely available on request from the ﬁrst author.

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design

89

Appendices A LHT Algorithm Pseudo-code The pseudo-code for the LHT algorithm implementation is given here. The description for each of the steps shown can be found in Section 5.2.1. LHT(M(δ)) 1 i ← 1; ok ← 1 2 while ok = 1 3 do M ← EXPAND(M) 4 if i = 1 5 then MA[1] ← S YMBOLIC -C OEFF(M) 6 MB[1] ← C ONSTANT-C OEFF(M) 7 metrics ← M ETRICS(MA[i]) 8 ordvect ← S IGMA -O RDERING(MA[i]) 9 if E XIST-P OLY C OEFF(MA[i]) = yes 10 then Hlist ← H ORNER -O RDERING(MA[i], metrics) 11 MHA ← H ORNER -FACTORIZATION(MA[i], Hlist) 12 MA[i], polylist ← P OLY-S UBSTITUTION(MHA) 13 metrics ← M ETRICS(MA[i]) 14 ordvect ← S IGMA -O RDERING(MA[i]) 15 for j ← 1 to LENGHT[ordvect] 16 affdir ← A FFINE -D IRECTION(MA[i], metrics, ordvect[ j]) 17 if affdir = 0 ⇒ skip parameter afﬁne factorization 18 L[i], MA[i], R[i] ← A FF -FACTOR(MA[i], affdir, ordvect[ j]) 19 short-hand: LMAR[i] ≡ L[i], MA[i], R[i] 20 if NOEMPTY(polylist) 21 then LMAR[i] ← P OLY-BACK S UBS(LMAR[i], polylist) 22 LMAR[i] ← EXPAND(LMAR[i]) 23 polylist ← empty 24 declist ← D ECOMPOSITION - LIST(MA[i]) 25 MA[i + 1], MB[i + 1] ← S UM -D ECOMP(MA[i], [], declist[1]) 26 short-hand: MAMB[i + 1] ≡ MA[i + 1], MB[i + 1] 27 j←1 28 while NOEMPTY(declist) 29 do j ← j + 1 30 declist ← C ONFLICT-A NALYSIS(MAMB[i + 1], declist[ j]) 31 MAMB[i + 1] ← S UM -D ECOMP(MAMB[i + 1], declist[ j]) 32 M ← MA[i + 1] 33 i ← i+1 34 if F ULLY-D EC(M) and N ONFULLY-D EC(MB[k]) 35 then M ← MB[k] 36 k is special index to get correct MB 37 elseif F ULLY-D EC(M) and F ULLY-D EC(MB[k]) 38 then ok ← 0

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B Symbolic Jacobian LFT Linearization The following proof provides an algorithmic implementation of the symbolic LFT linearization operation from Lemma 1. The input-output mappings for the symbolic well-posed lower LFT y = Fl (M, ∆)u where M = [M11 M12 ; M21 M22 ], ∆ = diag(∆1 , ∆2 (ρ)) and u = [ρ d] , are: y =M11 u + M12w

(5.13)

z =M21 u + M22w w =∆z

(5.14) (5.15)

Combining equations (5.14) and (5.15) yields: w = (I − ∆M22 )−1 ∆M21 u

(5.16)

which exists due to the well-posedness assumption on the LFT y = Fl (M, ∆)u. Since the coefﬁcient matrix terms M11 , M12 , M21 , M22 are constants, the symbolic ﬁrst-order Taylor approximations with respect to a general equilibrium point (yeq , ueq ) are: σy =M11 σu + M12 σw

(5.17)

σz =M21 σu + M22 σw ∂ (I − ∆M22 )−1 ∆M21 u σw = =L ∂u eq

(5.18) (5.19)

Obtain a symbolic lower LFT for L , i.e. σw = Fl (M J , ∆J )σu where M J = J M J ; M J M J ] is constant and ∆J contains all the ‘ﬁxed’ (at the general equi[M11 12 21 22 eq 2 (ρ) librium point yeq , ueq ) symbolic uncertain variables ∆1 |eq , ∆2 (ρ)|eq , ueq and ∂∆∂u in eq

diagonal format, see Figure 5.11:

σw

J

J M 12

J

M 22

M 11

J

M 21 σ _z

σu

eq

u

∆1

σ_ w

∆ 2( ρ)

∆2( ρ) δ ____ δu

eq

Fig. 5.11. Proof: symbolic linearized LFT for L , σw = Fl (M J , ∆J )σu

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design

91

The corresponding input-output mappings of the LFT for L are: J J σw =M11 σu + M12 σw¯

(5.20)

J J σz¯ =M21 σu + M22 σw¯ J σw¯ =∆ σz¯

(5.21) (5.22)

Substituting equation (5.20) into equation (5.17) yields: J J J J σy = M11 σu + M12 (M11 σu + M12 σw¯ ) = (M11 + M12 M11 )σu + M12 M12 σw¯

(5.23)

¯ ∆J )σu Combining with equations (5.21-5.22), a new lower symbolic LFT σy = Fl (M, is obtained which corresponds to the symbolic lower LFT for the symbolic linearization of equations (5.13-5.16). The coefﬁcient matrix M¯ corresponds to the formulae from equation (5.3) and the uncertain matrix ∆J is deﬁned above.

C Nested LFT Substitution As before, the following proof provides the algorithmic implementation for the nested LFT substitution of Lemma 2. First, the case for one uncertainty block is proved (this is in fact the well-known Red-Heffer product). This is extended to the case of a diagonal uncertainty matrix with two blocks, which generalizes the result due to the diagonal structure. Given the lower LFT Fl (M, ∆1 ), where ∆1 can also be represented by a lower LFT ∆1 = Fl (M ∆1 , ∆¯ 1 ) and where the coefﬁcient matrices M and M ∆1 are partitioned in the standard 2×2 block format, i.e. Figure 5.7 assuming ∆2 = 0. Substitute w1 = ∆1 z1 by w1 = Fl (M ∆1 , ∆¯ 1 )z1 and use the Red-Heffer product [147] ¯ ∆¯ 1 ), see Figure 5.12: to obtain a new lower LFT Fl (M, y

u M 11

M 12

M 21

M 22

∆ M 111 ∆ M 211

∆ M 121 ∆ M 221

z1

__

w1

y w1

__

z1

z1

__

M 12

__

__ __

__

M 11

u

__

M 21 M 22 __

__

w1

∆1

∆1

(a)

(b)

¯ ∆¯ 1 ) Fig. 5.12. Nested LFT proof: one block case - Fl (M,

The new coefﬁcient matrix M¯ = [M¯ 11 M¯ 12 ; M¯ 21 M¯ 22 ] is given by: ∆ ∆ ) = M11 + M12 ∆−1 M¯ 11 =Fl (M, M11 n M11 M21

(5.24)

92

A. Marcos, D.G. Bates, and I. Postlethwaite ∆ M¯ 12 =M12 ∆−1 n M12 ∆ ∆ ∆ ∆ −1 M¯ 21 =M21 (I + M22 ∆−1 n M11 )M21 = M21 (I − M22 M11 ) M21

(5.25) (5.26)

∆ ∆ ∆ M¯ 22 =Fu (M ∆ , M22 ) = M22 + M21 M22 ∆−1 n M12

(5.27)

∆ ∆n = I − M11 M22

(5.28)

Note that in the one-block case, M¯ ∆ from equation (5.7) is equal to M ∆1 . The case for two uncertain blocks Fl (M, diag(∆1 , ∆2 )) where it is desired to substitute ∆1 = Fl (M ∆1 , ∆¯ 1 ), is similar to the one-block case but with an intermediate step. The intermediate step augments the coefﬁcient matrix M ∆1 and the uncertain matrix ∆¯ 1 with appropriate channels and terms in order to diagonally include the uncertain block ¯ with ∆¯ = diag(∆¯ 1 , ∆2 ). As the original un∆2 , thus forming a new lower LFT Fl (M¯ ∆ , ∆) certain matrix is diagonal (i.e. wi = ∆i zi ), the row/column augmentation and uncertainty inclusion is straightforward, see Figure 5.13: w1

∆

M 121 ∆ M 221

∆ M 211

__

z1

z1

∆

M 111

__

w1

__

∆1

w2

∆2 (a)

∆

_ z1 z2

∆

z1 z2

M111 0 M121 0 ∆2 0

w1 w2

∆1 M21 0

_ ∆1

∆1 M22

_ w1

∆

w1 w2

_ z1 z2

∆

M111 0 0 0

M121 0 0 I

∆1 M21 0 0 I

∆1 M22 0 0 0

_ ∆1

z1 z2

_ w1 ∆2

w2

(b)

¯ Fig. 5.13. Nested LFT proof: two block case - (augmenting Fl (M ∆1 , ∆¯1 ) with ∆2 ) ≡ Fl (M¯ ∆ , ∆)

The augmented matrix M¯ ∆ is now in the appropriate 2×2 block format to perform a Red-Heffer product with the coefﬁcient matrix M, using equations (5.24-5.28), to obtain ¯ from equations (5.5-5.6). ¯ ∆) the desired result: Fl (M, The generalization of this result is direct, noting that the augmented matrix M¯ ∆ consists of blocks formed by identity and zero matrices of appropriate dimensions, see equation (5.7), with those terms in the position (ii, ii) (corresponding to the position in ∆ of the uncertain block to be substituted, e.g. ∆1 is in the (ii, ii) = (1, 1) position in ∆) equal to the terms from the coefﬁcient matrix to be substituted, i.e. M ∆1 .

6 Nonlinear LFT Modelling for On-Ground Transport Aircraft Jean-Marc Biannic1 , Andres Marcos2 , Declan G. Bates3 , and Ian Postlethwaite4 1 2 3 4

ONERA/DCSD, Toulouse, France [email protected] DEIMOS-SPACE, Madrid, Spain [email protected] University of Leicester, UK [email protected] University of Leicester, UK [email protected]

Summary. In this chapter, the challenging problem of aircraft on-ground modelling for control design and analysis is dealt with using a novel NDI-based force identiﬁcation approach and a general nonlinear symbolic LFT modelling framework. It is shown how the modelling framework is applicable to derive an exact LFT model of the aircraft-on-ground as well as some simpliﬁed design-oriented versions. An important contribution is the proposed force identiﬁcation method, which allows nonlinear ground forces to be replaced with saturation-type nonlinearities. As a result, the simpliﬁed model boils down to a reduced-order LFT plant where the ∆ block only contains time-varying or constant (but uncertain) parameters on the one hand, and saturationtype non-linearities on the other hand. Such a model is then very useful for applying modern analysis and synthesis techniques.

Notation θNW Π = [x y z]T Ω = [p q r]T Ξ = [φ θ ψ]T F = [Fx Fy Fz ]T Fa = [X Y Z]T M = [Mx My Mz ]T V = [Vx Vy Vz ]T BTMLGL/R ISVNW ISVbrk N1c T nL/R sang /cang

Nose wheel angle, rad Earth-based aircraft position, m Angular velocities, rad/s Euler angles, rad Body-axes total forces, N Aerodynamic forces N Body-axes total moments Nm Velocity vector at c.g. m/s Braking torque R/L Main Landing Gear Servovalve steering control current, mA Servovalve braking control current, mA Engine fan speed target Left/Right engine thrust, N sine/cosine of angle ang

D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 93–115, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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6.1 Introduction Fly-by-wire systems have become widespread on board transport aircraft over the past twenty years, allowing for increased piloting comfort, ﬂight envelope protection, automatic ﬂight/landing systems among many other beneﬁts. All these automated functions either help pilots accomplish their duties, or signiﬁcantly increase the safety of the aicraft. However, the ground motion of commercial aircraft is still achieved by manual control making use of devices such as rudder deﬂection, engines speed, braking and nose-wheel-steering systems. As a logical next step, for future transport aircraft generations, automated on-ground control systems will be developed. This will reduce pilot workload, increase safety during on-ground manœuvers, and eventually also reduce airport congestion. This development requires modelling approaches and identiﬁcation techniques that can cope in a systematic manner with the complexity of the aircraft dynamics on the ground. This is specially critical when modelling the interactions between the aircraft tyres and the ground under a variety of environmental condition such as icy or dry runways. In Chapter 5 a general modelling approach based on symbolic algebra and linear fractional transformations (LFTs) was presented. LFTs are the modelling paradigm in modern robust control, allowing a structured representation of the nonlinearities and uncertainties present in the system. However, current LFT modelling approaches typically result in LFT models of large complexity (i.e. of large dimension). In this chapter, the modelling framework from Chapter 5 is used to ﬁrst develop an exact LFT model of an on-ground-aircraft and subsequently to allow its simpliﬁcation to a model of manegeable size for control design and analysis. This application showcases how the gap between exact LFT modelling and modelling for control design & analysis is performed and how the different trade-offs between high-ﬁdelity and low-complexity are applied. Furthermore, an original nonlinear dynamic inversion (NDI) based identiﬁcation procedure is developed to obtain a simpliﬁed but accurate model of the highly nonlinear forces resulting from the interactions between the aircraft and the ground. Indeed, it is shown how such forces can be replaced by saturation-type nonlinearities using the proposed identiﬁcation approach. As a result, the structured matrix ∆ of the simpliﬁed LFT will only contain some time-invariant and time-varying uncertainties on the one hand, and some saturation-like nonlinear operators on the other hand. Such a model, despite its nonlinear nature, is then easily handled by modern analysis and synthesis techniques as proposed in some of the subsequent chapters of the book. The chapter is organized as follows. As a preliminary, the NDI-based force identiﬁcation method for LPV models is presented in Section 6.2. For completeness, in Section 6.3 the basic aircraft equations are summarised, together with the procedure for the development of an exact LFT model. In Section 6.4, based on a few simplifying assumptions, a smaller LFT model is derived, from which a full-motion control-oriented LPV model is ﬁnally obtained. At this stage, some inputs to the LPV model, namely the nonlinear ground-forces, are still unknown. Thus, using the identiﬁcation procedure of Section 6.2, Section 6.5 shows how to characterize these forces so that the outputs

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of the LPV model become as close as possible to the outputs of the nonlinear plant. In this and subsequent sections, the simpliﬁed full-motion LPV model is now restricted to a lateral/directionl LPV model for ease of presentation. The lateral/directional LPV model and ground forces models are then combined in Section 6.6 and the resulting model is evaluated in Section 6.7. Extensions to the longitudinal case are discussed in Section 6.8 and ﬁnally, some concluding remarks end the chapter.

6.2 NDI-Based Identiﬁcation for LPV Models During the application of the modelling approach from Chapter 5 to the aircraft-onground model (see subsequent sections) an off-line force identiﬁcation method for LPV systems based on nonlinear dynamic inversion [58] was required to estimate the aircraft ground forces. The developed method is presented in this section. Note that any nonlinear symbolic LFT representation (as those arising using the modelling approach from the last chapter) can be translated into a standard LPV model assuming the symbolic parameters are the scheduling parameters ρ of the LPV model and using the LFT feedback equation:

FU (M, ∆) = M22 + M21 ∆(I − M11 ∆)−1 M21

(6.1)

Indeed, after arranging the result from this LFT-to-LPV transformation the following LPV symbolic model is obtained: x˙LPV = A(ρ)xLPV + B1 (ρ)u + B2(ρ)ν

(6.2)

where it is assumed that everything is known (or measured) except for the input vector ν. The objective is to identify ν given all the above symbolic matrices (A( · ), B1 ( · ), B2 ( · )) and measurements (ρ, xLPV , u), together with the state vector xNL obtained from the nonlinear high-ﬁdelity model (either by experimentation or simulation). Assuming that B2 (ρ) is a nonsingular square matrix for all values of the time-varying vector ρ, the dynamic equation (6.2) may be inverted as follows: νˆ = B2 (ρ)−1 (x˙LPV − A(ρ)xLPV − B1 (ρ)u)

(6.3)

which reveals that the derivative signal x˙LPV can be fully controlled by an appropriate choice of νˆ . In other words, x˙LPV represents the desired signal x˙d from NDI theory (see chapters 8 and 11), which in this case drives the identiﬁcation procedure. Parallelling classical control ideas, it can be shown that using a sufﬁciently large λ (e.g. a proportional gain), and known values of the nonlinear xNL and LPV xLPV state vectors, it is possible to set, for continuous trajectories, the LPV model states arbitrarily close to those from the nonlinear plant provided that the input νˆ coincides with that of the LPV model in equation (6.2): x˙d = λ (xNL − xLPV )

(6.4)

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Therefore, the estimated input vector νˆ is calculated as follows: ⎤ ⎡ xNL − xLPV ⎦ νˆ = B2 (ρ)−1 [λI − A(ρ) − B1 (ρ)] ⎣ xLPV u

(6.5)

The application of this identiﬁcation method to the aircraft on-ground forces problem is described in detail in Section 6.5.

6.3 Exact LFT Modelling of the Aircraft-on-Ground The nonlinear model presented here characterizes the aircraft on-ground dynamics of a representative Airbus transport aircraft with two engines (see chapter 1 of this book) during on-ground rolling (i.e. taxi and after-touchdown). In this section, a general description is ﬁrst presented followed by its transformation into an exact symbolic LFT and a ﬁrst non-exact (simpliﬁed) LFT model. 6.3.1

A General Description

The open-loop nonlinear model can be represented by three main blocks of ODEs, see Figure 6.1. The equations of motion block, EoM, is generic for all aircraft (on-ground and airborne) and comprises the twelve standard aircraft degrees of freedom. The inputs

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are the total forces and moments (F and M) and the outputs are the linear and angular ˙ and the kinematic (Ξ) ˙ derivatives of the ˙ and navigation (Ψ) accelerations (V˙ and Ω) states. ⎤ ⎡ ⎤ ⎡ F V˙ m − Ω ∧V ⎥ ˙ ⎥ ⎢ −1 ⎢Ω ⎢ ⎥ = ⎢ I (M − Ω ∧ (I.Ω)) ⎥ (6.6) ˙ ⎦ ⎣Ξ⎦ ⎣ TBH .Ω ˙ TBE .V Ψ where TBH and TBE are transformation matrices from body-axes to local-horizon and Earth-frame respectively, and I is the inertial matrix. The total forces and moments

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FandM block has as inputs the aircraft states x, environment env and aerodynamic aero data, and the actuator inputs (θNW , BTMLGR/L , T nR/L ): ⎡ ⎤ ⎤ ⎡ ⎡ ⎤ ΣT nR/L Fx −mgsθ ⎣ Fy ⎦ = Fa + ⎣ mgsφ cθ ⎦ + ⎣ 0 ⎦ + [TWB .Fw ] (6.7) Fz mgcφ cθ 0 TW B is a transformation matrix from wheel-frame to body-frame, and Fw represents the nose wheel and landing gear contributions, which model the nonlinear interactions between the shock absorbers and the runway friction. These interactions are condensed in lateral and longitudinal forces due to wheel slip, rolling drag and braking forces [11, 18, 39]. The moments are given by, M = F · L, where L is the proper moment-arm. The actuator block Act, transforms the avionics commands from the pilot/on-groundautopilot (ISVNW , ISVbrk and N1c ) into the FandM actuator inputs. This block is formed by three subsystems: nose-wheel steering system, braking system and engine model. The latter is modelled by a ﬁrst quasi-steady stage followed by a dynamic model with amplitude and rate limits. The nose-wheel steering system calculates θNW using mechanical components, servovalves and pistons models for the hydraulic components: σ2 [K2 − |∆PNW |] (6.8) θ˙ NW = K1 σ1 [ISVNW ] 1 + K3 σ1 [ISVNW ] where σi represents saturation functions, Ki the different NW geometric and physical constants, and ∆PNW S is a nonlinear function of the pistons’ pressure and θNW . Similarly for the braking system (one equation per MLG bogie): βbrk ηSbrk [ISVbrk, Pbrk ] ∆Pbrk (6.9) P˙brk = Vbrk where βbrk is a compressibility coefﬁcient dependent on the braking pistons’ pressure difference ∆Pbrk (which is a function of Pbrk ), Vbrk is the piston swept volume, η is the ﬂow coefﬁcient, and Sbrk is a saturation function for the servovalve switching logic. A set of standard on-ground manœuvers are designed to evaluate both the ground and the aerodynamic parts of the model: • • • •

Manœuver 1 : 40◦ tiller step at T0 Manœuver 2 : Doublet in tiller 20◦ + 20◦ at T0 and T1 = T0 + 10 Manœuver 3 : Full acceleration until 150 kts then ± 5◦ doublet in pedals Manœuver 4 : Same as Man.3 plus doublet of ±2◦ in tiller

6.3.2

A Symbolic LFT Modelling Approach

The ﬁrst three steps in the modelling approach presented in the previous chapter are used here to obtain an exact nonlinear symbolic LFT model. The process is applied to each of the aircraft main blocks independently: EoM, FandM and Act blocks, but since a full and detailed presentation of the LFT modelling process is not possible due to space restrictions, only a general view of the different steps is given for the EoM block. Step 1 is direct in this case due to the standard manner of writing the ODEs for the aircraft motion (i.e. afﬁne in the states and inputs).

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Step 2 using the symbolization rule, given for this step, yields 24 symbolic parameters (states, inertial coefﬁcients and trigonometric functions) and no symbolic constants. Summing up the repetitions of the 24 ‘independent’ parameters yields a total value of 67. Step 3 an exact nonlinear symbolic LFT of order 43 with 23 symbolic parameters is obtained. Table 6.1 shows the results for the three aircraft blocks using the order-reduction LFT algorithms (LHT from Chapter 5, symtreed from [148] and ETD from [96]) before and after application of an additional ND numerical minimization technique [44] which also keeps the exactness of the models. All the min-ND LFT models for the EoM block result in the same number of repetitions for each of the 23 parameters (not a typical situation). Thus, any of the exact min-ND LFT models models can be chosen. Table 6.1. Exact LFT for EoM, FandM and Act blocks EoM block FandM block Act block no-min min-ND no-min min-ND no-min min-ND 177 134 (89) 28 26 symtreed 53 43 215 180 (120) 24 24 LHT 44 43 190 139 (87) 24 24 ETD 51 43

An important consideration in applying the modelling framework is to ‘cover’ complex functions by single symbolic parameters, for example this was necessary for the on-ground forces (subsequently, the symbolic parameter was substituted by the identiﬁed models). The example below shows how this functional ‘coverage’ can be performed: Example 1. The modelling of the local sideslip angle for the wheels (used to calculate the lateral contribution of the landing gear forces and moments) is given by: Vy + r LNW (6.10) − θNW = atan(β˜ NW ) − θNW βNW = atan Vx Vy − r LMLG βMLG = atan (6.11) = atan(β˜ MLG ) Vx Choosing ρ1 = atan(β˜ NW ), ρ2 = atan(β˜ MLG ) and using a ﬁctitious input an exact nonlinear state-space model is obtained: βNW −1 ρ1 θNW = (6.12) βMLG 0 ρ2 uf LFT modelling techniques could be used now on equation (6.12) to shift ρi to the ∆ matrix, where subsequently they can be substituted by approximated or identiﬁed models (see details in Chapter 5). 2 Repeating the above exact LFT modelling for each of the remaining aircraft blocks and ﬁnally combining the three exact nonlinear LFTs with lowest order yields the ﬁnal

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exact nonlinear symbolic LFT model. Table 6.2 shows the total order and the number of symbolic parameters and constants for each aircraft block, for the ﬁnal ”EXACT” and also for the ”NON-EXACT” (derived next) models. Note the large difference in orders. Table 6.2. Total symbolic LFT for aircraft-on-ground model: Exact and Non-exact EXACT NON-EXACT order no. ρ no. c order no. ρ no. c 8 5 2 EoM 43 24 0 41 19 10 FandM 134 27 15 14 12 0 Act 24 15 6 total 201 61 21 63 36 12

6.4 Simpliﬁcations and LPV Modelling In this section, Step 4 of the modelling approach from Chapter 5 is used to yield an intermediate model representing a compromise between the above exact, larger-order model and a ﬁnal control-oriented, low-order model (see subsequent sections). This non-exact model facilitates the understanding and manipulations needed to obtain the ﬁnal control-oriented model and also allows the identiﬁcation of problematic parts or non-appropriate simpliﬁcations in the case where the control law designed using the most simpliﬁed model is not valid for the full nonlinear model. It is stressed that the whole process is now highly automated due to the LFT and symbolic nature of the exact LFT model –and to the use of the LFT order-reduction software. In a ﬁrst step, the exact LFT model is simpliﬁed using standard on-ground-aircraft assumptions while in a second step, this ﬁrst approximated LFT model is further simpliﬁed to yield a design-oriented LPV model. 6.4.1

Simpliﬁcations of the Exact LFT Model

In Chapter 1 several assumptions are made based on the intended use of the aircraft model, i.e. design and analysis of steering and speed/braking on-ground controllers: A.1 No inertial cross-coupling terms (ξ2 = ξ4 = ξ6 = 0 ⇒ coupling of Mx and Mz dropped). A.2 Small-angle approximations and low speeds (less than 150 knots) ⇒ neglect products of angles and velocity terms. A.3 Runway is perfectly horizontal ⇒ the lift is quasi-constant and there are almost no variations in the vertical position of the center of gravity. A.4 Neglect compressibility effects of shock absorbers ⇒ quasi-steady pitch and roll. A.5 bicycle model ⇒ superimposed left and right MLGs ⇒ two points of contact with runway. Assumption 6.4.1 is quite standard in the study of on-ground vehicle behavior, although for manœuvers requiring severe braking (which induces a signiﬁcant pitch movement) or drastic differential thrusts (which results in rolling/yawing motion) this

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approximation might not be well suited. Therefore, in this ﬁrst intermediate model this assumption will not be considered. Using assumptions A.6.4.1, A.6.4.1 and A.6.4.1 (interpreted as the vertical position being constant ⇒ V˙z = Vz = 0 or Fz = 0), the reduced EoM model given in equation (6.13) is obtained. The resulting non-exact LFT has an order of 8 for a total of 5 ρ’s (Vx , Vy , p, φ, m−1 ) and 2 symbolic constants (ξ8 = (Ixx − Iyy )/Izz , ξ9 = 1/Izz). ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ q −1 ˙ Vx 0 0 ⎢ ⎥ 0 Vy m ⎢V˙y ⎥ ⎢ 0 −Vx 0 m−1 0 ⎥ ⎢ r ⎥ ⎢ ⎥=⎢ ⎥ ⎢ Fx ⎥ (6.13) ⎥ ⎣ r˙ ⎦ ⎣ξ8 p 0 0 0 ξ9 ⎦ ⎢ ⎣ Fy ⎦ ˙ ψ φ 1 0 0 0 Mz It is noted from the above simpliﬁcations that Fz , Mx and My can now be neglected. Nevertheless, some of their components are required to calculate the wheel forces Fw and thus when required these terms will be directly embedded. The simpliﬁed aerodynamic Fa , Ma and engine Feng , Meng forces and moments in FandM are: ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ Fxa + Fxeng S(−Cx + αCz ) 1 1 T nR ⎦ ⎦ ⎦ ⎣ ⎣ ⎣ Fya SCy qˆ + 0 0 (6.14) = T nL Mza + Mzeng cSC ¯ n −LyengR −LyengL where qˆ = 0.5ρairV 2 , ρair is the (constant) air density, c¯ the wing-chord, S the wingsurface and Cx , Cz , Cy , Cn are the stability-axes aerodynamic coefﬁcients. Furthermore, using assumptions A.6.4.1 and A.6.4.1, the gravity forces are assumed negligible in this ﬁrst non-exact model. We now describe the forces Fw generated by the friction of the tyres with the ground. Using assumption A.6.4.1 these forces can be split into nose-wheel (NW) and left/right main-landing-gear (MLG) components, which after some algebraic manipulations yields: ⎤ ⎡ θ a11 atanNW −θ ⎣ NW ⎦ Fxw a11 uf = a21 |NW −a21|NW atanNW − a21|MLG atanMLG φ Fyw ¯ NW Fz (6.15) ¯ −1 −θ FxMLG + ¯ NW 0 φ Fy a11 = −2sθNW GNW

2 βoptNW 2 + βoptNW

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βopt#2 2 β# + βopt#2

(6.16)

where atanNW and atanMLG were deﬁned in equations (6.10-6.11); βoptNW and βoptMLG correspond to optimal sideslip angles (for which maximum lateral forces are reached); and GNW , GMLG are referred to as the cornering gains. The yaw-moment generated by ground forces is easily obtained from the above wheel forces: Mzw = LxNW FyNW − LyMLG FxMLG + LxMLG FyMLG

(6.17)

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After applying the LFT order-reduction software, a non-exact FandM LFT is obtained with an order of 41, formed by 19 symbolic parameters and 10 constants (see Table 6.2). It is highlighted that the above representation is consistent with the mathematical tyre modelling of [11,18,39] where the behaviour is characterized using optimal sideslip angles and cornering gains models augmented by vertical load dependencies as illustrated in ﬁgure 6.2. Indeed, for small angles, linear variations are expected, allowing the introduction of constant cornering gains. In reality however, the situation is much more complex and the determination of these gains is not so simple. In fact they will not only depend on the sideslip angles, but also on the rolling speed, vertical loads (as illustrated also in ﬁgure 6.2) and the runway state. The latter depends on the quality of the surface and on weather conditions (dry, wet, icy), and will induce some large variations mainly on the optimum angles βoptNW and βoptMLG . Details of the identiﬁcation of these angles and cornering gains are given in Section 6.5. For the third aircraft block, the Act block, it was found that its complexity could be reduced using linear approximations based on a subset of the symbolic parameters. This was observed using a Monte Carlo analysis of the different components for the three subsystems in the block (i.e. simulations of the ρ’s from the exact LFT model of the engine, nose-wheel steering and braking subsystems). After this analysis, the NW steering system and the braking systems, one per bogie, are approximated as (K¯ i indicates non-symbolic constants): θ˙ NW =K¯1 ISVNW P˙brk =βbrk (K¯2 + K¯ 3 σ1 [Pbrk ] + K¯ 4 σ2 [ISVbrk ]) =ρ1 (K¯ 2 u f + K¯ 3 ρ2 Pbrk + K¯ 4 ρ3 ISVbrk )

(6.18) (6.19)

where the symbolic parameters in the derivative P˙brk represent normalized saturations, e.g. ρ2 = σ1P[Pbrk ] . Note that Pbrk is used to calculate the left/right braking torques brk BTMLG = Gbrk Pbrk , see Figure 6.1, where Gbrk = ρ4 is a braking disc gain with large variations. Therefore, the braking system approximation yields 8 symbolic parameters (three in P˙brk and one –per bogie– for Gbrk ). Finally, the engine model (one for each wing) is given below where LUTi represents look-up tables containing the engine dynamic/static information: ˙ = ρ29 N1c − ρ210N1 − 2ρ9ρ10 N1 ˙ (6.20) ¨ = LUT12 N1c − LUT12 N1 − 2LUT1LUT2 N1 N1 Combining all the Act subsystems, the non-exact Act LFT model is obtained for a total of 12 symbolic parameters and an LFT order of 14. Finally, the non-exact LFT models for each block are combined to yield the total non-exact nonlinear LFT model given in the last three columns of Table 6.2, which is now half the number of parameters and almost a third the order of the exact LFT model. Figure 6.3 shows the time responses of the exact and non-exact LFT models using manœuver 3 from Section 6.3. The responses shown are the pedal and N1c commands (top two plots), the longitudinal velocity and its error with respect to the nonlinear model (second row), and the lateral velocity and yaw rate (bottom plot). Note that the responses of the nonlinear (solid line), exact LFT (dashed line) and non-exact LFT (dotted line) are almost indistinguishable.

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6.4.2

Towards a Design-Oriented LPV Model

The complexity of the previous non-exact LFT model is still too high for control design and analysis. Thus, in this section the LFT model is further simpliﬁed and ﬁnally transformed into a purely LPV model (without the LFT structure) prior to the application of the force identiﬁcation method given in Section 6.5. It is noted that the previous

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non-exact model facilitates the simpliﬁcations performed in this section and also helps to validate the ﬁnal control designs. It is stressed again that the focus in this work is on developing a simpliﬁed model for aircraft on-ground control design, with emphasis on the use of symmetric thrust for longitudinal control and differential thrust and nose-wheel steering for lateral/directional control. Therefore, the braking system can be neglected (BTMLG = 0) while the engine thrusts T nR/L and nose-wheel deﬂection θNWc are assumed to be inputs. Due to the last assumption, the engine models are not necessary for the controloriented LFT model but a nose-wheel-steering system transforming θNWc to θNW is required, based on equation (6.8). Intensive simulations have shown that, for small commanded angles, the NW-steering system behaves like a ﬁrst-order linear plant whose time-constant can be approximated by 0.1 s; while for higher amplitude commandedangles, rate Lr and magnitude L p saturations appear. Also, a small time-delay τ and a constant offset |θ0 | ≤ 1 deg on the θNW are observed, so that the NW-steering system can be described, to a very high degree of accuracy, by : θ˙ NW (t) = satL p (λ (satLr (θNWc (t − τ)) − (θNW (t) + θ0 )))

(6.21)

In most practical cases, the magnitude saturation is never reached, and the time-delay can be efﬁciently approximated by a ﬁrst-order Pade function. The actuator is then further simpliﬁed as illustrated in ﬁgure 6.4. The aerodynamic forces/moments can be simpliﬁed using assumption A.6.4.1, yielding the following simpliﬁed aerodynamic coefﬁcients: ⎡ ⎡ ⎤ ⎤⎡ ⎤ 0 −Cx0 0 Fxa Vx V Sρ air a ⎣ Fya ⎦ = ⎣ 0 Cyβ cCy ⎦ ⎣Vy ⎦ ¯ r 0 0 2 Mza r 0 cCn ¯ β0 c¯2Cnr0 ⎡ ⎤⎡ ⎤ (6.22) 0 −Cx0 0 Wx SρairVa ⎣ 0 Cyβ0 VaCyδr ⎦ ⎣Wy ⎦ + 0 2 0 Cnβ0 VaCnδr δr 0

where C#0 are constants (derived from neural-network models, see chapter 1) and the V W aerodynamic sideslip angle was approximated by βa = Vy + Vy with Wy denoting lateral wind projection on body-axes (the longitudinal wind component comes from assuming only the state Vx is augmented by Wx ). Rather than using the detailed model given in (6.15), a simpler (forces Fxw , Fyw and moment Mzw ) wheel model is obtained by including FyNW , FyMLG and FxMLG as inputs-to-be-identiﬁed that capture all the force characteristics: ⎤ ⎡ ⎤⎡ ⎤ ⎡ 1 0 FyNW Fxw −θNW ⎣ Fyw ⎦ = ⎣ 1 0 1 ⎦ ⎣FxMLG ⎦ (6.23) Mzw FyMLG LxNW −LyMLG LxMLG Similarly, the equations of motion from (6.13) are simpliﬁed to a 3-DoF longitudinal/lateral/directional model (using now assumption A.6.4.1 which implies dropping ˙ and q-contributions). Simulation with this 3DoF model, using equations (6.22-6.23) Ψ

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and the engine contributions from (6.4.1), revealed that the gravity term along the longitudinal axis Fxg had to be considered to avoid steady-state errors on Vx and account for wheel lateral/longitudinal coupling.. After several tests, it was observed that this force could be nicely approximated by Fxg ≈ −gKg r where Kg denotes a constant term. Therefore, adding this new gravity force, we can split the almost ready LFT/LPV model [V˙x V˙y r˙] in two parts: one fully describing [V˙x p V˙y p r˙p ] and the other [V˙xw V˙yw r˙w ] which includes the only components left to be identiﬁed FyNW , FyMLG , FxMLG : ⎡ ⎤ ⎡ ⎤⎡ ⎤ V˙x p −ρsp Cx0 m−1 0 (g Kg + Vy ) Vx ⎣V˙y p ⎦ = ⎣ 0 ρsp Cyβ0 m−1 (ρsp c¯ Cyr0 m−1 − Vx )⎦ ⎣Vy ⎦ r r˙p ρsp c¯2 Cnr0 ξ9 0 ρsp c¯ Cnβ0 ξ9 ⎡ ⎤ (6.24) ⎡ ⎤ δr 0 m−1 −ρsp Cx0 m−1 0 ⎢ ⎥ −1 0 0 ρsp Cyβ0 m−1 ⎦ ⎢T nR/L ⎥ + ⎣ρsp Va Cyδr0 m ⎣ Wx ⎦ ρsp Va c¯ Cnδr ξ9 0 0 ρsp c¯ Cnβ0 ξ9 0 Wy ⎤⎡ ⎤ ⎡ ⎤ ⎡ −θNW m−1 FyNW m−1 0 V˙xw ⎣V˙yw ⎦ = ⎣ m−1 0 m−1 ⎦ ⎣FxMLG ⎦ r˙w FyMLG ξ9 LxNW −ξ9 LyMLG ξ9 LxMLG

(6.25)

Note that T nR/L in the above matrix denotes both T nR and T nL and the coresponding entries should therefore be duplicated (omitted for space reasons), while ρsp = 0.5SρairVa . In the remainder of the chapter the quasi-LPV model for the lateral equations will be used for clarity of presentation, the corresponding matrices are easily extracted from (6.24) and (6.25): ⎧ r Wy r˙ FyNW ⎪ ⎪ + Ba(θ) + Btyres ⎪ ˙ = Aa (θ) ⎨ Vy δr FyMG Vy (6.26) ⎪ r ⎪ β˜ NW = C (θ) r = 1 LNW 1 ⎪ ⎩ ˜ β Vx −L Vy Vy βMG MG 1 As is usual in LPV modelling, a vector θ of time-varying parameters has to be deﬁned. For this application the most natural choice is : θ = [Va Vx ]T

(6.27)

Note that these two parameters are linked and may even coincide when there is no wind.

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In this model, the ground forces are represented as external inputs which have now to be further detailed. As already observed, such forces mainly depend on the sideslip angles βNW and βMG . From equation (6.12), it is seen that the ﬁrst one is controllable by the nose-wheel angle θNW which itself is directly controlled by the pilot through the nose-wheel system (see ﬁgure 6.4).

6.5 On the Nonlinear Ground-Forces 6.5.1

Application of the NDI-Based Identiﬁcation Procedure

In this section, the identiﬁcation of the ground forces is described using the lateral/directional model from (6.26). As indicated in Section 6.4.1, the tyre models can be shown to be composed of cornering gains Gy (dependent on vertical forces) and sideslip angles β (dependent on runway conditions λrwy ∈ [0 , 1] with 1 being a dry runway with standard surface and 0 an icy one) – indices NW and MLG are omitted to simplify the notation: β2 (λrwy ) (6.28) Fy = Gy (Fz )β 2 OPT βOPT (λrwy ) + β2 To cover all these possible variations while limiting the complexity of the model, it is proposed to consider the cornering gains as uncertain, possibly time-varying, parameters. Then, equation (6.28) can be rewritten as : Fy = (1 + δGy )Gynom β

λ2rwy β2OPT 2 λrwy β2OPT + β2

= (1 + δGy )Fynom (β, λrwy )

(6.29)

The time-varying uncertain parameter satisﬁes |δGy (t)| ≤ 0.4 which means that the precision level on the cornering gains is rather poor (40%). It should also be emphasized that the value of such gains may also change signiﬁcantly from one landing to another. Combining the lateral LPV model (6.26) with the lateral forces described by (6.29), the objective now is to compute the nominal cornering gains Gynom and optimal sideslip angles βOPT such that the outputs of the LPV model match those of the nonlinear plant. A two-step identiﬁcation procedure is proposed: • First, using the proposed NDI-based identiﬁcation approach from Section 6.2, a set of lateral forces Fˆy is calculated for different runway conditions, rolling-speeds and external inputs so that the above objective is met. • Second, the forces are plotted versus the corresponding sideslip angles and then the values of the cornering gains and optimal angles are obtained. Therefore, using the on-ground manœuvers and different runway conditions, the estimated lateral forces FˆNW , FˆMLG are plotted versus βNW , βMLG respectively in ﬁgure 6.5. The cornering gains are ﬁrst identiﬁed by bounding the plots with linear functions (as illustrated by the dashed-lines on ﬁgure 6.5). From these bounds extremal values of the gains are easily found, from which a mean value is immediately deduced. These bounds and mean values are given in Table 6.3. Allowing a maximum error of 40 percent, the third column shows that this limit is not exceeded.

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4

6

x 10

18

x 10

16 5

14 Dry runway

Wet runway

FMG (N)

FNW (N)

dry runway

12

4

3

10

8 wet runway 6

2 Icy runway

4 1

icy runway

2

0

0

2

4

6

8

β

10 (deg)

12

14

16

18

20

0

0

1

2

NW

3

4 βMG (deg)

(a)

5

6

7

8

(b)

Fig. 6.5. (a) FˆNW versus βNW (b) FˆMLG versus βMLG Table 6.3. Extremal and mean values of the cornering gains min max mean error (%) Gˆ NW (N/rad) 1.1 105 2.5 105 1.8 105 38.9 Gˆ MG (N/rad) 4 105 5 105 4.5 105 11

Next, we must compute the optimal values βOPT of the sideslip angles βNW and βMLG for which the lateral forces are maximized. This is not an easy task as seen from equation (6.29), where these parameters appear as rather complex non-linear terms. Fortunately, from a control design and analysis perspective it is emphasized that an accurate model of the lateral forces beyond βNWOPT or βMLGOPT is not required. This is further supported by the fact that lateral control laws are designed so as to avoid large sideslip angles. 6.5.2

Introduction of Saturation-Type Non-linearities

Based on the above remarks, an efﬁcient simpliﬁcation of the lateral ground-forces is given. As illustrated by the two plots of ﬁgure 6.5 (see especially the dashed-lines), the lateral forces, until βOPT , are increasing functions of the sideslip angles. Thus, they can be approximated quite accurately by saturation-type nonlinearities: FyNW ≈ satLNW (λrwy ) (1 + δGNW (t))Gˆ NW .βNW (6.30) FyMLG ≈ satLMLG (λrwy ) (1 + δGMLG (t))Gˆ MLG .βMLG where LNW (λrwy ) and LMLG (λrwy ) denote maximum lateral forces values (reached at βNWOPT and βMLGOPT respectively); and βNW , βMLG are given by equation (6.10-6.11). As shown by the two ﬁgures, these values clearly depend on the runway state λrwy . Moreover, an additional uncertainty level may be introduced on LNW (λrwy ) which permits a reduction in the range of variations on Gˆ NW (see ﬁgure 6.5.a).

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6.6 Final LFT Modelling All the steps have now been introduced to complete the LFT modelling of the aircrafton-ground. This task can be easily performed thanks to the new version of the LFR Toolbox (v2.0) [96] which can be downloaded from [148] and also with the help of additional Simulink-based tools [25] (see also [24] for a more detailed description of the tools). 6.6.1

Main LPV Block

Using Equation (6.26), the quasi-LPV parts of the proposed model are easily converted into an LFT format. In this subsection, two LFT realization methods are evaluated. The ﬁrst method is a standard numerical approach (using standard tools of the LFR Toolbox [148], such as for example LFRT/abcd2lfr.m) and followed by a reduction step (LFRT/minlfr.m). The parameters of this LFT are: Vxn (normalized version of Vx ), Van (normalized version of Va ), δCn (LTI multiplicative uncertainty on Cn ), δCy (LTI multiplicative uncertainty on Cy ). The second method, see [150] and Chapter 5, is based on the symbolic tree decomposition algorithm [41] (LFRT/symtreed.m) applied to polynomial matrices in symbolic form. In this method, Va , Vx , invVx , δCn , δCy are now deﬁned as symbolic variables and once a reduced-order symbolic LFT is obtained, the variables Va and Vx are rewritten as linear functions of some normalized variables Van and Vxn : Va = λVa Van + Va Vx = λVx Vxn + Vx

(6.31)

where Va and Vx denote the mean values of Va and Vx respectively. Finally, invVx is deﬁned as the inverse of Vx . All the above elementary operations are easily achieved with the help of the LFRT/eval.m function. Remark: It should be emphasized here that invVx is ﬁrst considered as an independent variable. Thus, the rational expression is converted into a polynomial one. Note also at this stage that the normalization is no longer necessary and even has to be avoided! This operation would indeed destroy the factorized structure of the polynomial matrix which is exploited by the algorithm to generate a low-order LFT. The sizes of the lateral LFTs obtained are reported in table 6.4. Table 6.4. ∆-block sizes for the LPV part of the model Van Vxn δCn δCr size of ∆1 numerical approach 4 3 2 2 11 symbolic approach 3 3 1 1 8

The symbolic approach is obviously more powerful since the size of the corresponding LFT is smaller. It is worth pointing out that these two LFTs are equivalent from an input/output viewpoint. In other words, the second LFT is not an approximation of the

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ﬁrst one. In the remainder of the chapter the reduced-size LFT is used. Let us denote ∆1 as the corresponding ∆-block. According to the second line of the table 6.4, it can be written as : ∆1 = diag (Van (t)I3 ,Vxn (t)I3 , δCn , δCr ) (6.32) As illustrated by ﬁgure 6.6 the LFT model associated with the LPV part of the model has four inputs (Wy , δr , FNW and FMG ) and four outputs (x = [r Vy ]T , βNW and βMG ). ∆1 LTI/LTV block (V,Vx, δ Cn, δCr) Wy, δ r

F NW FMG

LFT model for the LPV part

M1 (s)

x

βNW βMG

Fig. 6.6. LFT representation of the LPV part

6.6.2

Other Nonlinear and LTV Elements

The next step in the LFT modelling procedure is to determine LFT models for the ground forces and for the atan functions (in equations (6.10) and (6.11)) which are used in the computation of the sideslip angles. LFT models for the lateral forces FNW and FMG are readily obtained from the equations (6.30). Using the function LFRT/lfr.m, the saturations (satLNW (.), satLMG (.)) and time-varying uncertainties (δGNW (t), δGMG (t)) are ﬁrst deﬁned as elementary LFR objects : >> >> >> >>

sat_NW = sat_MG = delta_NW delta_MG

lfr(’sat_NW’,’nlms’,1); lfr(’sat_MG’,’nlms’,1); = lfr(’delta_NW’,’ltv’,1); = lfr(’delta_MG’,’ltv’,1);

then, the equations (6.31) are simply translated as follows : >> F_NW = sat_NW*(G_NW0*(1+delta_NW)) >> F_MG = sat_MG*(G_MG0*(1+delta_MG)) and the global LFR object associated with the two lateral forces is ﬁnally obtained as the diagonal concatenation of F NW and F MG. This is achieved with the help of the LFRT/append.m function : >> F_tyres = append(F_NW,F_MG) This LFT is represented in ﬁgure 6.7. Its ∆-block, denoted ∆2 , is composed of two nonlinear elements (saturations) and two scalar LTV blocks associated with the uncertainties on the cornering gains : ∆2 = diag (satFNW , satFMG , δGNW (t) , δGMG (t))

(6.33)

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∆2 LTV/NL block δG

NW/MG

LFT model

βNW

for the tyre forces

F NW

βMG

M2 (s)

FMG

Fig. 6.7. LFT representation of the lateral forces

Finally, as is clear from equation (6.12), the trigonometric atan functions need to be approximated. The nonlinear simulations of Section 5 revealed that the sideslip angles βMG at the main landing gear never exceed about 10 deg. Then the following approximation holds : (6.34) βMG = atan(β˜ MG ) ≈ β˜ MG and only the atan function associated with βNW needs to be considered. To cover all cases, the approximation should be valid as long as |β˜ NW | ≤ 1.2 which corresponds to an output argument equal to 50 deg. As illustrated by ﬁgure 6.6, such an approximation (with very good precision level < 5%) can be achieved by a simple piecewise afﬁne function : |α| ≤ 0.4 → f (α) = α 2 sign(α) |α| ≥ 0.4 → f (α) = 23 α + 15

(6.35)

which may be conveniently rewritten as follows : α 2 2 f (α) = α + sat 3 15 0.4

(6.36)

where sat(.) denotes a normalized saturation nonlinearity. As a result, the atan function may now be easily described by a simple LFT object (see ﬁgure 6.9) whose nonlinear ∆-block is a single saturation operator. 6.6.3

Computing the Interconnection

The last phase of LFT modelling consists in connecting the above components together as illustrated by ﬁgure 6.10. Any linear interconnection of LFTs is well known to be an LFT itself. This step can be performed quite easily in a Simulink environment with the help of newly developed tools [25, 24] which provide (via a speciﬁc library) a simple interface between the new version of LFR Toolbox [148] and Simulink. Using these tools, the interconnection of LFTs is simply drawn (as it appears on ﬁgure 6.10, but without the ∆-blocks) and saved in a Simulink ﬁle. Let us name it ’LatLFR.mdl’. Then, using a speciﬁc function LFRT/slk/slk2lfr.m which can be viewed as a generalization of linmod.m1, the global LFT is readily obtained as follows : >> sysLFR_LAT1 = slk2lfr(’LatLFR’); 1

linmod.m is a standard Matlab function which is used to perform the linearization of a Simulink diagram.

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0.8

atan(α) (rad)

0.7

0.6 error < 2% 0.5

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

α

0.8

1

1.2

1.4

Fig. 6.8. Approximation of the atan function

∆3 NL block

~ β NW

static LFT model for the ATAN function

M3

βNW

Fig. 6.9. LFT representation of the atan function

This global LFT object has only two states and a 13 × 13 diagonal ∆-block which is structured as follows : (6.37) ∆ = diag(∆NL , ∆LTV , ∆LT I ) with :

∆NL = diag (satatan , satFNW , satFMG ) ∆LTV = diag(Van (t).I3, Vxn (t).I3 , δGNW (t) , δGMG (t)) ∆LT I = diag δCn , δCy

(6.38)

To conclude the LFT modelling, we ﬁnally add the actuator dynamics. This is easily achieved by redrawing the Nose-wheel system of ﬁgure 6.4 using the LFT format as shown in ﬁgure 6.11. The Nose-Wheel System is then simply plugged in at the third input of the interconnected LFT of ﬁgure 6.10. Let us denote by sysLFR NWS the LFR object associated with the nonlinear plant represented by ﬁgure 6.11. The aforementioned operation is then realized by the following command-line : >> sysLFR_LAT2 = sysLFR_LAT1*append(1,1,sysLFR_NWS);

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Wy, δ r

∆2

∆1 LTI/LTV block (V,Vx, δ Cn, δCr)

LTV/NL block

δG

NW/MG

θNW + −

βNW

x

LFT model

LFT model

for tyre forces

for LPV part

βMG

F NW , FMG

~ β NW

LFT model approx ATAN

∆3 NL block ∆ NL ∆ LTV ∆ LTI

Wy, δ r

θ NW

Global

β NW

LFT model

β MG x

Fig. 6.10. LFT representation of the global interconnection

∆4 NL block

LFT model θ NW c θ0

for the Nose Wheel

M4 (s)

θ NW

Fig. 6.11. LFT representation of the Nose-Wheel System

The new LFR object now has three states. The LTV and LTI blocks remain unchanged, while the non-linear block is augmented to : ∆NL = diag (satNW S , satatan , satFNW , satFMG )

(6.39)

The ﬁnal third-order lateral LFT model is illustrated in ﬁgure 6.12. Note that there are now four inputs. The ﬁrst two are associated with the control signals while the last two can be viewed as perturbations.

6.7 Simulation Results We now compare the outputs of our simpliﬁed LFT models with those of the full nonlinear system. In these simulations, note that the longitudinal outputs of the nonlinear

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w∆

control signals

δr

{ {

θ NW c

z∆

Lateral

r

LFT model

Vy

3rd−order Linear model

θ0

lateral states

β NW

(including Nose−Wheel)

Wy

perturbations

} }

β MG

sideslip angles

Fig. 6.12. LFT representation of the lateral model of the on-ground aircraft including the NoseWheel system 30

0.25 0.2

25

0.15 Vy (m/s)

r (deg/s)

20 15

0.1 0.05

10

0

5 0

−0.05 0

5

10 time (sec)

15

−0.1

20

5

0

5

10 time (sec)

15

20

0

5

10 time (sec)

15

20

1 0

0 (deg) MG

−5 −10

−2 −3

β

β

NW

(deg)

−1

−4 −15 −20

−5 0

5

10 time (sec)

15

20

−6

Fig. 6.13. Manœuver 1 : 40◦ step command on tiller - Nominal runway

system (Va and Vx ) are directly used within the LFT model for on-line computation of the ∆-block. In order to evaluate the accuracy of both the aerodynamic and ground models, two types of manœuvers are considered in the following tests. They correspond to manœuvers 1 and 4 deﬁned in the section 6.3.1 of this chapter. The ﬁrst one is a low-speed manœuver (below 10 kts) where high amplitude steps (40 deg for dry runway and 20 deg for wet runway) are applied on the tiller. In the second manœuver, full thrust is applied on the engines until the speed exceeds 140 kts. During this manœuver, a small amplitude (±2◦) doublet is applied on the tiller followed by a doublet on the pedals (±5◦ ).

Nonlinear LFT Modelling for On-Ground Transport Aircraft 14

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0.1

12

0 Vy (m/s)

r (deg/s)

10 8 6

−0.1 −0.2

4 −0.3

2 0

0

5

10 time (sec)

15

−0.4

20

5

0

5

10 time (sec)

15

20

0

5

10 time (sec)

15

20

1 0

(deg)

−1 −2 −3

β

MG

−5

β

NW

(deg)

0

−4

−10

−5 −15

0

5

10 time (sec)

15

−6

20

Fig. 6.14. Manœuver 1 : 20◦ step command on tiller - Wet runway 6

4

4

2

Vy (m/s)

r (deg/s)

0 2 0

−2 −4

−2 −4

−6 0

10

20 30 time (sec)

40

−8

50

6

4

4

2

10

20 30 time (sec)

40

50

0

10

20 30 time (sec)

40

50

0 βMG (deg)

βNW (deg)

2

0

0 −2 −4

−2 −4 −6

−6

−8

−8

−10

0

10

20 30 time (sec)

40

50

Fig. 6.15. Manœuver 4 : Doublet on tiller (± 20◦ ) followed by doublet on pedals (± 5◦ )

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Many more nonlinear simulations were performed to evaluate the model but are not reported here due to lack of space. It appeared that, in all cases, the simpliﬁed LFT model performed well. This is especially true when only considering the yaw rate outputs (r). This means that the proposed LFT model is perfectly well-suited for the development of multivariable lateral on-ground control laws using both rudder deﬂection and nose-wheel control. Moreover, the proposed LFT model is also adapted for some advanced analysis tasks such as robust performance analysis in the presence of multiple saturations and lateral wind.

6.8 Inclusion of the Longitudinal Dynamics In the above sections, we mainly focused on the lateral dynamics of the aircraft. But these are coupled with some longitudinal variables, and more speciﬁcally with Va and Vx . Consequently, to improve the accuracy of the lateral model, but also to enable the design and analysis of longitudinal control laws, an LFT model of the longitudinal dynamics is also required. For this purpose, the same modelling procedure as for the lateral case is applicable. Let us consider ﬁrst the following longitudinal equation, resulting from (6.6) : ρSV Tn 1 Cx0 (Vx + Wx ) + + (FxNW + FxMG ) V˙x = (gKg + Vy )r − 2m m m

(6.40)

where Kg and Cx0 are constants obtained through simulation tests. The longitudinal ground forces at the Nose-Wheel (FxNW ) are linked to the Nose-Wheel angle and the lateral forces (FyNW ) : FxNW = −θNW FyNW (6.41) At the main landing gear, the longitudinal forces are identiﬁed by the same inversion technique which was developed for the lateral case. Here it can be observed that such forces mainly depend on the longitudinal velocity Vx . The following approximation, strongly inspired by (6.30) can then be proposed : (6.42) FxMG =≈ satLx (λrwy ) (1 + δGxMG (t))Gˆ xMG .Vx MG

From the above equations, a global LFT model is rapidly obtained. Equation (6.40) is ﬁrst merged with the lateral LPV model (6.26) whose varying parameter θ has now three components: θ = [Va Vx Vy ]T (6.43) This augmented LPV model is then rewritten in the LFT format and combined with nonlinear LFT models of the ground-forces. Note here that equation (6.41) introduces an additional nonlinearity which will increase the size of the time-varying ∆-block. ∆NL = diag satatan , satFxNW , satFyNW , satFyMG (6.44) ∆LTV = diag δV (t).I3 , δVx (t).I4 , δVy (t) , δθNW (t) , δGyNW (t) , δGyMG (t) ∆LT I = diag δCn , δCx , δCy

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Let us ﬁnally check the validity of the full LFT model under high-speed conditions by considering the fourth manœuver again. From ﬁgure 6.16 it can be observed that the model performs very well. Note that the error on the longitudinal speed Vx , which is now a state of the full model, remains particularly small. Moreover, as in the previous cases, very small deviations are observed on the lateral variables (r and Vy ) as well.

60

10 NL LFT Vy, m/sec

Vx, m/sec

80

40 20 0 0

5 0 −5

20

40

−10 0

60

20

time, sec 4

60

40

60

10

(deg)

2

NW

0 −2

5 0

β

r, deg/sec

40 time, sec

−5

−4 −6 0

20

40 time, sec

60

−10 0

20 time, sec

Fig. 6.16. Manœuver 4 : Doublet on tiller (±2◦ ) followed by doublet on pedals (±5◦ ) on icy runway

6.9 Conclusion In this chapter, a complete methodology has been described to develop a simple LFT model for an aircraft-on-ground. An original NDI-based identiﬁcation procedure has been introduced by which the nonlinear ground forces could be drastically simpliﬁed. Interestingly, despite its simplicity, the proposed model performs very well on a large operating domain. It can, therefore, be used not only for the development of new onground control systems as proposed in chapter 7, but also for the application of robust and nonlinear analysis techniques as developed in chapter 9.

7 On-Ground Aircraft Control Design Using an LPV Anti-windup Approach Clement Roos1 , Jean-Marc Biannic2 , Sophie Tarbouriech3, and Christophe Prieur4 1 2 3 4

ONERA/DCSD and SUPAERO, Toulouse, France [email protected] ONERA/DCSD, Toulouse, France [email protected] LAAS-CNRS, University of Toulouse, France [email protected] LAAS-CNRS, University of Toulouse, France [email protected]

Summary. Based on the LFT model of the on-ground aircraft developed in Chapter 6, an antiwindup control technique is proposed to improve lateral control laws which have been designed using classical methods. The original idea of this work consists in taking advantage of a simpliﬁed representation of the nonlinear lateral ground forces, which are approximated by saturation-type nonlinearities. The anti-windup compensator is then implemented on the full nonlinear aircraft model using an on-line estimator of the ground forces. Simulations demonstrate the efﬁciency of the resulting adaptive controller.

Notation LFT LMI LPV LTI LTV NL sat(.) φ(.) θNW r, rc Ψ, Ψc Vx ,Vy Va Wy δr FˆyNW , FˆyMG GyNW , GyMG

Linear Fractional Transformation Linear Matrix Inequality Linear Parameter-Varying Linear Time-Invariant Linear Time-Varying Nonlinear Standard notation for saturations Standard notation for deadzones Nose-wheel deﬂection (rad) Yaw rate, commanded yaw rate (rad/s) Heading, commanded heading (rad) Longitudinal, lateral velocity (m/s) Aerodynamic speed (m/s) Lateral wind input (m/s) Rudder deﬂection (rad) Estimated lateral ground forces (N) Cornering gains (N/rad)

D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 117–145, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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βNW , βMG ε

Lateral sideslip angles (rad) Anti-windup controller input

7.1 Introduction Fly-by-wire systems are now commonly used onboard transport aircraft, allowing for automation of many parts of the ﬂight, including the landing phase. Immediately after touch-down, however, the motion is still controlled manually by the pilot who has to coordinate actions on rudder deﬂection, engines running speed, wheels brakes and nosewheel steering system. This piloting task is quite demanding, especially in bad weather conditions: indeed it should be emphasized that the aircraft behavior on ground significantly changes according to the runway state (dry, wet or icy). Moreover, in order to reduce congestion of most big airports, ground phases have to be constantly further optimized. Consequently, there is a real need to develop new control systems improving on-ground aircraft handling qualities. A preliminary solution based on a nonlinear dynamic inversion technique was recently proposed in [52] to control the lateral motion of an on-ground aircraft. Indeed, it is shown in that paper, and also in some other related works [51], that linear methods cannot be directly applied for this speciﬁc control application. This can be easily understood, since the considered model is affected by ground forces which exhibit highly nonlinear effects. In this chapter, an alternative solution to the on-ground control problem is proposed, which relies on the simpliﬁed LFT model developed in the previous chapter. The main point of this work consists of an original simpliﬁcation of the ground forces, which are approximated by saturation-type nonlinearities, where the saturation levels depend on the runway state. It is then shown in the present contribution that these saturation levels can be identiﬁed on-line, and that the resulting estimator admits an LFT-based expression. As a consequence, when combining the above two points, it appears that the aforementioned control issue falls within the scope of anti-windup techniques. This chapter should then be read as a non-standard application of anti-windup control, which is here an original alternative to dynamic inversion. A dynamic anti-windup design technique based on modiﬁed sector conditions [90] is ﬁrst proposed to optimize a newly introduced performance level for saturated systems [27]. The extension of the method to parameter-varying systems is also highlighted. Interestingly, the problem is shown to be convex for the considered application. As a result, the anti-windup gains are computed very easily. The second contribution of the proposed approach consists of an efﬁcient and direct use of an on-line estimation of the runway state, which enables a clever adaptation of the performance levels. The chapter is organized as follows. An LFT model of the aircraft, adapted from Chapter 6 for design purpose, is ﬁrst presented in Section 7.2, and the associated antiwindup design problem is described. Section 7.3 is then devoted to the presentation of some recent results regarding anti-windup design. It is followed in Section 7.4 by comments on how to extend the method to handle parameter-varying plants. Section 7.5 details both the design process on the simpliﬁed model and the adaptive controller implementation on the full nonlinear plant. At the end of this section, several nonlinear

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119

simulations are performed, which clearly demonstrate the signiﬁcant improvements induced by the anti-windup compensator. Some concluding remarks are ﬁnally presented in Section 7.6, which also provides directions for future works.

7.2 From the LFT Model to the Anti-windup Problem 7.2.1

Introduction to the On-Ground Control Problem

The design objective in this chapter consists of designing lateral control laws for the on-ground aircraft. Two types of maneuvers can be distinguished: • runway maneuvers mainly deal with lateral wind rejection, to ensure that the aircraft maintains a straight trajectory on the main runway while decelerating, • taxiway maneuvers aim at bringing the aircraft from the main runway to the parking area and are performed at lower speeds (below 40 kts). Only the second type of maneuver will be considered here. More precisely, the generic procedure depicted in Figure 7.1 will serve as a basis for the validation of the proposed control strategy. main runway 20 kts U-turn 30◦

10 kts

45 m

5 kts parking area

20 kts 30 kts

150 kts

Fig. 7.1. Generic maneuver

Particular attention will be paid to the three following sequences: • sequence 1: turn to take a 30◦ exit while decelerating from 30 kts to 20 kts, • sequence 2: make a 60◦ turn at 10 kts, • sequence 3: perform a U-turn at 5 kts on a 45 m wide runway. More generally, the design issue can be summarized as follows: Design challenge. Compute a (possibly nonlinear) controller, which ensures a good tracking of the yaw rate r and the heading Ψ:

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with as fast a response as possible, without overshoot (especially in heading), whatever the runway state (dry, wet or icy), for any aircraft longitudinal velocity between 5 and 40 kts.

7.2.2

An LFT Description of the On-Ground Aircraft

Recall the LFT representation of the lateral on-ground aircraft model depicted in Figure 12 of Chapter 6. Considering the design objectives detailed in Section 7.2.1, it can be further simpliﬁed, so as to obtain a synthesis model tractable for control law development. More precisely, the following assumptions are made: 1. Only low-speed maneuvers below 40 kts are considered, for which the rudder proves almost inefﬁcient. It is thus assumed that the lateral motion is only controlled via the nose-wheel steering system, and the δr input is removed. 2. The initial aerodynamic model is not appropriate below 70 kts to test the wind input Wy , which is removed too. It is then assumed that Va = Vx . 3. Neither uncertainties on the aerodynamic coefﬁcients nor on the cornering gains are considered at this stage (see Chapter 9 for comments about robustness analysis). The resulting simpliﬁed LFT representation of the lateral on-ground aircraft model is depicted in Figure 7.2. ∆NL ∆LTV ∆LT I

Simplified lateral θNW c

LFT model

r

Fig. 7.2. Simpliﬁed lateral LFT representation of the on-ground aircraft

The model has three states (r,Vy and the actuator state), a single input (θNW c ) and a single output (r). The associated 8 × 8 diagonal ∆-block is structured as follows: ∆ = diag(∆NL , ∆LTV , ∆LT I )

(7.1)

⎧ ⎪ ⎨ ∆NL = diag (satNW S , satatan , satFNW , satFMG ) ∆LTV = Vx (t).I4 (7.2) ⎪ ⎩ ∆LT I = ∅ It is assumed that both r and Vy are available for feedback. This is not restrictive, since r is actually available and Vy can be easily estimated from r and Ψ. where:

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7.2.3

121

General Structure of the Proposed Solution

The huge variation in the ground forces with respect to the runway state (see Figure 5 of Chapter 6) lead us to think that standard robust control methods would here yield very conservative results. On the other hand, the saturations that appear in the simpliﬁed LFT model depicted in Figure 7.2 strongly suggest the use of anti-windup techniques instead, which are thus further investigated in this chapter. A two-step design procedure is proposed. A nominal controller depending on the aircraft longitudinal velocity Vx is ﬁrst designed, so as to ensure good stability and performance properties when no saturations are active. It is classically composed of both an inner loop for yaw rate control and an outer loop for heading control. The second step then consists in designing an anti-windup compensator, which also depends on Vx and acts on the nominal controller to reduce the negative effects of the saturations. The design of this controller on the simpliﬁed lateral LFT model, as well as its implementation on the full nonlinear model, are extensively detailed in Section 7.5.

7.3 Anti-windup Design 7.3.1

Introduction

Anti-windup design aims at compensating for the performance degradation due to actuator saturation. Nominal control laws are ﬁrst designed for the linear unsaturated system. Additional feedbacks are then introduced to counter the adverse effects of saturations and to recover, as much as possible, the nominal performance level. More specifically, once the difference between the inputs and the outputs of the nonlinearities has been computed, the anti-windup strategy consists of the design of new gains acting either on the inputs of the nominal controller or on the inputs of the nonlinearities. The general anti-windup scheme is depicted in Figure 7.3. + −

linear model

NL

−

+

controller

Fig. 7.3. General anti-windup scheme

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The ﬁrst results on anti-windup consisted of ad hoc methods intended to work with standard PID controllers [67, 8], which are commonly used in current industrial controllers. Various systematic approaches and techniques for anti-windup compensation are, however, now available in the literature. Major improvements in this ﬁeld have indeed been achieved in the last decade, see for example, in [16, 31, 114, 121, 122, 170, 227,228]. An interesting review of anti-windup strategies was also presented during the ACC03 Workshop T-1: Modern anti-windup synthesis, proposed by A.R. Teel and his co-workers. More recently, a speciﬁc anti-windup structure was proposed by [239] to avoid interactions between robustness properties of the controller and its sensitivity to the actuator saturations. Several results on the anti-windup problem are concerned with achieving global stability properties. However, since global results may not be obtained for open-loop (strictly) unstable linear systems in the presence of actuator saturation, local results have to be developed. In this context, a key issue is the determination of stability domains for closed-loop systems (estimates of the basin of attraction). Note, however, that with very few exceptions, most of the local results available in the literature do not provide explicit characterization of the stability domain. Many LMI-based approaches now exist to adjust the anti-windup gains in a systematic way (see [230] for a quick overview). Most often, they are based on the optimization of either a stability domain [32, 90] or a nonlinear L2 -induced performance level [102, 144, 231]. More recently, based on the LFT/LPV framework, extended antiwindup schemes were proposed [144, 203, 239]. In these contributions, the saturations are viewed as sector nonlinearities and the anti-windup control design issue is recast into a convex optimization problem under LMI constraints. Following a similar path, alternative techniques using less conservative representations of the saturation nonlinearities are proposed by [27, 90, 102, 226] and will be used throughout this chapter, where a new approach is presented to compute dynamic anti-windup controllers. 7.3.2

Saturations and Sector Conditions

Consider the nonlinear operator Φ(.) in IR m , which is characterized as follows: T Φ(z) = φ(z1 ) . . . φ(zm )

(7.3)

where φ(.) is a normalized deadzone nonlinearity. More precisely, each element φ(zi ), i = 1, ..., m, is deﬁned by: ⎧ ⎨ 0 if |zi | ≤ 1 (7.4) φ(zi ) = zi − 1 if zi > 1 ⎩ zi + 1 if zi < −1 By deﬁnition, Φ(.) is a decentralized and memoryless operator. Remark 1. It is important to underline that every system which involves saturationtype nonlinearities can be easily rewritten with deadzone nonlinearities. Considering a saturation function sat(z), the resulting deadzone function φ(z) is obtained from φ(z) = z − sat(z), as depicted in Figure 7.4.

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φ(z)

sat(z) z0 -z0

-z0 z0

z0

z

z

-z0

Fig. 7.4. Saturation and deadzone nonlinearities

Moreover, by re-scaling the appropriate inputs and outputs of the considered saturated plant, it can be assumed without loss of generality that these deadzone functions are normalized. It will thus always be assumed in the sequel that saturations have been preliminarily converted into normalized deadzones. Remark 2. Other ways to mathematically represent saturations can be considered to derive constructive conditions of stability/stabilization based on the use of Lyapunov functions. Hence, the exact representation through regions of saturation consists in dividing the state space into 3m regions [224, 110]. Such a representation is mainly used for stability analysis purposes, and generally for the case of systems with few inputs, due to the complexity of the resulting conditions. Modelling based on linear differential inclusions (LDI) which leads to a polytopic approach can also be used [101, 99]. The main drawback of LDI modelling is that the conditions allowing the computation of the anti-windup gains involve BMIs [32]. Let us thus deﬁne the following polyhedral set: S1 = {z ∈ IR m , ω ∈ IR m ; −1 ≤ zi + ωi ≤ 1, i = 1, ..., m}

(7.5)

Lemma 1. [225] If z and ω are elements of S1 , then the nonlinear operator Φ(.) satisﬁes the following inequality: Φ(z)T S−1(Φ(z) + ω) ≤ 0 for any diagonal positive deﬁnite matrix S ∈ IR

m×m

(7.6)

.

Remark 3. It should be pointed out that condition (7.6) is more generic than the classical sector condition (see for instance [116, 100]) given as: Φ(z)T S−1[Φ(z) − Λz] ≤ 0 , 0 < Λ ≤ Im

(7.7)

where Λ is a diagonal matrix. It can indeed be assumed that ω = −Λz, which leads to less conservative results. This is illustrated in Figure 7.5 for the case of a single deadzone nonlinearity: penalizing areas are introduced inside the nominal region [−1, 1] if the classical condition is applied. Moreover, this condition is only valid on a ﬁnite segment [−M, M] to be deﬁned a priori. It is also worth emphasizing that unlike (7.7), the modiﬁed sector condition (7.6) allows the formulation of stability/stabilization conditions directly in LMI form. Moreover, Lemma 1 allows us to easily deal with nested saturations [225].

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penalizing areas

φ(z)

φ(z)

Λ -1

-1 1 M

z

classical condition

1

z

modified condition

Fig. 7.5. Classical and modiﬁed sector conditions for a deadzone nonlinearity

Remark 4. Particular formulations of Lemma 1 can be found in [90] (concerning systems with a single saturation function) and in [226] (concerning systems presenting both amplitude and rate limited actuators). 7.3.3

Stability Analysis for Saturated Systems

Let us now consider a generic system described as follows: ⎧ ⎨ x˙ = A x + B1 r + B2 Φ(z) z p = C1 x + D11 r + D12 Φ(z) ⎩ z = C2 x + D21 r

(7.8)

where x ∈ IR n , r ∈ IR p , z p ∈ IR q and z ∈ IR m denote the state of the system, the exogenous inputs, the exogenous outputs and the saturation inputs respectively. For the sake of simplicity, it is assumed in the sequel that p = q. The A matrix in equation (7.8) describes the nominal (linear) behavior of the system. Implicitly, it includes stabilizing control laws and therefore this A matrix, without loss of generality, is assumed to be Hurwitz. It is also assumed that D22 = 0, which means that nested saturations are not considered here. By adapting the results in [90] (and therefore by using Lemma 1) in the context of stability analysis (r = 0), the following proposition can be stated: Proposition 1 (Stability analysis). If there exist matrices: • Q = QT ∈ IR n×n • S = diag(s1 , . . . , sm )

(7.9)

• Z ∈ IR m×n such that the following LMI conditions hold (where Zi and C2i denote the ith rows of Z and C2 respectively): AQ + QAT B2 S − Z T <0 (7.10) SBT2 − Z −2S

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach

then the ellipsoid:

T Q ZiT + QC2i Zi + C2i Q 1

125

> 0, i = 1...m

EQ−1 = x ∈ IR n ; xT Q−1 x ≤ 1

(7.11)

(7.12)

deﬁnes a domain of attraction of system (7.8) with r = 0. Remark 5. The result of Proposition 1 is stated as an LMI feasibility problem. It should be emphasized that there exist many ways of converting it into an optimization problem, since in general the objective is to maximize the region of stability. A standard objective consists in maximizing the size of the ellipsoid EQ−1 , which is known to be a convex problem with respect to the decision variables. It can indeed be stated as a linear objective minimization problem under LMI constraints. Alternative approaches consist, for example, in optimizing the shape of the ellipsoid, so that it contains the farthest point in a given direction u of the state space [89]. This problem reduces to the maximization of a linear objective under LMI constraints: Q βu max β such that >0 (7.13) βuT 1 Other criteria can also be used (see for example [2, 101]). 7.3.4

Performance Analysis of Saturated Systems

The above stability analysis is performed on an autonomous nonlinear system. In practice, however, the considered system is generally affected by some exogenous input signals such as perturbations (wind, turbulences) or commanded inputs, and thus r = 0. In terms of performance analysis, a classical problem is to evaluate the tracking error of the system for a given class of exogenous inputs. A popular approach consists of considering the class of ﬁnite energy (L2 bounded) input signals. The performance level is thus evaluated through the L2 -induced norm, for which LMI characterizations are well-known. Unfortunately, the class of ﬁnite energy signals is quite large and may not be wellsuited to saturated systems, for which the performance level is expected to depend on the shape (and especially the amplitude) of the input signal. For this reason, it is suggested to restrict the class of input signals. In practice, the time-domain behavior of a closed-loop system is generally evaluated through step responses, but the associated step input signals are not L2 -bounded. Consequently, in our proposed analysis approach, slowly exponentially decreasing signals r ∈ IR p are considered instead: ∀t ≥ 0 , r(t) = r0 e−εt

(7.14)

where ε is a given positive integer. Each elementary input signal ri (t), i = 1, . . . , p , can be interpreted as a step input bounded by |r0i | provided that ε is small enough compared to the system dynamics (see Figure 7.6).

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step signal

ri(t) r0i

slowly decreasing signal t

0

Fig. 7.6. Step input approximation

Remark 6. A more general class of slowly exponentially decreasing signals can be considered by setting: (7.15) ∀t ≥ 0 , ri (t) = r0i e−εit where εi , i = 1 . . . p , are given positive integers. This allows us to impose speciﬁc dynamics for each component of the signal r. p

Let us now deﬁne the associated class Wε (ρ) as the set of signals r ∈ IR p satisfying (7.14) with r0 ≤ ρ. Interestingly, such signals are easily generated by a stable autonomous system R(s) with non-zero initial states r0 , whose linear equations are deﬁned as follows: ! r˙ = −εr (7.16) R(s) : r(0) = r0 R(s) can be integrated into system (7.8) and an augmented state vector ξ is deﬁned by: r ν= ∈ IR na (7.17) x The corresponding augmented plant is ﬁnally obtained as follows: ⎧ ⎨ ν˙ = A ν + Bφ Φ(z) z = Cφ ν ⎩ z p = C p ν + D pφ Φ(z) where:

−εI p 0 0 A = ; Bφ = B2 B1 A Cφ = D21 C2 C p = D11 C1 ; D pφ = D12

(7.18)

(7.19)

By combining the stability result of Proposition 1 and Remark 5, a method can be derived to compute the maximum amplitude ρ such that, for any exogenous signal r ∈ Wεp (ρ), system (7.8) with zero initial conditions (x(0) = 0) remains stable despite the saturation effects. The method consists of computing the invariant set which contains T any state ν = rT 0 with r ≤ ρ for a maximized value of ρ.

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Beyond stability considerations, the next step is to evaluate the impact of the input signal r on the exogenous output z p . More precisely, the objective is to calculate a bound on the energy of this output. A solution is proposed in the following proposition: Proposition 2 (Performance analysis). Consider system (7.8) with the above notation in mind. If there exist matrices: • Q = QT ∈ IR na ×na • S = diag(s1 , . . . , sm ) • Z ∈ IR

(7.20)

m×na

and positive scalars γ and ρ such that the following LMI conditions hold: ⎞ ⎛ ρI p Q ⎝ 0 ⎠>0 ρI p 0 Ip ⎛

⎞

A Q + Q A T Bφ S − Z T Q C p T ⎝ SB T − Z −2S SD pφ T ⎠ < 0 φ C pQ D pφ S −γI p

(7.21)

Q ZiT + QCφiT Zi + Cφi Q 1

(7.22)

> 0, i = 1...m

(7.23)

then, for all ρ ≤ ρ and all exogenous inputs r ∈ Wεp (ρ), system (7.8) is stable for all initial condition x0 in the domain E (ρ) deﬁned as follows: " T r r p n E (ρ) = x ∈ IR ; ∀r ∈ Wε (ρ), P ≤1 (7.24) x x where P = Q−1 . Moreover, the output energy satisﬁes: ∞ 0

z p (t)T z p (t) dt ≤ γ

(7.25)

Sketch of proof: The above proposition is a rather straightforward extension of Proposition 1. As for the stability result, it is based on a quadratic approach, (i.e. the search for a quadratic Lyapunov function V (ν) = νT Pν, where P = PT > 0). The main difference is observed in inequality (7.22), which implies γV˙ + zTp z p < 0 and thus (7.25) by integration. 2 7.3.5

Full-Order Anti-windup Design

Let us now focus on the design issue. Consider the nonlinear interconnection of Figure 7.7. The saturated plant G(s) to be controlled is written in a standard LFT form:

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⎧ Φ(z) ⎪ ⎪ ⎨ x˙G = AG xG + BG u G(s) : ⎪ ⎪ ⎩ z = CG xG + DG Φ(z) u y

(7.26)

where u and y denote the control inputs and the measured outputs respectively. Remember that the nonlinear operator Φ(.) is deﬁned by (7.3) and (7.4). anti−windup

J(s) v1

R(s)

r

Φ

v2

+ K(s)

G(s)

z

u

+

plant

+ yr

−

zp

K(s) linear feedback

y = [yTr . . . ]T L(s)

yrlin

nominal closed−loop plant

M(s)

Fig. 7.7. Standard interconnection with a general anti-windup architecture

Remark 7. It is assumed in the sequel that DG =

0

0

, which is generally fulDG21 DG22 ﬁlled in practice. If it is note, this will be due to the presence of nested saturations, which requires a speciﬁc treatment as proposed in [225].

Suppose that a nominal linear controller has been ﬁrst designed, so as to stabilize the plant G(s) and ensure good performance properties in the linear region. To mitigate the adverse effects of saturations, additional signals v1 and v2 are injected both at the input and output of the controller. A state-space representation of the resulting controller K(s) is then given by: x˙K = AK xK + BK y + v1 (7.27) K(s) : u = CK xK + DK y + v2 v The signals v1 and v2 are obtained as the outputs v = 1 ∈ IR nv of the dynamic v2 anti-windup controller J(s) to be determined:

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach

J(s) :

x˙J = AJ xJ + BJ Φ(z)

129

(7.28)

v = CJ xJ + DJ Φ(z)

where the input signal Φ(z) can be interpreted as an indicator of the saturation activity. Finally, as is illustrated in Figure 7.7, the reference signal r is generated by a stable autonomous plant R(s), whose linear equations are described by equation (7.16). Let us now deﬁne the following augmented state vector ξ, obtained by merging the states of the reference model (r), the nominal (linear) closed-loop system (xL ), the openloop plant (xG ) and the nominal controller (xK ): ⎡ ⎤ r ⎢ xL ⎥ ⎢ ξ=⎣ ⎥ (7.29) ∈ IR nM xG ⎦ xK The resulting system M(s) connected with the anti-windup compensator is illustrated in Figure 7.8 and can be deﬁned as follows: ⎧ ˙ ⎪ ⎨ ξ = A ξ + Bφ Φ(z) + Ba v M(s) : (7.30) z = Cφ ξ ⎪ ⎩ p z p = C p ξ + D pφ Φ(z) + D pa v = yr − yrlin ∈ IR where yr corresponds to the ﬁrst elements of the output vector y = [yTr . . . ]T .

Φ

M(s) J(s)

v

z zp

Fig. 7.8. A synthetic view of Figure 7.7

Finally, by adding the state xJ ∈ IR nJ of the anti-windup compensator, the following augmented state vector is deﬁned: ξ ν= (7.31) ∈ IR n xJ and therefore the global nonlinear closed-loop plant P(s) reads: ⎧ A BaCJ B φ + B a DJ ⎪ ⎪ ˙ Φ(z) ⎪ ⎨ ν = 0 AJ ν + BJ P(s) : z = Cφ 0 ν ⎪ ⎪ ⎪ ⎩ z p = C p D paCJ ν + D pφ + D paDJ Φ(z)

(7.32)

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Let the state vector ν be partitioned as ν = [rT ζT ]T to distinguish more clearly the reference r from the other states ζ = [xTL xTG xTK xTJ ]T ∈ IR n−p . The anti-windup design problem to be solved can then be summarized as follows: Problem 1 (Anti-windup design). Compute a dynamic anti-windup controller J(s) (i.e. matrices AJ , BJ , CJ and DJ ) and a domain E (ρ) as large as possible such that, for a given p positive scalar ρ and any reference signal r ∈ Wε (ρ), the following properties hold: • the nonlinear closed-loop plant (7.32) remains stable for all initial condition ζ0 inside E (ρ), • selected outputs yr of the plant remain as close as possible to the linear reference yrlin (associated with the nominal non-saturated behavior), i.e. the energy of the error signal z p is minimized. On the basis of the above problem statement, the following result, adapted from [27] and from Proposition 1, can now be stated: Proposition 3 (Performance characterization). Consider the nonlinear interconnection of Figure 7.8 with a given anti-windup controller J(s). Let u(ρ) = [ρI p 0]T ∈ IR n×p . If there exist matrices: • Q = QT ∈ IR n×n • S = diag(s1 , . . . , sm ) • Z ∈ IR m×n

(7.33)

and positive scalars γ and ρ such that the following LMI conditions hold (where Zi and Cφi denote the ith rows of Z and Cφ respectively) 1 : Q >0 (7.34) u(ρ)T I p ⎛ ⎞ T A BaCJ A BaCJ ⎟ Q+Q ⎜ 0 A 0 AJ J ⎜ ⎟ ⎜ ⎟ T ⎜ ⎟<0 (7.35) B φ + B a DJ ⎜ S −Z −2S ⎟ ⎜ ⎟ BJ ⎝ ⎠ D pφ + D paDJ S −γI p C p D paCJ Q Q > 0, i = 1...m (7.36) Zi + Cφi 0 Q 1 p

then, for all ρ ≤ ρ, and all reference signals r ∈ Wε (ρ), the nonlinear interconnected system (7.32) is stable for all initial condition ζ0 in the domain E (ρ) deﬁned as follows: " T r r p n−p E (ρ) = ζ ∈ IR ; ∀r ∈ Wε (ρ), P ≤1 (7.37) ζ ζ where P = Q−1 . Moreover, the output energy satisﬁes: 1

For compactness, the symmetric terms in the matrix inequalities are replaced by ”” throughout.

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach ∞ 0

z p (t)T z p (t) dt ≤ γ

131

(7.38)

Let us now focus on the anti-windup design issue stated in Problem 1. In this case, the decision variable Q introduced in Proposition 3 and the state-space matrices of J(s) have to be computed simultaneously. As a result, inequality (7.35) becomes a BMI and is thus no longer convex. However, in the full-order case (i.e. nJ = nM ), the constraints (7.34)-(7.36) exhibit particular structures which can be exploited to derive a convex characterization. Proposition 4 (Full-order anti-windup design). Consider the nonlinear interconnection of Figure 7.8. Let Γ = diag(Na , Im , N pa ), where Na and N pa denote any basis of the null-spaces of BTa and DTpa respectively. Let u(ρ) = [ρI p 0]T ∈ IR nM ×p . There exists an anti-windup controller J(s) such that the conditions of Proposition 3 are satisﬁed if and only if there exist matrices: • X = X T ,Y = Y T ∈ IR nM ×nM • S = diag(s1 , . . . , sm ) • W = U V ∈ IR m×(nM +nM )

(7.39)

such that the following LMI conditions hold: u(ρ)T Xu(ρ) < I p ⎛

AT X + XA Cp −γI p

AY + YAT T ⎝ SBT − V Γ φ C pY ⎛ X ⎝ InM Y Ui Vi + Cφi Y

(7.40)

<0

⎞ −2S ⎠ Γ < 0 0 −γI p ⎞ ⎠ > 0, i = 1...m 1

(7.41)

(7.42)

(7.43)

Proof. Following a scheme proposed by [76], it sufﬁces to rewrite inequalities (7.34)(7.36) of Proposition 3 by capturing the decision variables AJ , BJ , CJ and DJ into a single matrix Ω, which can then be eliminated using the projection lemma. 2 Remark 8. The matrix Q of Proposition 3 is obtained from X and Y via the following relation [76]: −1 Y InM InM X Q= where M T N = InM − XY (7.44) 0 M N 0 The decision variable Q being ﬁxed, inequality (7.35) is convex with respect to the state-space matrices AJ , BJ , CJ and DJ of the anti-windup controller, which can thus be easily computed. Moreover, using a suitable change of variables (see Proposition 5), it can be observed that S and Z do not have to be ﬁxed during the reconstruction phase. This offers additional degrees of freedom that can be used, for example, to add some further constraints on the controller matrix AJ .

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Reduced-Order Anti-windup Design

The case where the matrix AJ is entirely ﬁxed also deserves some attention. This is indeed a convenient way to control precisely the dynamics of the anti-windup controller. Moreover, as stated in Proposition 5 below, when CJ is also ﬁxed, the anti-windup problem becomes convex in the reduced-order case. Proposition 5. The BMI constraint (7.35) of Proposition 3 is convex as soon as the matrices AJ and CJ of the anti-windup controller are ﬁxed. Proof: It immediately follows from a classical change of variables B˜ J = BJ S and D˜ J = 2 DJ S, which is here valid since S > 0. Based on this result, the following algorithm is introduced: Algorithm 1 (Fixed-dynamics anti-windup design) 1. Choose appropriate AJ and CJ matrices, which deﬁne respectively the state and the output matrices of the anti-windup controller J(s) to be computed, 2. Fix ρ and minimize γ under the LMI constraints (7.34), (7.35) and (7.36) w.r.t. the variables Q, S, Z, B˜ J , D˜ J , 3. Compute BJ and DJ by inverting the aforementioned change of variables. The main difﬁculty in the above algorithm consists in choosing the matrices AJ and CJ correctly. This choice may appear more intuitive and natural by considering the following decomposition: n1

n2 Mi1 Mi2 +∑ 2 2 s + λ s + 2η i i ωi + ωi i=1 i=1

J(s) = M0 + ∑

(7.45)

where DJ = M0 and BJ contains the collections of matrices Mi1 and Mi2 . For this decomposition, the ﬁxed matrices AJ and CJ can be chosen as: AJ = diag (−λ1 , . . . , −λn1 , A1 , . . . , An2 ) CJk = 1 . . 1( [1 0] . . . [1 0] , k = 1 . . . nv % .&' % &' ( n1

(7.46)

n2

0 1 Ai = (7.47) , i = 1 . . . n2 −ω2i −2ηi ωi From this observation, the ﬁrst step of Algorithm 1 simply boils down to the choice of a list of poles for the anti-windup controller, whose matrices AJ and CJ are then immediately deduced from (7.46) and (7.47). where:

Remark 9. The poles of the reduced-order anti-windup controller can be chosen by selecting some of those obtained in the full-order case. Typically, the slow and fast dynamics are eliminated. Alternatively, an iterative procedure starting from the static case can be used. The list of poles is then progressively enriched until the gap between the full and reduced-order cases becomes small enough. Note that the order of the reducedorder controller is given by nJ = n1 + 2n2. The two parameters n1 and n2 should then be chosen sufﬁciently small to ensure that nJ < nM .

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7.4 Extension to Parameter-Varying Systems 7.4.1

Description of the Problem

The nonlinear interconnection considered in this section and depicted in Figure 7.9 is similar to the one described in Section 7.3.5, except that the saturated plant G(s) now depends on real time-varying parameters δ1 (t), . . . , δk (t). More precisely, for each t ≥ 0, θG (t) is a block-diagonal operator specifying how these parameters enter the plant dynamics, i.e. wG = θG (t)zG , where: (7.48) θG (t) = diag δ1 (t)Ir1 , . . . , δk (t)Irk The operator θG (t) is normalized, i.e. θG (t)T θG (t) ≤ Ir , where r = r1 + . . . + rk . The set of all normalized operators with structure (7.48) is denoted by ΘG , and thus θG (t) ∈ ΘG ∀t ≥ 0. anti−windup

v zJ

wJ

J(s)

Φ

θJ

plant

linear feedback

z

v1 R(s)

approximation of step input

r y

v2

+

K(s) zK +

wK

u

G(s)

y zG

wG

yr +

zp

−

θG

θK y = [yTr . . . ]T r wL

L(s)

yrlin zL

θL nominal closed−loop plant

Fig. 7.9. Description of the general parameter-dependent anti-windup architecture

Similarly, the nominal controller K(s), the nominal closed-loop plant L(s) and the anti-windup controller J(s) all depend on real time-varying parameters. The associated normalized block-diagonal operators are denoted by θK (t), θL (t) and θJ (t) respectively.

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In this context, the augmented plant M(s) introduced in Section 7.3.5 can now be described by: ⎧ ξ˙ = A ξ + Bφ Φ(z) + Ba v + Bθ wM ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z = Cφ ξ (7.49) M(s) : z p = C p ξ + D pφ Φ(z) + D pa v + D pθ wM ⎪ ⎪ ⎪ zM = Cθ ξ + Dθφ Φ(z) + Dθa v + Dθθ wM ⎪ ⎪ ⎪ ⎩ wM = θM (t) zM where: θM (t) = diag(θL (t), θG (t), θK (t)) ∈ ΘM ⊂ IR

pM ×pM

(7.50)

The parameter-dependent anti-windup controller J(s) to be computed is then given by: ⎧ ⎪ ⎨ x˙J = AJ xJ + BJφ Φ(z) + BJθ wJ J(s) : v = CJa xJ + DJaφ Φ(z) + DJaθ wJ (7.51) ⎪ ⎩ zJ = CJθ xJ + DJθφ Φ(z) + DJθθ wJ The global nonlinear closed-loop plant P(s) including J(s) is ﬁnally obtained as follows: ⎧ ν˙ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z = P(s) : z p = ⎪ ⎪ ⎪z = ⎪ θ ⎪ ⎪ ⎩ wθ =

A ν + Bφ Φ(z) + Bθ wθ Cφ ν C p ν + D pφ Φ(z) + D pθ wθ Cθ ν + Dθφ Φ(z) + Dθθ wθ

(7.52)

θ(t) zθ

where: θ(t) = diag(θM (t), θJ (t)) ∈ ΘP ⊂ IR

pP ×pP

(7.53)

and: ⎛

A

⎜ Cφ ⎜ ⎝ Cp

Cθ

7.4.2

⎛

A BaCJa Bφ + Ba DJaφ ⎜ 0 AJ BJφ Bφ Bθ ⎜ ⎜ Cφ 0 0 0 0 ⎟ ⎟=⎜ D pφ D pθ ⎠ ⎜ C D C D + D p pa Ja pφ pa DJaφ ⎜ ⎝ Cθ DθaCJa Dθφ + Dθa DJaφ Dθφ Dθθ 0 CJθ DJθφ ⎞

⎞ Bθ Ba DJaθ 0 BJθ ⎟ ⎟ ⎟ 0 0 ⎟ D pθ D pa DJaθ ⎟ ⎟ Dθθ Dθa DJaθ ⎠ 0 DJθθ

Parameter-Varying Anti-windup Design

Let Sx be the convex set of positive deﬁnite scaling matrices that commute with every operator θ of a given set Θx :

Sx = {L = LT > 0 : Lθ = θL ∀θ ∈ Θx }

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where x ∈ {P, M} in the sequel. On the basis of the above context, Proposition 3 is adapted to deal with time-varying parameters. Inequality (7.35) is simply replaced by: ⎞ ⎛ A Q + QA T ⎜ SB T − Z −2S ⎟ ⎟ ⎜ φ ⎟ ⎜ ⎜ (7.54) C pQ D pφ S −γI p ⎟ ⎟<0 ⎜ ⎟ ⎜ T T L B 0 L D −L ⎠ ⎝ θ pθ Cθ Q Dθφ S 0 Dθθ L −L where L ∈ SP . Let us now focus on the anti-windup design issue. In this case, the analysis variable Q and the state-space matrices of J(s) have to be optimized simultaneously. As a result, inequality (7.54) is no longer convex, unless the following assumptions are satisﬁed: • full-order anti-windup controllers are designed, i.e. nJ = nM , • the structures of θJ (t) and θM (t) are the same, i.e. θJ (t) = θM (t) ∀t ≥ 0. Proposition 4 can then be adapted to deal with time-varying parameters. Let Γ = diag(Na , Im , N pa , I2pM , Nθa ), where Na , N pa and Nθa denote any basis of the null-spaces of BTa , DTpa and DTθa respectively. Inequalities (7.42) and (7.43) are replaced by: ⎛

AY + YAT ⎜ T ⎜ SBφ − V −2S ⎜ ⎜ C pY D pφ S −γI p ⎜ ΓT ⎜ ⎜ RBTθ 0 RDTpθ −R ⎜ ⎜ T Bθ 0 DTpθ −InM ⎝ CθY Dθφ S 0 Dθθ R

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟Γ < 0 ⎟ ⎟ −T ⎟ ⎠ Dθθ −R

(7.55)

⎛

⎞ AT X + XA ⎜ ⎟ Cp −γI p ⎟ ⎜ ⎜ ⎟ ⎜ BTθ X DTpθ −InM ⎟ ⎜ ⎟<0 ⎜ ⎟ 0 T D −T TC ⎝ ⎠ θ θθ Cθ 0 Dθθ −InM −R

(7.56)

where R, T ∈ SM .

7.5 Application to the On-Ground Control Problem 7.5.1

Anti-windup Design on the Simpliﬁed LFT Model

Using the theoretical background described in Sections 7.4 and 7.5, a controller can now be designed on the simpliﬁed lateral LFT model of the on-ground aircraft presented in Section 7.2.2, so as to meet the design speciﬁcations described in Section 7.2.1.

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Inner Loop Let us ﬁrst focus on the yaw rate control, also called the inner control loop. The saturations are temporarily ignored and a nominal PI controller Kr (s) depending on the aircraft longitudinal velocity Vx is designed by a modal technique using the Matlab LFR Toolbox [148, 97]. The resulting closed-loop plant is depicted in Figure 7.10, where K is a third-order rational function in Vx . It behaves like a well-damped second-order system with a pulsation ωr = 2.15 rad/s for any aircraft longitudinal velocity between 5 and 40 kts. Note that this is the minimum order which guarantees that ωr remains constant on the whole speed range. Kr (s) Hr

+ rc

−

1 s

+ K

Vx

+

θNW c

on−ground

r

aircraft

Vx

Fig. 7.10. Structure of the nominal parameter-varying PI controller

The reference yaw rate r0 is tracked without steady-state error. Nevertheless, an important overshoot occurs when the saturations are active, as shown by the dotted line on Figure 7.16. This overshoot cannot be eliminated unless the pulsation of the PI controller is signiﬁcantly reduced, resulting in a excessively slow response time. The next step thus consists of designing a parameter-varying anti-windup compensator, which is a function of Vx , to take into account the adverse effects of the saturations. As pointed out in the previous section, this issue reduces to the minimization of a linear objective (the performance index) under LMI constraints. Several static and dynamic controllers were thus computed for various runway conditions, i.e. for various saturation levels (see Figure 5 of Chapter 6), leading to the following conclusions: • a static controller is sufﬁcient to meet the design requirements, • an anti-windup action on either satatan or satFMG does not bring a signiﬁcant improvement, and • the anti-windup gains do not vary signiﬁcantly with the runway conditions. A simple anti-windup gain matrix Jr optimized for wet runway conditions and acting on both satNW S and satFNW thus proves to be a good compromise. Each of these two gains is a second-order rational function in Vx .

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Outer Loop The yaw rate commands rc can be directly sent to the inner loop by the pilot through the use of a dedicated control device, such as a side-stick tiller. But they can also be provided by an autopilot in charge of controlling the aircraft heading: this is the outer loop, which is built following the same procedure as the inner loop. The nominal controller KΨ (s) is a ﬁrst-order system, whose gain, zero and pole are ﬁrst, second and secondorder rational functions of Vx respectively. This guarantees that the dominant mode is about 1.2 rad/s when no saturations are active. An anti-windup controller JΨ can then be added to counter the effect of saturations. The closed-loop plant with both inner and outer controllers is shown on Figure 7.11. + ε

Jr

−

Vx

rautopilot Ψc

KΨ (s)

+ rc

Kr (s)

+

θNW c

on−ground

r pilot

aircraft

Vx Vx

Ψ

Vx

1 s

outer loop controller

inner loop controller

Fig. 7.11. Overall structure of the closed-loop plant

The LFT representation of the overall controller is ﬁnally depicted in Figure 7.12.

Vx

ε

Controller

Ψc / rc

LFT model

θNW c

r

Fig. 7.12. LFT representation of the parameter-dependent anti-windup controller

r

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On-Line Estimation of the Ground Forces

The next step is to implement this anti-windup controller on the full nonlinear model. The main difﬁculty here is that the saturation levels depend on the ground forces, whose characteristics are strongly effected by the runway state and the vertical payload (the higher the payload, the higher the saturation levels). It is thus necessary to identify the ground forces on-line to detect whether the saturations are active or not. An identiﬁcation procedure is proposed, which is based on the dynamic inversion of the lateral LPV model of the on-ground aircraft deﬁned by equations (24)-(26) of Chapter 6. More precisely, using the notation below: • xNL = [rNL VyNL ]T : state vector of the full nonlinear model • xLPV = [rLPV VyLPV ]T : state vector of the LPV model and following along the same lines as in Section 2 of Chapter 6, the lateral ground forces can be estimated by: xNL − xLPV FˆyNW −1 ≈ Btyres [αI − Aa (Vx )] (7.57) xLPV FˆyMG where α is a positive scalar which is sufﬁciently large compared to the aircraft dynamics. Note that the aerodynamic effects have been neglected, which is a reasonable assumption, since only maneuvers below 40 kts are considered. Combining equations (10), (11) and (34) of Chapter 6, the sideslip angles are then given by: βNW = arctan β˜ NW − θNW (7.58) βMG = β˜ MG where:

β˜ NW = Cβ (Vx ) xNL β˜ MG

(7.59)

The saturations are active as soon as the estimated ground forces are less than their linear aproximation. The saturations levels can thus be estimated via two signals εFNW and εFMG as follows: εFNW = sign (βNW ) max |GyNW βNW | − |FˆyNW |, 0 (7.60) εFMG = sign (βMG ) max |GyMG βMG | − |FˆyMG |, 0 Let us now combine this estimator with the nose-wheel steering system model described by Figure 7.13. Using LFT modelling, the global estimator depicted in Figure 7.14 is ﬁnally obtained. The outputs εNW S , εFNW and εFMG of this system are standard indicators of saturation levels, which exhibit non-zero values when the corresponding saturations are active. Such signals can then be directly used by an anti-windup control device. 7.5.3

LFT-Based Implementation of the Controller

The resulting adaptive controller is composed of the parameter-varying anti-windup controller computed in Section 7.5.1 and of the aforementioned estimator block. It is implemented on the full nonlinear model as shown on Figure 7.15.

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139

εNW S + +

θNW c

_

_

1 s

λ

θNW

Fig. 7.13. Estimation of the nose-wheel steering system saturation level

Vx

⎫ εNW S ⎪ ⎪ ⎬ εFNW ε ⎪ ⎪ ⎭ εF

Estimator xNL

LFT model

θNW c

MG

Fig. 7.14. On-line estimation of the saturation levels

Vx

Vx ΨNL

δr Ψc / rc

Controller LFT model

θNW c

NL model

xNL

Estimator LFT model

ε

rNL

Fig. 7.15. Controller integration on the full nonlinear model

It is worth pointing out that this controller is given in an LFT format and can thus be easily implemented on-line. Moreover, both its order and the repetition of Vx in the ∆-block remain low, as is summarized in Table 7.1. Let us now present a few simulation results, so as to demonstrate: • the relevance of the multidimensional anti-windup approach, • the robustness of the anti-windup controller to a payload variation, • the ability of the on-line estimator to identity the lateral ground forces, and thus the runway state.

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C. Roos et al. Table 7.1. Complexity of the adaptive controller Controller

Order

Size of the ∆-block

Inner loop (nominal)

1

3

Inner loop (anti-windup)

0

4

Outer loop (nominal)

2

5

Estimator

2

4

Total

5

16

Figure 7.16 ﬁrst shows the response of the on-ground aircraft to a yaw rate step input on a wet runway and for a longitudinal velocity Vx ≈ 8 m/s. The input signal is drawn as a thin solid line. Without anti-windup compensation (dotted line), the PI controller ensures that there is no steady-state error but a strong overshoot can be observed, and it takes almost 8 seconds to reach the desired value. If the anti-windup gain associated with the rate limitation of the nose-wheel steering system is employed (dashed line), the overshoot is signiﬁcantly reduced. Finally, with full anti-windup compensation (solid line), the reference input is perfectly tracked in little more than 3 seconds with a negligible overshoot, which conforms to the design speciﬁcations expressed in Section 7.2.1. It appears that each gain term brings its own contribution, thus showing the relevance of the proposed multidimensional anti-windup scheme. 14

12

yaw rate (deg/s)

10

8

6

4

2

0 0

1

2

3

4

5

6

7

8

9

10

time (s)

Fig. 7.16. Step responses with no, partial and full anti-windup compensation

Figure 7.17 shows the responses to a yaw rate step input on a wet runway for several values of the aircraft weight. The solid line corresponds to the nominal weight, whereas the dashed and dotted lines correspond to 30 % lower and upper weights respectively. The aircraft behavior does not vary signiﬁcantly, which shows the good robustness of the proposed control system to strong variations in the aircraft weight, and thus in the vertical payload.

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141

10 9 8 7

yaw rate (deg/s)

6 5 4 3 2 1 0 −1

0

1

2

3

4

5

6

7

8

9

10

time (s)

Fig. 7.17. Step responses for different values of the aircraft weight

Figure 7.18 ﬁnally shows the responses to a yaw rate or a heading step input for various runway conditions (dry, wet or icy) and with fully active anti-windup conpensation. The aircraft longitudinal velocity is the same as before, i.e. Vx ≈ 8 m/s. The adaptive controller performs very well, since the time response is satisfactory whatever the runway conditions. 10

35

9 30 8 25

6

heading (deg)

yaw rate (deg/s)

7

5 4 3

20

15

10

2 5 1 0 0 −1

0

1

2

3

4

5

time (s)

6

7

8

9

10

−5

0

2

4

6

8

10

12

time (s)

Fig. 7.18. Step responses for various runway conditions

7.5.4

Results

Let us now consider the generic maneuver illustrated on Figure 7.1 to demonstrate the efﬁciency of the parameter-dependent anti-windup controller, and especially the three sequences detailed in Section 7.2.1. The emphasis is placed on the yaw rate control, which is more commonly used in practice by pilots. Each simulation proceeds as follows from a piloting point of view: • a yaw rate step input with a suitable amplitude is ﬁrst applied to start the turn, • the yaw rate input is set back to zero when needed to reach the desired heading.

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The basic idea underlying this piloting strategy is to consider only simple commanded inputs, which can be easily applied by a human pilot. Simulations are performed for various runway conditions: in the sequel, solid, dashed and dotted lines correspond to dry, wet and icy runways respectively. For each of the ﬁrst two sequences, ﬁve graphs are plotted (see Figures 7.19 to 7.22): • • • • •

the heading responses, the commanded yaw rate inputs and the yaw rate responses, the commanded and achieved nose-wheel deﬂections, the on-ground aircraft trajectories in the horizontal plane, the inputs ε of the anti-windup controller, i.e. the difference between the inputs and the outputs of the saturations.

Although the aircraft is commanded in yaw rate, the heading responses are plotted ﬁrst, since the control objectives are deﬁned in terms of heading. Sequence 1: turn to take a 30◦ exit while decelerating from 30 kts to 20 kts. 35

30

heading (deg)

25

20

15

10

5

0

−5

0

2

4

6

8

10

12

time (s)

Fig. 7.19. Heading responses (sequence 1) 14

20

12

nose−wheel deflection (deg)

15

yaw rate (deg/s)

10

8

6

4

10

5

0

2 −5

0 0

2

4

6

time (s)

8

10

12

−10

0

2

4

6

8

10

12

time (s)

Fig. 7.20. Commanded and achieved yaw rates (upper left) and nose-wheel deﬂections (upper right), aircraft trajectories (lower left), and saturation activity (lower right)

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach

143

0.2

160

0.15

FMG

140

0.1

sat

120

0.05

100

x (m)

0 0

2

4

0

2

4

6

8

10

12

6

8

10

12

80

6

60

4

sat

NWS

40

20

2 0 −2 −4

0

−6

−5

0

5

10

15

20

25

30

35

40

45

50

time (s)

y (m)

Fig. 7.20. (continued)

Sequence 2: make a 60◦ turn at 10 kts. 70

60

heading (deg)

50

40

30

20

10

0

−10

0

2

4

6

8

10

12

14

time (s)

Fig. 7.21. Heading responses (sequence 2) 50

18

16

40

nose−wheel deflection (deg)

14

yaw rate (deg/s)

12

10

8

6

4

30

20

10

2

0 0

−2

0

2

4

6

8

time (s)

10

12

14

−10

0

2

4

6

8

10

12

14

time (s)

Fig. 7.22. Commanded and achieved yaw rates (upper left) and nose-wheel deﬂections (upper right), aircraft trajectories (lower left), and saturation activity (lower right)

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C. Roos et al. 0.6

60

sat

FMG

50

0.4

0.2

40

x (m)

0

30

0

2

4

6

0

2

4

6

8

10

12

14

8

10

12

14

10

20

NWS

5

sat

10

0

−5

0 −10

−5

0

5

10

15

20

25

30

35

40

45

50

time (s)

y (m)

Fig. 7.22. (continued)

The anti-windup signal becomes non-zero as soon as satFNW or satNW S is active. The orders that are sent to the control effectors are thus alleviated when the sideslip angles become too high, which prevents the aircraft from slipping on the ground. Note that in most cases, the closed-loop plant is unstable without anti-windup compensation. Sequence 3: perform a U-turn at 5 kts on a 45 m wide runway. At the beginning of the maneuver, there is a 30◦ angle between the aircraft body and the runway axis. The U-turn is then performed, which consists in making a 240◦ turn on the right. The aircraft then returns to its initial position, maintaining a 30◦ angle with the runway axis. The overall maneuver lasts 34.0 s on a dry runway, 34.3 s on a wet runway and 43.0 s on an icy runway.

50

40

x (m)

30

20

10

0

−30

−20

−10

0

10

20

y (m)

Fig. 7.23. Aircraft trajectories (sequence 3)

30

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach

145

It appears that for the three considered sequences, the heading is perfectly controlled whatever the runway state and the aircraft longitudinal velocity, which conforms to the design speciﬁcations and proves the effectiveness of the proposed control strategy.

7.6 Conclusions In this chapter, an original parameter-varying anti-windup control design technique is developed and successfully applied in a non-conventional context. In most anti-windup applications indeed, the saturations typically appear on the actuators dynamics and are well identiﬁed. But in this work, they are used to represent some nonlinear ground forces, whose magnitudes strongly depend on the runway state. As a result, such saturations are a priori unknown and have to be identiﬁed on-line. As shown in this contribution, this operation can be handled by an LFT-based estimator. An adaptive LPV anti-windup controller is ﬁnally obtained, which is easily implemented thanks to its LFT structure. Nonlinear simulations prove the efﬁciency of the proposed methodology to handle low-speed lateral maneuvers whatever the runway state (from nominal to icy conditions). Thanks to the anti-windup action, the nose-wheel control system is fully exploited at all times, without any risk. This results in highly optimized yaw rate and heading responses. This approach can be easily extended to handle a higher speed range. In this case, the aerodynamic forces cannot be neglected anymore and the on-ground aircraft has to be controlled with both the rudder and the nose-wheel steering system. The crucial point is thus to ﬁnd a suitable way to share the control action between these two devices according to the aircraft velocity, either using an allocation module or performing a multi-inputs design. Note that the LFT model developed in Chapter 6 can be directly exploited, since it already takes into account the aerodynamic effects. As a ﬁnal point, we emphasize that a similar methodology would be applicable to control the longitudinal dynamics of the on-ground aircraft. The longitudinal ground forces can indeed be modelled by saturation-type nonlinearities too. However, special attention would need to be paid to the inﬂuence of the vertical payload on the saturation levels.

8 Rapid Prototyping Using Inversion-Based Control and Object-Oriented Modelling Gertjan Looye DLR German Aerospace Center Institute of Robotics and Mechatronics Oberpfaffenhofen D-82234 Wessling, Germany [email protected] Summary. Object-oriented modelling allows for efﬁcient construction of multi-disciplinary system models in a physically-oriented way. As a unique feature, the modelling approach allows for automatic generation of regular, static, as well as inverse simulation models. Inverse models form the basis of various commonly used nonlinear controller synthesis methods, such as Feedback Linearisation. The resulting multivariable control laws are easily tuned to meet performance speciﬁcations and avoid the need for gain scheduling. In combination with automatic inversion, these synthesis methods are ideally suited for control law rapid prototyping, allowing experimentation with command variables, control selection and allocation, etc. in very short automated design cycles. After selection of the ﬁnal architecture, the control laws may be developed further in a detailed design stage. In this chapter, we demonstrate this approach on the aircrafton-ground nonlinear control problem. Keywords: Flight dynamics modelling, Object-oriented modelling, Inversion-based control, Nonlinear Dynamic Inversion, Rapid prototyping.

8.1 Introduction Many nonlinear control law synthesis methods, like Nonlinear Dynamic Inversion (NDI), Feedback Linearisation (FL), Model Following Control (MFC) and Inverse Feed Forward Compensation (IFFC) are based on inverse model equations. These equations directly compensate for nonlinear dynamic behaviour of the system, whereas desired command response behaviour is imposed using a linear feedback controller and command ﬁlters. The main advantage of these methods, disregarding any modelling errors, is that the control laws automatically achieve full decoupling of command variables (CVs) and automatically adapt as a function of all known parameters (e.g. ﬂight condition), thus avoiding the need for gain scheduling. This implies that, apart from the effort of model inversion and coding, functional control laws are obtained for the complete operating envelope in one shot. During the last decade, the methodology of object-oriented modelling for implementation of multi-disciplinary models has matured and found application in many ﬁelds of engineering, such as mechatronics, electronics, automotive and aerospace engineering. The main point in object-oriented modelling is that physical objects and phenomena, D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 147–173, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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and their interactions, may be implemented one-to-one into model components, and components interconnections respectively. Model implementation takes place at a physical level, rather than in the form of sorted and solved differential (algebraic) equations. The step from user-implemented model equations to differential equations for simulation is performed automatically by a model compiler . To this end, reliable algorithms and tools are readily available. The way differential equations are generated depends on the inputs and outputs selected by the user. This implies that the model implementation is independent of its causality in simulation. In the frame of this work, the most interesting aspect is that this automation step allows for the generation of regular, just as well as inverse, simulation models. The straight-forward nature of inversion-based control design and the capability of automatic model inversion allow for a highly interesting approach in the early stages of ﬂight control law design: rapid prototyping. This is of great interest in various aspects: • early availability of an initial set of control laws, as soon as a ﬁrst model is available; • quick comparison of different command variables, up to the stage of pilot-in-theloop ﬂight simulations; • quick re-design in case of model data updates (as is frequently the case in the aircraft pre-design phase); • early release of initial control laws for other engineering departments; • possibility of simulation-based formulation of requirements for control sizing in preliminary aircraft design; • testing of individual control law functions in the detailed design phase within or around a preliminary, but fully functional, control system. Since synthesis techniques like Feedback Linearisation have already proven their value in many applications, these control laws may be naturally developed further in the detailed design phase, once key decisions like control variable selection and control allocation have been made, and the aircraft conﬁguration has been ﬁnalised. In this chapter the above approach is described and demonstrated on the aircraft-onground nonlinear control problem. In Section 8.2 object-oriented modelling of ﬂight dynamics is discussed. Sections 8.3 and 8.4 describe the generation of simulation and inverse models, and discusses how the latter may be applied within various control law structures. Section 8.5 describes the proposed rapid prototyping design steps. Section 8.6 describes their application to lateral-directional ground control laws. Conclusions are drawn in Section 8.7.

8.2 Object-Oriented Modelling of Aircraft Flight Dynamics In this section a brief overview of object-oriented modelling and its application to model development for the aircraft-on-ground is given. 8.2.1

Object-Oriented Modelling

Implementation of models and model components is usually performed in the form of ordinary differential equations (ODEs). This is done in software code (e.g. C, FORTRAN), but also more and more in the form of block-diagrams. The latter allows for

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149

easy organisation of generic components in libraries from which models may be constructed graphically. The ODEs originate from physical equations from which derivatives of the selected state variables and output variables have been solved. System components are interconnected using input-output-based signal ﬂows. The basic principle of object-oriented modelling is that model implementation already takes place at the level of the original physical equations behind system components, rather than at the level of sorted and solved differential equations. This approach has the following advantages: • system components may be interconnected according to physical interactions (energy ﬂows, constraints), and are not limited to input-output relations. This allows for true one-to-one implementation of physical objects and phenomena into model software objects; • the above advantage allows for development of engineering discipline-speciﬁc component libraries, based on a common language base; • model code in executable form for simulation (i.e. differential equations) is generated automatically, based on user-speciﬁed inputs and outputs at the highest hierarchical level of the model. The second advantage provides an ideal basis for multi-disciplinary modelling, since each component of a model may be constructed using methods that are common place within the involved engineering area (i.e. an electronic circuit can be implemented as a circuit diagram, a mechanical systems can be constructed using multi-body formalisms, a control law may be implemented as a block diagram, etc.). This greatly improves model visibility, especially to engineers from other disciplines. The third advantage implies that model implementation does not yet freeze model causality. This feature will be exploited in this chapter.

Example 1. The bicycle model Throughout this chapter the concepts behind object-oriented modelling and automatic equation solving will be illustrated for a bicycle approximation of the aircraft-onground model [109]. The bicycle model may be divided into three components: the frame, the rear wheel, and the steerable front wheel. In order to keep the example simple, the acceleration in body x-direction is also neglected (i.e. ub is constant) and mass is assumed to be concentrated in the frame, resulting in the following set of equations: m (v˙b + ubrb ) − Fyext = 0

(8.1)

Izz r˙b − Mzext = 0

(8.2)

where Fyext and Mzext are the external force and moment along the body yb and zb axes respectively. For both wheels the equation for the lateral tyre force has the following structure:

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G. Looye

Fytyre = Gtyre βtyre , with: vb + Lxwheel rb − θsteer βtyre = ub

(8.3) (8.4)

where Gtyre is the (linearised) cornering gain for a given tyre type, vertical load and taxi way surface condition, βtyre is the tyre slip angle and Lxwheel is the body x-coordinate of the wheel axle.

R e a r W h e e l

e x te n d s W h e e l (e q n . 3 ,4 )

q

s te e r

b

w h e e l

L x

= 0 , = b

w h e e l

F ro n t W h e e l

e x te n d s W h e e l (e q n . 3 ,4 )

q M L G

= L x

, M L G

= q

s te e r

b

w h e e l

L x

w h e e l

,

N W

= b

N W

,

= L x

N W

A e r o d y n a m ic s (e q n . ....)

F ra m e

(e q n . 1 ,2 )

Fig. 8.1. Bicycle model in object-oriented form

In the case of object-oriented modelling, these equations are sufﬁcient to start implementing the bicycle model. Fig. 8.1 shows the basic model structure. The frame equations of motion (8.1), (8.2) are coded literally into the object Frame. The equations (8.3) and (8.4) can ﬁrst be coded in a class Wheel. This class is then instantiated as a rear and front wheel in the model by setting appropriate values for Gtyre , Lxwheel , and θsteer (for the rear wheel θsteer = 0, for the front wheel, θsteer = θNW ). The connectors contain two so-called ﬂow variables (Mz and Fy ) and two across variables (rb and vb 1 ). When connecting components, ﬂow variables are summed to zero and across variables are set equal between the connectors involved. This is automatically done by the model compiler, as will be discussed further on. As shown in Fig. 8.1, this approach allows for very easy addition of other components. Even if the mass is no longer concentrated in the frame (i.e. distributed over the wheels as well), the model structure remains the same and only appropriate equations of motion have to be added to the Wheel model class.

As can be seen from the example, an object-oriented approach allows for model implementation at an earlier stage in the modelling process in the form of physical 1

In practice, inertial position and the orientation matrix of the local body axes are used instead.

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equations, before these are transformed into differential equations. In the case of implementation in ODE or block diagram form, the user has to do considerably more work before implementation can start, like deriving differential equations, translating equations into declarative form to make sure all unknowns appear on the left hand side, and sorting them in the right order. Object-oriented modelling is freely available in the form of the modelling language Modelica. Initiated in 1996, and speciﬁed and maintained by the international Modelica association [171], Modelica is especially suitable for implementation of largescale multi-engineering models. Component libraries for multi-body systems, block diagrams, electronics, hydraulics, power-trains, vehicle dynamics, etc. are available2 , as well as commercial environments for graphical construction and compilation of models. In the context of this work, Dymola (Dynamic modelling laboratory [1]) has been used. In aircraft dynamics modelling there is a strongly growing need for multi-disciplinary model integration, since interactions between various disciplines, like ﬂight mechanics, structural dynamics and systems, tend to grow with each new aircraft design. For this reason, DLR has developed the Flight Dynamics Library (FlightDynLib) [172, 137]. This library is fully compatible with Modelica libraries for other engineering disciplines, thus allowing the multi-disciplinary ideas behind object-oriented modelling to be fully exploited. In constructing this library, a new physically-oriented generic aircraft model structure has been adopted. This structure will be discussed for the aircraft on-ground model. 8.2.2

Object-Oriented Modelling of the Aircraft-on-Ground

The aircraft-on-ground model as described in Chapter 1 has been implemented using components from the Flight Dynamics Library. The main tree of the library is depicted to the left in Fig. 8.2. Generic components, like environment or sensor models, are readily at hand. For aircraft-speciﬁc components, base classes are available that provide standard interfaces and variables. These base classes may be inherited by components describing aircraft-speciﬁc model equations. The basic model structure is depicted to the right in Fig. 8.2. The environment objects (lower right) include a world (in this case, Earth), atmosphere, and terrain model. The world model provides the Earth-Centred Inertial (ECI) [220] (2000 printing) as an inertial reference frame for all model components, as well as a geodetic reference in the form of the World Geodetic System 1984 (WGS-84), based on the Earth-Centered Earth-Fixed (ECEF) frame. The world model contains ﬁeld functions for the gravity acceleration, the EGM-96 Mean Sea Level (MSL) surface, and WMM-2005 magnetic ﬁeld. Double-clicking on the object allows a number of parameters to be set, to represent whether the Earth is rotating or assumed to be in rest, the type of gravity model to be used, etc. The atmosphere model in Fig. 8.2 in this case represents the MSL-referenced International Standard Atmosphere (ISA). Again, atmospheric conditions and geodetic wind ﬁelds with respect to the Earth surface are described using ﬁeld functions. 2

For a more complete list, see [171]

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Fig. 8.2. Structure of the aircraft-on-ground model, alongside the Flight Dynamics Library top level structure (left)

The depicted terrain model contains a parameterised model of the Earth’s surface, based on JAR-AWO speciﬁcations [111]. This model is normally used for automatic landing certiﬁcation. More detailed models may be linked from simulator terrain databases in the future. Earth-ﬁxed navigational equipment (e.g. VOR, DME, ILS systems at speciﬁed locations) are incorporated in separate objects (not shown), which are naturally attached to the terrain model. Note that the environment models have no connection with the aircraft model, which physically makes sense. This allows multiple aircraft to be implemented, all using the same environment objects. Components of the aircraft model may individually call to the environment models. For example, each landing gear may obtain its own height above terrain or compression by a simple function call to the terrain model. The environment models also make sure that all components use the same ﬁeld functions and parameters. For example, clicking on the atmosphere model, the ISA standard atmosphere may be selected with a higher than nominal sea level temperature T0 . All model components that request their local atmospheric conditions are then provided with ISA conditions based on this T0 . The core of the model structure is of course the component that represents the actual aircraft, see Fig. 8.2. The backbone of the aircraft model is composed of the kinematics and airframe blocks. The ﬁrst deﬁnes a ”North-East-Down” (NED) local vertical frame

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with its origin moving with a ﬁxed position in the aircraft, preferably the centre of gravity (CoG). The object further deﬁnes a body-ﬁxed reference frame (x-axis towards the nose, z-axis down, origin in CoG) to which the airframe is attached. The attitudes and inertial positions of both reference systems are available in the two connectors (top: body, below: NED). Attitude descriptions may be based on quaternions or Euler angles, and position integration may be based on a ﬂat or elliptic Earth. The airframe object basically describes the mechanical equations of motion. In the case of the on-ground model, the airframe is assumed to be a rigid-body. However, the airframe may be fully ﬂexible as well [140,194], or may be constructed from the Modelica multi-body systems (MBS) library. Connection between the airframe and kinematics objects (see Fig. 8.2) ensures that the reference systems in both connectors merge, i.e. from then on the airframe is moving freely with respect to the inertial reference, with its kinematics described in the corresponding object. The airframe object has a second connector on top, intended for interconnection of, for example, external force model components, sensor models, etc. Besides kinematic variables, each connector also describes (generalised) forces and moments along the local reference systems axes, declared as ﬂow variables. The airframe equations of motion are primarily driven by aerodynamic, propulsion, and gear forces and moments. These are computed in corresponding model components in Fig. 8.2. These components are aircraft-speciﬁc, in this case containing the neuralnetwork based model described in Chapter 1. Computation of key variables like the angle of attack, side slip, true airspeed, etc. is inherited from an aerodynamics base class. Local mean wind and turbulence are obtained from the localWind object. Besides the airframe, each component is described in its own local reference frame. An attachment object, as for example between an engine and the airframe models, transforms kinematics and forces and moments between its connectors. In the case of a rigid attachment, relative positions and attitudes may be entered as parameters. In the case of a ﬂexible airframe it is sufﬁcient to enter the grid point number on the structural model. The attached components then automatically move and rotate with the local structure point. The actuation system component in Fig. 8.2 describes actuators and hydraulic / electric systems. These may be constructed from hydraulics and electronics libraries (see for example [15]), but for the aircraft-on-ground model, only the co-ordination of control surface movements is described. Finally, the thin bar at the top of Fig. 8.2 represents a so-called data bus. This bus includes signals that one would typically ﬁnd on avionics buses in the aircraft, like the readings of all sensors, command signals to engine and control surface actuators, gear status, etc. For this reason, the sensor, actuator, and engine models have been attached to the bus object. The bus is also accessible from outside and allows direct connection to elements from the Modelica block diagram library. This enables a control system composed using this library to easily communicate with the aircraft components. Landing gears Of course, the most interesting aspect of the aircraft-on-ground model is the landing gear. The physical equations are provided in Chapter 1. Fig. 8.3 shows the gear

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constructed as a simple multi-body system. Most important is the gear compression, which determines internal gear load, which in turn heavily inﬂuences tyre forces. In contrast to Chapter 1, here the gear model uses the terrain model to the right in Fig. 8.2, so that the compression of each gear results from location and orientation of the airframe, and the local terrain elevation above the WGS-84 ellipsoid. The principle of computation is illustrated in Fig. 8.4. The center line of the gear in its uncompressed position is extended to the ground: Rterrain = Rgear + T0,gear [0, 0, lgear ]T h = fW GS84 (Rterrain )

(8.5) (8.6)

hterrain = fterrain (Rterrain ) (constraint) 0 = h − hterrain

(8.7) (8.8)

whereby Rgear is the inertial position of a ﬁxed reference point on the gear (in this case, the bottom of the unloaded gear), Rterrain is the inertial position of the crossing between the gear extension and the ground, and lgear is the length from the gear reference point to the ground. In case lgear < 0, this variable is the amount of compression (note that tyre compression is neglected, see Chapter 1). The transformation matrix from gear body into inertial axes is T0,gear . The geodetic height h (in WGS-84 co-ordinates) at Rterrain is computed from the function fW GS84 , which is available in the world model. The terrain elevation at Rterrain is hterrain , computed from the terrain model function fterrain (in the terrain model). By applying the constraint (8.8), lgear can be computed. The equations have been implemented as such in the Ground Reference object in Fig. 8.3. The model compiler will automatically build in a numerical nonlinear equation solver to solve for lgear .

Fig. 8.3. Landing gear model, constructed from multi-body system components

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155

rn e ) e (a irb o a irfra m n g ro u n d ) a irfra m e (o

lgear R

g e a r

R

te rra in

h

lgear te rra in

h

te rra in

W G S -8 4 e llip s o id

Fig. 8.4. Computation of landing gear compression

8.3 Translation of Object-Oriented Models In object-oriented modelling, model objects are deﬁned with the help of mathematical equations. Since the equations are not declarative, as in programming languages, the ﬁnal model comprises one large system of (nonlinear) equations that may be collected from all components and interconnections into the following form: ˙ 0 = F(X(t), X(t), w(t), p,t)

(8.9)

˙ occur in the model, and Here X(t) ∈ RnX are variables of which time derivatives X(t) w(t) ∈ Rnw contains any other variables, like local ones, inputs, and outputs. The vector p ∈ Rn p contains any parameters that may be set prior to, but remain constant during, simulation. In the rest of this chapter, the time (t) argument will be omitted if time dependency is obvious. Numerical integration algorithms as used in simulation tools require the model equations to be translated into an algorithm that computes state derivatives x(t) ˙ and the unknown part of w(t) (these are outputs and intermediate variables) from the state vector x(t) and the known part of w(t) (these are inputs). The state variables are automatically selected or may be imposed by the user (usually from X(t)). The algorithm must be provided in the form of computer code, like C or FORTRAN. After compilation this code may be called by the numerical integration algorithm. The way in which the equations are solved depends on which variables in w(t) are known or unknown. This is ﬁxed at the moment when the user has speciﬁed the model inputs (u ∈ Rnu ) and outputs (y ∈ Rny ). Mathematically, the system of equations (8.9) is then to be translated into the form of an ordinary differential equation (ODE) in state space form: x(t) ˙ = f (x(t), u(t), p,t) (8.10) y(t) = h(x(t), u(t), p,t) For simplicity of notation, it is here assumed that intermediate variables in w(t) have been eliminated. Although the second equation is an algebraic one, it only computes output variables y(t), on which the ﬁrst equation does not depend.

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Sometimes it is not possible to explicitly solve for outputs and state derivatives. The equations are then translated into the form: ˙ y(t), u(t), p,t) 0 = F1 (x(t), x(t),

(8.11)

Assuming that x(t) ˙ and y(t) can be solved unambiguously, this equation is a so-called differential algebraic equation (DAE) of differential algebraic index 1 [175], which may be time integrated using a dedicated DAE solver. Nowadays powerful symbolic algorithms are available that allow the translation of a physical model into the form of (8.10) or (8.11) to be performed automatically [55, 56, 181].

Example 2. The bicycle model (cntd.) In order to simulate the bicycle model, the inputs and outputs need to be selected ﬁrst. The input obviously is the nose wheel steering angle, and the yaw rate is chosen as output: θNW = u y = rb

(8.12) (8.13)

The model compiler will ﬁrst collect all equations and identify unknown variables. The collected equations are as follows: 0 0 0 0 0 0 0 0 0 0 0 0

= = = = = = = = = = = =

m (v˙b + ubrb ) − Fyext Izz r˙b − Mzext FyNW − GNW βNW FyMLG − GMLG βMLG MzNW − FyNW xNW MzMLG − FyMLG xMLG βNW − (vb + xNW rb ) /ub − θNW βMLG − (vb + xMLG rb ) /ub (−Fyext ) + FyNW + FyMLG (−Mzext ) + MzNW + MzMLG y − rb θNW − u

(from frame) (from frame) (from nose gear) (from main gear) (from nose gear) (from main gear) (from nose gear) (from main gear) (connector equation) (connector equation) (output equation) (output equation)

(8.14)

Comparing this system of equations with (8.9), the vector arguments are: X = x = [vb , rb ]T x˙ = [v˙b , r˙b ]T

(8.15) (8.16)

w = [y, u, θNW , Fyext , FyNW , FyMLG , Mzext , MzNW , MzMLG , T

βNW , βMLG ] p = [ub , m, Izz , GNW , GMLG , xNW , xMLG ]T

(8.17) (8.18) (8.19)

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Since u and x are known from input and integration of x˙ respectively, 12 variables (2 in x, ˙ the remaining 10 in w) need to be solved from the 12 equations above. In this case, this is very easily done by the model compiler using graph-based methods, resulting in equations of the form (8.10).

8.4 Inverse Model Generation As already remarked in Section 8.2, in object-oriented models causality is not yet ﬁxed in the implementation. This allows for automatic generation of inverse simulation models, where physical inputs and outputs have been reversed. Starting with the bicycle model, this section sketches the underlying procedures for model inversion. More details and examples can be found in [141, 229].

Example 3. The bicycle model (cntd.) For inversion of the bicycle model, the input and outputs are now selected as follows: rb = u y = θNW

(8.20) (8.21)

The collected model equations (8.14) remain the same, only the last two are replaced with (8.20) and (8.21). The number of unknowns and equations does not change. However, equation (8.20) no longer contains unknowns, since rb is a state variable. The model compiler will therefore differentiate the equation: r˙b = u˙ (8.22) Since this requires availability of the derivative of the new input u, the user is requested to facilitate this. One option is to declare u˙ as input, rather than u, or to generate a differentiable u from e.g. a ﬁlter with relative degree 1. In this example, the following ﬁlter is added: r˙c = −1/τu rc + 1/τuucmd u = rc so that:

(8.23) (8.24)

u˙ = r˙c

(8.25)

where τu is the ﬁlter time constant, rc is the commanded yaw rate, and ucmd is the commanded input. The differentiated equation (8.22) and the ﬁlter add three equations, but just two unknowns: u˙ and r˙c . One solution now is to introduce a so-called dummyderivative for u [155]: instead of u, ˙ u is used, making u and u independent variables: r˙b = u u = r˙c

(8.26) (8.27)

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The total number of equations now amounts to 15, from which 15 unknown variables can be solved, resulting in the following set of differential equations: x˙ = f (x, ucmd , p)

(8.28)

θNW = h(x, ucmd , p)

(8.29)

where x = [vb , rb , rc ]. The obtained inverse model may be connected with the original model, as depicted in Fig. 8.5. Because of (8.26), the depicted relation between u and the 1 J us + 1 1 u

c m d

r

c o m m a n d filte r

.

u ' c

r

G

in v e r s e b ic y c le m o d e l (d y n .)

c

s r

b ic y c le m o d e l

N W

b

Fig. 8.5. Controlled bicycle model

output rb is an integration. In combination with the command ﬁlter (with differentiable output rc ), the transfer function between ucmd and r becomes: 1 ucmd τu s + 1

r=

(8.30)

The selection of τu thus determines the command response behaviour, which is independent of the parameters p. In Fig. 8.6 an example response of the conﬁguration in Fig. 8.5 is shown3 . To the left the command response of rb perfectly matches the response of the command ﬁlter rc to the step input on ucmd . To the right the lateral acceleration (ny = − (v˙b + ub rb ) /g, g = 9.81 m/s2 ) response in the CoG and the deﬂections of θNW are shown for future reference. Yaw rate

Lateral acceleration, NWS input

0.01

0.025 u

0.008

(rad)

c

b

0.006

NW

r

c

−ny (−), θ

r , r (rad/s) c b

r

0.004 0.002 0 0

0.02 −ny

0.015

θNW 0.01 0.005

2

4

6 time (s)

8

10

0 0

2

4

6

8

10

time (s)

Fig. 8.6. Example time response of the structure in Fig. 8.5 3

The following parameter values were used: m = 70 · 103 kg, Iz = 4.0 · 106 kgm2 , ub = 20 m/s, xNW = 10 m, xMLG = −2.5 m, GNW = −4.0 · 105 N/rad, GMLG = −1.4 · 106 N/rad, τu = 1 s

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159

In order to make a comparison with inversion-based controller synthesis methods, the translation for inverse models is sketched starting from (8.10), rather than (8.9): x˙ = f (x, uc , uo , p) ycmd = hcmd (x, p)

(8.31) (8.32)

The inputs u have been divided into uc (available to the control laws) and uo (any other inputs, like disturbances, ﬂap settings, etc.). The outputs y have been grouped into ycmd (command variables, to be tracked by the controller), and other outputs yo (sensor outputs, other variables for assessment, etc.). The outputs yo are computed from: yo = ho (x, u, p)

(8.33)

Model inversion involves assigning ycmd as inputs and uc as outputs. For now it is assumed that the dimensions of both variables are the same and, as already apparent from (8.32), do not algebraically depend on each other. As in the bicycle example above, immediately the problem arises that (8.32) no longer contains unknowns. Manual nonlinear controller synthesis methods like FBL, and a model compiler both address this problem in the same way: by differentiating each of the individual output equations. This can be described using Lie derivatives [214, 104] for the ith entry as follows: y˙cmdi =

∂hcmdi (x, p) · f (x, uc , uo , p) = L f hcmdi (x, p) ∂xT

(8.34)

In case L f hcmdi (x, p) does not explicitly depend on one or more control inputs, differentiation proceeds: (r )

i = Lrfi hcmdi (x, p) ycmd i

(8.35)

where (ri ) is the rith time derivative. This procedure is repeated for each of the entries in ycmd until each of the entries of uc can be solved. If done manually, this procedure may be quite tedious, especially if relative orders of ycmd are high. In the case of object-oriented modelling, the procedure is performed automatically. The algorithms by Pantelides [181] are used to determine the minimum number of times each of the equations (8.32) has to be differentiated until uc can be solved for. Eventually, the model differential equations can be written in the following form: x˙ = finv (x, ν, uo , p) uc = hinv (x, ν, uo , p)

(8.36) (8.37)

where finv arises from f by substitution of uc (in the case that uc cannot be solved explicitly, a solution is found numerically). The vector ν contains differentiated entries of ycmd : , (rnyc ) T (r1 ) (r2 ) ν = ycmd , y , · · · , y (8.38) cmdn cmd2 1 yc

The model translator will make the user aware that derivatives of ycmd are expected as inputs, rather than ycmd directly. These derivatives may then for example be generated using command ﬁlters with relative degrees of r1 , ..., rnyc respectively.

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The inversion procedure described above is very similar to the one for synthesis of Feedback Linearisation control laws [214,104]. The principal difference is that in these references the differential equations depend on the controls uc in an afﬁne way, (see Chapter 11). In equations (8.36) and (8.37) the differential equations for x(t) have been retained. In order to make sure the inverse model is stable, finv must be investigated. For this task, methods as described in [214, 104] may be used. So far, the inverse and forward systems connected in series are still two independent entities, each including differential equations for nearly the same state vector. Modelling errors, disturbances, integration errors, etc. will eventually cause the internal states x˙ in (8.37) to deviate, and worse, the rith derivative of the system output ycmdi to deviate from the commanded νi . For this reason, the inverse control laws are mostly combined with a feedback controller and inverse model states are obtained from measurement or a reference model. The differences between various inversion-based synthesis methods, like FBL, MFC and IFFC, arises mainly with respect to these aspects. This will now be shown for the bicycle model.

Example 4. The bicycle model (cntd.) Fig. 8.7 shows various control structures around the core of the reference ﬁlter/model, inverse model, and system (Fig. 8.5). The ﬁrst variant adds a feedback control law, regulating the error between the actual yaw rate rb and rc , commanded by the reference ﬁlter. The inverse model includes its own state equations4, which is not recommended because these will easily drift away. For this reason, the next variant generates reference values for states in the command model, extending the command ﬁlter with desired dynamics for the lateral velocity vb , using a state equation for vbc . In order to make sure vbc ≈ vb , the error in between is also regulated by the feedback controller. The advantage is that the internal or zero dynamics of the system (in this case, vb ) are not left on their own. Both controller structures have the disadvantage that the feedback controller still may require scheduling as a function of parameters p (8.19) or, in the case of an aircraft, the ﬂight condition. The third variant therefore uses feedback around both the system and the inverse model. Furthermore, the internal states are directly obtained from measurement. The latter provides the most accurate inversion, but requires the full state vector to be measured, or estimated. The latter structure is the one behind Feedback Linearisation (FBL) or Nonlinear Dynamic Inversion (NDI).

As can be seen from the example, states are preferably not integrated within the inverse model itself, but rather obtained from the reference model (MFC) [93], or from measurements (FBL, NDI). If requested by the user, a model compiler can easily declare states as inputs instead. The model differential equations are then automatically removed. 4

In fact, this is comparable with inverting a (proper) transfer function of the linearised system and adding this as a dynamic compensator.

Rapid Prototyping Using Inversion-Based Control . r u

c m d

u ' c

c o m m a n d filte r

r c

G

in v e r s e b ic y c le m o d e l (d y n .)

+

. r

c m d

r b

K (s ) -

In v e rse fe e d -fo rw a rd

u

b ic y c le m o d e l

N W

+

161

re fe re n c e m o d e l

u ' c

x c

G

in v e r s e b ic y c le m o d e l (s ta t.)

x c

r b

r b

b ic y c le m o d e l

N W

+ K (s ) x +

M o d e l fo llo w in g

u

r c m d

c o m m a n d filte r

r

.

u '

c

c

+

-

G

in v e r s e b ic y c le m o d e l (s ta t.)

K (s )

N W

b ic y c le m o d e l

x

N D I, F B L

Fig. 8.7. Controlled bicycle model: various inverse controller structures

8.5 Rapid Prototyping Design Process As discussed in the previous section, once the system model is available in an objectoriented form the generation of an inversion-based controller requires little effort, since algebraic and coding work is automated. Of course, the generated control laws are not ready for hardware implementation, but they are fully functional over the operating envelope and can (besides selection of a few ﬁxed feedback gains) readily be used as a prototype design in a manned or off-line simulation environment. In Section 8.1 several examples of the practical usefulness of this approach in ﬂight control law design have been listed. Fig. 8.8 shows the basic steps for inversion-based design up to detailed design. Although the model is assumed to be available, model construction has been added as a ﬁrst step. The reason is that not necessarily all dynamics in the model are to be inverted. For example, actuator models are often represented by low order transfer functions. The internal states are usually not available as measurements from the real actuator, or sometimes not even of physical origin. It is common practice to residualise the transfer function to a static gain, since the bandwidth is usually considerably higher than that of the aircraft dynamics. Also, model complexity directly inﬂuences inverse control law complexity. For this reason, if complex features can be reasonably replaced with simpliﬁed versions, this should be taken into consideration. The next step in Fig. 8.8 is command variable selection. In most cases this selection is not free, since commonality with other aircraft series is required. In new applications, like manual on-ground control, this section may not be obvious from the start. The same

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holds for selection of control effectors to be used. In case of redundancy, equations or algorithms have to be added in order to allow the model compiler to appropriately allocate and co-ordinate control deﬂections. Analyses such as those described in Chapter 13 are an excellent basis on which to specify such equations. The implementation of the linear feedback control law and command ﬁlters basically decides which type of nonlinear control law is generated (see for example Fig. 8.7). In this chapter, NDI is always used, thus avoiding the need for gain scheduling of the linear control law as a function of ﬂight condition and other known parameters. In the Flight Dynamics Library some control structures are available that may be copied to the model and interconnected prior to inverse model generation. After the inverse model code has been generated, it may be implemented in the preferred simulation environment with the aircraft simulation model. The preliminary design may be evaluated in batch or pilot-inthe-loop simulations. Once the basic conﬁguration has been frozen, the detailed design may start, involving feedback signal synthesis (estimation of all states used in the inverse model equations), tuning of free controller parameters, robustness assessment, etc. In this phase also the coding must be reviewed. For example, variables computed from environment models are to be replaced with measured ones, iterative equation solving should be prevented, etc. An application example of the rapid prototyping process up to manned simulations is described in [217]. Ref. [139] describes a detailed design for a (ﬂight tested) automatic landing system, with emphasis on robustness aspects. . C o n s tr u c t a ir c r a ft m o d e l S e le c tio n o f c o m m a n d v a r ia b le s ( y

c m d

A llo c a tio n o f c o n tr o ls ( u c)

)

B a s ic d e c is io n s

A d d lin e a r c o n tr o lle r / c o m m a n d filte r s ( fr o m lib r a r y ) In v e r s io n o f A /C m o d e l

A u to m a tic W o r k in g d e s ig n

P r e lim in a r y a n a ly s is ( v a lid a te /c o m p a r e b a s ic d e c is io n s ) n o

W o r k in g p r o to ty p e fo r o th e r d e p a rtm e n ts

B a tc h / r e a l- tim e s im u la tio n

o k y e s

D e ta ile d d e s ig n .

Fig. 8.8. Flight control law design process from rapid prototyping to detailed design

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163

The basic decisions in Fig. 8.8 require physical insight by the designer. In the ﬁrst place, selected control effectors should actually be suitable to perform the tracking of command variables. For example, for roll rate control effectors should be used that primarily generate moments around the aircraft longitudinal axis. Also, when selecting lateral load factor as a command variable in ﬂight, the designer should be aware that the rudder generates a yawing moment, as well as a small side force. As a result, the model compiler will exploit the latter when generating inverse model equations. Although tracking performance may look good at ﬁrst sight, internal dynamics will be unstable, since (at least in the CoG) the lateral acceleration response to rudder input is nonminimum phase5 . In that case, the designer has to adapt the aerodynamic model to make sure the rudder no longer inﬂuences side force, or the command variable selection must be reviewed. Similar practical issues are discussed in Ref. [217].

Example 5. The bicycle model (cntd.) Making an improper selection of command variable can once more be nicely illustrated on the bicycle model. Suppose the lateral load factor in the CoG is chosen as: θNW = u y = −ny = (v˙b + ubrb ) /g

(8.39)

where g is the gravity acceleration. The – sign has been added in order to obtain command responses in the same direction as for yaw rate control. The model compiler will be quick to ﬁnd out that from the connector equation for lateral force it follows that: (v˙b + ub rb ) /g = (GNW βNW + GMLG βMLG ) /mg

(8.40)

Since βNW depends on the control effector θNW , inversion may be done without differentiation of the equation. Fig. 8.9 shows the time responses to a step command on nyc . To the left it appears that ny is perfectly following its command. To the right, the response of rb and θNW are shown. It immediately becomes clear that the internal dynamics of the combined system are badly damped. The reason for this is quite obvious and could have been thought of in advance. Being positioned at the front of the aircraft, the nose wheel steering is primarily intended to generate yawing moments around the vertical axis to initiate and to maintain a turn. The lateral force is necessary to achieve this, but at the same time inﬂuences the total lateral force balance. The inverse controller is just using this effect to achieve its tracking task, thereby generating large yawing moments that overly excite the yaw response. One solution in this case is to change the command variable (e.g. ny ≈ −ub rb /g), as is depicted in Fig. 8.10. Note that a ﬁrst order command ﬁlter is required and that the response of course closely resembles Fig. 8.6. The direct feed through of the lateral force generated by θNW can be seen in the ny response to the left. In the mean time, the yaw rate increases, taking over the main effect on ny due to centrifugal acceleration.

5

A nice example is given in [167].

164

G. Looye Lateral acceleration

Yaw rate, NWS input

0.012

0.02

0.01

0.015 rb (rad/s), θNW (rad)

0.008 −ny

c

−ny, −ny (−)

rb

0.006

c

−n

y

0.004 0.002

θ

NW

0.01 0.005 0 −0.005

0 0

2

4

6

8

10

−0.01 0

2

4

time (s)

6

8

10

time (s)

Fig. 8.9. Example time response of the IFFC bicycle model with ny as command variable Lateral acceleration 5

0.008 −ny (−)

y

rb (rad/s), θNW (rad)

−n

c

−ny (appr.)

0.006

−n

y

0.004 0.002 0 0

2

4

6

Yaw rate, NWS input

−3

0.01

8

10

x 10

rb

4

θ

NW

3 2 1 0 0

time (s)

2

4

6

8

10

time (s)

Fig. 8.10. Example time response of the IFFC bicycle model with approximate ny as command variable

8.6 Aircraft-on-Ground Control Design The ﬁrst step in development of on-ground control laws is of course to prepare the aircraft simulation model. To this end, inputs and outputs identical to those of the Aircraft block on the right in Fig. 8.11 are added to the object-oriented implementation in Fig. 8.2. This is done by extracting and inserting signals from and into the data bus. The model compiler then produces simulation code, which replaces the afore mentioned Aircraft block in the original implementation (Chapter 1). In order to properly initialise the simulation model, trim computation is needed. This can be done in two ways: by generating static simulation code from the object model after specifying trimming conditions as inputs and unknown control deﬂections as outputs, or by automatically generating a script based on the differential equations generated for simulation. This script speciﬁes which entries of x, x, ˙ y and u are known or unknown (making sure the numbers of equations and unknowns ﬁt) and calls a nonlinear equation solver to ﬁnd the trim settings [138]. For the aircraft-on-ground model the latter version was used, allowing for accurate computation of equilibrium conditions on the ground as well as in the air.

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165

Fig. 8.11. Generation of simulation code for the aircraft on ground model. Note that inputs and outputs match those in the Simulink environment (Chapter 1) and are inserted into and taken from the bus via the avionics object.

The preliminary design of on-ground control laws will be discussed along the lines of Fig. 8.8. The ﬁrst step involves the creation of an appropriate model for inversion. The following changes as compared with the original model are made: 1. Removal of actuator dynamics on the nose wheel steering system. It is assumed that θNWc = θNW i.e. the nose wheel steering angle θNW directly follows its commanded value θNWc . In the simulation model, from which the actuator dynamics of course are not removed, the command input is the servo valve control current (ISVNW ). For computing the appropriate input from θNWc the nose wheel steering control law provided within the BSCU is used (see Chapter 1); 2. The computation of the lateral tyre force of (only) the nose wheel is simpliﬁed as follows: FyNW ≈ GyNW (t)βNW where GyNW (t) is the current cornering gain of the combined nose wheels (depends on momentary vertical loading, see Chapter 1). The reason for this step is that the actual nonlinear function has a maximum at βNWOPT , causing inversion problems if the demanded side force FyNW is larger than FyNWMAX (Chapter 1); 3. Limits on control inputs that are to be computed must be removed from the model to be inverted. Otherwise, no solution exists once control deﬂections beyond these limits are required; 4. Wind and other disturbance models are removed. The next step in Fig. 8.8 is to make a selection of command variables. Note that the objective here is to control the lateral dynamics only. A number of options were

166

G. Looye

C o n tr ib u tio n to y a w in g m o m e n t

considered: (1) yaw rate, (2) lateral acceleration in the CoG, and (3) lateral acceleration in the cockpit. In this section, the yaw rate design will be elaborated. Other variants have been evaluated at low speeds using the inverted bicycle model for controller design, but applied to the full model in desktop simulations [174]. For turning the aircraft, the two most obvious controls are rudder δR and nose wheel steering θNW . The ﬁrst is mostly effective at higher dynamic pressure, the second mostly at lower speeds. Other means, like differential braking have been left out for the moment. In order to invert the model, some rule must be provided for how to distribute the available control power between both effectors. As a ﬁrst step, the distribution is based on calibrated airspeed, as shown in Fig. 8.12. If Vcas < Vlow = 30 m/s only nose wheel steering is used, if Vcas > Vhigh = 80 m/s only rudder is used for yaw control, and in between a linear blend based on generated yawing moments is applied. Of course, the values of Vlow and Vhigh can be optimised or a more advanced allocation structure may be selected in the detailed design phase.

1

G

0

@

V

N W

lo w

V

R

h ig h

V

c a s

Fig. 8.12. Initial allocation of nose wheel steering and rudder as a function of airspeed

In order to implement the control architecture, a structure as depicted in Fig. 8.13 has been implemented. To the left, the control allocation can be recognised. The outputs are δR and θNW . The allocation block obtains its required variables from the aircraft data bus. To the right the linear PI control law as well as the command ﬁlter can be recognised. As in the case of the bicycle model example, the yaw rate has relative order one after removal of actuator and hydraulic system dynamics. Initially, the yaw rate r = rc is set as input and inserted into the data bus. After connection with the aircraft model, the compiler will notify the designer that r needs to be differentiated once in order to solve the inversion problem. As in the case of example 3, the command shaping ﬁlter: 1 ucmd rc = (8.41) τr s + 1 is added, where ucmd is the new command input. The model compiler may then differentiate rc . The feedback controller is added in the usual way for Feedback Linearisation, see Figure 8.7. The error between commanded and actual yaw rate is minimised using a PI controller. Since this system is to produce references for the differentiated

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167

command variable, a small trick is applied in Fig. 8.13: the derivative of the yaw rate command is taken from the command ﬁlter and fed to the data bus via an integrator. The PI controller output may then be added to r˙c before the integrator input. During compilation, this integrator is automatically removed when differentiating rb , since this variable is an enforced state in the implemented airframe equations of motion.

G

N W

C o n tr o l a llo c a tio n : N W S / r u d d e r 1

@

G

N W

@

S R

u 1

.

V

c a s

u = r b

r

+

lin e a r c o n tr o lle r

R

u ' P I

r

c

u

c o m m a n d filte r

+ -

.

r c

S

1

1 /J

c m d

+ r

-

se n s

A ir c r a ft b u s a c c e s s

C o n n e c t w ith A /C b u s

Fig. 8.13. Implementation of inverse control law in Modelica

The complete block in Fig. 8.13 is added to the prepared aircraft-on-ground model and connected to the aircraft data bus (Fig. 8.2). Note that the structure itself is aircraft independent. For this reason it has been added to the Flight Dynamics Library, along side a similar block for inversion-based pb , qb and β control in ﬂight. Of course, parameters must be set depending on the speciﬁc aircraft type and control speciﬁcations. As shown in the FBL variant in Fig. 8.7, model states are obtained from measurement. To this end, the model compiler simply replaces aircraft states with inputs, while removing unnecessary state equations. The integrators in the PI controller as well as in the command in Fig. 8.13 have to be retained. Since the control law is to be embedded in the discrete overall control system (Chapter 1), the integrator states need to be discretised. This is done by the model compiler by adding an in-line integration algorithm [57]. In this case, a simple explicit Euler integration method with the required 40 ms sampling period has been included. After automatic model inversion and coding, ﬁrst simulation runs can already be performed. To this end, step inputs on the yaw rate command (ucmd ) are given starting at four different trimmed airspeeds: well below Vlow (10 m/s), well over Vhigh (85 m/s), and two in between (40 m/s and 60 m/s), see Fig. 8.12. The yaw rate responses are of most interest of course. These are depicted in Fig. 8.14. They match the expected ﬁrst order behaviour as in Fig. 8.6 quite well. At this point, only hand-picked values have been set for the linear control law and command ﬁlter parameters. In subsequent design steps, these values may be tuned to optimise robustness and performance, for example using multi objective optimisation [139, 174]).

168

G. Looye Yaw rate step response 1.2

1

yaw rate (deg/s)

0.8

0.6

0.4

0.2

10 40 60 85

0

−0.2 0

2

4

6

8

10 Time (s)

12

14

16

m/s m/s m/s m/s

18

20

Fig. 8.14. Yaw rate responses for different velocities

Next, it is interesting to look at the moment distributions (Figs. 8.15 – 8.18). At 10 m/s the aerodynamic moment is zero, the turn is initiated and maintained using the nose wheel steering only. At 40 m/s some moment is generated aerodynamically, at Aero and NWS moments: Vground = 10 m/s

4

14

x 10

12

10

Moment (Nm)

8

NWS Aero

6

4

2

0

−2 0

2

4

6

8

10 Time (s)

12

14

16

18

20

Fig. 8.15. Moment distribution at 10 m/s

60 m/s the distribution is relatively balanced, whereas starting from 85 m/s steering is only performed aerodynamically. Since no caster has been included in the original model (Chapter 1), the control law is actively turning the nose wheel to follow its local track. Note that at low speeds the rudder slightly deﬂects, see Fig. 8.19. This is caused by the fact that the rudder is compensating for a small aerodynamic

Rapid Prototyping Using Inversion-Based Control

Aero and NWS moments: Vground = 40 m/s

4

12

169

x 10

10

NWS Aero

Moment (Nm)

8

6

4

2

0 0

2

4

6

8

10 Time (s)

12

14

16

18

20

Fig. 8.16. Moment distribution at 40 m/s

Aero and NWS moments: Vground = 60 m/s

4

9

x 10

8

7

Moment (Nm)

6

5

4

3

2

1

0 0

NWS Aero

2

4

6

8

10 Time (s)

12

14

16

18

20

Fig. 8.17. Moment distribution at 60 m/s

yawing moment caused by slipping and yaw rate, so that the total aerodynamic moment is zero. Alternatively, the moment distribution may be applied to the yawing moment delivered by the rudder only. However, in the case of the aircraft-on-ground model the aerodynamics have been provided as a ”black box” from which the rudder contribution can only be derived indirectly, which has not been done for this ﬁrst design. In Figs. 8.20 – 8.23 the actual and commanded nose wheel steering deﬂections are shown. Note that the curves match well, justifying the removal of NWS actuator dynamics from the inverse model. However, in the case of high command inputs measures have to be taken in order to prevent saturation of the nose wheel steering system.

170

G. Looye

Aero and NWS moments: Vground = 85 m/s

4

16

x 10

14

12

Moment (Nm)

10 8

6

NWS Aero

4 2

0 −2 0

2

4

6

8

10 Time (s)

12

14

16

18

20

16

18

20

Fig. 8.18. Moment distribution at 85 m/s

Rudder deflection 1

0

−1

rudder (deg)

−2

−3

−4

−5

10 40 60 85

−6

−7 0

2

m/s m/s m/s m/s

4

6

8

10 Time (s)

12

14

Fig. 8.19. Rudder deﬂections at various airspeeds

In Figs. 8.20 – 8.23 an interesting effect in the model can be observed. Starting from 10 m/s, the nose wheel steering angle is initially positive and remains so in order to maintain the turn. The corresponding slip angle of the nose wheel has opposite sign. At higher speeds the nose wheel steering angle becomes strongly negative in the longer term. This is explained by the over all side slip angle of the aircraft, required to generate the centrifugal force to turn. In order to generate the required yawing moment, this slip angle is partly compensated for, so that the nose wheel is moving outward with respect to the turn. Above 80 m/s the nose wheel should rotate with the local velocity vector at the nose of the aircraft. In Fig. 8.23 it can be observed that the NWS angle starts with a

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171

Nose wheel steering: Vground = 10 m/s 2.5 2 1.5

Angle (deg)

1 0.5 0 −0.5

NWS angle NWS angle command NW slip angle Sideslip angle CoG

−1 −1.5 −2 0

2

4

6

8

10 Time (s)

12

14

16

18

20

18

20

Fig. 8.20. Nose wheel steering and slip angles at 10 m/s

Nose wheel steering: Vground = 40 m/s 2

NWS angle NWS angle command NW slip angle Sideslip angle CoG

1.5

1

Angle (deg)

0.5

0

−0.5

−1

−1.5

−2 0

2

4

6

8

10 Time (s)

12

14

16

Fig. 8.21. Nose wheel steering and slip angles at 40 m/s

small offset. This offset was obtained from trim computation and compensates a slight asymmetry in the aerodynamics. As claimed in the process picture in Fig. 8.8, the control law obtained from automatic inversion is working properly. This has been proven at a range of taxi speeds and is further conﬁrmed by interactive real-time simulations using a desktop simulator [137]. This simulator allows the aircraft to be taxied around at an airport. The control laws in their present form allow for take-off simulations from brake release until the aircraft is airborne, even under strong and turbulent cross-wind conditions. This provides a good basis from which to start a detailed design.

172

G. Looye

Nose wheel steering: Vground = 60 m/s 1

NWS angle NWS angle command NW slip angle Sideslip angle CoG

0.5

Angle (deg)

0

−0.5

−1

−1.5

−2 0

2

4

6

8

10 Time (s)

12

14

16

18

20

18

20

Fig. 8.22. Nose wheel steering and slip angles at 60 m/s Nose wheel steering: Vground = 85 m/s 0.5

0

−0.5

Angle (deg)

−1

NWS angle NWS angle command NW slip angle Sideslip angle CoG

−1.5

−2

−2.5

−3

−3.5 0

2

4

6

8

10 Time (s)

12

14

16

Fig. 8.23. Nose wheel steering and slip angles at 85 m/s

8.7 Conclusions In this chapter it has been shown that inversion-based control methods, in combination with an object-oriented model implementation allow for automatic synthesis of fully functional preliminary nonlinear control laws. The resulting fast design cycle can be exploited to make quick simulation-based comparisons between for example various control conﬁgurations. The preliminary control laws are easily adapted as more model data becomes available and can be used as a platform for detailed design of individual functions. The methodology has been demonstrated on the aircraft-on-ground problem, showing functionality of the preliminary control laws in manoeuvering over the full speed range from brake release at stand-still to take-off.

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173

Once key decisions have been made and the aircraft conﬁguration is frozen, the detailed design may proceed based on the same methodology. In this phase issues such as robustness, feedback signal processing, system failures, coding, etc. are addressed. It is important to note that key decisions, especially in the ﬁrst steps in Fig.8.8, are made by the designer. As demonstrated via a simple example, it is the designer’s physical insight, not the successful inversion by the model compiler, that is the key to a successful design.

Acknowledgements Dipl.-Ing. Philipp Nagel made major contributions to the DLR participation in GARTEUR AG(17) in the form of his Master’s Thesis project. In particular, he made the bicycle model approximation work on the full nonlinear model. The author wishes to thank Philipp for the pleasant and fruitful co-operation.

9 Robustness Analysis Versus Mixed LTI/LTV Uncertainties for On-Ground Aircraft Clement Roos1 and Jean-Marc Biannic2 1 2

ONERA/DCSD and SUPAERO, Toulouse, France [email protected] ONERA/DCSD, Toulouse, France [email protected]

Summary. In this chapter, the robustness properties of the adaptive anti-windup controller designed in Chapter 7 are evaluated, with special emphasis on the high uncertainty level affecting the nonlinear ground forces. It is ﬁrst shown how to convert the initial nonlinear problem into a fairly standard robustness analysis problem versus mixed time-invariant/time-varying uncertainties. An original approach, based on the notion of semi-positive realness, is then introduced to solve the problem. The application of this method to the on-ground aircraft ﬁnally reveals the good robustness properties of the lateral controller despite uncertainties and saturation effects. It also gives some relevant information that enables further improvements to be made to the design.

Notation KYP LFT LMI LTI LTV µ r, rc Ψ Vx ,Vy GyNW , GyMG ε δ× τ×

Kalman-Yakubovich-Popov Linear Fractional Transformation Linear Matrix Inequality Linear Time-Invariant Linear Time-Varying Structured singular value (s.s.v.) Yaw rate, commanded yaw rate (rad/s) Heading (rad) Longitudinal, lateral velocity (m/s) Cornering gains (N/rad) Anti-windup controller input Standard notation for uncertainties Standard notation for delays

9.1 Introduction An adaptive anti-windup controller was designed in Chapter 7 in order to control the lateral motion of an on-ground aircraft. The nonlinear simulations presented in that chapter demonstrated the excellent performance of the proposed control system despite some signiﬁcant variations in the external conditions, such as the runway state. Nevertheless, the robustness properties of the proposed solution have not been fully checked. For example, as mentioned in Chapter 6, the nonlinear ground forces are poorly known D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 175–194, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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in practice and for many different reasons can even change from one landing to another. As a result, the robustness of the controller - which implements an on-line estimator of such forces - must be evaluated for the case where deviations appear between the actual ground forces and their estimations. This is a difﬁcult nonlinear robustness analysis problem. Fortunately, it is shown in Section 9.2 that the saturation-type nonlinearities which appear in the nonlinear LFT representation of the on-ground aircraft can be replaced by time-varying gains. This is the key point of this chapter, since it allows us to convert the initial nonlinear problem into a more standard robustness analysis problem versus mixed time-invariant (LTI) and time-varying (LTV) uncertainties. A signiﬁcant part of this chapter is thus devoted to the development of a new approach for robustness analysis versus such uncertainties. The basic idea consists of an extension of the µ-analysis framework, which is restricted to LTI uncertainties, to handle LTV uncertainties as well. For a better understanding of the proposed technique, let us ﬁrst recall that the most standard approach to evaluate the robustness margin versus LTI uncertainties consists in evaluating an upper bound µ of the structured singular value µ as a function of frequency. This can be achieved by computing frequency dependent scaling matrices D(ω) and G(ω). The robustness margin is then obtained as the inverse of the maximal value of µ over the frequency range. As is well-known, using constant scaling operators D and G, similar computations can be performed to check robustness versus LTV uncertainties. The mixed case is then logically handled by combining both classes of scaling operators. Such a problem, which admits an LMI-based formulation, exhibits some nice convex properties. However, it remains quite difﬁcult to solve, since it involves both an inﬁnite number of constraints and an inﬁnite number of variables. In this chapter, it is proposed to simplify the class of frequency dependent scaling operators associated with the LTI uncertainties by choosing semi-constant ones. More precisely, D(ω) and G(ω) are chosen constant on IR + . Surprisingly, as shown in [199], such a restriction does not lead to very conservative results. Moreover, for this particular choice, it is shown that the mixed-µ upper bound characterization of [244] with semi-constant scalings can be rewritten as an extended positive realness test (semi-positive realness). This test, for which an LMI formulation based on a generalized version of the KYP lemma can be developed [106, 105], consists of checking the positive realness of a complex-valued system on IR + . As a ﬁnal point, in this new LMI characterization, constant scaling operators associated with LTV uncertainties are easily incorporated. An LMI-based characterization of the robustness margin versus mixed LTI/LTV uncertainties is then obtained, which involves a ﬁnite number of constraints and a ﬁnite number of variables. Moreover, frequency-gridding techniques are avoided, and it is thus impossible to miss a narrow and high peak on the µ plot. The chapter is organized as follows. It is ﬁrst shown in Section 9.2 how to convert the initial nonlinear problem into a more standard robustness analysis problem. Section 9.3 is then dedicated to robustness analysis theory. As brieﬂy mentioned above, the proposed technique is based on the notion of semi-positive realness, which is discussed in Section 9.3.2. The main analysis results are then developed in the remainder of Section 9.3. Finally, the application to the on-ground control problem is detailed in Section 9.4, and some concluding comments end the chapter.

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177

9.2 Description of the Robustness Analysis Problem 9.2.1

A Robustness Analysis Problem Versus LTI/LTV Uncertainties

Based on the simpliﬁed LFT model of the on-ground aircraft developed in Chapter 6, a parameter-varying anti-windup controller was designed in Chapter 7 to control the lateral motion of the aircraft during taxiway maneuvers. From various nonlinear simulations, it appeared that the heading could be perfectly controlled whatever the runway state and the aircraft longitudinal velocity, which conformed to the design speciﬁcations and thus proved the effectiveness of the proposed control strategy. Nevertheless, an important issue now arises, which deals with robustness analysis. More precisely, the aim of the present chapter is to determine to what extent the stability of the resulting closed-loop plant can be guaranteed, and which performance levels can be obtained. Basically, the following question should be answered. Does the on-ground aircraft remains stable: • whatever the runway state and • for any variation of the longitudinal velocity between 5 and 40 kts, even if it is affected by structured model uncertainties • on the cornering gains and • on the input of the anti-windup controller ? It was indeed shown in Chapter 6 that the cornering gains depend on several parameters, which were not considered in the simpliﬁed LFT model used for controller design. These simpliﬁcations can be conveniently modelled by time-varying uncertainties. Moreover, the ground forces, and thus the input ε of the anti-windup controller, were assumed to be perfectly known during the design process. This is no longer true, however, when the contoller is implemented on the full nonlinear model. Taking into account uncertainties on ε thus allows us to check the robustness of the closed-loop plant to an imperfect estimation of the ground forces. A ﬁrst idea consists in exploiting the framework presented in Sections 3 and 4 of Chapter 7 was to perform robustness analysis in the presence of both saturations (or equivalently deadzones) and time-varying parameters. However, this can lead to excessive conservatism when applied to the on-ground aircraft model, which not only depends on time-varying parameters but also on time-invariant ones, as is summarized in Table 9.1. Table 9.1. LTI and LTV parameters considered for robustness analysis Parameter

Name

Type

Aircraft longitudinal velocity

Vx

LTV

Uncertainties on the cornering gains

δGNW , δGMG

LTV

Uncertainty on the input of the AW controller

δε

LTI

Delay on the input of the AW controller

τε

LTI

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Such an observation reveals that there is a real need to develop speciﬁc tools dedicated to robustness analysis in the presence of mixed LTI/LTV parameters or uncertainties. These are further detailed in Section 9.3, and then applied to the on-ground aircraft in Section 9.4. 9.2.2

A Linear Representation of the Aircraft Nonlinear LFT Model

As a preliminary, a suitable representation of deadzone nonlinearities has to be determined, which complies with the aforementioned framework. Noting that any deadzone function φ satisﬁes the following sector condition: φ(z) [φ(z) − z] ≤ 0 ∀z ∈ IR

(9.1)

a natural solution consists in converting such a nonlinearity into a simple time-varying gain δ(t) ∈ [0, 1], i.e. φ(z) = δ(t)z. This is illustrated by Figure 9.1, where the deadzone function is obviously included in the grey sector covered by the time-varying gain, whatever the value of z0 . φ(z)

-z0 z0

z

Fig. 9.1. Representation of a deadzone function with a time-varying gain

The four nonlinearities of the simpliﬁed lateral model depicted in Figure 2 of Chapter 7 are transformed in this way. The keypoint of this chapter thus consists in modelling the nonlinear part of the closed-loop LFT representation of the on-ground aircraft by a linear time-varying one, which can be more easily analyzed using the method developed in Section 9.3. Assessing the robustness to the resulting time-varying gains, denoted by δNW S , δatan , δFNW and δFMG in the sequel, thus indirectly allows us to check the robustness to the runway state. The closed-loop LFT model used for robustness analysis is depicted in Figure 9.2. It is obtained from the LFT representations of the open-loop aircraft and the anti-windup controller shown in Figures 2 and 11 of Chapter 7. Note that only the inner loop is considered here. The model has four states (r,Vy , the actuator state and the controller state), a single input (rc ) and a single output (r). The associated 19 × 19 diagonal ∆-block is structured as follows: (9.2) ∆ = diag(∆LTV , ∆LT I )

Robustness Analysis Versus Mixed LTI/LTV Uncertainties

179

∆LTV ∆LT I

Closed−loop rc

LFT model

r

Fig. 9.2. Closed-loop LFT representation of the on-ground aircraft

and:

∆LTV = diag (δNL (t) , Vx (t).I11 , δGNW (t) , δGMG (t)) ∆LT I = diag (δε , τε )

(9.3)

where the following notation is used for the sake of conciseness: δNL (t) = diag (δNW S (t) , δatan (t) , δFNW (t) , δFMG (t))

(9.4)

Note that there is no longer any nonlinear element in the ∆-block.

9.3 Robustness Analysis Versus LTI/LTV Uncertainties The robustness analysis problem described in Section 9.2 clearly motivates the develoment of a new approach to robustness analysis versus mixed LTI/LTV uncertainties. After a short introduction, the notion of semi-positive realness is deﬁned in Section 9.3.2. It is then exploited in Section 9.3.3 to provide an alternative formulation of a mixed-µ upper bound. Special attention is paid to this speciﬁc point, since it allows us to easily derive a robustness test for systems with both LTI and LTV uncertainties, as shown in Section 9.3.5. 9.3.1

Introduction

The structured singular value µ was introduced in [50] as a tool for the robustness analysis of systems subject to structured complex uncertainties, and a detailed tutorial introduction to this theory can be found in [178]. This approach was later extended to the case where there exist both real and complex perturbations, such as parametric uncertainties and neglected dynamics [59]. The exact computation of µ is NP hard [29], but several polynomial-time algorithms now exist to compute reliable upper and lower bounds. Much attention has been paid to the evaluation of the gap between the actual value of µ and its upper bounds (see [160, 223] and references therein). The robustness margin is deﬁned as the inverse of the maximum s.s.v. over the frequency range. A guaranteed value can thus be determined by computing a µ upper

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bound as a function of the frequency. This can be easily achieved by optimizing at each point of a frequency gridding some scaling matrices whose structures ﬁt the uncertainty block, using the convex characterization of [59, 244]. However, this technique is unreliable in the case of narrow and high peaks on the µ plot (which is often the case for ﬂexible systems), since it becomes possible to miss the critical frequency if the gridding is not ﬁne enough, and thus to over-evaluate the robustness margin. Several approches have been proposed to overcome this problem [26, 65, 130, 145]. Whatever the method, the resulting value of the robustness margin is pessimistic, since only an upper bound of the s.s.v. is computed. Nevertheless, conservatism usually remains reasonable in practice and can be estimated by computing a µ lower bound [64, 243]. µ-analysis is now recognized as an efﬁcient tool for robustness analysis, but it can only handle LTI uncertainties. This can sometimes be restrictive, since many systems also depend on time-varying parameters, such as the on-ground aircraft model described in Section 9.2. Such an issue has been under consideration for a long time and some necessary and sufﬁcient stability conditions for systems subject to arbitrarily fast time-varying structured uncertainties have been proposed, exploiting either the LFT [159, 208] or the IQC [157] framework. As in the time-invariant case, robustness can be evaluated by optimizing some scaling operators. The sole difference is that these now have to be constant on the whole frequency range. The case of time-varying uncertainties with bounded rates of variation has also been adressed [37, 113, 189]. Taking into account mixed LTI/LTV uncertainties can logically be achieved by combining both classes of aforementioned scaling operators [133,180], but only a few computationally tractable algorithms have been proposed to compute a guaranteed robustness margin. This is indeed a more complex issue than the standard LTI case, because of the constant scaling matrices associated with the time-varying uncertainties, which make it impossible to independently solve the problem at each frequency (see nevertheless [66] for a frequency-based technique, which exploits some results of [26]). A classical approach in the state-space domain consists of using the well-known KYP lemma, but only LTV uncertainties can be considered. In this context, an improved state-space approach is further investigated in the remainder of Section 9.3 to evaluate the robustness properties of a plant in the presence of both LTI and LTV uncertainties. It is based on the notion of semi-positive realness and the suitable application of a generalized version of the KYP lemma. 9.3.2

Semi-positive Realness

In this section, the notion of semi-positive realness is introduced and characterized by a generalized version of the positive real lemma. I n×n , B ∈ C I n×p , C ∈ C I p×n and D ∈ C I p×p . Deﬁnition 1 (Semi-positive realness). Let A ∈ C The complex-valued system (Σ) characterized by its transfer matrix G(s) = C(sI − A)−1 B + D is said to be semi-positive real if:

G( jω) + G( jω)∗ ≥ 0 where IR + stands for [0, +∞) including +∞.

∀ω ∈ IR +

(9.5)

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Remark 1. Semi-positive realness is relevant only for complex-valued systems. This notion is indeed equivalent to positive realness for real-valued systems, since in this speciﬁc case G(− jω) = G( jω). Let us now recall the standard version of the KYP lemma, which provides an LMI characterization of positive realness for real-valued systems (see e.g. [193] for a proof). Proposition 1 (Positive real lemma). Let (Σ) be a real-valued system. Assume that the pair (A, B) is controllable. The following statements are equivalent: 1. G( jω) + G( jω)∗ ≥ 0 ∀ω ∈ IR with det( jωI − A) = 0 T A P + PA PB − CT n×n T s.t. 2. ∃P = P ∈ IR ≤0 BT P − C −D − DT Let us then consider the following generalized version. Proposition 2 (Semi-positive real lemma). Let (Σ) be a complex-valued system. Assume that the pair (A, B) is controllable. The following statements are equivalent: 1. G( jω) + G( jω)∗ ≥ 0 ∀ω ∈ IR + with det( jωI − A) = 0 I n×n where R = R∗ and S = S∗ ≥ 0 s.t. 2. ∃Z = R + jS ∈ C ∗ A Z + Z ∗ A Z ∗ B − C∗ ≤0 B∗ Z − C −D − D∗

(9.6)

The next lemma is introduced as a preliminary to the proof of Proposition 2: I n and z ∈ C I n . The following statements are equivalent: Lemma 1. Let w ∈ C

1. ∃δ ∈ IR + s.t. w = δz 2. w∗ Sz + z∗Sw ≥ 0 ∀S = S∗ ≥ 0 w∗ Rz − z∗ Rw = 0 ∀R = R∗

(9.7) (9.8)

Proof: First note that 1. ⇒ 2. is obvious. To show the converse, let w = (wi )i∈[1,n] , z = (zi )i∈[1,n] , and let Rα,k = R∗α,k be deﬁned as follows: Rα,k (p, m) = Apply (9.8) with

α=1 and k ∈ [1, n]

α if (p, m) = (1, k) or (k, 1) 0 otherwise

α= j . Combine the resulting equations to show k ∈ [2, n]

that w1 /z1 = wk /zk ∀k ∈ [1, n]. Thus, there exists δ ∈ IR s.t. w = δz. Finally combine 2 this relation with (9.7): it is straightforward that δ ∈ IR + .

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Proof of Proposition 2: First note that G( jω) can be put under an LFT form as follows: 1 ω

w

- jA

B

z

u

- jC

D

y

Fig. 9.3. LFT representation of G( jω)

where y = G( jω)u = − jCw + Du. The semi-positive realness of G is thus equivalent to: 1 0 jC∗ w ξ∗ ξ ≥ 0 ∀ξ = , w = z, ω ∈ R+ − jC D + D∗ u ω Using the relation z = − jAw + Bu and Lemma 1, w = ω1 z, ω ∈ IR + can be rewritten as: ∗ ∗ (R − jS)A + A (R + jS) (S + jR)B ξ ξ ≥ 0 ∀R = R∗ , ∀S = S∗ ≥ 0 B∗ (S − jR) 0 Under the assumption that the pair (A,B) is controllable, the S -procedure is ﬁnally applied (see [106] and references therein for an introduction to the S -procedure). 2 Remark 2. This result can also be obtained from Theorem 4 in [105], which is part of a more general work on the KYP lemma. 9.3.3

Mixed-µ Upper Bound Computation

The aim of this section is now to apply the semi-positive real lemma to the µ-analysis framework, so as to derive a convex state-space formulation of a mixed-µ upper bound and thus a guaranteed robustness margin. Robust performance is treated in Section 9.3.4 and a few comments on the method are made in Section 9.3.5. A straightforward application to robustness analysis in the presence of both LTI and arbitrarily fast time-varying uncertainties is then presented in Section 9.3.6. ∆

w

- M(s) (a)

∆

z

w yr

-

M(s)

z y

(b)

Fig. 9.4. Standard interconnection structures for robust stability (a) and robust performance (b) analysis

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183

Let us consider the standard interconnection structure of Figure 9.4.a, where M(s) is a stable real-valued LTI system and ∆ a structured perturbation, i.e. a block diagonal matrix containing real scalars (associated with parametric uncertainties) as well as complex scalars and full complex blocks (which represent neglected dynamics). The structure of ∆ is ﬁxed a priori and B∆ = {∆ : σ(∆) < 1} denotes the unit ball of perturbations with this admissible structure. Moreover, all perturbations are assumed to be time-invariant (see Section 9.3.6 for comments on the mixed LTI/LTV case). The s.s.v. µ and its connection with the robustness margin are ﬁrst recalled below. Deﬁnition 2 (Structured singular value). Let M be a given complex matrix. If no perturbation ∆ makes I − ∆M singular, then the s.s.v. µ∆ (M) is deﬁned as µ∆ (M) = 0. Otherwise, it is deﬁned as the inverse of the size of the smallest perturbation ∆ satisfying det(I − ∆M) = 0: µ∆ (M) =

1 min{k ∈ IR + : ∃∆ ∈ kB∆ s.t. det(I − ∆M) = 0}

(9.9)

Deﬁnition 3 (Robustness margin). Assume that M denotes the value M( jω) of the transfer matrix M(s) at s = jω. The robustness margin kmax is obtained as the inverse of the maximal s.s.v. µ∆ (M( jω)) over the frequency range: kmax =

1 max µ∆ (M( jω))

(9.10)

ω∈IR +

Robust stability of M(s) is thus guaranteed for all uncertainties ∆ ∈ kmax B∆. To compute a guaranteed robustness margin of the plant M(s) despite uncertainties ∆, the issue is thus to minimize (an upper bound of) the s.s.v. µ over the frequency range. The mixed-µ upper bound of [244] is introduced in this context. Proposition 3. Assume that M(s) is asymptotically stable. Let β be a positive scalar. If there exist frequency dependent scaling matrices D(ω) = D(ω)∗ > 0 and G(ω) = G(ω)∗ , whose structures ﬁt the one of ∆, s.t. ∀ω ∈ IR + : M ∗ ( jω)D(ω)M( jω) + j(G(ω)M( jω) − M ∗ ( jω)G(ω)) ≤ β2 D(ω)

(9.11)

then µ∆ (M( jω)) ≤ β ∀ω ∈ IR + . Remark 3. See [244] for a detailed characterization of the admissible scaling matrices D(ω) and G(ω). Let us now assume that the D, G scaling matrices are constant on IR + , i.e. D(ω) = D and G(ω) = G ∀ω ∈ IR + . The following proposition then states the main result of Section 9.3 and provides a state-space formulation of a mixed-µ upper bound under the previous assumption: Proposition 4. Assume that M(s) = C(sI − A)−1 B +D0 is asymptotically stable. Let β be a positive scalar. If there exist constant scaling matrices D = D∗ > 0 and G = G∗ ,

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whose structures ﬁt that of ∆, and an extended Lyapunov matrix Z = R + jS, where R = R∗ and S = S∗ ≥ 0, s.t.: ⎞ ⎛ ∗ A Z + Z∗A Z ∗ B − jC∗ G C∗ D ⎝ B∗ Z + jGC − β2 D + j(GD0 − D∗ G) D∗ D ⎠ ≤ 0 (9.12) 0 0 −D DC DD0 then µ∆ (M( jω)) ≤ β ∀ω ∈ IR + . Proof: By applying the Schur complement, inequality (9.11) with constant scaling matrices on IR + can be rewritten as: . / β2 D − j(GM( jω) − M ∗ ( jω)G) M ∗ ( jω)D ≥0 DM( jω) D which is equivalent to the semi-positive realness of: / . 2 β D − jGM(s) 0 2 H(s) = D DM(s) 2 A state-space representation of H(s) is given by: ⎛ ⎞ B 0 A ⎜ ⎟ H(s) ↔ ⎝ − jGC β2 D2 − jGD0 0 ⎠ D DD0 DC 2 The semi-positive real lemma stated in Proposition 2 is ﬁnally applied to H(s).

(9.13) 2

The issue is thus to minimize the value of β w.r.t. the constant scaling matrices D and G and the extended Lyapunov matrix Z. Inequality (9.12) corresponds to a generalized eigenvalue problem, which can be efﬁciently solved with an LMI solver [79]. Let β∗ be the minimum value of β satisfaying inequality (9.12). The interconnection M(s) − ∆ remains stable ∀∆ ∈ β1∗ B∆ and thus β1∗ is a guaranteed robustness margin. 9.3.4

Performance Analysis

With reference to Figure 9.4.b, the issue is now to compute the L2 -induced norm of the transfer function Tyr →y between the reference input yr and the output y despite structured uncertainties ∆, i.e. to minimize (an upper bound of) γ under the L2 -induced norm constraint: (9.14) Tyr →y iL2 ≤ γ ∀∆ ∈ B∆ Proposition 5. Assume that M(s) is asymptotically stable. Let D(ω) = D∗ (ω) > 0 and G(ω) = G∗ (ω) be some frequency dependent scaling matrices, whose structures ﬁt that of ∆. Let D (ω) = diag(D(ω), I), G (ω) = diag(G(ω), 0) and Iγ = diag(I, γ2 I). As a sufﬁcient condition, (9.14) is satisﬁed if ∀ω ∈ IR + : M ∗ ( jω)D (ω)M( jω) + j(G (ω)M( jω) − M ∗ ( jω)G (ω)) ≤ Iγ D (ω)

(9.15)

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185

Assume now that the D , G scaling matrices are constant on IR + . The semi-positive real lemma can then be applied to provide a state-space formulation of a γ upper bound. Proposition 4 is restated and inequality (9.12) is replaced with inequality (9.16) below: ⎞ ⎛ ∗ Z ∗ B − jC∗ G C∗ D A Z + Z∗A ⎝ B∗ Z + jG C − Iγ D + j(G D0 − D∗0G ) D∗0 D ⎠ ≤ 0 (9.16) D D0 −D DC The resulting optimization problem corresponds to the minimization of a linear objective γ under the LMI constraint (9.16), which can be efﬁciently solved with an LMI solver [78]. 9.3.5

Comments on the Method

The minimum value of β satisfying (9.11) is just an upper bound of µ, but the conservatism of the sufﬁcient condition in Proposition 3 is usually very reasonable in practice, see e.g. [63]. Conservatism is also introduced in Proposition 4, since the D, G scaling matrices are constant on IR + . Nevertheless, the semi-positive real lemma appears especially relevant compared to the original positive real lemma from the perspective of computing a reliable mixed-µ upper bound. It deals indeed with complex-valued systems, which allows I p×p and G ∈ C I p×p , as in the classical formulation of Proposition 3. us to consider D ∈ C On the other hand, applying the standard positive real lemma requires the real part of H(s) in Section 9.3.3 to be positive real, and thus constrains D and G to lie in IR p×p and jIR p×p respectively, which signiﬁcantly increases conservatism (such scaling matrices are commonly used in robustness analysis versus fast time-varying uncertainties - see Section 9.3.6). Moreover, the µ plot corresponding to a rigid system usually exhibits a single peak. It can thus be expected that the constant D, G scaling matrices computed with the LMI formulation of Proposition 4 are adapted to this worst case frequency, and that the associated µ upper bound is almost non-conservative compared to the one of Proposition 3. This appears to actually be the case when realistic applications are considered [199]. Conservatism is expected to be slighly more important when a ﬂexible system is considered, due to the possible presence of several separate peaks on the µ plot. In any event, the computation of the mixed-µ upper bound in (9.12) is based on LMI optimization and can become cumbersome for high order systems, which makes this method attractive mainly for low order systems which are also typically rigid ones. Apart from its low conservatism, the result of Proposition 4 has some other interesting features: • The problem of Proposition 3 has both an inﬁnite number of frequency dependent constraints and an inﬁnite number of variables, whereas the state-space formulation of Proposition 4 consists of a unique LMI constraint, while gridding-based techniques are avoided which means it is impossible to miss a narrow and high peak on the µ plot, and thus to over-estimate the robustness margin. • Beyond the computation of a mixed-µ upper bound, this framework can be more widely exploited to develop a whole robustness analysis and robust synthesis process in the presence of mixed LTI/LTV uncertainties, that cannot be easily addressed

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by classical methods based on a frequency approach (see Section 9.3.6 for robustness analysis and [200] for robust synthesis). 9.3.6

Extension to Mixed LTI/LTV Uncertainties

The semi-positive real lemma can be further exploited to extend the results of Proposition 3 to robustness analysis versus mixed LTI/LTV uncertainties. Such an extension is important, since it allows us to consider an important class of nonlinear systems. With reference to Figure 9.4.a, the issue is to compute a guaranteed robustness margin for the plant M(s) in the presence of both LTI and arbitrarily fast time-varying uncertainties. The uncertainty block ∆ is structured as follows: ∆ = diag(∆LTV (t), ∆LT I ), and ˜ no assumption is made on the rate of variation of ∆LTV . Let D(ω) = diag(DTV , DT I (ω)) ˜ and G(ω) = diag(GTV , GT I (ω)) be some scaling matrices, whose structures ﬁt that of ∆, and where: ⎧ DTV ∈ IR pv ×pv DTV = DTTV > 0 ⎪ ⎪ ⎨ ∗ I pi ×pi DT I (ω) = DT I (ω) > 0 DT I (ω) ∈ C ∗ GTV = GTV GTV ∈ jIR pv ×pv ⎪ ⎪ ⎩ ∗ I pi ×pi GT I (ω) = GT I (ω) GT I (ω) ∈ C Note that there is no need to consider a wider class of DTV and GTV scalings [159]. Proposition 6. Assume that M(s) is asymptotically stable. Let β be a positive scalar. If ∗ > 0 and G(ω) ∗ , whose structures ˜ ˜ ˜ ˜ = G(ω) there exist scaling matrices D(ω) = D(ω) ﬁt that of ∆, s.t. ∀ω ∈ IR + : ˜ ˜ ˜ ˜ M ∗ ( jω)D(ω)M( jω) + j(G(ω)M( jω) − M ∗ ( jω)G(ω)) ≤ β2 D(ω)

(9.17)

then the interconnection M(s) − ∆ is stable ∀∆ ∈ β1 B∆. Assume now that DT I and GT I are constant on IR + . Proposition 4 can then be adapted by replacing D and G with D˜ and G˜ to provide a state-space formulation of a guaranteed robustness margin in the presence of mixed LTI/LTV uncertainties. Remark 4. Note that unlike LTI uncertainties, no additional conservatism is introduced in the case of LTV uncertainties, since DTV and GTV are already assumed to be constant on IR + in Proposition 6. The state-space approach presented in Sections 9.3.3 and 9.3.4 can thus be easily extended to the mixed LTI/LTV case. This is not true for classical methods based on a frequency gridding because of the constant scalings DTV and GTV , which make it impossible to independently solve the problem at each frequency.

9.4 Application to the On-Ground Aircraft The theoretical results presented in Section 9.3 are now exploited to perform both a stability and a performance analysis on the closed-loop LFT representation of the onground aircraft described in Section 9.2.

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187

As a preliminary, the considered longitudinal velocity range [5, 40 kts] is split into two parts: • [10, 40 kts] corresponds to all taxiway maneuvers, which aim at bringing the aircraft from the main runway to the parking area, • [5, 10 kts] corresponds to speciﬁc maneuvers performed in the parking area, notably U-turns. During the ﬁrst type of maneuvers, the aircraft velocity is likely to vary considerably, and it is relevant to consider Vx as a time-varying parameter. On the contrary, when maneuvers are performed on the parking area, it can be considered as a ﬁrst approximation that the aircraft velocity remains constant, and Vx is thus removed from the ∆-block. Robustness analysis is performed separately on these two intervals. 9.4.1

Stability Analysis

Neither uncertainties on the cornering gains nor on the input of the anti-windup controller are considered at this stage. The ﬁrst issue is to determine whether the stability of the closed-loop plant can be guaranteed whatever the runway state and for any variation of the longitudinal velocity within the interval [10, 40 kts]. More precisely, as explained in Section 9.2, the four initial deadzones are replaced by time-varying gains δNL (t). The maximum value kmax is then computed, which ensures stability for all δNL (t) such that: 0 ≤ δNL (t) ≤ kmax .I4 ∀t ≥ 0

(9.18)

The associated ∆-block is thus structured as follows: ∆ = diag (δNL (t) , Vx (t).I11 ) where:

(9.19)

0 ≤ δNL (t) ≤ kmax .I4

(9.20)

Vx (t) ∈ [10, 40 kts]

The value kmax = 0.75 is obtained. It is less than 1, which means that stability cannot be guaranteed for any values of the input signals of the deadzone nonlinearities, as is illustrated by Figure 9.5. φ(z) kmax z -z0 z0 zmax

z

Fig. 9.5. Existence of an upper bound on the inputs of the deadzone nonlinearities

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There exists indeed an admissible upper bound zmax on these signals, which is deﬁned by: z0 (9.21) zmax = 1 − kmax The behavior of the closed-loop on-ground aircraft was then analyzed during several standard maneuvers, notably those described in Section 2 of Chapter 7. It was then possible to determine, for each of the four deadzones, the maximum amplitude zsat of the signals z which actually enter the nonlinearity, and thus the upper limit ksat of the sector which includes the deadzone function for all z ∈ [-zsat , zsat ]. As an example, Figures 9.6 and 9.7 show the inputs of the four nonlinearities and the heading Ψ during a 60◦ turn performed at 10 kts. 2

0.6 0.5

1.5

FMG

dz

dz

atan

0.4 1 0.5

0.3 0.2 0.1

0

0 0

2

4

6

8

−0.1

10

1.5

15

1

10

0

0

2

0

2

4

6

8

10

4

6

8

10

5

NWS

0.5

0

dz

dz

FNW

−0.5

−5 −0.5 −1

−10 0

2

4

6

8

−15

10

time (s)

time (s)

Fig. 9.6. Input signals of the four deadzone nonlinearities 70

60

heading (deg)

50

40

30

20

10

0

−10

0

1

2

3

4

5

6

7

time (s)

Fig. 9.7. Heading response

8

9

10

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The solid lines correspond to the aircraft behavior obtained with the anti-windup controller designed in Section 5 of Chapter 7. It can be observed that the rate limitation of the nose-wheel steering system is by far the most active nonlinearity. This leads to a high value of the corresponding zsat , which is not desirable from a stability analysis point of view. This problem can be addressed by introducing a feedforward ﬁlter, namely a well-damped second order transfer function, so as to render the pilot inputs smoother. The result is shown using dashed lines in Figures 9.6 and 9.7. As expected, the peak on the nose-wheel steering system input is ﬂattened, leading to a more balanced activity of the nonlinearities. Moreover, the impact on the heading Ψ remains moderate, since the response time is only slightly increased. After having considered the most demanding maneuvers, the obtained values of ksat are 0.72, 0.49, 0.63 and 0 for satNW S , satatan , satFNW and satFMG respectively. It can be checked that ksat < kmax for each of the four nonlinearities, which ensures that these are now correctly modelled by (9.18) for any variation of the longitudinal velocity between 10 and 40 kts. Note that such a result could not be achieved without the use of the feedforward ﬁlter. The next step consists in determining to what extent the stability of the closed-loop plant can be guaranteed in the presence of structured model uncertainties. It is ﬁrst assumed that: GyMG (t) = (1 + δGMG (t)) GyMG0 (t) , |δGMG (t)| ≤ 0.1 (9.22) GyNW (t) = (1 + δGNW (t)) GyNW0 (t) , |δGNW (t)| ≤ 0.1 where GyMG and GyNW represent the cornering gains, which can vary by ± 10% around their nominal values GyMG0 and GyNW0 . The ∆-block is now structured as follows: ∆ = diag (δNL (t) , Vx (t).I11 , δGNW (t) , δGMG (t)) where:

⎧ 0 ≤ δNL (t) ≤ diag(0.72, 0.49, 0.63, 0) ⎪ ⎪ ⎨ Vx (t) ∈ [10, 40 kts] ⎪ ⎪ ⎩δ (t), δ (t) ∈ [−0.1, 0.1] GNW

(9.23)

(9.24)

GMG

A new analysis is performed and shows that stability is preserved. In addition, either an uncertainty δε or a delay τε is introduced on the input of the anti-windup controller, so as to take into account an imperfect estimation of the ground forces: ε(t) = (1 + δε ) ε0 (t) (9.25) ε(t) = ε0 (t − τε ) Both δε are τε are assumed to be time-invariant parameters, and the corresponding ∆block is given by: ∆ = diag (δNL (t) , Vx (t).I11 , δGNW (t) , δGMG (t) , δε /τε )

(9.26)

190

where:

C. Roos and J.-M. Biannic

⎧ 0 ≤ δNL (t) ≤ diag(0.72, 0.49, 0.63, 0) ⎪ ⎪ ⎪ ⎪ ⎨ Vx (t) ∈ [10, 40 kts] ⎪ δGNW (t), δGMG (t) ∈ [−0.1, 0.1] ⎪ ⎪ ⎪ ⎩ δε ∈ [−δεmax , δεmax ] / τε ∈ [0, τεmax ]

(9.27)

The maximum admissible values δεmax and τεmax of δε and τε , beyond which stability can no longer be guaranteed, are summarized in Table 9.2. Table 9.2. Maximum admissible values of δε and τε Parameter

Maximum value

Uncertainty δε

0.13

Delay τε

0.10 s

These results are conclusive, since the on-ground aircraft is shown to be stable whatever the runway conditions and for any variation of Vx between 10 and 40 kts despite model uncertainties on both the cornering gains and the input of the anti-windup controller. Nevertheless, this global stability analysis can be conservative, since the range of variation of Vx is quite large. Another approach consists in considering that each maneuver is performed within a limited velocity range, or even with a constant velocity. In this perspective, it becomes possible to remove Vx from the ∆-block and to perform a new stability analysis. Thus:

where:

∆ = diag (δNL (t) , δGNW (t) , δGMG (t) , δε /τε )

(9.28)

⎧ 0 ≤ δNL (t) ≤ diag(0.72, 0.49, 0.63, 0) ⎪ ⎨ δGNW (t), δGMG (t) ∈ [−0.1, 0.1] ⎪ ⎩ δε ∈ [−δεmax , δεmax ] / τε ∈ [0, τεmax ]

(9.29)

For the sake of conciseness, results are only presented for one of the most demanding maneuvers, namely a 60◦ turn performed at 10 kts. As expected, results are much less conservative, since stability is preversed as long as δε ≤ 0.95 or τε ≤ 0.44 s. It can thus be concluded that there is no need to compute a very precise estimation of the ground forces, which validates the use of the fairly simple estimator introduced in Chapter 7. The low-speed maneuvers (U-turns) performed on the parking area with Vx less than 10 kts can be analyzed following the same procedure. Table 9.3 shows for several values of Vx the maximum admissible values of δε and τε , for which the closed-loop stability is ensured. It appears once again that there is no need to compute a very precise estimation of the ground forces.

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Table 9.3. Maximum admissible values of δε are τε during U-turns

Parameter

9.4.2

Maximum value Vx = 5 kts

Vx = 7 kts

Vx = 9 kts

Uncertainty δε

0.54

0.60

0.72

Delay τε

0.26 s

0.30 s

0.36 s

Performance Analysis

Now that the stability of the on-ground aircraft has been assessed, it is interesting to determine what performance levels can be guaranteed. More precisely, the yaw rate response r obtained in the presence of uncertainties on both the cornering gains and the input signal ε of the anti-windup controller is compared to the nominal yaw rate response r0 . An upper bound of the L2 -induced norm of the transfer function between the reference yaw rate rc and the error signal r − r0 is then computed. The aircraft longitudinal velocity is ﬁxed in the sequel and the ∆-blocks of the uncertain and nominal closed-loop plants are respectively given by: ∆ = diag (δNL (t) , δGNW (t) , δGMG (t) , δε ) (9.30) ∆0 = δNL (t) where: ⎧ 0 ≤ δNL (t) ≤ diag(0.72, 0.49, 0.63, 0) ⎪ ⎪ ⎨ δGNW (t), δGMG (t) ∈ [−0.1, 0.1] ⎪ ⎪ ⎩ δ ∈ [−δ , δ ] ε

εmax

(9.31)

εmax

The LFT interconnection for performance analysis is depicted in Figure 9.8. The optimization problem of Section 9.3.4 is then solved and the performance index γ is plotted on Figure 9.9 as a function of the maximum value of |δε |. The solid curve is obtained when the cornering gains are allowed to vary by ± 10% around their nominal values, whereas the dashed curve corresponds to δGNW (t) = δGMG (t) = 0 ∀t ≥ 0. The performance index does not grow very fast with δε , which means that the yaw rate response is not altered too much if δε remains reasonably low. It can thus be expected that the heading response of the on-ground aircraft is not very sensitive to the presence of uncertainties in the ground forces modelling and estimation. This can actually be observed when simulations are performed on the full nonlinear model. Figure 9.10 shows for instance the aircraft behavior during a 60◦ turn performed at 10 kts for various values of δε . The heading responses corresponding to δε = −0.5, δε = 0 and δε = 1, i.e. to an anti-windup signal ε = 0.5 ε0 , ε = ε0 and ε = 2 ε0 respectively, are plotted in solid lines. They can hardly be distinguished, which shows that the closed-loop plant is almost insensitive to a bad estimation of the saturation levels, and thus of the ground forces.

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If δε = −0.9, i.e. ε = 0.1ε0 , only a slight overshoot can be observed (dashed line), even though the anti-windup signal is ten times lower than its nominal value. The aircraft time-domain behavior is thus satisfactory even if the anti-windup signal is very different from its optimal value computed in Chapter 6, which conﬁrms the good robustness properties of the proposed anti-windup controller. Finally, note that without anti-windup compensation, i.e. ε = 0, the on-ground aircraft remains stable, but the heading response exhibits strong oscillations (dotted line). 9.4.3

Comments on Conservatism

The proposed approach can be conservative for several reasons. • The ﬁrst source of conservatism is inherent to the method, which is based on a suitable use of a µ upper bound, and not of µ itself (see Section 9.3.5 for a detailed discussion). However, numerous examples [199] reveal that conservatism remains very reasonable in practice, especially for non-ﬂexible systems, which is the case for the considered on-ground aircraft. • Deadzone functions are then approximated by time-varying gains, as depicted in Figure 9.5, so as to convert the difﬁcult initial nonlinear problem into a more standard robustness analysis problem versus mixed LTI/LTV uncertainties. However, for each of the four deadzones, the upper limit ksat of the associated sector can be chosen as small as possible according to nonlinear simulations performed for a set of standard maneuvers. To a certain extent, this allows us to minimize conservatism. • The aircraft longitudinal velocity is ﬁnally considered as an arbitrarily fast timevarying parameter, although its rate of variation is of course bounded in practice.

∆

Uncertain closed−loop

rc

LFT model

∆0

Nominal closed−loop +

LFT model −

r − r0

Fig. 9.8. LFT interconnection for performance analysis

Robustness Analysis Versus Mixed LTI/LTV Uncertainties

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maximum amplitude of |δε|

Fig. 9.9. Performance level as a function of the maximum value of |δε |

But if Vx is considered as an LTI parameter instead, the maximum value of kmax for which stability is preserved is kmax = 0.77. This is very close to the value of 0.75 computed in Section 9.4.1, which shows that this assumption is almost nonconservative. It thus appears that the conservatism of the proposed robustness analysis method remains moderate. It is also worth pointing out that very satisfactory performance results have been obtained, which demonstrates the effectiveness of the proposed anti-windup approach. From a theoretical point of view, good stability margins and performance levels can indeed be guaranteed. Moreover, simulations on the full nonlinear on-ground aircraft model reveal that the heading can be perfectly controlled even if the lateral ground forces are poorly estimated. 100 90 80

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Fig. 9.10. Heading response for various values of δε

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9.5 Conclusions In this chapter, a method is proposed to evaluate the robustness properties of a linear plant in the presence of both time-invariant and arbitrarily fast time-varying uncertainties. It is then shown that the nonlinear closed-loop LFT representation of the on-ground aircraft obtained in Chapter 7 can be easily adapted, so as to comply with the proposed analysis tools. A stability and performance analysis can thus be performed, which reveals the good robustness properties of the lateral controller. Stability of the on-ground aircraft is indeed guarateed whatever the runway state, for any variation of the longitudinal velocity within quite a large interval, and despite model uncertainties. It should ﬁnally be emphasized that the proposed metodology could be exploited to perform a whole robustness analysis for a higher velocity range by considering additional time-invariant uncertainties on the stability derivatives Cn and Cy .

Part III

Applications to the ADMIRE Benchmark

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10 An LPV Control Law Design and Evaluation for the ADMIRE Model Maria E. Sidoryuk1, Mikhail G. Goman2, Stephen Kendrick3, Daniel J. Walker3 , and Philip Perfect3 1 2 3

Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia 140160 De Montfort University, Leicester, England LE1 9BH [email protected] University of Liverpool, Liverpool, England L69 3GH [email protected]

Summary. This chapter presents the design and evaluation of an LPV control law for the ADMIRE model over a speciﬁed wide ﬂight envelope, including subsonic, transonic and supersonic regions. The design of the LPV control law is based on the parameter-dependent Lyapunov function approach with gridding of the parameter space. It is demonstrated that by using a linear piece-wise interpolation of the aircraft model the LPV approach allows the design of a controller for the whole ﬂight envelope (including the transonic region) with satisfactory performance and robustness characteristics. The longitudinal LPV controller provides an automatic transition from the α-demand system at Mach numbers M < 0.58 to the nz -demand system at M > 0.62; in the intermediate region a mixed control principle is implemented. A thorough evaluation of the designed LPV controllers is performed using a number of methods, including time and frequency domain criteria, linear and nonlinear simulation tests, and also piloted simulation in real time on the HELIFLIGHT simulator at the University of Liverpool. The performed evaluation clearly demonstrates that the designed LPV control laws satisfy most of the design requirements. Ways of further improving the performance of the LPV controller are discussed at the end of the chapter. Keywords: gain-scheduled design, LPV systems, LMI formulation, parameter-dependent Lyapunov function approach, design criteria, control law evaluation, piloted simulation.

Nomenclature A(ρ), B1 (ρ), B2 (ρ), = matrices of the LPV model C1 (ρ), D11 (ρ), D12 (ρ) C2 (ρ), D21 (ρ) AK (ρ), BK (ρ), = matrices of the LPV control law CK (ρ), DK (ρ) H, M = ﬂight altitude and Mach number Yβ , Zα , Zq = side and normal aerodynamic force partial derivatives Lβ , L p , Lr = rolling, yawing and pitching aerodynamic moment Nβ , N p , Nr partial derivatives Mα , Mq D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 197–229, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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Yδ∆e ,Yδr ,Zδe Lδ∆e , Lδr Nδ∆e , Nδr Mδe ny , nz n zα , n zq , n zδ e p, q, r q¯ pW V x α, β θ, φ γ δe , ∆δe δc , δr ρ

= side and normal load factors and their partial derivatives = body-axis angular rates = dynamic pressure = stability axes roll rate = ﬂight velocity = state vector = angle of attack and sideslip angle = body axis pitch and bank Euler angles = ﬂight path angle or induced L2 norm on p.200 = symmetric and differential elevons deﬂections = canard and rudder deﬂections = ﬂight regime parameter vector

Abbreviations ADMIRE = Aero Data Model In Research Environment FCS = Flight Control System FQL = Flying Quality Level LPV = Linear Parameter Varying LMI = Linear Matrix Inequality

10.1 Introduction The control law design challenge for the ADMIRE model [115] aims to investigate nonlinear control law design techniques, as alternatives to classical design methods, which have the potential for application to real engineering design problems. The design study should demonstrate their strengths and weaknesses in terms of stability, performance and robustness properties that they provide to the aircraft closed-loop system. Another important point to be cinsidered is the time required for design and also for post design evaluation of the control laws. The candidate control law design method should meet the realistic design requirements normally used in engineering practice and should be applied to a wide ﬂight envelope including subsonic, transonic and supersonic regions. With this ambitious aim in mind this chapter presents an application of the modern technique for self-scheduled control law design known as a Linear Parameter Varying (LPV) approach. A potential difﬁculty for application of the LPV design approach to the ADMIRE model is connected with its rather wide ﬂight envelope, which includes a substantial range of Mach number from subsonic speeds M = 0.2 − 0.3 to supersonic speeds M = 1.4, giving the ADMIRE aerodynamic characteristics signiﬁcant variations along this ﬂight envelope. Another difﬁculty for the LPV approach is linked with the need to meet a set of practical design requirements and to design a unique controller with automatic

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transition from the α-demand control system at low Mach numbers to the nz -demand control system at high Mach numbers. There are two conceptually different methods for synthesizing robust gain scheduled controllers for LPV systems . The ﬁrst one deals with single and parameter-dependent Lyapunov function approaches [19, 7], and the second one is based on linear fractional transformations [6]. Both these methods generalize the application of H∞ control theory [107], widely used for linear time invariant (LTI) systems, to linear parameter-varying (LPV) systems. These methods have been the subject of a signiﬁcant number of theoretical and applied research during recent years, for example, the reader can be referred to [5, 13, 17, 23, 77, 182, 183, 191, 232, 238, 240]. The design of gain-scheduled controllers for LPV systems is based on a search for adequate Lyapunov functions that establish stability and performance bounds for the closed-loop system. The design approach based on parameter-dependent Lyapunov functions allows the inclusion of information regarding the parameter rate variation in the synthesis technique leading to a less conservative solution. Both techniques allow the design problem to be expressed in terms of Linear Matrix Inequalities (LMIs) and be solved using efﬁcient optimization software which is now widely available [79]. A wide variety of modiﬁcations of the main techniques have been proposed during the last ﬁve years to further reduce the conservatism of the designed controller [48,210,216]. One of the main modiﬁcations is based on the incorporation of D-K iterations, which is the standard procedure in the LTI H∞ control design, into the LPV design framework [17,216]. This allows different parts of the closed-loop transfer function to be scaled on the basis of µ-analysis [12,50,177] giving iteratively a controller which more closely satisﬁes the design goals. Another promising approach is linked with incorporation of blending methodologies into the linear parameter varying control synthesis [210, 238]. In this approach the gain-scheduled controllers can be improved by construction of a global Lyapunov function, which interpolates a number of Lyapunov functions designed locally. Stability and performance properties in the interpolation zone require special consideration. Design procedures guaranteeing robust stability and performance in the interpolation zone are proposed in [210, 238]. In this study a relatively simple approach based on parameter-dependent Lyapunov functions [7, 19] is implemented to demonstrate the signiﬁcant potential of these new methods based on the LPV control theory for the purpose of fast conceptual design of aircraft control laws in a wide ﬂight envelope while meeting realistic design requirements.

10.2 Parameter-Dependent Lyapunov Function Approach For the application of the LPV synthesis, the aircraft dynamics should be described by a linear parameter dependent system. Suppose that the control design problem is formulated for the following state-space linear parameter dependent system: x˙ = A(ρ)x + B1(ρ)w + B2 (ρ)u z = C1 (ρ)x + D11w + D12 (ρ)u y = C2 (ρ)x + D21(ρ)w

(10.1)

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where y, z, w, and u are measurements, errors, disturbances and control signals. Matrices A ∈ Rn×n , D12 ∈ R p1 ×m2 , D21 ∈ R p2 ×m1 deﬁne the problem dimension. The time-varying parameter ρ(t) ∈ P ⊂ Rs and its rates of variation for each parameter component are bounded by known bounds: ρi (t) ∈ [ρi , ρ¯ i ] ρ˙ i (t) ∈ [νi , ν¯ i ], i = 1, ..., s. The induced L2 norm of w to z is deﬁned as γ=

z2 ρ∈P ,w∈L2 ,w2 =0 w2 sup

The goal of the LPV control design is to ﬁnd a control law depending on the parameter vector ρ x˙K = AK (ρ)xK + BK (ρ)y (10.2) u = CK (ρ)xK + DK (ρ)y that stabilizes the closed-loop system (10.1)-(10.2) exponentially and makes the induced L2 norm from the disturbance signal w to the error signal z less than γ for all ˙ at zero state initial conditions. Such a controller K(ρ) can admissible trajectories (ρ, ρ) be constructed from parameter-dependent symmetrical matrices R(ρ) > 0, S(ρ) > 0 that are solutions of the following optimization problem [5, 240]: min γ

R,S∈Rn×n

subject to

where

⎛

ˆ + RAˆ T − ∑i [νi , ν¯ i ] ∂R − γBˆ 2 Bˆ T AR 2 ∂ρi ⎜ ⎝ ˆ C1 R Bˆ T1 ⎛ SA˜ + A˜ T S − ∑i [νi , ν¯ i ] ∂S − γC˜2C˜2T ∂ρi ⎜ ⎝ B˜ T1 S C˜1T RI > 0. I S

⎞ RCˆ1T Bˆ 1 ⎟ −γI Dˆ T11 ⎠ < 0. Dˆ 11 −γI ⎞ SB˜ 1 C˜1T ⎟ −γI D˜ 11 ⎠ < 0. D˜ T11 −γI

(10.3)

(10.4)

(10.5)

ˆ ˆ ˆ ˆ Bˆ 2 = B2 D+ 12 , A = A − B2C1 , B1 = B1 − B2 D11 + ˆ Cˆ1 = (I − D12 D+ )C , D = (I − D D 11 12 12 )D11 12 1 ˜ ˜ ˜ ˜ C , A = A − B , C = C C˜2 = D+ C 1 2 1 1 − D11C2 21 2 ˜ 11 = D11 (I − D21D+ ) ), D B˜ 1 = B1 (I − D21D+ 21 21

and D+ denotes the pseudoinverse of D. Notation [νi , ν¯ i ] means that both lower and upper rate bounds are included in the sum. The LMIs (10.3)-(10.5) must be satisﬁed for all allowed values of the parameter

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201

vector ρ and its rate of change. The controller is generated from R, S matrices, the plant matrices and vector ρ˙ : AK = N −1 (Aˆ k − SB2Cˆk − Bˆ kC2 R − SAR)M −T BK = N −1 Bˆ k , CK = Cˆk M −T , DK = 0 where 1 dM T T T ˆ Aˆ k = S dR dt + N dt − A − γ [SB1 + Bk D21 C1 ]

BT1 C1 R + D12Ck

Bˆ k = −(γC2T + SB1DT21 )(D21 DT21 )−1 Cˆk = −(DT12 D12 )−1 (γBT2 + DT12C1 R), MN T = I − RS.

(10.6) (10.7)

The solvability conditions (10.3)-(10.5) are obviously inﬁnite-dimensional. To make the optimization problem computationally tractable, the scheduled parameters ρ are discretized using some grid of points [240], replacing the inﬁnite type conditions by some ﬁnite number of constraints. R(ρ) and S(ρ) are represented by some set of basic functions gi (ρ) and hi (ρ) Nx

Ny

i=1

i=1

R(ρ) = ∑ gi (ρ)Ri , S(ρ) = ∑ hi (ρ)Si , where gi (ρ) and hi (ρ) are continuously differentiable functions. There is no analytical method giving a choice for the best set of basic function. Most often the basic functions in different designs are polynomials or afﬁne functions of the scheduling parameters. Following from (10.6)-(10.7), an implementation of an LPV controller requires not only the measurement of the parameter ρ in real-time, but also the measurement of its ˙ Because the parameter derivatives are usually not available several time derivative ρ. simple approaches are used (for example, [5]) to restrict R or/and S dependence on the ˙ parameters in order to remove the dependence on ρ: 1. 2. 3. 4.

R := R(ρ), S := S0 R := R0 , S := S(ρ) R := R(ρ), S := S(ρ), ρ˙ = 0 R := R0 , S := S0 , ρ˙ unbounded

Each restriction can lead to a certain amount of conservatism. In this work the ﬁrst simplifying assumption is used. The main drawback of this approach is that the calculation time rises very quickly with the number of parameters and the number of grid points.

10.3 ADMIRE Model and Design Requirements The ADMIRE model represents a generic small ﬁghter aircraft with a delta-canard conﬁguration, which is described in detail in [115]. The designed control system should provide the ADMIRE model with good handling qualities (FQL 1) and satisfy the basic set of requirements from [115, 168] such as, the maximum roll rate should be of the

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order of 300 deg/s, the maximum normal load factor of the order of 9 g, and the maximum angle of attack at low Mach numbers should be 30 deg. There is also a typical constraint requirement on the controller gains [115] to reduce coupling with structural mode dynamics. The main objective in this study was to demonstrate that the LPV approach allows the design of a controller for the whole ﬂight envelope while meeting the design requirements. An LPV controller explicitly depends on the ﬂight parameters and therefore it is automatically gain scheduled. The LPV controller architecture is similar to the structure of the classical ADMIRE Flight Control System (FCS), implemented in Simulink. It contains separately the longitudinal controller and the lateral-directional controller. The longitudinal controller is responsible for the short-period dynamics. There is no special design made for speed control, simply the speed controller from the ADMIRE Simulink model is implemented along with the LPV longitudinal control law. Below Mach number M < 0.58 the longitudinal LPV controller is designed as an α-demand control system (instead of pitch rate control in the original ADMIRE FCS) and at Mach numbers above M > 0.62 it is converted to an nz -demand control system. In the region 0.58 ≤ M ≤ 0.62 a transitional version of the LPV controller is implemented. The longitudinal LPV control law requires the output measurements for angle of attack α , pitch rate q and normal load factor nz = −az /g (az is vertical acceleration). A combined symmetrical elevon deﬂection (the inboard and outboard elevons are identically deﬂected) and symmetrical canard deﬂection are used as control effectors in the LPV controller. The canard is mostly used for trimming and then in the post-design control allocation scheme at low Mach numbers. The lateral-directional controller is designed to control the variation of sideslip angle β and the stability axis roll rate pW , which is a projection of angular velocity vector on the plane of symmetry OXZ; it can be expressed through the body axis roll and yaw rates and angle of attack: pW = p cosα + r sin α. It is supposed that the pilot pedal command is transformed into the commanded sideslip angle βcom and the pilot lateral stick input is transformed into the stability axis roll rate command pW com . The output measurements used for lateral-directional control are the roll and yaw rigid body axis rates p, r, sideslip angle β and angle of attack α. The differential elevons (inboard and outboard together) and the rudder are used as the control effectors in the lateral-directional controller. The thrust vectoring control can also contribute to the control task, however it is not used in this study.

10.4 Obtaining an LPV Model The synthesis of an LPV controller requires an LPV model approximating the ADMIRE nonlinear dynamics. The closed-loop performance essentially depends on the accuracy of this LPV model. The conventional approach to generate an LPV model of an aircraft is based on the Jacobian linearization at trim points for the straight-and-level ﬂight. The alternative quasi-LPV model represents dynamics for trim and transient conditions, thus capturing more accurately the nonlinear dynamics; such LPV representations are

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obtained in [182]. These models are especially important for consideration of nonlinear high angle of attack aircraft manoeuvres. The main difﬁculty of the design problem for the ADMIRE model is its rather wide ﬂight envelope: the Mach number varies from low subsonic speeds M = 0.3 to supersonic regimes with M = 1.4; the altitude varies from H = 100 m up to H = 6000 m. The design challenge includes a special interest in the transonic region (0.9 < M < 1.1). Manoeuvring at high angles of attack, however, is not a primary consideration in this study. Therefore, a more simple LPV aircraft model based on Jacobian linearization at trim points for the straight-and-level ﬂight is implemented. The ADMIRE aerodynamics in the transonic region is not very smooth at some states, posing difﬁculties for obtaining an adequate LPV model. To obtain the ADMIRE LPV model, a trimming analysis for the straight-and-level ﬂight conditions at different points of the ADMIRE ﬂight envelope has been performed. At each point, the aerodynamic derivatives have been evaluated and used to construct the linearized models. The linear models were reduced to the size of ﬁve states corresponding to the longitudinal short period mode ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ Zα (ρ) Zq (ρ) Zδe (ρ) α˙ α ⎣ q˙ ⎦ = ⎣ Mα (ρ) Mq (ρ) ⎦ (10.8) + ⎣ Mδe (ρ) ⎦ δe q nzα (ρ) nzq (ρ) nzδe (ρ) nz and the third order lateral-directional mode (the spiral motion mode is here omitted): ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ Yβ (ρ) sin α0 (ρ) − cos α0 (ρ) Y∆δe (ρ) Yδr (ρ) β β˙ ∆δe ⎣ ⎦ ⎣ ⎦ ⎣ p˙ ⎦ = ⎣ Lβ (ρ) L p (ρ) ⎦ Lr (ρ) p + L∆δe (ρ) Lδr (ρ) , (10.9) δr Nr (ρ) Nβ (ρ) N p (ρ) N∆δe (ρ) Nδr (ρ) r r˙ where the parameter α0 is an equilibrium angle of attack, δe and ∆δe are symmetric and differential elevon deﬂections, δr is rudder deﬂection; parameter α0 is parameterized throughout the ﬂight envelope. A detailed study of the longitudinal and the lateral-directional dynamics has been performed to determine the most simple and adequate parameterization scheme. Different parameter sets, and linear and quadratic ﬁtting techniques based on the least square regression have been tested for approximation of the aerodynamic derivatives and the output coefﬁcients. The following combinations of characterizing ﬂight regime parameters have been considered: • • • •

dynamic pressure q¯ and q/V ¯ ; Mach number M and altitude H; dynamic pressure q¯ and Mach number; dynamic pressure q¯ and altitude H.

In the transonic region aerodynamic derivatives Mα and Mδe have rather irregular variations. The derivative Mα changes its sign when Mach number equals M0 = 0.925. The longitudinal short-period mode is aperiodically unstable at M < M0 and stable at M > M0 . Following this analysis it was found reasonable to construct an LPV ADMIRE model using two different ﬂight envelope regions. The ﬁrst subregion covers altitudes 100 < H < 6000 m at subsonic speeds M0 < 0.9 and the second subregion covers the

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same altitudes 100 < H < 6000 m at transonic and supersonic ﬂight speeds M0 > 0.95; in the intermediate region the obtained LPV approximations are interpolated using a linear or spline method. Fig. 10.1 shows results of a piecewise linear ﬁtting of the longitudinal derivatives ¯ − 1. and nz partial derivatives using two scaled parameters ρ1 = q¯q¯0 − 1 and ρ2 = q¯q/V 0 /V0 It shows data predicted by the LPV model and the trim data for different ρ1 values. The LPV model data were obtained for two groups of Mach number grid points M = 0.3, 0.4, ...0.9 and M = 0.95, 1.0, 1.1, 1.2 for the same set of altitude points H = 100, 1000, 2000, .., 6000 m. The lateral-directional LPV model was obtained in the whole ﬂight envelope using the same scaled parameters as in the longitudinal motion case, the altitude grid points were the same as in the longitudinal case and grid points for Mach number had a constant step increment M = 0.05. The results of the LPV model ﬁtting are shown in Fig. 10.2. 10.4.1

LPV Model for Switching Between α- and nz - Demand Systems

The LPV model used for the design of the control law also includes the aircraft outputs reﬂecting the type of control demands. Following design speciﬁcations the FCS should be of the α-demand type at low speeds with M < 0.58 and of the nz -demand type at higher speeds with M > 0.62. The controlled output y = c[α q]T + dδe is represented as a piecewise function of parameters, where the coefﬁcients c, d at M < 0.58 shape the α-demand system c = [1 0], d = 0 and at M > 0.62 they shape the nz -demand system c = [nzα (ρ) nzq (ρ)], d = nzδ (ρ). The values of c and d are interpolated in the intermediate region 0.58 < M < 0.62. The coefﬁcients nzq and nzδ are relatively small in their absolute value in comparison with the nzα value, which varies in the range (20, 39) when altitude H changes from Z

Z

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Fig. 10.1. Piecewise linear ﬁtting of the longitudinal derivatives using two scaled parameters q/V ¯ ρ1 = q¯q¯0 − 1, ρ2 = q¯ /V − 1 (−1 < ρ2 < 1), o - trim data, + - LPV ﬁtting 0

0

An LPV Control Law Design and Evaluation for the ADMIRE Model Yβ 0

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Fig. 10.2. Linear ﬁtting of the lateral-directional derivatives using scaled parameters ρ1 = ρ2 =

q/V ¯ q¯0 /V0

205

q¯ q¯0

− 1,

− 1 (−1 < ρ2 < 1), o - trim data, + - LPV ﬁtting

100 to 6000 m at M = 0.6. Because the α and nz outputs have different magnitudes the angle of attack output is scaled by a factor knα = 30 (approximately the mid nzα value in the transitional zone) to have a smooth variation of a generalized control output.

10.5 LPV Control Law Design To formulate a design problem in the form of an interconnection diagram is a usual approach in the standard H∞ -optimization and LPV design methods. Note that the LPV control law design is an extension of the H∞ control optimization problem applied to parameter-dependent systems. The block diagram or interconnection diagram presented in Fig. 10.3 illustrates the interconnections used in the design of the longitudinal and the lateral-directional controllers. This diagram allows the design objectives to be formulated in the presence of model uncertainties, external disturbances and imposed constraints using a number of weighting transfer functions presented as blocks in the interconnection structure. The presented diagram is rather typical and it allows a sufﬁcient degree of ﬂexibility. This diagram integrates the performance and robust stability objectives into a single control design framework. It consists of the aircraft model A/C LPV model, actuator dynamics Actuator, sensor dynamics presented by block Delay and other weighting block-functions for input commands, noise, wind gust, actuator activity and closed-loop system performance. Performance requirements are formulated through the choice of a desired aircraft handling qualities model, Ideal model, and performance weighting functions Wper f applied to the output signals of the system. Multiplicative uncertainty, represented in the block diagram by the uncertainty weighting function Wdel and the

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Fig. 10.3. Interconnection diagram for LPV control law design

uncertainty set ∆, is used to describe the errors in the linearization process and system modelling. The actuator deﬂections and deﬂection rates are weighted by function Wact to ensure that they do not exceed their physical capabilities. Sensor noise function Wnoise , pilot commands activity function Wcom , and all other block functions with different content are presented both in the longitudinal and the lateral-directional control law designs. More detailed information separately for the longitudinal and the lateraldirectional control law designs is given in the following sections. 10.5.1

Longitudinal Control Law Design

For the longitudinal motion the primary performance objective is to provide Level I handling qualities in the aircraft’s pitch axis throughout the whole ﬂight envelope. As was mentioned, the control variable in the longitudinal channel at M < 0.58 is angle of attack α and at M > 0.62 is the normal load factor nz with interpolation of these signals in the intermediate zone of Mach number. A usual approach in the H∞ control design is to specify the required closed-loop dynamics by an appropriate reference model transfer function. For the α- and nz -demand systems the desired time responses to a step pitch stick input, which is converted to an adequate input signal for the αcom or nzcom , respectively, are usually approximated by the following second order transfer function: ω20 nz α . (10.10) = 2 αcom nzcom s + 2ζω0s + ω20 To meet Level 1 handling quality requirements throughout the ﬂight envelope, the parameters of this reference model change depending on the ﬂight regime. The following

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scheduling of the longitudinal reference model parameters are made: ζ=0.8 for the whole ﬂight envelope, ω0 is a linear function of dynamic pressure q¯ with ω0min = 2.5 at q¯ corresponding to ﬂight condition H = 100 m and M = 0.3 and ω0max = 6.5 at q¯ at H = 100 m and M = 1.1. With ω0 linearly depending on dynamic pressure q, ¯ the transfer function (10.10) is a quadratic function of q. ¯ Since the LPV aircraft model is given in a piecewise linear form, the transfer function (10.10) is rewritten to represent the state-space matrices as linear functions of the dynamic pressure q, ¯ which is scaled and replaced by parameter ρ1 : ∆ω ∆ω a11 a12 x1 b x˙1 ρ1 + ρ1 u, = 1+ 1+ x˙2 −a12 a22 x2 −b ω0c ω0c y = x1 + x2

−a11a22 + ω20c , b = ω20c /(a11 − ω −ω a22 − 2a12), k = 4.3333, ω0c = ω0min − ∆ωρ1min , ∆ω = ρ0max − ρ 0min . 1max 1min Measurement signals used for feedback in the longitudinal controller are angle of attack α, pitch rate q and normal load factor nz . It is usual practice in conventional engineering design approaches to use α, q as feedback signals in α-demand system and q, nz in nz -demand systems. To design the LPV controller with a transition from αdemand system to nz -demand system, the LPV controller structure employs at M < 0.58 mainly α and q signals and at M > 0.62 mainly nz and q signals. This is achieved via a special choice of measurement noise weighting function Wnoise . The noise weighting function for nz is large at M < 0.58 in comparison with noise weights for α, q, and an opposite ratio is used for ﬂight regimes with M > 0.62 - the noise weighting function for α is large in comparison with the weighting function for nz , q. Selection of weighting functions is an iterative procedure due to the conﬂicting requirements for robustness and performance. In the present design the noise weighting function is selected as where a11 = −2ζω20c /(1 + k), a22 = a11 k, a12 = −

0 = 30α, Wnoise (s) = diag(Wn0α ,Wnq (s),Wnz ), α where at M < 0.58

0.3s + 1 ; Wn0α = 0.15; Wnq (s) = 0.005 0.02s +1

at M > 0.62

Wn0α = 10;

Wnz = 1

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and in the transitional intermediate zone 0.58 ≤ M ≤ 0.62 interpolation between these two sets of weighting functions is performed. The pitch rate noise weighting function supposes that at low frequencies there is an error of a level of 0.005 rad/s and at high frequencies the error level increases up to 0.083 rad/s. This guarantees low controller gain in the pitch rate channel, thus helping to meet the design constraints. The performance variable is constructed as the difference between the ideal response and the actual signal. It is weighted by a low-pass ﬁrst order ﬁlter Wper f (s) = 25

0.033s + 1 33s + 1

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This weighting function enforces the commanded response to follow the ideal response with a static accuracy of 0.0013 rad in the α-demand system and of 0.04 g in the nz demand system. At frequencies higher than 5 rad/s the accuracy requirements are relaxed. The range of activity for pilot commands is deﬁned by the weighting function Wcom = 4. The actuator dynamics are deﬁned in [115] using a ﬁrst order transfer function 1/(0.05s + 1). To take into account the time delay needed for controller computation and to implicitly include the time delay from sensors, a Pad´e approximation of a pure time delay has been added. The ﬁnal actuator model for controller design is this given as follows: 1 − τs 2 , τ = 0.07 Actuator = (1 + τs )(0.05s + 1) 2 Sensor models have been taken as unity matrices to minimize the controller order, which is equal to the total order of the augmented plant. The control activity is speciﬁed by weighting the control effector deﬂection amplitude and its rate of deﬂection. The weighting function Wact = diag(Wde f l ,Wrate ) is used for this purpose in the α-demand system as Wact = diag(0.15, 0.15) and in the nz -demand system as Wact = diag(0.1, 0.1). These control activity weights are often selected to be inversely proportional to the available maximum actuator deﬂections and maximum deﬂection rates. In our design they were at a lower level because of conﬂicting requirements and an intention to use canard at low Mach numbers, where the elevon’s rate of deﬂection becomes extremely critical. Multiplicative uncertainty and vertical gust weighting functions are taken as frequency independent to reduce the controller order: Wpert = 0.02 and Wgust = 0.02. These weights are rather small, but satisfactory for design purposes; higher weights lead to an undesirable increase in the controller gains. The augmented LPV plant, required for application of the optimization procedure, is created from the interconnection diagram, Fig. 10.3, by gathering together all inputs and outputs and forming a singe transfer function with the use of special functions for connecting parameter-dependent blocks [79]. The synthesis of controller matrices presented in (10.6) is made by determining matrices R(ρ) and S, which satisfy the afﬁne matrix inequalities over the entire ﬂight envelope with minimum possible γ value. The LMI problem is solved on the selected grid of points in the parameter space using the MATLAB LMI Toolbox [79]. The following grid in the ﬂight envelope was used: H = 100, 1000, 3000, 6000 m H = 3000, 6000 m H = 7000 m

M = 0.25, 0.3, 0.4, 0.5, 0.58, 0.6, 0.62, 0.7, 0.8, 0.9, 0.95, 1.0, 1.1, 1.15 M = 1.2 M = 1.0

(the last three grid points were added after the closed-loop system evaluation of the preliminary design). The following four basic functions ρ0 = 1, ρ1 =

q¯ q/V ¯ − 1, ρ2 = − 1, ρ3 = M/M0 − 1 q¯0 q¯0 /V0

(10.11)

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have been used for plant and controller (matrix R) paramteterization. It was found that addition of function ρ3 in the parameterization of the controller decreases its conservatism, because the augmented LPV plant is constructed from three different twoparameter linear plant approximations which are interpolated over two Mach number ranges - 0.58 < M < 0.62 and 0.9 < M < 0.95. Different bounds on rate of parameter variations were tested to synthesize the LPV controller. However, a satisfactory design solution was obtained only when the rate of parameter variation was set to zero; This can probably be explained by the fairly large size of the parameter envelope. The result of the optimization procedure is a linear 8th -order parameter varying controller that provides the closed-loop system with robust performance γ = 3.48 for the overall ﬂight envelope. Robust performance is achieved if γ is less than 1 and condition γ > 1 indicates that some of the weighting functions do not adequately reﬂect the actual requirements and should be changed. It is also an indication that the design is too conservative, because some weighting functions have been chosen too small in order to satisfy different constraints, for example, bounding controller gains. The design can be made less conservative by considering a D-K iteration procedure [12] for scaling different parts of the closed-loop transfer function. Alternatively, several LPV controllers can be designed for smaller size subregions and blended into one controller covering the whole ﬂight envelope. However, from the post design evaluation it was found that the properties of the relatively simple LPV controller as designed above are mostly quite satisfactory. q (deg/s)→ δ

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Fig. 10.4. Frequency responses of the longitudinal controller

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Frequency responses of the controller transfer functions for several ﬂight envelope test points (H = 1, 3, 6 km and M = 0.5, 0.7, 0.9, 1.1) are shown in Fig. 10.4. One can see that at ﬂight regimes with the α-demand system the controller magnitude in the nz channel is very low (≈ 1.e − 4), and at ﬂight regimes with the nz -demand system the α component is very low (1.e − 5). This change of controller type has been achieved by special shaping of the noise weighting functions as described above. The controller has a uniﬁed structure allowing the transition between two types of control input and this makes the controller design procedure simpler. Note that design constraints for high frequency gains are satisﬁed completely in the whole ﬂight envelope. For minimizing dynamic coupling between the rigid body motion and structural modes the gain from the pitch rate to control surfaces is expected to be less than 9 dB for frequencies above 5 Hz. At most ﬂight regimes this condition is satisﬁed with a signiﬁcant margin. 10.5.2

Lateral-Directional Control Law Design

The lateral-directional controller has been designed to control sideslip angle β and stability axis roll rate pW with requirements to follow the appropriate control commands. The controller is fed by inputs β, α, p, r, βcom and pWcom and its two outputs are sent to actuators for differential elevons and rudder. The design of the lateral-directional controller is made across the same ﬂight envelope as in the design of the longitudinal controller. The handling qualities for the lateral-directional motion are also deﬁned through the reference models for sideslip and stability axis roll rate dynamics. Usually the reference model for sideslip angle is deﬁned similar to the angle of attack case as a second order transfer function. However, to decrease the controller order the reference models for sideslip and stability axis roll rate were taken as the following ﬁrst order transfer functions: pW 1 1 β , . (10.12) = = βcom 0.5s + 1 pWcom 0.4s + 1 The Ideal Model block in the interconnection diagram (Fig. 10.3) is represented now as a diagonal augmentation matrix with these two transfer functions. The weighting functions for the pilot command range Wcom and the noise weighting function Wnoise are represented as simple diagonal augmentation matrices = diag(0.1, 1) Wcom = diag(W βcom ,WpW com )

Wnoise = diag Wnβ ,Wn p ,Wnr = diag (0.02, 0.05, 0.05). The closed-loop system performance in sideslip β and stability axis roll rate pW channels is weighted by function Wper f , which is a diagonal augmentation matrix with two transfer functions weighting sideslip and stability axis roll rate tracking errors: ⎛ ⎞ s s +1 +1 ⎟ ⎜ 40 40 ⎜ ⎟ Wper f (s) = diag Wβ per f ,Wp per f = diag ⎜180 , 50 ⎟. ⎝ ⎠ s s +1 +1 0.04 0.04

An LPV Control Law Design and Evaluation for the ADMIRE Model β → ∆δ

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Fig. 10.6. Frequency responses of the lateral-directional controller (rudder channel)

The control activity for the differential elevon and rudder actuator deﬂections and rates of deﬂection is speciﬁed by the diagonal weighting function Wact

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elev elev rud rud Wact = diag Wde ,W ,W ,W rate rate = diag(1.25, 0.625, 1.25, 0.625). de f l fl Robustness in the closed-loop system is achieved via a multiplicative uncertainty deﬁned by the uncertainty block ∆ and perturbation weight Wdel = diag(0.02, 0.02). The gust weighting function was not taken into account in the design of the lateraldirectional controller. The actuator models for both control effectors are taken as 1/(0.05s + 1). The optimization procedure used to design the lateral-directional controller was similar to the procedure used in the design of the longitudinal controller. Three basic functions ρ0 , ρ1 , ρ2 (10.11) have been used for controller parameterization. The result of the optimization procedure is a 9th - order LPV controller that provides closed-loop robust performance γ = 3.5 throughout the whole ﬂight envelope. Frequency responses of controller transfer functions for H = 1, 3, 6 km and M = 0.5, 0.7, 0.9, 1.1 are shown in Figs. 10.5 and 10.6.

10.6 Evaluation of the LPV Control Law Performance and Robustness 10.6.1

Linear Analysis and Robustness Evaluation

Analysis of Stability Margins and Closed-Loop System Eigenvalues The closed-loop system eigenvalues have been calculated and analyzed at all grid ﬂight envelope points and also at some intermediate points. For the longitudinal motion the closed-loop pole pcl = −1.86 was the nearest one to the imaginary axis from all the poles analyzed throughout the ﬂight envelope. In some intermediate ﬂight regime points the closed-loop poles were very slightly closer to the imaginary axis. There were no unstable eigenvalues at all. Note that when unstable poles were found at the ﬁrst design stage near the ﬂight envelope boundaries, the controllers were redesigned after adding new grid points in the problematic region. The maximum modulus of the closed-loop eigenvalues throughout the ﬂight envelope lie in a range of 60 ÷ 200. These values are acceptable and reasonably low for the real-time implementation. For the lateral-directional motion the nearest closed-loop poles to the imaginary axis are placed in a range of pcl = −2.0 ÷ −1.5; only near the ﬂight envelope boundary at H = 6 km and M = 0.3 does there exist a pole pcl = −0.83. The maximum absolute values of the closed-loop system eigenvalues are about 170. Standard requirements for the gain/phase margins are speciﬁed by the Nichols plot (see Figs. 10.7 and 10.8), where the trapezoidal area deﬁnes the exclusion zone. For the short-period longitudinal motion the Nichols plot is constructed from the transfer function obtained by breaking the closed loop at the symmetrical elevon input (here one general control signal is considered without a control allocation scheme). The closedloop system includes the aircraft short-period model for the selected ﬂight condition, the actuator, the controller, and the pitch rate, normal acceleration and angle of attack q,n α ) as described in [115]: sensor models (Wsen z , Wsen q,nz Wsen =

0.0001903s2 + 0.005346s + 1 1 α ; Wsen = . 0.02s + 1 0.0004942s2 + 0.03082s + 1

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Fig. 10.7. Nichols plot for the longitudinal control law (trapezoidal region is the exclusion zone)

For ﬂight regimes H = 1, 3, 6 km and M = 0.5, 0.7, 0.9, 1.1 the Nichols plots are shown in Fig. 10.7. At a ﬂight regime with H = 1 km and M = 1.1 with high dynamic pressure the Nichols diagram approaches a corner of the exclusion zone and for ﬂight points with lower altitudes and higher Mach numbers, near the ﬂight envelope dynamic pressure boundary, the Nichols curve passes through the exclusion zone. Analysis of the ﬂight regime points inside the ﬂight envelope and points which are close to other ﬂight envelope boundaries have demonstrated sufﬁcient phase margins. The gain/phase margin can be increased by tuning the weighting functions in the interconnection structure (Fig. 10.3), which provides the augmented LPV plant for the optimization process. This is an indirect approach and due to many conﬂicting requirements it is sometimes not very effective. However, there are several other ways in which to increase the phase margin, the main one being to increase the assumed time delay in the block Delay; adjustments to the perturbation and/or actuator weights can also be used to increase the phase and gain margins. This means that the design of an LPV controller in terms of shaping the weighting functions in interconnection structure in Differential elevons loop is broken

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Fig. 10.9. Interconnection diagram for µ-analysis

Fig. 10.3 is an iterative and empirical process and it can be more successful in more experienced hands. The Nichols plots for the lateral-directional motion for two control channels (the closed-loop system is broken separately for elevons and for rudder) are shown in Fig. 10.8. These diagrams were computed for ﬂight regimes with H = 1, 3, 6 km and M = 0.5, 0.7, 0.9, 1.1. The roll and yaw rate sensor models were the same as the pitch rate sensor model, the sideslip angle sensor model is identical to the angle of attack sensor model. From the results presented, it is clear that the lateral-directional gain-phase margins are entirely satisfactory. µ-Analysis Structured singular value analysis or so called µ-analysis [12] is now a widely used method of robust stability and performance analysis of multi-dimensional systems with parametric and modelling uncertainties. In this work µ-analysis is applied considering uncertainties in aerodynamic derivatives and multiplicative uncertainty on aircraft inputs. Robust stability and performance are tested by computation of the µ parameter in accordance with the interconnection diagram presented in Fig. 10.9. The uncertainties of aerodynamic coefﬁcients and velocity presented in Fig. 10.9 by block ∆a are considered as perturbations of magnitude equal to 10% of their nominal values. The multiplicative perturbation is a complex uncertainty with weight Wdel = 0.1. For evaluation of the robust performance the reference model at each particular ﬂight regime is obtained from the ideal parameter-dependent model (10.10) by freezing its parameters. The performance frequency content was the

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same as in the design process (see Fig. 10.3). It is expected that static errors should be less than 0.5 deg for α and less than 0.1 g for nz . The noise weighting function is taken as Wnoise = diag(Wnα ,Wnq ,Wnnz ) = diag(1/57.3, 0.1/57.3, 0.001). The lower and upper µ bounds for the robust stability and robust performance values at ﬂight regimes H = 4 km and M = 0.5, 0.7, 0.9, 1.1 are presented in Fig. 10.10. The robust stability test results are represented by dashed curves. In all these tests µ < 1 and the closed-loop system is robustly stable in the presence of the considered parameter uncertainties and multiplicative perturbations. Robust performance is achieved at M = 0.5, where the α-demand system is activated, but is not achieved at higher Mach numbers with the nz -demand system. To meet robust performance requirements the selected weighting functions in the interconnection diagram in Fig. 10.9 should be appropriately modiﬁed. Analysis shows that it can be achieved by lowering the performance requirements or the uncertainty values. Note that robust stability is guaranteed in the whole ﬂight envelope for multiplicative and added parametric uncertainties to a much greater extent than was required in the design. 10.6.2

Off-Line Nonlinear Simulation Tests

Performance and robustness properties of the closed-loop system with the designed LPV controller have been also analyzed in direct numerical simulation of the ADMIRE nonlinear dynamics by performing pull-up, push-over and velocity vector roll manoeuvres at different ﬂight regimes for the nominal case and for the case with uncertain parameter scattering. One can not expect that the LPV aircraft model obtained by Jacobian linearization at level ﬂight trim points as applied in this work will adequately capture

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nonlinear dynamics. To effectively capture the nonlinear dynamics, for example, at high angles of attack or intensive rotation in roll the applied linear design approach should borrow some features from the nonlinear dynamic inversion method or use a quasi-LPV model. Nevertheless, the simulation results show that the designed LPV controller provides very satisfactory closed-loop system performance and robustness for most ﬂight regimes in the ﬂight envelope under consideration. At low Mach numbers (M < 0.62) the LPV longitudinal controller signal is allocated between the symmetrical elevons (joint deﬂection of the outboard and inboard surfaces) and the canard using a standard pseudo-inverse matrix technique. The canard amplitude activity is weighted as 25 ÷ 40% in comparison with the elevon activity. The canard control derivatives required for the control allocation scheme are parameterized against the ﬂight regime similar to the parameterization of the ADMIRE LPV model. The Nominal Case Pull-up/push-over manoeuvres The simulation results for pull-up and push-over manoeuvres with a 3 second long 50% amplitude step longitudinal stick input at H = 3 km and M = 0.5, 0.9, 1.1 are shown in Fig. 10.11. These results demonstrate good handling qualities across the ﬂight envelope. Note that one curve at M = 0.5 corresponds to the α-demand system with αdem = 10 deg, and two other curves correspond to the nz -demand system with nzdem = 5 at higher Mach numbers. The simulation results demonstrate that the speciﬁed performance is achieved at high Mach numbers for the nz -demand system and rather good handling for the α-demand system.

An LPV Control Law Design and Evaluation for the ADMIRE Model n , cosθ

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As recommended in [115] the ADMIRE closed-loop dynamics have been evaluated in 10 second long numerical simulations for two types of control with full amplitude longitudinal stick for step inputs and 3 second long ramp inputs at H = 1, 4, 6 km and

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M = 0.5, 0.7, 0.9, 1.1 and an additional point at H = 1km and M = 0.3. The results of simulation of the pull-up and push-over manoeuvre with full step input are presented in Figs. 10.12 and 10.13; the results with the ramp control input are presented in Fig. 10.14. n

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The Mach number in these simulations necessarily varies during manoeuvre execution. Nevertheless, simulation results show smooth transitions in manoeuvre parameters for ﬂights in subsonic, transonic and supersonic regions. A particular focus on the transonic region shows that there is no visible pitch-up divergence effect during transition through this region. Analysis of simulation results shows that the designed LPV controller is certainly adequate for control in the transonic region. Note that they also show good transient dynamics around the speciﬁed ﬂight envelope even including regimes, for example, at H = 7800 m, which were not included in the speciﬁed ﬂight region. Simulation of ﬂight manoeuvres with transition from the nz -demand to the α-demand control system did not show any particular problems affecting closed-loop stability and performance. Dotted curves correspond to simulation results at M = 0.7 with the PID prototyping ADMIRE control laws. Roll manoeuvres Fig. 10.15 demonstrates simulation results for a velocity-vector roll manoeuvre with 10 seconds control input for maximum angular velocity of 300 deg/s at several ﬂight regime points H = 6 km and M = 0.5, 0.9, 1.1. At all regimes the controller tracks the demand in roll rate quite accurately. At low Mach numbers (M = 0.3) the demand value was decreased to 250 deg/s to avoid saturation of the elevons. It should be noticed that

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there is an oscillatory character of variation of practically all parameters in Fig. 10.15. The origin of these oscillations may be linked with the effect of the gravitational force or an onset of oscillatory instability at intensive velocity roll rotation in the closed-loop system with the LPV control laws. Higher control demands for velocity-vector roll rate lead to saturation of elevons and rudder deﬂection which potentially can induce aircraft departure at subsonic ﬂight regimes. The LPV control law does not take into account nonlinear inertia moments, which become very signiﬁcant at rapid velocity roll rotation. The effect of gravitational force can contribute to these oscillations as well. Further improvement of the LPV control laws can be achieved by taking into account these nonlinear factors. However, the improved design of the LPV control law and the closed-loop dynamics analysis needs special consideration. The Uncertainty Case As recommended in [115] the pull-up and push-over manoeuvres with full pitch stick input have been simulated at ﬂight regimes with H = 4 km and M = 0.5, 0.7, 0.9, 1.1 in the presence of pitching moment uncertainty, speciﬁed by a scatter in the aerodynamic derivative −0.1 < ∆Cmα < 0.1. Fig. 10.16 shows the normalized α or nz responses for different ﬂight regimes with a superimposed template for time response criterion, proposed in [115]. The left plot shows results for pull-up manoeuvres and the right plot for push-over manoeuvres. Note that the considered ﬂight regimes correspond to intermediate points for the ﬂight regime grid used in controller design. Nevertheless, the performance is quite satisfactory, so the simulation results conﬁrm the robustness properties of the LPV control law. The

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Fig. 10.16. Normalized α or nz responses for full stick pull-up (left plot) and push-over (right plot) manoeuvres in the presence of uncertainty in |∆Cmα | ≤ 0.1 at H = 4 km and M = 0.5, 0.7, 0.9, 1.1

only exclusion is the pull-up manoeuvre at M = 0.5 with ∆Cmα = 0.1 when the time response goes outside the template (the LPV model does not take into account nonlinear variations of aerodynamic characteristics with angle of attack). All other time responses including the regime with M = 0.7, where transition from nz -demand to α-demand control type is implemented, are inside the prescribed template. Normalized nz responses for a pull-up manoeuvre with 50% pitch stick input in the presence of uncertainty for a combination of two and three different parameters in the longitudinal aerodynamics have been simulated at ﬂight regime H = 2 km and M = 0.7. These results are presented in Fig. 10.17. For a combination of two uncertain parameters the size of uncertainty is decreased by 20%, and for a combination of three uncertain parameters, the size of uncertainty is decreased by 40% [115]. As all responses in Fig. 10.17 are inside the required template, one can conclude that the designed longitudinal LPV controller is sufﬁciently robust.

Fig. 10.17. Normalized nz responses for pull-up manoeuvre (50% pitch stick input) with variation of up to three uncertain parameters at H = 2km and M = 0.7

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Fig. 10.18. Roll manoeuvre (100% side stick step input) with uncertainty of dihedral derivative |∆Clβ | ≤ 0.04 at H = 6 km and M = 0.5, 0.7, 0.9, 1.1

A velocity-vector roll manoeuvre with 5 seconds 100% side stick step input has been simulated with uncertainty in the dihedral derivative |∆Clβ | ≤ 0.04 for a number of ﬂight regimes with H = 6 km and M = 0.5, 0.7, 0.9, 1.1. The simulation results presented in Fig. 10.18 show very high performance and robustness properties in angular velocity p reaching the maximum level of 300 deg/s with a very small overshoot. Response to Vertical Wind Gust The simulation of the closed-loop system dynamic responses to a vertical wind gust with amplitude of 5 m/s are shown in Fig. 10.19. Several ﬂight regime points have been considered from the ﬂight envelope H = 1, 3 km and M = 0.5, 0.7, 0.9. In the ﬂight regimes where the control system is operating as an nz -demand control system the dynamic responses are satisfactory and the maximum change in the ﬂight path angle is less than one degree. In the subsonic region where the α-demand system is activated the dynamic overshoot in the ﬂight path angle exceeds the required limit of one degree in the ﬁrst three seconds. Unfortunately, in the performed design there was no special consideration of dynamic responses to wind gust disturbances, which required an adequate shaping of the frequency dependent gust weighting function Wgust . The D-K iterative procedure with appropriate scaling of different parts of the closed-loop transfer function on the basis of µ-analysis can be a useful tool for solving this problem.

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Fig. 10.19. Dynamic responses of ﬂight path angle to vertical wind gust with amplitude 5 m/s at H = 1, 3 km and M = 0.5, 0.7, 0.9

10.7 Real-Time Pilot-in-the-Loop Simulation of the LPV Control Law Real-time piloted simulation using ground-based ﬂight simulators is a key stage in the development of ﬂight control laws and although formal simulator-based assessment introduces issues well beyond the scope of the Action Group, it was felt to be worthwhile porting the ADMIRE model and the LPV control law to the HELIFLIGHT simulator at the University of Liverpool in order to make some preliminary tests with the system in real-time. The team involved has experience of running a wide variety of aircraft (mainly rotorcraft) models on HELIFLIGHT; this however was the ﬁrst time that a multivariable gain-scheduled control system had been tested. The aim was to test the basic functionality of the control law. 10.7.1

The HELIFLIGHT Motion Simulator

The HELIFLIGHT ﬂight simulator at the University of Liverpool became operational in 2001. It has a six degree-of-freedom motion base using Stewart platform geometry. It is interfaced to a specialized ﬂight dynamics modelling software package (FLIGHTLAB) with a real time interface (PILOTSTATION). The simulator has a generic cockpit that enables rotorcraft and ﬁxed-wing aircraft to be simulated. Its main features are: • Four axis dynamic control loading (Loadcue); • Three-channel collimated visual display for forward view, plus two ﬂat panel chin windows, providing a wide ﬁeld of view visual system (Optivision), each channel running a visual database;

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• Re-conﬁgurable head-up-display (HUD); • Re-conﬁgurable computer-generated head-down instrument display panel. Fig. 10.20 shows the simulator cockpit room, capsule and simulator control room. Further details of the facility and of some of the projects that have made use of it are given by Padﬁeld & White (2003) [179]. The original conﬁguration of the system was designed to enable the real-time execution of FLIGHTLAB models, via the PILOTSTATION interface. However, work has recently been undertaken to extend the system architecture to allow real-time simulation of SIMULINK models. The preferred approach when running SIMULINK models in real-time has been to use xPC Target from The MathWorks. However, certain technical issues made it preferable for us to adopt a slightly different strategy to implement the LPV control law. This involved use of a third-party real-time blockset by Daga (2004) [43]. The blockset contains a real-time block, based on an S-function written in C++ which allows Simulink models to be executed in real-time without any of the auto-coding that is normally required for Real-Time Workshop applications. This blockset is based on the simple concept that, to make the Simulink model run in real-time, the cycle time should be lower then the desired simulation step time. If this assumption is not valid, real-time simulation is not possible. The real-time block holds the execution of the Simulink simulation if the cycle time is lower then the simulation step time, and the block waits for the time needed to ﬁll the simulation step, leaving the remaining CPU time to other Windows processes. That has proved to be simple and effective in this case. 10.7.2

Real-Time Implementation and Simulation Architecture

The simulation architecture consists of three main systems; the computer that runs the Simulink model, a communications bridge computer and the HELIFLIGHT systems. When real-time simulation was required, the Simulink model was executed in the conventional manner and the only changes to the standard ADMIRE model setup were the inclusion of UDP send and receive blocks to allow the model to receive pilot inputs from the cockpit controls and send the simulation data to the simulator using a local

Fig. 10.20. HELIFLIGHT pod and simulator control room

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Fig. 10.21. Simulator Architecture

area network. The UDP simulation data was converted into a HELIFLIGHT compatible data form and broadcast to the HELIFLIGHT systems using a communication bridge computer and a C++ coded application. The HELIFLIGHT systems consisted of the motion base, cockpit controls (main stick, throttle and pedals), and visual environment. Fig. 10.21 shows the hardware architecture. 10.7.3

Simulator Data from the ADMIRE LPV Control Law

A test was performed to investigate the stability and performance of the system during moderate and large amplitude manoeuvering in the transonic regime. A variety of step and singlet type inputs were imposed on lateral and longitudinal controls and on pedals. The aim was to investigate the control power, the primary responses and any major cross-couplings. An open-loop control strategy was used and the simulation was performed ﬁxed-base, because the simulator motion drive laws have not yet been tuned for fast jet work. The cockpit inceptors connected to the ADMIRE model in this test were a conventional centre-stick for lateral and longitudinal control, conventional pedals, and a throttle lever situated to the pilot’s left. The sensitivities were adjusted to enable the control law to achieve the maximum required limits in terms of load factor, velocityvector roll rate and sideslip. Data were recorded at two ﬂight conditions (M=0.8 and M=1.1 at a pressure altitude of 5000m). Figs. 10.22 - 10.25 show approximately 50 seconds of data during which two pullups were performed using 50% and 100% longitudinal stick. Fig. 10.22 shows the tracking performance of the control law. The load factor (nz ) response is precise and lags the command by under 0.5sec. The roll rate response (p) caused by small lateral inputs is shown next. (Note that in this case, the command is for velocity vector roll rate.) The maximum sideslip excursion is 2 deg. True air speed ranged between 207 and 262 m/s, Mach number between 0.62 and 0.85. Figs. 10.23 - 10.24 show various other simulation outputs, including pitch and roll angles, rates and accelerations, angle of attack, Mach number and altitude. Fig. 10.25 shows control surface and throttle deﬂections. During these pull-ups, ﬂight path angle (γ) (see Fig. 10.23) peaks at 28 deg and later at 41 deg. The incidence lag Tθ2 (i.e. the lag of γ on θ) is in the

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Fig. 10.22. Pull up manoeuvres of 5g and 8g: (i) normal load factor (-) and commanded value (–), (ii) roll rate (-) and scaled lateral stick input (- -), (iii) side-slip angle (-) and scaled pedal input (- -), (iv) true air speed (M = 0.8 and H = 5km)

Fig. 10.23. Pull up manoeuvres (5g and 8g): (i) ﬂight path angle and pitch attitude, (ii) pitch rate, (iii) pitch acceleration, (iv) angle of attack (M = 0.8 and H = 5km)

region of 0.75 sec. The longitudinal stick input at t = 15.5 sec generates a peak pitch acceleration of 24 deg/s2 (0.42 rad/s2 ) and a steady-state normal load factor of approximately 4.8g. This gives a value for the control anticipation parameter (CAP) of

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Fig. 10.24. Pull up manoeuvres (5g and 8g): (i) roll angle, (ii) roll acceleration, (iii) Mach number, (iv) altitude (M = 0.8 and H = 5km)

Fig. 10.25. Pull up manoeuvres (5g and 8g): (i) right canard, (ii) right outboard elevator, (ii) left outboard elevator, (iv) rudder, (v) throttle (M = 0.8 and H = 5km)

0.12 rad/s2 /g. For the larger step input at t = 36 sec the peak pitch rate acceleration is 76 deg/s2 and a steady-state normal load factor of approximately 8.5g; the corresponding value of the CAP is 0.18 rad/s2 /g. The acceleration sensitivity nα is deﬁned as the steady-state normal load factor per unit steady-state angle of attack. For the two doublet inputs under consideration, nα = 34 g/rad. Plotting the results of this albeit fairly

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Fig. 10.26. Compliance with short period frequency requirements from MIL-F-8785C

Fig. 10.27. 360 deg roll manoeuvre: (i) normal load factor (-), commanded value (–), (ii) roll rate (-) and scaled lateral stick input (- -), (iii) side-slip angle (-) and scaled pedal input (- -), (iv) true air speed (M = 1.1 and H = 5 km)

basic analysis on the chart shown in Fig. 10.26 with boundaries from MIL-F-8785C superimposed suggests that the short period handling qualities may be adversely affected by the relatively low initial pitch acceleration. However the LPV control law provides a non-classical - indeed, a highly augmented - response-type, so application of classical criteria may be somewhat misleading. Figs. 10.27 and 10.28 show data corresponding

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Fig. 10.28. 360 deg roll manoeuvre: (i) roll angle, (ii) roll acceleration, (iii) Mach number, (iv) altitude (M = 1.1 and H = 5 km)

to a 360 deg roll manoeuvre during which the roll rate reached 300 deg/s. Sideslip angle is maintained within 1.5 deg.

10.8 Summary and Discussion A parameter varying control law for the ADMIRE model in a wide ﬂight envelope including subsonic, transonic and supersonic regimes has been designed using an LPV control design approach. The design of the LPV control law is based on the application of the parameter-dependent Lyapunov function method via gridding of the parameter space. The control law design has been applied to the ADMIRE model linearized at the level ﬂight conditions throughout the ﬂight envelope and presented as a LPV system using linear piece-wise interpolation. The single longitudinal LPV controller for the whole ﬂight envelope is automatically reconﬁgured at M = 0.58 − 0.62 from an α-demand system at low Mach numbers to a nz -demand system at high Mach numbers. The longitudinal and lateral-directional controllers are represented as linear dynamical systems (10.2) of 9th -order, whose matrices depend on ﬂight regime parameters. The designed LPV controllers satisfy most of the imposed design requirements for handling quality, disturbance rejection, robustness to parametric uncertainties, gain and phase margins and high frequency gain limitation. Off-line nonlinear simulation tests have demonstrated good performance properties of the LPV controllers for all recommended manoeuvres including tests with aerodynamic derivative uncertainties. Most of the design requirements speciﬁed in [115] are satisﬁed, except the wind gust response requirement at low Mach numbers. This design problem, in the authors’ opinion, can be

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solved by application of the D-K iterative procedure with appropriate scaling of different parts of the closed-loop transfer function on the basis of µ-analysis and introducing a frequency dependent gust weighting function. To exclude the possibility of aircraft departure during high angle of attack manoeuvring at low Mach numbers occuring due to saturation of control effectors, the LPV control laws require an additional functionality for angle of attack and velocity-vector roll rate limitation. This part of the control law design is also feasible using an LPV approach, but has not been addressed in this study. Following extensive evaluation tests including time and frequency domain criteria, linear and nonlinear off-line simulation, and real time piloted simulation on the HELIFLIGHT simulator at the University of Liverpool we can conclude that the designed LPV controllers provide acceptable handling quality, stability and robustness characteristics including ﬂight in the transonic region. The presented LPV control design method proves to be appropriate for fast conceptual ﬂight control law design and can be easily adopted in engineering practice.

Acknowledgements The ﬁrst two authors are grateful to Yoge Patel for cooperation in the area of LPV control law design during the research project with DERA/QinetiQ Ltd, Bedford and also to Peter Messer and Jeff Knight at De Montfort University for their support given to GARTEUR FM(AG17) activities. The authors from University of Liverpool are grateful to Mr Gary Ireland and Mr Steve Hodge for their work on the development of the HELIFLIGHT Simulink interface.

11 Block Backstepping for Nonlinear Flight Control Law Design John W.C. Robinson Department of Autonomous Systems, Swedish Defence Research Agency (FOI), 164 90 Stockholm, Sweden [email protected]

Summary. In this chapter we describe the block backstepping approach to ﬂight control law design. Block backstepping is a Lyapunov based technique for controller design which is particularly well suited to the rigid body control problem where the main means of control is through the moments, which is the case in most aircraft. The resulting controller has semi-global (in the state space) stabilising properties and has a moderate number of parameters that can be used for tuning. We illustrate the theory by simulations of the ADMIRE model with a block backstepping controller in demanding manoeuvres such as high-alpha ﬂight and high-rate velocity vector rolls. Keywords: Nonlinear control, ﬂight control systems, backstepping.

11.1 Introduction Theory and methods for design of nonlinear ﬂight control systems in ﬂy-by-wire aircraft have by now evolved to the point that they are a viable alternative also for the practising engineer in industry. For agile manoeuvrable aircraft, such as military ﬁghters, nonlinear methods offer the potential to operate the aircraft closer to its physical limits which in turn makes it possible to better exploit the performance capabilities of the airframe. In civilian applications nonlinear methods can be used to more precisely utilise the control resources, thus offering better overall design economy. Nevertheless, the most signiﬁcant driver for nonlinear ﬂight control system development has traditionally been requirements imposed in extreme conditions, such as high angle of attack operation or manoeuvres involving rapid transition through the transonic region. In these situations the standard gain scheduled linear design methods will often encounter difﬁculties and this has been the main reason to look for better tools. With today’s short turn-around times in aircraft system design, a new motive for using nonlinear design methods has emerged. The standard linear design methods in general require a signiﬁcant amount of manual work in each iteration of the aircraft design cycle. This is a consequence of the fact that methods based on linearisation normally require that a controller is designed for a large number of different points in the ﬂight envelope. These controllers are then “glued” together using an appropriate scheduling strategy to form the overall controller. A nonlinear design does not in general require this since it relies on a nonlinear description of the aircraft dynamics which is (more or less) globally valid. Moreover, the ability to quickly obtain a controller design D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 231–257, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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that covers most of the ﬂight envelope is often more important than having a design that maximises performance, at least initially. Optimisation of the ﬂight control system can then be pushed down in the work ﬂow to later stages when the overall design is more ﬁnalised. In the present chapter we will describe and apply a method for ﬂight control law design that has its roots in classical Lyapunov design techniques but exploits the special structure of certain systems to yield a recursive design technique known as backstepping [125, 118]. Backstepping relies on the system to be controlled to be in strict feedback form which, under some simplifying assumptions, holds for the standard rigid body (Newton-Euler) equations when interpreted in vector form. The result of applying block backstepping to these vector equations is a controller with semi-global stabilising properties and a moderate number of parameters to be used for tuning. Block backstepping is related to nonlinear dynamic inversion via time-scale separation (NDI-TSS) but includes additional terms in the control law which makes the time-scale separation assumption unnecessary. Backstepping has been used in a number of variants in ﬂight control applications such as ﬂight path angle control [94], adaptive longitudinal control [61], neuro-adaptive robust 6-DOF control [131], and intelligent UAV control [236]. Only a few studies, however, have treated the block backstepping case for full three-axis (6-DOF) control, [81, 196]. Apparently, the ﬁrst application of block backstepping in aircraft control was the design of an autopilot for a missile operated in bank-to-turn mode [218]. 11.1.1

Outline

In the following sections we shall ﬁrst review some basic facts from ﬂight mechanics and establish the notation to be used henceforth. After this we describe the block backstepping approach to controller design and point out certain key properties of the solution. In particular, we shall show how it is related to the standard nonlinear dynamic inversion controllers often used in ﬂight control law design. The controller performance will then be illustrated on the ADMIRE model in a set of manoeuvres that highlight its properties under varying conditions. Finally, we sum up the results and make some concluding remarks. 11.1.2

Notation

Vectors are indicated with bold type as v and are considered as column vectors with components indicated using ordinary type. Transposition is marked with superscript T , so that for instance v = [u, v, w]T . The Euclidean norm of a vector v is denoted by v and the A-weighted norm vA of v is given by v2A = vT Av, where A is a symmetric positive deﬁnite square matrix of compatible dimensions.

11.2 Dynamics The dynamical relations needed for the controller design shall now be derived. We begin with the classical rigid body mechanics relations and develop these in a few steps to arrive at a form of the dynamics suitable for the formulation of the control problem.

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Rigid Body Mechanics

Our starting point will be the Newton-Euler equations for rigid body motion expressed in a Cartesian coordinate system B ﬁxed in the aircraft body [220] (2003 printing), [28]. To make things simple we shall assume that the system B has the standard vehicle orientation and that the mass distribution of the aircraft is constant. The Newton-Euler equations in B are 1 fb + v × ω, m ˙ = J−1 (mb − ω × Jω), ω v˙ =

(11.1) (11.2)

where v = [u, v, w]T is the velocity, ω = [p, q, r]T is the angular velocity, fb = [ fx , fy , fz ]T represents the sum of all external forces acting on the body, m is the total mass, J ∈ R3×3 is the inertia matrix (which in our applications is always invertible) and mb ∈ R3 is the sum of all external moments. In order to temporarily suppress the physical signiﬁcance of the quantities above and instead emphasise the control theoretic aspects of the problem we shall rewrite (11.1), (11.2) in a variant of nonlinear controlled afﬁne form. This shall be done in two steps where we ﬁrst transform the linear velocity dynamics to aerodynamic coordinates and then introduce some simplifying assumptions about the forces fb and moments mb . 11.2.2

Aerodynamic Coordinates

For most aircraft the changes in normal (i.e. cross track) acceleration can be much larger than the time derivative of the airspeed and it is therefore natural to try to describe the dynamics in a way that reﬂects this. We shall therefore make the standard change of coordinates in the force equation (11.1) to spherical coordinates deﬁned by α = arctan(w/u),

(11.3a)

β = arcsin(v/V ), V = u2 + v2 + w2 ,

(11.3b)

u = V cos(α) cos(β),

(11.4a)

v = V sin(β), w = V sin(α) cos(β),

(11.4b) (11.4c)

(11.3c)

with inverse

where V is the airspeed, α is the angle of attack and β is the sideslip angle. For simplicity we shall assume that V > 0 and α ∈ [−π, π), β ∈ [−π/2, π/2] at all times. By time differentiation of (11.3a), (11.3b) and using (11.1) together with (11.4a)–(11.4c) we obtain the force equation relations in the form ⎡ ⎤ p ˙α f − cos(α) tan(β) 1 − sin(α) tan(β) ⎣ ⎦ q , = α + (11.5) fβ sin(α) 0 − cos(α) β˙ r

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where fz cos(α) − fx sin(α) , mV cos(β) 1 fβ = fy cos(β) − fx cos(α) sin(β) − fz sin(α) sin(β) . mV

fα =

(11.6a) (11.6b)

The relation for V˙ has been left out since we shall henceforth assume that V˙ is small and that V therefore can be considered as a scheduling variable. The body force components fx , fy , fz are made up of aerodynamic forces, thrust and gravity, and the former can be expressed [220] in terms of standard aerodynamic coefﬁcients CT ,CC ,CN , dynamic pressure qa and reference area Sref . In the mathematical discussions below we shall assume that the functions fx , fy , fz are deﬁned over the same domain as our state space description (to be introduced) even though in reality they are only deﬁned over a subset of it, corresponding to realistic ﬂying conditions. We shall also assume that they are smooth and bounded functions of their arguments. How the translation of the theory to real world application is achieved is discussed in the sequel. 11.2.3

Forces, Moments and Auxiliary Variables

To apply (block) backstepping the aircraft rigid body control problem must be cast in a strict feedback form. In order to do so, we assume that the aircraft is essentially moment controlled, i.e. that the control effectors (control surfaces, thrust vectoring) mainly produce moments and that their contribution to the forces fα and fβ in (11.6) can be neglected. We shall also neglect the dependence on p, q, r and regard fα and fβ as functions of the two aerodynamic angles α and β only. All dependency on other variables, such as airspeed V , gravity, dynamical pressure, and engine induced ﬂow effects etc. can be modelled parametrically in terms of (measurable) parameters. Our state feedback control solution moreover assumes that all the ﬁve state variables α, β, p, q, r are directly accessible for measurement. 11.2.4

Velocity Vector Roll

The ﬁve equations obtained from (11.5) and (11.2) completely describe the motion of the aircraft in body coordinates if V is constant and fb and mb are speciﬁed. For V˙ = 0, the force vector fb in (11.1) is perpendicular to the vector v and represents the normal acceleration, which then by assumption is essentially determined by α, β. As a rough approximation, valid for most aircraft conﬁgurations of interest here, the size of the normal acceleration is determined mostly by controlling α with β ≈ 0 and the orientation of the normal acceleration in an Earth ﬁxed coordinate system E is controlled by rotating the aircraft around the velocity vector (bank-to-turn operation). For this reason we introduce the conical rotation rate Ω as Ω=

vT ω V

(11.7)

which is just the magnitude (with sign) of the component of ω along the velocity vector v, and accordingly the conical rotation angle ξ deﬁned cyclically through

Block Backstepping for Nonlinear Flight Control Law Design

ξ˙ = Ω

235

(11.8)

where ξ ∈ [−π, π). The relation (11.8) can, using the aerodynamic angles α, β and the deﬁnition (11.7), be expressed as ξ˙ = p cos(α) cos(β) + q sin(β) + r sin(α) cos(β)

(11.9)

and this is the relation for the rolling motion that we are going to use. To solve for the position and orientation of the aircraft relative to E, a set of six kinematic relations have to be added to the equations (11.1) and (11.2). The conical rotation angle captures the most important degree of freedom of these kinematics with regard to the formulation of the control problem. 11.2.5

Control Afﬁne Form

The dynamical relations developed in the previous sections can now be collected and put in the standard control afﬁne form used in nonlinear control [30]. We shall have reason to consider generic controlled systems of the form x˙ = f(x) + g(x)y, y˙ = h(y) + ku,

(11.10) (11.11)

where x ∈ Rn , y ∈ Rm , u ∈ Rm , and f, h are two smooth vector ﬁelds on Rn and Rm , respectively, g is a smooth matrix valued function on Rn with values in Rn×m and k ∈ Rm×m is a constant full rank matrix. In our ﬂight control application we have m = n = 3 and the two (partial) state vectors x and y are given by ⎡ ⎤ ⎡ ⎤ p α (11.12) x = ⎣ β ⎦, y = ⎣ q ⎦, r ξ and from (11.5), (11.9), (11.2) and the assumptions in Sec. 11.2.3 we have ⎡ ⎡ ⎤ ⎤ − cos(α) tan(β) 1 − sin(α) tan(β) fα sin(α) 0 − cos(α) ⎦ , f(x) = ⎣ fβ ⎦ , g(x) = ⎣ cos(α) cos(β) sin(β) sin(α) cos(β) 0 and

h(y) = −J−1 (y × Jy),

k = J−1 .

(11.13)

(11.14)

We take the vector of body moments to be our primary control variable, i.e. we set u = mb , and assume that the translation to actual control effector settings is solved as a separate problem. (In aircraft control this is frequently the case, and the translations from moment commands to e.g. control surface settings is made as part of the control allocation procedure.) We shall refer to the matrix g(x) in (11.10) as the virtual control gain matrix. (The name will be clariﬁed in connection with the development of the backstepping controller.) The system (11.10)–(11.14) is now in the strict feedback form required for (block) backstepping [125].

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11.2.6

Virtual Control Gain Matrix

For future developments it will be useful to make some observations about the virtual control gain matrix g(x) in the form (11.13) that it appears in the ﬂight control application. We begin by noting that the ﬁrst two rows of g(x) are always orthogonal and span a subspace of R3 which is orthogonal to the one-dimensional subspace spanned by the third row. Since the subspace spanned by the third row of g(x) is, in view of (11.4a)– (11.4c), identical to the subspace [v] spanned by the velocity vector v, it follows that the ﬁrst two rows of g(x) span [v]⊥ . With these observations it is easy to write g(x) factored as a singular value decomposition g(x) = UΣVT , where

⎡ U = I,

Σ=⎣

1 cos(β)

0 0

(11.15) 0 1 0

⎤ 0 0⎦ 1

and the orthogonal matrix V is given by ⎡ ⎤ − cos(α) sin(β) sin(α) cos(α) cos(β) ⎦. cos(β) 0 sin(β) V=⎣ − sin(α) sin(β) − cos(α) sin(α) cos(β)

(11.16)

(11.17)

In particular, this shows that g(x) in (11.13) is always invertible and gives a simple formula for the inverse as g(x)−1 = VΣ−1 . (11.18) It also shows that the problem of controlling α, β has a natural decoupling from the problem of controlling ξ since the two problems take place in [v]⊥ and [v], respectively. 11.2.7

Tracking Problem

When x, y are given by (11.12) it follows from the third row of (11.10) that an equilibrium point for (11.10), (11.11) must necessarily correspond to zero conical rotation rate, i.e. Ω = 0. A set point control formulation based on (11.10)–(11.14) is thus clearly inadequate for ﬂight controller design and we must therefore extend the dynamical description to a tracking formulation. In connection to the generic system (11.10), (11.11) it is thus convenient to introduce a generic vector xc ∈ Rn with smooth time varying reference signals as components and deﬁne the error vector x˜ as (11.19) x˜ = x − xc . The dynamics (11.10), (11.11) can now be extended to include time varying reference signals and be written in the form x˙˜ = ˜f(˜x,t) + g˜ (˜x,t)y y˙ = h(y) + ku,

(11.20) (11.21)

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237

where the time varying vector ﬁeld ˜f and virtual control gain matrix function g˜ are given by ˜f(˜x,t) = f(˜x + xc ) − x˙ c , g˜ (˜x,t) = g(˜x + xc ). (11.22) In the special case of our ﬂight control application the vector xc of reference signals is given by ⎡ ⎤ αc (11.23) xc = ⎣ βc ⎦ . ξc With (11.20), (11.21) the state tracking problem for the error vector x˜ based on the aircraft variables (11.23) has been reformulated in the guise of a time varying set point control problem. The system (11.20), (11.21) based on the aircraft dynamics (11.12)– (11.14) has a rich set of equilibrium points useful for ﬂight controller design. For instance, when αc , βc and ξ˙ c = Ωc are all constant an equilibrium point for (11.20), (11.21) corresponds to a velocity vector roll with constant conical rotation rate. In this case, if [α0 , β0 , Ω0 , p0 , q0 , r0 ]T is an equilibrium point then we see from (11.10) (since g˜ (˜x,t) has full rank) that the triple α0 , β0 , Ω0 uniquely determines the triple p0 , q0 , r0 .

11.3 Controller The controller structure will be motivated by a simple intuitive argument and the main result stated (the proof is given in the appendix). A number of properties of the controller will be discussed and, in particular, its relation to NDI-TSS will be clariﬁed. 11.3.1

Underlying Idea

The basic idea of block backstepping 1 is very simple and can most easily be described if we consider the generic set point control problem for the (autonomous) system (11.10), (11.11) but with h(y) = 0 and k = I (integrator backstepping). When the problem is to stabilise an existing equilibrium at 0 ∈ Rn+m for (11.10), (11.11) the approach can be broken down to a two step, or cascade, procedure where the ﬁrst step amounts to trying to ﬁnd a behaviour for y which would stabilise the x-system according to some desired smooth dynamics (11.24) x˙ = fdes (x), where fdes (0) = 0. The second step is to ﬁnd a control law u(x, y) which synthesises this behaviour for the x-system while at the same time maintaining some form of stability for the y-system (the input integrator). The key problem in backstepping is to construct what is known as a “virtual” control law yd (x) for y which, when substituted for y in (11.10), synthesises the desired dynamics (11.24). This is done in the ﬁrst step described above. Written out in terms of yd (x) this means that the dynamics for the x-system in (11.10) become 1

The name backstepping can be explained by the way the operations involving yd (x) appear when written out using block diagrams, see [117, Fig. 14.15].

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x˙ = f(x) + g(x)y = f(x) + g(x)yd (x) + g(x)(y − yd (x)) = fdes (x) + g(x)ε(x), where the error ε(x) is deﬁned as ε(x) = y − yd (x) and the virtual control law yd (x) satisﬁes (11.25) fdes (x) = f(x) + g(x)yd (x). Under suitable conditions on fdes (x) and yd (x) there will exist a feedback law u(x, y) to replace u in (11.11) which will achieve the objectives of keeping (x, y) small and maintaining stability of y so that x is driven to 0. Thus, the main design problem in backstepping lies in constructing the virtual control law yd (x) for the x-system; a control law for the whole system u(x, y) can then be obtained by a “standard recipe”. However, in order to carry out this program successfully some tools from Lyapunov stability theory are needed. 11.3.2

Basic Stability Result

We shall now present the fundamental stability result of block backstepping that we will use in the controller design. It mechanises the idea outlined in the previous section using Lyapunov techniques and the central part is the Lyapunov function for the desired dynamics to be introduced next. Lyapunov Function Returning to the case of the generic time varying system (11.20), (11.21) we assume that a desired (autonomous) dynamical behaviour as in (11.24) has been chosen for the variable x˜ . We moreover assume that there exists a corresponding continuously differentiable function V : Rn → [0, ∞) which is positive deﬁnite (V (˜x) > 0 for x˜ = 0), vanishes at 0, is radially unbounded (V (˜x) → ∞ as ˜x → ∞) and for which it holds that d V (˜x) = ∇V (˜x)T fdes (˜x) ≤ −W (˜x) < 0 (11.26) dt along the solution trajectories to (11.24) in Rn \ {0}, for some continuous positive definite function W deﬁned on Rn with W (0) = 0. The conditions imposed on V are standard requirements for a Lyapunov function for the system in (11.24). Existence of a Lyapunov function V is sufﬁcient to guarantee global asymptotic stability of the solutions to (11.24) throughout Rn according to standard stability results [117, Ch. 4]. The time invariant Lyapunov function V will be used to construct a stabilising control for the time varying system (11.20), (11.21). Integrator Block Backstepping The basic stability result we shall rely on is the following. It is formulated here in a slightly generalised form compared to the standard formulation [117, Sec. 14.3] where we extend the results to the time varying case in (11.20), (11.21).

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239

Theorem 1. Consider the generic system in (11.20), (11.21) for the special case h(y) = 0 and k = I. Assume that y˜ d (˜x,t) is a smooth virtual control law such that ˜f(˜x,t) + g˜ (˜x,t)˜yd (˜x,t) = fdes (˜x),

(11.27)

for all x˜ ,t, where fdes is the vector ﬁeld in (11.24) with Lyapunov function V as in (11.26). Then, the following control law u(˜x, y,t) =

∂˜yd (˜x,t) ˜ ∂˜y (˜x,t) (f(˜x,t) + g˜ (˜x,t)y) + d ∂˜x ∂t − g˜ (˜x,t)T ∇V (˜x) − Wy (y − y˜ d (˜x,t)),

(11.28)

where Wy is a positive deﬁnite symmetric matrix, drives the tracking errors x˜ and y − y˜ d (˜x,t) for the closed loop system to zero asymptotically as t → ∞. The proof is given in the appendix. In our ﬂight control application g˜ (˜x,t) is invertible and therefore a unique virtual control law as in (11.27) always exists. In this case it is also easy to give sufﬁcient conditions under which both y and y˜ d (˜x,t) remain bounded. For instance, if αc , βc and ξ˙ c = Ωc are all constant (velocity vector roll) then both ˜f(˜x,t) and g˜ (˜x,t) are time invariant, and therefore so is y˜ d (˜x,t), and it follows that y converges (and is hence bounded). 11.3.3

Block Backstepping Control Law

For brevity, the basic stability result Thm. 1 was formulated and proved for the case h(y) = 0 and k = I in (11.21). However, the general case is obtained by a simple change of variables in (11.21) as ˜ x, y,t) = k−1 u(˜

∂˜y (˜x,t) ∂˜y (˜x,t) d (˜f(˜x,t) + g˜ (˜x,t)y) + d ∂˜x ∂t

− g˜ (˜x,t)T ∇V (˜x) − Wy (y − y˜ d (˜x,t)) − h(y) . (11.29)

The stabilising properties of the control law (11.29) are guaranteed only when the generic model (11.20), (11.21) is valid globally. In our ﬂight control setting (11.12)– (11.14) this is not the case, and in practice the domain over which the dynamical model is deﬁned is much smaller. Theoretically this is not a problem since the global assertion of the theorem can by a standard device be replaced by a local assertion, by bounding the starting points to lie within a level surface of the Lyapunov function in question [117, Ch. 4]. In practice, however, it means that one will have to do an elaborate evaluation of the stability region based on bounds on state variables, available control authority, rate and position limitations of effectors, and the requirements for tracking performance. Based on this, a command generator can be designed which translates the pilot’s input to realizable reference signals, thus effectively resulting in a so called model following controller design. Some observations about how this can be done are presented below.

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11.3.4

Properties of the Control Law

Virtual Control Law In our ﬂight control application (11.12)–(11.14) the virtual control law y˜ d (˜x,t) in (11.27) can be expressed explicitly in terms of g˜ (˜x,t)−1 as y˜ d (˜x,t) = g˜ (˜x,t)−1 (fdes (˜x) − ˜f(˜x,t)).

(11.30)

and the inverse g˜ (˜x,t)−1 is given by (11.18). Further, since the only explicit time dependence in y˜ d (˜x,t) is through the time varying reference signal xc , the virtual control law can be written in terms of the original variables x, y used in (11.10), (11.11) by deﬁning yd (x, xc , x˙ c ) = y˜ d (˜x,t). This representation is useful also to make explicit the dependence on xc and its derivatives in the control law (11.29). Indeed, for the closed loop system obtained by applying u(˜x, y,t) in (11.29) to (11.20), (11.21) we have ∂˜yd (˜x,t) ˜ ∂˜y (˜x,t) (f(˜x,t) + g˜ (˜x,t)y) + d ∂˜x ∂t d d = y˜ d (˜x,t) = yd (x, xc , x˙ c ) dt dt ∂yd (x, xc , x˙ c ) ∂yd (x, xc , x˙ c ) ∂yd (x, xc , x˙ c ) (f(x) + g(x)y) + = x˙ c + x¨ c . (11.31) ∂x ∂xc ∂˙xc Moreover, if we rewrite (11.30) in terms of the original variables x, y used in (11.10), (11.11) we have yd (x, xc , x˙ c ) = g(x)−1 (fdes (x − xc ) − f(x) + x˙ c ),

(11.32)

and it follows that ∂yd (x, xc , x˙ c ) ∂fdes (x − xc ) = g(x)−1 , ∂xc ∂xc ∂yd (x, xc , x˙ c ) = g(x)−1 . ∂˙xc

(11.33) (11.34)

The representations (11.32)–(11.34) are convenient when implementing the block backstepping control law (11.29). Closed Loop System and Error Dynamics ˜ x, y,t) in (11.29) can The closed loop error dynamics resulting with the control law u(˜ straightforwardly be written down in terms of the original variables x, y using (11.10), (11.11) and (11.27) as d (x − xc ) = fdes (x − xc ) + g(x)(y − yd (x, xc , x˙ c )), dt d (y − yd (x, xc , x˙ c )) = −g(x)T ∇V (x − xc ) dt −Wy (y − yd (x, xc , x˙ c )),

(11.35)

(11.36)

Block Backstepping for Nonlinear Flight Control Law Design

241

from which the closed loop dynamics immediately follow. The special cross coupling in these equations, in particular the effect of the Lyapunov gradient term −g(x)T ∇V (x − xc ) in (11.36), is not present in the error dynamics for the NDI-TSS control law (see below) and this explains their different stability properties.2 Tuning Parameters and Desired Dynamics There are a number of parameters in the control law (11.29) which can be used for tuning. These parameters are implicit in the desired dynamics fdes , the accompanying Lyapunov function V and the matrix Wy . The relative scaling between V and Wy determine their relative inﬂuence in (11.36). As for the desired dynamics term fdes in (11.26), (11.27), a limitation of the backstepping procedure is that even though it can be chosen freely, it has to be ﬁrst order. For this reason the desired dynamics are often chosen to have a decoupled ﬁrst order linear form, given as fdes (˜x) = −D1 x˜ ,

(11.37)

where D1 ∈ Rn×n is a diagonal positive deﬁnite matrix. For this choice, a quadratic Lyapunov function V such as 1 V (˜x) = ˜x2D2 , (11.38) 2 is natural, where D2 ∈ Rn×n is another diagonal positive deﬁnite matrix. The difference in time scales between the two equations (11.35), (11.36) is to a large extent determined by how the desired dynamics fdes are chosen relative to V and Wy . In block backstepping there is no requirement for a certain time-scale separation between the inner and outer loop, like in NDI-TSS, but in practice it is useful to ensure a certain separation to avoid too “sluggish” a behaviour and balance this against available control effector resources (and their limitations). Trim Points The equilibrium points of the dynamics (11.20), (11.21) do not appear explicitly in the block backstepping tracking control formulation employed here. However, they appear implicitly in the sense that they determine a particularly interesting set of values for reference signals xc . In the ﬂight control case, where the interesting equilibrium points are those that represent trim points, the trim points correspond to reference signals xc which are such that αc , βc and ξ˙ c = Ωc are constant (velocity vector roll). 11.3.5

Generalisations

Including Actuator Dynamics Including linear and nonlinear actuator dynamics in the design is conceptually straightforward since the closed loop error system (11.35), (11.36) resulting when applying 2

The presence of the Lyapunov gradient term on the right in (11.36) has interpretations in terms of control Lyapunov function (CLF) based design. It adds a correction to the inﬁnitesimal increment of y − yd (x, xc , x˙ c ) in (11.36) which helps drive the error x − xc in (11.35) towards the stabilising set [42].

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˜ x, y,t) in (11.29) to (11.20), (11.21) can be interpreted as the (comthe control law u(˜ pleted) ﬁrst step in an augmented block backstepping design. In the case of ﬁrst order actuator dynamics the whole procedure becomes especially simple since it can then be ˜ x, y,t) in (11.29) is designed. performed with only one extra backstepping step once u(˜ To see this, assume for a moment that it is not u that is to be controlled but rather its derivative, so that we may augment the system (11.20), (11.21) with the equation u˙ = ν.

(11.39)

If we further introduce the notation y˜ = y − y˜ d (˜x,t), ˜h(˜x, y˜ ,t) = h(˜y + y˜ d (˜x,t)), we can write the open loop error dynamics obtained from (11.20), (11.21) in time varying form as d x˜ = dt y˜ fdes (˜x) + g˜ (˜x,t)˜y 0 ˜ x, y˜ ,t) − ∂˜yd (˜x,t) ˜f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)) − ∂˜yd (˜x,t) + k u, (11.40) h(˜ ∂˜x ∂t where we have used (11.27) and (11.31). ˜ x, y,t) in (11.29) can be interpreted as a virtual control law Now, the control law u(˜ u˜ d (˜x, y˜ ,t) for the system in (11.40), with Lyapunov function for the desired dynamics ˜ x, y,t) applied) given by the function Va in (the dynamics obtained in (11.40) with u(˜ (11.52) in the proof of Thm. 1. The integrator block backstepping result in Thm. 1 can therefore be applied once more to yield a closed loop control law ν(˜x, y˜ , u,t) which will simultaneously globally stabilise the origin in (˜x, y˜ ) and the error variable u− u˜ d (˜x, y˜ ,t). The result so obtained can immediately, by a change of variables, be generalised to the case where (11.39) is replaced by actuator dynamics as u˙ = p(u) + q(u)ν,

(11.41)

Rm

where p is a smooth vector ﬁeld on and q is a smooth matrix valued function on Rm m×m with values in R such that q(u) is invertible everywhere. In the original variables x, y a closed loop control law ν˜ (x − xc , y − yd (x, xc , x˙ c ), u,t) which will simultaneously stabilise the origin in (˜x, y˜ ) and the error variable u − u˜ d (x − xc , y − yd (x, xc , x˙ c ),t) can thus, following (11.29), be deﬁned through p(u) + q(u)ν˜ (x − xc , y − yd (x, xc , x˙ c ), u,t) = − [0, kT ]∇Va (x − xc , y − yd (x, xc , x˙ c )) d + u˜ d (x − xc , y − yd (x, xc , x˙ c ),t) dt − Wu (u − u˜ d (x − xc , y − yd (x, xc , x˙ c ),t)),

(11.42)

where we have used an identity analogous to the one provided in the leftmost equality of (11.31). Here, Wu is a positive deﬁnite symmetric matrix in Rm×m . For higher order dynamics, the above procedure can simply be repeated.

Block Backstepping for Nonlinear Flight Control Law Design

243

Handling Actuator Limitations As mentioned above, any practical application of the block backstepping control law ˜ x, y,t) in (11.29) requires that one somehow deals with the restrictions imposed by u(˜ the control actuators. In ﬂight control applications the most common restrictions are position and rate limits. A standard way of doing this is to ﬁlter the pilot inputs commands through a command generator, which then issues the actual commands to the controller. Command generator design is outside the scope of this chapter but we shall offer a few remarks related to it. ˜ x, y,t) are restricted to the In order to practically apply the control law the values of u(˜ ˜ x, y,t) set Ux,y of allowed controls at the state x, y. Simply “truncating” the values of u(˜ to Ux,y can lead to instability problems and is therfore often not a viable alternative. For ﬁxed values of the state variables x, y implicit in the model (11.20), (11.21), the ˜ x, y,t) in (11.29) can, in view of (11.31), be interblock backstepping control law u(˜ preted as a map cx,y : R3n → Rm deﬁned by ˜ x, y,t). cx,y : (xc , x˙ c , x¨ c ) → u(˜

(11.43)

If cx,y is injective, it becomes bijective when restricted to the pre-image c−1 x,y (Ux,y ) (i.e. values (xc , x˙ c , x¨ c ) such that cx,y (xc , x˙ c , x¨ c ) ∈ Ux,y ). A simple command generator could then simply operate by truncating (xc , x˙ c , x¨ c ) to c−1 x,y (Ux,y ), which would mean that the commands actually issued to the control actuators will always be admissible with respect to position. However, the map (11.43) is clearly not injective in general without restricting the domain, and determining restricted domains over which it is injective can be complicated. One situation in the ﬂight control application when an explicit sufﬁcient condition for the requirement on cx,y to be injective over c−1 x,y (Ux,y ) can be given is the special case where xc is constant, the desired dynamics fdes are of the linear form (11.37) and the Lyapunov function V is of the quadratic form (11.38). In this case the virtual control expressed in the original variables x, y as in (11.32) takes the form yd (x, xc , x˙ c ) = g(x)−1 − D1 (x − xc ) − f(x) and therefore ∂ g(x)−1 − D1 (x − xc ) − f(x) (f(x) + g(x)y) ∂x − g(x)T D2 (x − xc ) − Wy y − g(x)−1 − D1 (x − xc ) − f(x) − h(y) .

˜ − xc , y,t) = k−1 u(x

˜ − xc , y,t) can be written as an afﬁne function of the It follows that the control law u(x vector of xc and the injectivity requirement on cx,y can therefore be determined by a rank condition on a matrix. Other aspects of the problem of incorporating actuator limitations in a backstepping design are discussed in [60]. Robustness Since the backstepping technique is a Lyapunov based design technique there are standard ways to incorporate nonlinear robustness (in terms of stabilisability) into the design

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by modifying the nominal Lyapunov function or control so that they together cover the whole family of systems obtained by perturbing the nominal system with various types of uncertainties [117, 192]. In particular, so called matched uncertainties are relatively easy to handle in a standard fashion [117, 192]. Matched uncertainties are those that can be expressed as a perturbation in the range of the control gain matrix. In the ﬂight control control application the matched uncertainties correspond to moment uncertainties. 3 11.3.6

Relation to NDI-TSS

Nonlinear dynamic inversion via time-scale separation is related to block backstepping but employs a simpler control law. This means that NDI-TSS must rely on a time-scale separation assumption 4 to guarantee stability [206, 205], a requirement which is not necessary for backstepping. Underlying Idea of NDI-TSS The idea behind nonlinear NDI-TSS is similar to that of block backstepping and starts from the same basic generic system [206]. We shall consider the tracking formulation here and the generic system is then given by (11.20), (11.21) but we shall write it in terms of the original variables x, y. Slow subsystem. The ﬁrst step in NDI-TSS is conceptually identical to that in backstepping and consists in determining a virtual control law yd (x, xc , x˙ c ) such that f(x) + g(x)yd (x, xc , x˙ c ) − x˙ c = fdes (x − xc )

(11.44)

where fdes represents the desired dynamics for the error x − xc . Since no Lyapunov function appears explicitly in the NDI-TSS approach the ﬁrst step is simpler to perform however. This is related to the fact that in NDI-TSS the virtual control gain matrix g(x) is assumed everywhere invertible (as in our ﬂight control application (11.10)– (11.14)), so that in principle any dynamics can be synthesised. Because of the assumed invertibility of the virtual control gain matrix the virtual control law in NDI-TSS is most often explicitly deﬁned as yd (x, xc , x˙ c ) = g(x)−1 (fdes (x − xc ) − f(x) + x˙ c ). The desired dynamics fdes are normally selected to be of the linear decoupled ﬁrst order type in (11.37). With the relation (11.44) for the virtual control law and desired dynamics the error dynamics for the x-system can be written as 3

4

A certain robustness to force uncertainties is displayed in the simulated examples below since all control surface force contributions are neglected in the design, and therfore enter as uncertainties. The stability proof [206, 205] assumes also that the reference signals are constant and that the fast system dynamics inversion is perfect and instantaneous. The latter means, in practice, that any actuator dynamics must be very fast relative to the fast subsystem dynamics.

Block Backstepping for Nonlinear Flight Control Law Design

d (x − xc ) = f(x) + g(x)y − x˙ c dt = fdes (x − xc ) + g(x)ε(x, xc , x˙ c ),

245

(11.45)

where the error ε(x, xc , x˙ c ) is given by ε(x, xc , x˙ c ) = y − yd (x, xc , x˙ c ).

(11.46)

Fast subsystem. In the second step in NDI-TSS one proceeds analogously as in the ﬁrst step to derive the closed loop control law uDI (x, xc , x˙ c , y) from a relation describing the desired behaviour as y˙ = h(y) + kuDI (x, xc , x˙ c , y) (f)

= fdes (y − yd (x, xc , x˙ c )),

(11.47)

(f)

where fdes represents some desired dynamics for the fast subsystem. It is important to note that in the NDI-TSS approach one synthesises desired dynamics for the variable y rather than for the error y − yd (x, xc , x˙ c ), which is what is effectively done in the backstepping approach (cf. (11.35), (11.36)). Thus, one does not fully handle the effect of the time variability of the term yd (x, xc , x˙ c ). Indeed, in the stability proof [206, 205] of NDI-TSS it is explicitly assumed that “perfect fast subsystem inversion” holds, which is the same as assuming that the error time derivative d ˙ c )) is zero. dt (y − yd (x, xc , x NDI-TSS Control Law The dynamics for the NDI-TSS error variables x− xc and y − yd (x, xc , x˙ c ) can readily be written down using (11.45), (11.47) and it is easy to see that the type of cross coupling that is present in the error dynamics (11.35), (11.36) for the block backstepping control law is not present in the NDI-TSS case. The difference can also be clearly seen if the NDI-TSS control law is written out explicitly. From (11.47) we obtain the NDI-TSS control law uDI (x, xc , x˙ c , y) explicity as (f) uDI (x, xc , x˙ c , y) = k−1 fdes (y − yd (x, xc , x˙ c )) − h(y) . (f)

Most often the desired dynamics fdes are chosen as linear decoupled ﬁrst order dynamics of the type in (11.37). In this case the NDI-TSS control law can be written (f) (11.48) uDI (x, xc , x˙ c , y) = k−1 − Wy (y − yd (x, xc , x˙ c )) − h(y) , (f)

where Wy ∈ Rm×m is a diagonal matrix with positive elements along the diagonal. It is now easy to compare the NDI-TSS control law uDI (x, xc , x˙ c , y) in (11.48) with the ˜ x, y,t) in (11.29) (after the latter has been rewritten block backstepping control law u(˜ in terms of the original variables x, y). Apparently there are two “missing terms” in the NDI-TSS control law: The time derivative dtd yd (x, xc , x˙ c ) commented on above and the Lyapunov gradient term −g(x)T ∇V (x − xc ) which was discussed in connection with the error system dynamics (11.35), (11.36).

246

J.W.C. Robinson

11.4 Application: ADMIRE In this section, we shall illustrate the performance of the block backstepping controller with some examples using the ADMIRE model. The examples are meant to mimic some of the conditions used in the design challenge described in Chap. 4 including robustness tests. However, all examples are cast directly in terms of tracking problems for the state variables α, β and ξ since pilot command converters and manoeuvre load limiter functions are not included in the implementation of the block backstepping controller used here. All manoeuvres start form an approximately trimmed straight level ﬂight condition. Unless otherwise stated, all conﬁguration parameters for the model have their nominal values at all times, with the exception that zcg = 0. For deﬁnitions and notation used in the ADMIRE model below we refer to Chapter 3. The throttle is kept constant at the position corresponding to straight level trim in all manoeuvres, except the decelerating turn where it is set to idle at the beginning of the manoeuvre. 11.4.1

Controller Implementation

˜ x, y,t) in (11.29) for the aircraft states in The controller implements the control law u(˜ (11.12) as well as an extra backstepping step for compensation for actuator dynamics based on (11.41). The total control law is thus of the form (11.42). A linear model5 for the actuator states is used in the controller design, with the same parameters as in the linear actuator dynamics in ADMIRE. The state variables α, β, p, q, r and the ﬂight state dependent parameters of the model (Mach number, Euler angles, etc.) are obtained from the ADMIRE sensor model outputs. A single set of gain constants are used for all manoeuvres and ﬂight points in the envelope, thus no gain scheduling is employed, and the parameters are as follows. The synthesised dynamics for the x-system as in (11.10), (11.13) are of the linear ﬁrst order form in (11.37) with the matrix D1 given by D1 = diag(5.5, 3, 2), and the Lyapunov function is of the quadratic form (11.38) with the matrix D2 given by D2 = diag(25, 20, 15). The matrix Wy in (11.29) is chosen as Wy = diag(20, 55, 30). For the part of the control law that compensates for the actuator dynamics a scalar weighting of 70 has been introduced on the second term in the Lyapunov function Va in (11.52) and the matrix Wu in (11.42) is selected as Wu = diag(30, 30, 30). 5

The actuators in ADMIRE have position and rate limiters on the input followed by linear ﬁrst order dynamics, but these nonlinearities are not accounted for in the controller designed here.

Block Backstepping for Nonlinear Flight Control Law Design

247

Control allocation is ﬁxed and is done according to δna δne δei δey δai δay δr

= 0, = δeg , = 1.1δeg , = 0.9δeg , = 0.9δag , = 1.1δag , = δrg ,

(11.49)

where δag , δeg , δrg , are generic aileron, elevator and rudder deﬂections and δna , δne , δei , δey , δai , δay and δr are the symmetric/asymmetric versions of the physical control surface deﬂections in the ADMIRE model. (The generic control surface deﬂections are what is computed by the part of the control allocation that translates the commanded moments to control surface deﬂections.) The translation from commanded moments, which is the primary output of the controller, to generic control surface deﬂections is done using a built-in algebraic equation solver in Simulink operating on simpliﬁed versions of the ADMIRE aerodynamic moment coefﬁcient functions. The Jacobian terms in the control law are calculated using an analytical description of the aerodynamic forces together with the representation on the right hand side of (11.31), but omitting the rightmost term (the one linear in x¨ c ). This analytical description is based on a weighted least squares ﬁtting against the tabulated aerodynamic data in ADMIRE using an essentially polynomial basis. In some of the velocity vector roll manoeuvres we employ coordinated sideslip and roll rate commands, since this may extend the interval of trimable roll rates for the ADMIRE airframe compared to operation at β ≈ 0, as can be inferred from some the ﬁgures in Chapter 13. 11.4.2

Decelerating Turn

In this manoeuvre, the aircraft is banked quickly to the left and the throttle is set to idle at the beginning of the manoeuvre. At the same time, the angle of attack is commanded up in a step like fashion.6 Following the initial sharp increase, the angle of attack command is piecewise linear in a manner which will yield approximately constant (high) vertical load factor nz . After the initial roll command, the roll command is slowly adjusted in a piecewise linear way that would keep the aircraft on approximately the same altitude under nominal conditions. Both the angle of attack and roll commands are “pre programmed” (i.e. do not depend on the response of the aircraft). The manoeuvre begins at t = 0.5 s at an altitude of 3000 m and Mach 1.2, and lasts until t = 10 s, at which time the aircraft has reached below Mach 0.9. The result is displayed in Figure 11.1. In the nominal case, without sensor error on the measurement of angle of attack, the manoeuvre is easy for the controller to realize with fast response and good tracking of 6

The step is not instantaneous, but with ﬁnite derivative, and this can be thought of as the result of passing the stick command through a command ﬁlter.

248

J.W.C. Robinson

Mach number

α [deg], nz [−]

10

5

α α_cmd nz

0 2

4

6 time [s]

1

0.9

0.8

10

8

1.1

2

4

6 time [s]

8

10

6 time [s]

8

10

6 time [s]

8

10

6 time [s]

8

10

6 time [s]

8

10

8

10

3200

Altitude [m]

Euler ang. φ [deg]

0 −20 −40 −60

3100

3000

2900

−80 2

4

6 time [s]

2800

10

8

2

4

Mach number

z

α [deg], n [−]

10

5

α α_cmd nz

0 2

4

6 time [s]

8

1.1

1

0.9

0.8

10

2

4

3200

Altitude [m]

Euler ang. φ [deg]

0

−20

−40

−60

3100

3000

2900

−80

2

4

6 time [s]

8

2800

10

2

4

5 α α_cmd nz

0

2

4

6 time [s]

8

Mach number

α [deg], n [−] z

10 1.1

1 0.9 0.8

10

2

4

3200

Altitude [m]

Euler ang. φ [deg]

0

−20

−40

−60

3100 3000

2900

−80

2

4

6 time [s]

8

10

2800

2

4

6 time [s]

Fig. 11.1. Decelerating turn. Commanded αc , actual α, load factor nz , Mach number, Euler bank angle φ and altitude. Top four panels: No sensor error. Middle four panels: Sensor error δα = −2◦ . Bottom four panels: Sensor error δα = 2◦ .

Block Backstepping for Nonlinear Flight Control Law Design

249

the angle of attack, resulting in an almost constant vertical load factor, and essentially constant altitude. The effects of passage through the transonic region are hardly visible. When sensor errors are present on the measured value of the angle of attack the effects on angle of attack tracking are noticeable, with a small variation in load factor (about 1g) as a result. The effects are not emphasised by the transonic passage however. 11.4.3

Velocity Vector Roll

The velocity vector roll is executed at the three Mach numbers 0.7, 0.9 and 1.1, and at two different altitudes, 3000 m and 4000 m. At the highest Mach number the sideslip is commanded to 0 but for the other two Mach numbers a sideslip command of −6◦ is used. Uncertainties in Cnβ as well as xcg are introduced in some cases. The commanded conical rotation (roll) rate is 300◦ /s for all Mach numbers. Altitude 4000 m, Mach number 0.7 In this case the manoeuvre is executed with nominal parameter values as well as with uncertainty in Cnβ of ±0.04. A sideslip command of −6◦ is used. The result is displayed in Fig. 11.2. In the nominal case the roll tracking and tracking of sideslip angle are both good. The commanded angle of attack is 6◦ , and the response to this command is fast but shows a “steady state” error of about 1◦ due to modelling errors in the controller. (These are less pronounced for higher angles of attack, cf. Fig. 11.5.) The effects of uncertainties in Cnβ are small. Altitude 3000 m, Mach number 0.9 In the second subsonic case the manoeuvre is executed with nominal parameters as well as with uncertainty in xcg of ±0.15 m. Again, the sideslip command is −6◦ . The result is shown in Fig. 11.3. In the nominal case, the behaviour is very similar to that seen for Mach 0.7 and altitude 4000 m with good tracking in roll and sideslip, and an error in angle of attack tracking. The effects of uncertainties in xcg are small. Altitude 3000 m, Mach number 1.1 In the supersonic case the manoeuvre is executed with nominal parameters as well as with uncertainty in Cnβ of ±0.04. A sideslip command of 0 is used. The result is shown in Fig. 11.4. The results are qualitatively similar to those in the two other versions of the velocity vector roll manoeuvre and the effects of the uncertainty in Cnβ are minor despite the fact that the manoeuvre takes the aircraft through the transonic region. The maximum sideslip angle error is about 1◦ .

J.W.C. Robinson 8

2

6

0

4 2

α α_cmd n

0

−4

−6

z

2

4

6 time [s]

8

−8

10

350

0.8

300

0.75

Mach number

250 200 150

100

dξ/dt dξ_cmd/dt

50

z

α [deg], n [−]

0

2

4

6 time [s]

8

0.5

10

0

4

α α_cmd n 4

6 time [s]

10

2

4

6 time [s]

8

10

β β_cmd

−2 −4 −6

z

2

8

0.6

6

0

6

0.55

2

−2

4

0.7

8

2

2

0.65

β [deg]

dξ/dt [deg/s]

−2

β β_cmd

−2

β [deg]

z

α [deg], n [−]

250

8

−8

10

2

4

6

8

10

2

4

6 time [s]

8

10

0.8

Mach number

0.75

200

dξ/dt dξ_cmd/dt

100

α [deg], nz [−]

0

2

4

6 time [s]

0.7

0.65

0.6

0.55

8

0.5

10

8

2

6

0

4 2

α α_cmd nz

0 −2

β [deg]

dξ/dt [deg/s]

300

2

4

6 time [s]

β β_cmd

−2 −4

−6 8

−8

10

2

4

6

8

10

2

4

6 time [s]

8

10

0.8

Mach number

dξ/dt [deg/s]

300

200

dξ/dt dξ_cmd/dt

100

0

2

4

6 time [s]

8

10

0.7

0.6

0.5

Fig. 11.2. Velocity vector roll at altitude 4000 m, Mach 0.7 and βc = −6◦ . Commanded αc , actual α, load factor nz , commanded βc , actual β, commanded roll rate Ωc = ξ˙ c , actual roll rate Ω = ξ˙ and Mach number. Top four panels: No uncertainty in Cnβ . Middle four panels: δCnβ = −0.04. Bottom four panels: δCnβ = 0.04.

Block Backstepping for Nonlinear Flight Control Law Design

251

8 β β_cmd

4

α α_cmd nz

2

0 2

4

6 time [s]

−2

β [deg]

α [deg], n [−] z

0 6

−4

−6 8

10

−8

2

4

6

8

10

2

4

6 time [s]

8

10

1

Mach number

dξ/dt [deg/s]

300 200

dξ/dt

100

dξ_cmd/dt 0

2

4

6 time [s]

8

0.9

0.8

0.7

10

8

β β_cmd

−2

β [deg]

z

α [deg], n [−]

0 6 4

α α_cmd nz

2

0 2

4

6 time [s]

−4 −6

8

−8

10

2

4

6

8

10

2

4

6 time [s]

8

10

1

0.95

Mach number

dξ/dt [deg/s]

300

200

100

0

dξ/dt dξ_cmd/dt 2

4

6 time [s]

0.9 0.85 0.8 0.75

8

0.7

10

8

β β_cmd

β [deg]

α [deg], nz [−]

0 6 4

α α_cmd nz

2

0

2

4

6 time [s]

−2 −4 −6

8

−8

10

2

4

6

8

10

2

4

6 time [s]

8

10

1

Mach number

dξ/dt [deg/s]

300

200 100

dξ/dt

0.9

0.8

dξ_cmd/dt

0

2

4

6 time [s]

8

10

0.7

Fig. 11.3. Velocity vector roll at altitude 3000 m, Mach 0.9 and βc = −6◦ . Commanded αc , actual α, load factor nz , commanded βc , actual β, commanded roll rate Ωc = ξ˙ c , actual roll rate Ω = ξ˙ and Mach number. Top four panels: No uncertainty in xcg . Middle four panels: δxcg = −0.15 m. Bottom four panels: δxcg = 0.15 m.

252

J.W.C. Robinson 2

4

α α_cmd nz

2

0 −2

β β_cmd

1

β [deg]

α [deg], n [−] z

8 6

2

4

−1

8

6 time [s]

0

10

−2

2

4

6

8

10

2

4

6 time [s]

8

10

Mach number

dξ/dt [deg/s]

300

200

dξ/dt

100

dξ_cmd/dt 0

2

4

6 time [s]

8

1.1 1

0.9 0.8

10

8

2

4

α α_cmd nz

2 0 −2

β β_cmd

1

β [deg]

α [deg], n [−] z

6

2

4

−1 8

6 time [s]

0

−2

10

2

4

6

8

10

2

4

6 time [s]

8

10

Mach number

dξ/dt [deg/s]

300

200

100

dξ/dt

1.1

1 0.9

dξ_cmd/dt 0

2

4

8

6 time [s]

0.8

10

2

8

4 α α_cmd nz

2

0

−2

β β_cmd

1

β [deg]

α [deg], nz [−]

6

2

4

6 time [s]

0

−1

8

−2

10

2

4

6

8

10

2

4

6 time [s]

8

10

350

250

Mach number

dξ/dt [deg/s]

300

200

150 100

dξ/dt dξ_cmd/dt

50

0

2

4

6 time [s]

8

10

1.1 1 0.9 0.8

Fig. 11.4. Velocity vector roll at altitude 3000 m, Mach 1.1 and βc = 0. Commanded αc , actual α, load factor nz , commanded βc , actual β, commanded roll rate Ωc = ξ˙ c , actual roll rate Ω = ξ˙ and Mach number. Top four panels: No uncertainty in Cnβ . Middle four panels: δCnβ = −0.04. Bottom four panels: δCnβ = 0.04.

25

1

20

0

15

−1

10

β [deg]

α [deg], nz [−]

Block Backstepping for Nonlinear Flight Control Law Design

α α_cmd nz

5

β β_cmd

−2

−3

−4

0

2

253

4

−5

10

8

6 time [s]

2

4

6

8

10

2

4

6 time [s]

8

10

Mach number

dξ/dt [deg/s]

200 150

100 dξ/dt

50

1

0.8

0.6

dξ_cmd/dt

4

2

Rudder [deg]

Outer elevons [deg] Inner elevons [deg]

Canards [deg]

0

0.4

10

drc dlc

0

−20

1

2

3

4

5

6

7

1

2

3

4

5

6

7

1

2

3

4

5

6

1

2

3

4

5 time [s]

6

8

9

10

8

9

10

7

8

9

10

7

8

9

10

7

8

9

10

8

9

10

8

9

10

20

0 drie

−20

dlie

20 0

droe −20

dloe

20

0

drc

100

dlc 0

−100

1

Outer elevon rates [deg/s] Inner elevon rates [deg/s]

8

20

−20

Canard rates [deg/s]

6 time [s]

2

3

4

5

6

drie

100

dlie

0

−100

1

2

3

4

5

6

7

droe

100

dloe

0 −100

1

2

3

4

5 time [s]

6

7

Fig. 11.5. Extreme velocity vector roll at altitude 3000 m, Mach 1.1 and βc = −3.5◦ . Top four panels; Commanded αc , actual α, load factor nz , commanded βc , actual β, commanded roll rate Ωc = ξ˙ c , actual roll rate Ω = ξ˙ and Mach number. Middle four panels; control surface positions. Bottom four panels; control surface rates.

254

J.W.C. Robinson

11.4.4

Extreme Velocity Vector Roll

In order to demonstrate behaviour at more extreme conditions7 a version of the velocity vector roll is also performed where the commanded angle of attack is 15◦ and the initial Mach number is 1.1. The altitude is 3000 m and the commanded conical rotation (roll) rate is 180◦ / s. A coordinated roll and sideslip command is used where the commanded sideslip angle is −3.5◦. The result is shown in Fig. 11.5. The tracking in angle of attack, roll rate and sideslip angle are all good and the manoeuvre results in a peak vertical load factor of over 20g. As a result, the airspeed is reduced dramatically and goes from Mach 1.1 to Mach 0.5 in a period of a little over 8 s. The rapid passage through the transonic region leaves virtually no trace in the tracked variables, however, and tracking performance remains very good throughout the manoeuvre. 11.4.5

Pull-Up Manoeuvre

In this manoeuvre a step like command in angle of attack is issued at the beginning of the manoeuvre, and the angle of attack command is then gradually increased in a piecewise linear manner in order to keep the load factor or pitch rate approximately constant. The manoeuvre is executed at three different Mach numbers, 0.5, 0.7 and 0.9, and two different altitudes, 1000 m and 4000 m. In the two versions of the manoeuvre starting at Mach 0.7 and 0.9 the angle of attack command is given so as to produce approximately constant vertical load factor. In the version of the manoeuvre starting at Mach 0.5 the angle of attack command is given so as to produce approximately constant pitch rate. For the highest Mach number the manoeuvre is performed with nominal parameters as well as with uncertainty in Cmα of ±0.1. The result is shown in Figs. 11.7, 11.6.

10

1

30

1.5 2 time [s]

2.5

10 0

1

2

0.6 0.4

q Mach

0.2

1

2 1.5 time [s]

2.5

3

4

5

0.8

α α_cmd nz

20

0.8

0 0.5

3

Mach number

α [deg], nz [−]

z

20

0 0.5

α [deg], nz [−]

q [rad/s], Mach number

α α_cmd n

30

3 time [s]

4

5

0.7 0.6

0.5

0.4 0.3

1

2

3 time [s]

Fig. 11.6. Pull-up at altitude 1000 m, Mach 0.5 and 0.7. Commanded αc , actual α, load factor nz , pitch rate q and Mach number. Top two panels: Mach 0.5. Bottom two panels: Mach 0.7. 7

The commanded angle of attack results in values for the vertical load factor which are outside structural limits for manned aircraft but fall in the region conceivable for highly manoeuvrable UAVs.

Block Backstepping for Nonlinear Flight Control Law Design

1

Mach number

10

z

α [deg], n [−]

15

α α_cmd nz

5 0 1

2

3

4 time [s]

5

6

0.8

0.7

0.5

7

1

2

3

4 time [s]

5

6

7

1

2

3

4 time [s]

5

6

7

1

2

3

4 time [s]

5

6

7

1

Mach number

α [deg], nz [−]

0.9

0.6

15 10 α α_cmd nz

5

0

1

2

3

4 time [s]

5

6

0.9

0.8

0.7 0.6 0.5

7

1

Mach number

15

10

z

α [deg], n [−]

255

α α_cmd nz

5 0 1

2

3

4 time [s]

5

6

7

0.9

0.8 0.7 0.6

0.5

Fig. 11.7. Pull-up at altitude 4000 m, Mach 0.9. Commanded αc , actual α, load factor nz and Mach number. Top two panels: No error in Cmα . Middle two panels: δCmα = −0.1. Bottom two panels: δCmα = 0.1.

The tracking performance in angle of attack is good in the nominal case for all Mach numbers and the uncertainty in Cmα has only a small effect on performance.

11.5 Conclusions We have in this chapter outlined a design procedure based on the block backstepping approach. This approach offers a route to fast controller design which can yield a a controller that covers most of the design envelope with a single set of tuning parameters. Indeed, in our simulation examples we used a single, moderately large, set of constants, and at least one of the manoeuvres displayed was extreme. Part of the explanation for the good performance is the fact that block backstepping can easily accommodate an accurate nonlinear model of the plant, including actuator dynamics. In addition, the resulting control law does not make any special assumptions about the plant, except the condition of strict feedback form (i.e. moment control, in the ﬂight control applications). This is in contrast to the NDI-TSS approach which employs a correct model for the plant but which relies on a simplifying assumption of time-scale separation to guarantee stability. The time-scale separation assumption leads to a requirement for higher loop gains and fast actuator dynamics, not only for performance, but also for stability. We have shown that this can be traced back to the absence of terms in the NDI-TSS

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control law, terms that are present in the block backstepping control law. The added complexity of the block backstepping control law extremely modest however.

Acknowledgement The work was partly sponsored by FOI project E21308. The author wishes to thank the reviewer for insightful remarks and helpful suggestions, and Mr. F. Berefelt, FOI, for many stimulating discussions which led to a considerably improved presentation.

Appendix Stability Proof for Integrator Block Backstepping As mentioned above, the standard proof for set point control block backstepping can be found in several places in the literature, e.g. [117, Sec. 14.3]. The formulation here is slightly extended to cover the time varying case encountered in our tracking problem. By deﬁning the error variable y˜ = y − y˜ d (˜x,t) we obtain from (11.20), (11.21) for the special case h(y) = 0 and k = I the controlled time varying system x˙˜ = ˜f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)), ∂˜y (˜x,t) ∂˜y (˜x,t) ˜ y˙˜ = u − d . f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)) − d ∂˜x ∂t

(11.50) (11.51)

The object is to show that with u in (11.51) replaced by the control law u(˜x, y,t) in (11.28) the point (˜x, y˜ ) = 0 in the closed loop system thereby obtained from (11.50), (11.51) becomes globally asymptotically stable.8 For this we introduce the Lyapunov function candidate Va as 1 Va (˜x, y˜ ) = V (˜x) + ˜y2 , (11.52) 2 where V is the Lyapunov function in (11.26). For any choice of continuous control u we have along the solution trajectories to (11.50), (11.51) that d Va (˜x, y˜ ) = ∇V (˜x)T x˙˜ + y˜ T y˙˜ dt = ∇V (˜x)T ˜f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)) ∂˜y (˜x,t) ∂˜y (˜x,t) ˜ f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)) − d + y˜ T u − d ∂˜x ∂t T ˜ T ˜ = ∇V (˜x) f(˜x,t) + g(˜x,t)˜yd (˜x,t) + y˜ u ∂˜y (˜x,t) ∂˜y (˜x,t) ˜ + y˜ T g˜ (˜x,t)T ∇V (˜x) − d f(˜x,t) + g˜ (˜x,t)(˜y + y˜ d (˜x,t)) − d ∂˜x ∂t = ∇V (˜x)T fdes (˜x) + y˜ T u ∂˜y (˜x,t) ∂˜y (˜x,t) ˜ y˜ T g˜ (˜x,t)T ∇V (˜x) − d f(˜x,t) + g˜ (˜x,t)y − d , (11.53) ∂˜x ∂t 8

For the closed loop system the point (˜x, y˜ ) = 0 is an equilibrium by (11.27) and the properties of V in (11.26) and fdes (x) in (11.24).

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where we have used (11.27). If we now choose u in (11.53) to be identical with the control law u(˜x, y,t) in (11.28) we obtain d Va (˜x, y˜ ) = ∇V (˜x)T fdes (˜x) − ˜y2Wy ≤ −W (˜x) − ˜y2Wy < 0, dt

(11.54)

for all t and x˜ ∈ Rn \ 0, y˜ ∈ Rm \ 0. It follows that the Lyapunov function candidate Va in (11.52) is indeed a (time invariant) Lyapunov function for the (time varying) resulting closed loop system obtained from (11.50), (11.51) by inserting u(˜x, y,t) in (11.28) in place of u in (11.51). Since W (˜x) + ˜y2Wy is zero if and only if (˜x, y˜ ) = 0 it follows moreover from standard stability results for nonautonomous systems (see e.g. [136, Corollary 5.6]) that the limit set of the solutions to the closed loop system consists of the single point {0}. 2

12 Optimisation-Based Flight Control Law Clearance Prathyush P. Menon, Declan G. Bates, and Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK {ppm6, dgb3, ixp}@le.ac.uk

Summary. This chapter describes the development of evolutionary and deterministic global optimisation methods for the clearance of nonlinear ﬂight control laws for highly augmented aircraft. The algorithms are applied to the problem of evaluating a nonlinear handling qualities clearance criterion for the ADMIRE benchmark model. An optimisation-based approach for computing worst-case pilot input demands is also presented. Hybrid versions of the global algorithms, incorporating local gradient-based optimisation, are shown to signiﬁcantly reduce computational complexity while at the same time improving global convergence properties. The proposed approach to ﬂight clearance is shown to have signiﬁcant potential for improving both the reliability and efﬁciency of the current industrial ﬂight clearance process. Keywords: Global optimisation, nonlinear systems, robustness analysis, ﬂight control, simulation.

12.1 Introduction Modern high performance aircraft are often designed to be naturally unstable due to performance reasons and, therefore, can only be ﬂown by means of a ﬂight control system (FCS) which provides artiﬁcial stability. As the safety of the aircraft is dependent on the controller, it must be proven to the clearance authorities that the controller functions correctly throughout the speciﬁed ﬂight envelope in all normal and various failure conditions, and in the presence of all possible parameter variations and all possible pilot inputs. This task is a very lengthy and expensive process, particularly for high performance aircraft, where many different combinations of ﬂight parameters (e.g. large variations in mass, inertia, centre of gravity positions, highly nonlinear aerodynamics, aerodynamic tolerances, air data system tolerances, structural modes, failure cases, etc.) must be investigated so that guarantees about worst-case stability and performance can be made [68]. The aircraft models used for clearance purposes describe the actual aircraft dynamics, but only within given uncertainty bounds. One reason for this is the limited accuracy of the aerodynamic data set determined from theoretical calculations and wind tunnel tests. These parameters can even differ between two aircraft of the same type, due to production tolerances. Moreover, the employed sensor, actuator and hydraulic models are usually only approximations, where the nonlinear effects are not fully modelled because they are either unknown or would make the model unacceptably complex. D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 259–300, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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The goal of the clearance process is to demonstrate that a set of selected criteria expressing stability and handling requirements is fulﬁlled. Typically, criteria covering both linear and nonlinear stability, as well as various handling and performance requirements are used for the purpose of clearance. The clearance criteria can be grouped into four classes, (i) linear stability criteria, (ii) aircraft handling/pilot induced oscillation (PIO) criteria, (iii) nonlinear stability criteria, and (iv) nonlinear handling criteria. This chapter focuses on the evaluation of a nonlinear handling criterion, which is described in detail in a later section. Details of other criteria typically employed in the ﬂight clearance process can be found in [68]. In the clearance process, for each point of the ﬂight envelope, for all possible aircraft conﬁgurations and for all combinations of parameter variations and uncertainties, violations of the clearance criteria and the worst-case result for each criterion must be found. Based on the clearance results, ﬂight restrictions are imposed where necessary. Faced with limited time and resources, the current ﬂight clearance process employed by the European aerospace industry uses either a gridding or sampling approach, whereby the various clearance criteria are evaluated for all combinations of the extreme points of the aircraft’s uncertain parameters, or using Monte Carlo simulation, [68]. This process is then repeated over a gridding of the aircraft’s ﬂight envelope. Clearly, the effort involved in the resulting clearance assessment increases exponentially with the number of uncertain parameters (gridding) or desired conﬁdence level of the results (Monte Carlo). Another difﬁculty with these approaches is the fact that there is no guarantee that the worst-case uncertainty combination has in fact been found, since (i) it is possible that the worst-case combination of uncertain parameters does not lie on the extreme or sampled points, and (ii) only a few selected points in the aircraft’s ﬂight envelope can be checked. This paper presents a new approach to the clearance problem based on the use of hybrid optimisation techniques, which will be shown to have the capability to signiﬁcantly improve both the reliability and efﬁciency of the current ﬂight clearance process. The search for worst-case pilot control inputs is another important part of the clearance process for any new ﬂight control system. The deﬁnition of “worst-case”, of course, depends on the particular clearance criterion that is being considered. For highly agile combat aircraft, a key consideration is the identiﬁcation of so-called departure susceptibility - the computation of pilot inputs that will excite the nonlinear aircraft dynamics to such an extent as to lead to loss of stability and/or controllability (in the sense of handling qualities). For ﬂight control laws equipped with a Manoeuvre Load Limiter (MLL), on the other hand, pilot inputs that test the robust functionality of the envelope protection system are required to be computed. Extensive non-linear ofﬂine simulations are widely used in the aerospace industry for investigating the stability, performance and departure susceptibility of new aircraft, thereby ensuring the carefree maneuvering capability of the designed ﬂight control law, [68, 36]. Numerous simulations are generally required to test the effectiveness of the control law and departure susceptibility of the aircraft. Two commonly used methods are: (1) A set of pre-deﬁned control inputs, such as a pitch doublet, roll doublet, rudder step, and 360o roll are applied to the nonlinear closed loop aircraft model and the responses are analysed. The ofﬂine simulations are repeated at different trim

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points over the ﬂight envelope. Composite sets of simulations are repeated for different conﬁgurations of the aircraft. (2) Pilot/Engineer-In-Loop (PIL/EIL) real-time simulations [190] are carried out on a digital simulator platform equipped with the nonlinear aircraft model, in an attempt to depart the aircraft by ﬂying in different parts of the ﬂight envelope and by giving rapid stick and pedal inputs of large amplitude. This method obviously completely depends on heuristics and experience gained by the test pilots and engineers. In both of the above approaches, the computational effort required is significant and interpretation of the results requires expert human intervention. Hence, the question, “Is there a control input combination that when applied to an aircraft in an equilibrium/trim state violates a deﬁned clearance criterion?” is an extremely difﬁcult and expensive one to answer. The presence of signiﬁcant levels of uncertainty in the aircraft model obviously complicates the situation still further. In this chapter, an optimisation-based approach to this problem is developed which will be shown to have signiﬁcant advantages over current industrial approaches. The chapter is organised as follows. Section II describes the aircraft simulation model and ﬂight clearance criterion used in this study. Section III reviews previous research on optimisation-based ﬂight clearance, and introduces the various optimisation algorithms considered in this study. In Section IV, the results of applying these algorithms to the ﬂight clearance problem are described. Section V describes the application of two of the most promising optimisation methods to the problem of clearing continuous regions of the ﬂight envelope. Optimisation-based methods for the computation of worst-case pilot inputs as part of the ﬂight clearance process are described in Section VI. Finally, some conclusions are presented in Section VII.

12.2 ADMIRE Aircraft Model and Clearance Criterion The aircraft model used in the present study is the ADMIRE (Aero-Data Model In a Research Environment) [74], a nonlinear, six degree of freedom simulation model developed by the Swedish Aeronautical Research Institute (FOI) using aero data obtained from a generic single seated, single engine ﬁghter aircraft with a delta-canard conﬁguration. ADMIRE is augmented with a full-authority ﬂight control system and includes engine dynamics and detailed nonlinear actuator models. The model includes a large number of uncertain aerodynamic, actuator, sensor and inertia parameters, whose values, within speciﬁed ranges, can be set by the user. The aircraft dynamics are modelled as a set of twelve ﬁrst order coupled nonlinear differential equations, given as follows: x(t) ˙ = f (x(t), u(t), ∆)

(12.1a)

y(t) = h(x(t), u(t))

(12.1b)

where x(t) is the state vector with twelve components, i.e., velocity, angle of attack (AoA or α), sideslip angle, and angular rate, attitude and position vectors. ∆ represents the uncertain aircraft parameters - Table 12.1 shows the uncertain parameters considered in this study. y(t) is the output vector, and u(t) is the control input vector, whose

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components are left and right canard deﬂection angle, left and right inboard/outboard elevon deﬂection angle, leading edge ﬂap deﬂection angle, rudder deﬂection angle, landing gear status (extract/retract), and vertical and horizontal thrust vectoring. Table 12.1. Aircraft Model Uncertain Parameters [75] Parameter

Bound

Description

∆mass

[-0.1, +0.1]

variation in aircraft mass (9100 kg) [%]

∆xcg

[-0.075, +0.075]

variation in position of center of mass [m]

∆Cm

δe

[-0.05, +0.05] uncertainty in pitching moment due to elevator [1/rad]

∆Iyy

[-0.2, +0.2]

the inertia uncertainty (81000 kg · m2 ) [%]

∆Cmα

[-0.05, +0.05]

uncertainty in pitching moment due to AoA [1/rad]

The control input is determined by u(t) = g(y(t), yREF (t))

(12.2)

where g( · , · ) is a ﬂight control law which is provided with the ADMIRE model, and yREF (t) is the reference demand that consists of the pilot inputs such as pitch stick demand, roll stick demand, rudder pedal demand, and thrust demand. Equations (12.1) and (12.2) together represent the closed loop dynamics of the aircraft with the ﬂight control law in the loop. For longitudinal control, canards and symmetric elevons are used. Rudder and differential elevons are used for lateral/directional control. For this purpose, an elevon rudder interconnect (ERI) is provided. The inner and outer elevon surfaces have the same deﬂection both for the pitch and roll control. Also, the right and left canard deﬂections are the same. The function of the longitudinal controller is pitch rate control (qcom ) below Mach number 0.58 and load factor control (nzcom ) above Mach number 0.62. A blending function is used in the region in between, in order to switch between the two different modes. The longitudinal controller also contains a speed control (Vtcom ) functionality. The lateral controller enables the pilot to perform roll control around the velocity vector of the aircraft (pwcom ) and to control the sideslip angle (βcom ). The longitudinal ﬂight control laws are designed at 29 different design points over the aircraft’s ﬂight envelope using standard linear design methods, and the control law gains are then interpolated (scheduled) with Mach number and altitude using the classical industrial approach. The ﬂight control system also uses a control selector, which allocates the control signals to the seven control surface actuators corresponding to the applied demand signal. The augmented ADMIRE operational ﬂight envelope is deﬁned up to Mach 1.2 and altitude 6000 meters [74]. For further details on the ﬂight envelope limitations for variations in Mach number and the details of the ﬂight control design the reader is referred to Ref. [74]. The model also contains rate limiting and saturation blocks [201], as well as nonlinear stick shaping elements in its forward path. The saturation limits in the input signal path represent the limitations of the forces that may be applied by the pilot. For further details of the ADMIRE model and its associated ﬂight control system, the reader is referred to Chapter 3.

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The clearance criterion considered in this study is the AoA limit exceedence criterion [68, 75, 72]. For this criterion, it is required to identify the ﬂight cases where, for the pull-up manoeuvre deﬁned in Fig. 12.1, the maximum overshoot occurs in AoA. In particular, the combination of uncertainties that yields the largest exceedence of the deﬁned limits must be identiﬁed. The test aims to assess the effectiveness of the AoA limiting scheme in the ﬂight control system, in terms of the peak overshoot in AoA that occurs in response to the speciﬁed manoeuvre. Fig. 12.1 shows the speciﬁed pitch stick command, a rapid pull in longitudinal stick to a deﬁned level (40N) at a 640N/sec stick rate with stick hold for 10 seconds. The present analysis aims to estimate the clearance criterion [68]: 50 45

Pitch Stick Input [N]

40 35 30 25

Pull Rate 640N/sec

20 15 10 5 0

0

1

2

3

4

5

6

7

8

9

10

time [sec] Fig. 12.1. Pitch Stick Pull Command

αmax = max(α(t)) for t ≤ 10 [sec]

(12.3)

for all possible combinations of aircraft parametric uncertainty.

12.3 Optimisation-Based Flight Clearance In this chapter, the ﬂight clearance problem deﬁned in Section 12.2 is formulated as a maximisation problem and solved using different local, global and hybrid optimisation schemes. The optimisation problem itself is to ﬁnd the combination of real parametric uncertainties that gives the worst violation of the criterion deﬁned in Eq. (12.3). Since this and many other clearance criteria must be checked over a huge number of envelope points and aircraft conﬁgurations, it is imperative to ﬁnd the most computationally efﬁcient approach to this problem. Previous efforts to apply optimisation methods to this problem have revealed that the nonlinear optimisation problems arising in ﬂight

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clearance, while having relatively low order, often have multiple local optima and expensive function evaluations, [68], Chapter 7. Therefore, the issue of whether to use local or global optimisation, and the associated impact on computation times is a key consideration for this problem. In [68], Chapter 21, local optimisation methods such as SQP (Sequential Quadratic Programming), and L-BFGS-B (Limited memory Broyden-Fletcher-Goldfarb-Shanno method with bounded constraints) were used to evaluate a range of linear clearance criteria for the HIRM+ (High Incidence Research Model) aircraft model. In [68], Chapter 22, global optimisation schemes such as genetic algorithm’s (GA’s), adaptive simulated annealing (ASA) and multi coordinate search (MCS) were also applied to evaluate nonlinear clearance criteria for the same aircraft model. Also, in [75, 72], global optimisation methods such as GA and ASA were applied to the ADMIRE model with a different ﬂight clearance criterion. While the results of this study were quite promising, the very slow convergence of the global optimisation methods considered meant that the computational complexity of the proposed approach was very high. In [68], a suggestion was made to make use of hybrid optimisation schemes in the ﬂight clearance problem, to provide computationally feasible convergence to a true global solution. As far as can be established, however, there has so far been no work done in this direction, apart from the work reported here and in [161, 162, 163, 164, 165]. Also, all previously reported work on optimisation-based ﬂight clearance has focussed on the clearance of ﬂight control laws at single points, rather than over continuous regions of the aircraft ﬂight envelope, [163, 164]. Finally, to the best of the authors’ knowledge, this study reports the ﬁrst results of the application of deterministic global optimisation methods to the problem of ﬂight clearance, [163]. To date, very little research has been reported in the literature on the computation of worst-case pilot inputs. In [202], the departure susceptibility of the X-31 Enhanced Fighter Manoeuvrability demonstrator aircraft was evaluated. Genetic Algorithms (GA’s) and a high-ﬁdelity nonlinear simulation model were used to search for pilot inputs that maximised a cost function associated with aircraft departures - the absolute sum of certain states of the system, such as attitude rates, AoA and sideslip angle. This methodology was subsequently developed in [166] using a multi-modal genetic search approach with a reﬁned energy-like cost function, and applied to a full nonlinear simulation model of the Indian Light Combat Aircraft, [36]. Neither of the above studies, however, considered any form of uncertainty in the aircraft simulation model. A different, but related, approach to the same problem is reported in [75]. In this study, a particular sequence of pilot inputs called the Clonk manoeuvre, which was developed by Saab using piloted simulation testing to detect the departure susceptibility of the Gripen aircraft, was applied to the ADMIRE aircraft simulation model, [74]. Global optimisation methods were then used to compute the worst-case combination of uncertain aircraft parameters for this sequence of control inputs. Many different classes of optimisation algorithms are available in the literature. Some of these algorithms use the gradient information of the cost function to ﬁnd the search direction while determining the optimum. Other algorithms use only the cost function value. Many of these non-gradient-based search and optimisation techniques make use of heuristic search directions, in an efﬁcient and intelligent way. The search

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265

space, or design space, can be convex or non-convex. Depending on this, the optimisation algorithms provide a global or a local optimum. Obviously, if the search space is convex, both local and global optimisation algorithms will converge to the true global solution. In the case of a non-convex search space, gradient-based optimisation algorithms provide a local solution, rather than the true global solution. The performance of a given optimisation algorithm is generally problem dependent, and there is no unique optimisation algorithm for general classes of problems which will guarantee computation of the true global solution with reasonable computational complexity. In this study, therefore, a number of different optimisation algorithms are applied, evaluated and compared, in order that the most promising methods for the problem of ﬂight clearance may be uncovered. These algorithms are now brieﬂy described. 12.3.1

Local Gradient-Based Optimisation Methods

Sequential Quadratic Programming The Sequential Quadratic Progamming (SQP) method is one of the most effective methods for medium-size non-linearly constrained optimisation problems. It can be seen as a generalisation of Newton’s method for unconstrained optimisation in that it ﬁnds a step away from the current point by minimising from a sequence of quadratic programming subproblems. SQP methods are standard general purpose algorithms for solving smooth and well-scaled nonlinear optimisation problems when the functions and gradients can be evaluated with high precision. They generally require few iterations and function evaluations. In many situations, the local gradients will not be available analytically and in all such situations numerical approximations of gradients have to be computed and this might cause slower and less reliable performance, especially when the function evaluations are noisy. In SQP, a second order information of the objective function and constraints are approximated. A quadratic approximation of the Lagrange function and an approximation of the Hessian matrix are deﬁned by a quasi-Newton matrix. The SQP algorithm replaces the objective function with a quadratic approximation and replaces the constraints with linear approximations. The quasi-newton matrix is updated in every iteration using the standard Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula. The present study make use of the SQP method implemented as the function “fmincon” provided in [154]. 12.3.2

Evolutionary Global Optimisation Methods

Genetic Algorithms Genetic Algorithms (GA’s) are general purpose stochastic search and optimisation algorithms, based on genetic and evolutionary principles [82]. This approach assumes that the evolutionary processes observed in nature can be simulated on a computer to generate a population, or a set, of ﬁttest candidates. A ﬁtness function is deﬁned to assign a performance index to each candidate - this function is speciﬁc to the problem and is formed from the knowledge domain. In genetic search techniques, a randomly sourced population of candidates undergoes a repetitive evolutionary process of reproduction through selection for mating according to a ﬁtness function, and recombination

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via crossover with mutation. A complete repetitive sequence of these genetic operations is called a generation. To use this evolutionary method, it is necessary to have a method of encoding the candidate as an artiﬁcial chromosome as well as a means of discriminating between the ﬁtness of candidates. The reader is refereed to Ref. [82, 46] for general information on GA’s and the different genetic operators associated with this approach. Details of the particular algorithm, operators, etc, used in this study are provided later in this chapter. In recent years, several researchers have applied genetic methods to a wide variety of problems in aircraft design optimisation, structural optimisation, and ﬂight control problems in the aerospace science ﬁeld. There are also many different applications in other ﬁelds of science. Krishnakumar and Goldberg [123] report an early work related to ﬂight controls in 1992. GA’s were used to design a lateral autopilot controller and a wind shear controller. The results were then compared with designs based on gradient based optimisation methods. Using GA’s, Schmitendorf, et.al. [204] designed a ﬁxed order dynamic compensator with the objective of pole-placement at deﬁned points. Krishnakumar, et.al. [124] used GA’s to design decentralised controllers by matching performance with existing centralized controllers. Classical control was used with GA’s in [47] and [186]. GA’s have also been used for a modiﬁed LQR design by Marrison and Stengel, in [153]. [222] explains the design of H ∞ controllers using genetic methods. Mulgund, et.al. in [173] developed a software tool for optimisation of large-scale air combat tactics using stochastic search and optimisation techniques such as GA’s. GA’s and Simulated Annealing (SA) have also been used for linear robustness analysis with real parameter variations, a well known NP hard problem, by Zhu, et.al. [247]. The results are then compared with those obtained from a branch and bound technique. Fleming and Purshouse, in Ref. [70], provide a comprehensive review of various applications of GA’s in the control engineering ﬁeld. Differential Evolution Differential evolution (DE) is a relatively new global optimisation method, introduced by Storn and Price in [221]. It belongs to the same class of evolutionary global optimisation techniques as GA’s, but unlike GA’s it does not require either a selection operator or a particular encoding scheme. Despite its apparent simplicity, the quality of the solutions computed using DE is claimed to be generally better than those achieved using other evolutionary algorithms, both in terms of accuracy and computational overhead [221]. This method also starts the optimisation from randomly generated multiple candidate solutions. In DE, however, a new search point in each iteration is generated by adding the weighted vector difference between two randomly selected candidate points in the population, with yet another third randomly chosen point. The vector difference determines the search direction and a weighting factor decides the step size in that particular search direction. The DE methodology consists of the following four main steps 1) Random initialization, 2) Mutation 3) Crossover 4) Evaluation and Selection. There are different schemes of DE available based on the various operators that are employed. The one used in the present studies is referred as “DE/rand/1/bin” [221].

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DE has been applied to problems in different ﬁelds of science, with promising results. For example in engineering design, the DE methodology has been successfully applied to ﬁnd the optimal solution for a mechanical design example, formulated as a mixed integer discrete continuous optimization problem, [127]. In [198], the DE method has been applied and compared with other local and global optimization schemes in an aerodynamic shape optimization problem for an aerofoil. 12.3.3

Hybrid Optimisation

For non-convex problems such as ﬂight clearance, with the use of local optimisation alone the chances of getting locked into a local optimum are high, particularly since there is little information available with which to choose a good initial starting point. If the initial guess is close to the true worst-case, however, local optimisation methods can converge to the global optimum extremely quickly. Global optimisation methods, on the other hand, have a high probability of converging to the global solution, if allowed to run for a long enough time with a sufﬁcient initial population and reasonably correct probabilities. Their rate of convergence is very slow, however, and moreover, there is still no guarantee of convergence to the true global solution. In order to try to get the best from both schemes, several researchers have proposed combining the two approaches, [46, 241, 135]. In such hybrid schemes there is the possibility of incorporating domain knowledge, which gives them an advantage over a pure blind search based on evolutionary principles such as GA’s. Most of these hybrid schemes apply a technique of switching from the global scheme to the local scheme after the ﬁrst optimisation algorithm ﬁnishes its search or optimisation. In Ref. [135], some guidelines are provided on designing more sophisticated hybrid GA’s, along with experimental results and supporting mathematical analysis. In a similar way, the conventional DE methodology was augmented by combining it with a downhill simplex local optimisation scheme in [197]. At each iteration, local optimisation was applied to the best individual in a current random set. This hybrid scheme was applied to an aerofoil shape optimization problem and was found to signiﬁcantly improve the convergence properties of the method. There are many local optimisation schemes available and there are also different ways of hybridizing the algorithms. However, the common aim of these schemes is to provide faster convergence to the true global solution. 12.3.4

Deterministic Global Optimisation

A signiﬁcant drawback of all of the probabilistic methods described above is that no formal proofs of convergence are available, and hence multiple trials are required to provide (statistical) conﬁdence measures that the global solution has been found. To avoid this problem, a deterministic global optimisation algorithm known as DIRECT (DIviding RECTangles), which was introduced in [112], is also considered for the ﬂight clearance problem. This algorithm is a modiﬁed version of a class of Lipschitzian optimisation schemes, which, when run for a long enough time, have been proved to converge to the global solution [69]. To the best of our knowledge, this study is the ﬁrst time that a deterministic, as opposed to probabilistic, global optimisation algorithm has been applied to the ﬂight clearance problem. The DIRECT algorithm has previously been

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successfully applied to several different classes of optimisation problems. In Ref. [246], DIRECT optimisation is applied to a realistic slider air-bearing surface (ABS) design, an important engineering optimisation problem in magnetic hard disk drives, in which the cost function evaluation also requires substantial computational time. Fast convergence of the algorithm and a favourable comparison with adaptive simulated annealing were demonstrated in this study. In Ref. [34] the minimization of the cost of fuel and/or electric power for the compressor stations in a gas pipeline network is attempted using the DIRECT algorithm and a hybrid version of DIRECT algorithm with implicit ﬁltering. Again, the application is a complex and realistic one, and the reported results are very promising. The hybridisation step signiﬁcantly improved the convergence to the solution, particularly in the presence of noisy data, which is handled using the implicit ﬁltering method. In Ref. [248] the DIRECT optimisation method is used for the parameter identiﬁcation problem in a complex Systems Biology application, the biochemical control for MPF activation in frog egg extracts. The DIRECT optimisation is used to search for the globally optimal kinetic rate constants for a proposed mathematical model of the control system that best ﬁts the experimental data set. The improvement obtained over the locally optimised parameter set was clearly demonstrated.

12.4 Flight Clearance Results - Single Flight Condition The analysis results reported in this chapter were all generated using Matlab version 6.5.2 and Simulink Release 13. All the worst-case analysis results presented were generated with the ADMIRE model trimmed at Mach 0.4 and altitude 3000 meters in straight and level ﬂight. Once the trim is achieved, the pull up manoeuvre shown in Fig. 12.1 is applied and the cost function is given by Eq. 12.3, i.e., maximum AoA. The combination of uncertain parameters considered are given in Table 12.1. The worst-case analysis results, together with the details of the local, global and hybrid optimisation algorithms used to generate them, are described in the following sections. 12.4.1

Local Optimisation - Worst-Case Analysis Results

Use of a local optimisation method based on gradient estimation, speciﬁcally the function “fmincon” provided in [154], is ﬁrst considered for the clearance problem. Local optimisation methods can, of course, get locked into a local optimum in the case of nonconvex and/or multimodal surfaces, however, they are also much more computationally efﬁcient than global optimisation approaches. Whether a local method converges to a local or global optimum completely depends on the initial starting point in the search space, and the convexity of the search space. Crucially however, in typical ﬂight clearance problems very little information is available as to where to start the optimisation - the number of uncertain parameters and strong nonlinearity of the system mean that even advanced knowledge of ﬂight mechanics provides little insight into how to choose initial values for the uncertain parameters. The function “fmincon” ﬁnds the constrained minimum of a scalar function of several variables starting at an initial estimate. In the present analysis, constraints are due only to the upper and lower bounds of the uncertainty in the variables. A medium scale optimisation scheme is chosen where the

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269

gradients are estimated by the function itself using the ﬁnite difference method. The function uses the sequential quadratic programming(SQP) method - for further details of the “fmincon” optimisation strategy, the reader is referred to [154]. Table 12.2. Result for local optimisation only case Starting Point 0 ∆mass ∆0xcg ∆C0 m ∆0Iyy ∆C0 m δe

α

Convergent Point , ∗ ∆mass ∆∗xcg ∆C∗ m ∆∗Iyy ∆C∗ m δe

α

Number of Simulations

[0 0 0 0 0]

[0.100 0.0750 0.050 0.06084 0.050]

375

[0.100 0.0750 0.050 0.200 0.050]

[0.100 0.0750 0.050 0.18309 0.050]

366

[-0.100 -0.0750 -0.050 -0.200 -0.050] [0.100 0.0750 0.050 -0.12634 0.050]

322

In the present study, a number of starting points for “fmincon” were chosen, based on nominal, minimum and maximum values of the uncertain parameters. The algorithm calls the simulation model to evaluate the cost function for a particular point over the search space. The uncertain parameter values are supplied by the algorithm at each iteration and the cost function is evaluated and returned. The iterations continue until the speciﬁed termination criterion (either maximum number of function evaluations or convergence accuracy) is met. Typical calculation results for our problem are shown in Table 12.2 - note that in the table the last column shows the total number of simulations, i.e., the number of cost function evaluations. Later, it will be shown, via exhaustive global optimisation trials, that for this particular example the parameter combination in the second row is (as far as can be established) the global solution. The objective function value about the global solution is 36.0908◦. As expected, however, for each different initial guess for the values of the uncertain parameters, “fmincon” converges to a different point in the uncertain parameter space. These results show, therefore, that using local optimisation methods in isolation allows little conﬁdence to be established that the true worst-case violation of the clearance criterion has been found. 12.4.2

Genetic Algorithm: Worst-Case Analysis Results

Variable Representation The genetic representation, i.e. the chromosome, for the clearance problem considered here, is the real uncertain parameter set. Each of the uncertainties corresponds to one gene. A binary coded string is generated to represent the chromosome, where each of the parametric uncertainties is bounded as shown in Table 12.1. The level of accuracy for each parameter was chosen to be 10−6, so that a change in the least weighted bit will give an accuracy level of 10−6 to each uncertain parameter. The number of bits required to represent a variable depends on the upper and lower bounds of the optimisation variables and the accuracy level required. All the variables are represented in a binary vector format. The length of the chromosome is 105 bits, consisting of 5 genes each of 21 bits. The binary values for each uncertain parameter are converted into the

270

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real values and these real values are assigned to the respective uncertain parameter variables in the ADMIRE model immediately after the trim condition is achieved, and prior to applying the stick command shown in Fig. 12.1. After simulation, each chromosome is assigned a ﬁtness value, and the ﬁtness function is the nonlinear response criterion given in Eq. (12.3). Initialization The GA search starts from an initial random number of candidates of a given size Nsize . For the present study, the number Nsize is kept ﬁxed at 50. If the population size is reduced below a certain level, the population loses diversity over the search space [82] and the quality of the ﬁnal solution falls or takes longer to compute. Selection The manner in which the candidates in the current iteration (generation) are qualiﬁed for producing successive generations depends on the selection scheme. There are many different selection schemes available [82, 46], the basic ones being roulette wheel selection, tournament selection and stochastic universal sampling. In this analysis, use is made of the roulette wheel selection scheme. Parents are selected according to their ﬁtness. The chromosomes having higher ﬁtness have more chance to be selected. This is analogous to a roulette wheel containing all the chromosomes in the population. The size of a section in the roulette wheel is proportional to the value of the ﬁtness function of every chromosome - the bigger the value is, the larger the section is. In the roulette wheel selection, the probability Pi of the ith candidate being selected from the total numNsize ber of candidates N p is given by: Pi = f itnessi / ∑ j=1 f itnessi . Thus, there is a high probability for a candidate having higher ﬁtness to be selected to appear in the next generation. Crossover Crossover is a recombination operator that ensures the mixing up of the information content in two binary coded chromosomes. Usually, two parent chromosomes are selected randomly to interchange the information content, and thereby produce new offspring that contain information content from both the parents. A probability of crossover is deﬁned which determines the maximum allowed number of pairs for crossover operation. In general, the probability of crossover is kept high. A simple single point crossover scheme is employed in this study with a probability of 0.9 as recommended in [82]. The information between the parents is exchanged at a randomly chosen crossover point over the length of bits. Mutation Mutation introduces random variations in the population over the search space, by randomly ﬂipping a bit value in the case of binary coded GA. In this study, the operation is done with a very low probability of 0.05. The probability is kept very low so that the acquired genetic information is not lost due to the mutations of the chromosomes.

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However, to maintain adequate diversity over the search space some possibility for mutations to occur, albeit with a low probability, is necessary, [70]. A binary uniform mutation scheme is used which randomly selects an individual and sets it to a random value by ﬂipping a randomly selected single bit. Replacement Strategy An elitist strategy is followed such that over each generation the best candidate in the current population moves into the new generation population by replacing the worst candidate of the population. This ensures the presence of a better candidate in the new generation and thereby increases the average ﬁtness of the population over generations. Termination Criterion Many different termination criteria can be employed. In the present study, an adaptive termination criterion is used that is dependent on improvement in the solution accuracy over a ﬁnite number of successive generations. The algorithm terminates the search for the best solution, if there is no improvement on the best solution achieved better than the deﬁned accuracy level, here kept ﬁxed at 10−6 , for a deﬁned successive number of generations. This number of generations is ﬁxed at 15. Hence, the total number of function evaluations required in different trials will be different. As the method is stochastic, there is no rule of thumb to determine upper or lower bounds on the number of simulations required, other than a statistical estimate from many different trials. Results Fig. 12.2 shows the number of simulations versus the best ﬁtness over 100 trials. The statistics of the results, from the 100 independent trials, are given in Table 12.3. The main disadvantage of GA’s is the slow convergence to the true global solution. A large number of simulations, an average of 4485 simulations in this case, is required to obtain the global, or near global solution. The global solution found in this example is the following: , ∆∗mass , ∆∗xcg , ∆C∗ m , ∆∗Iyy , ∆C∗ mα , Max(α(t)) δe (12.4) = [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908o] The reliability in ﬁnding the exact global solution is also not very good. The left histogram of Fig. 12.3 shows the percentage distribution of the maximum value of AoA achieved over 100 trials. The right histogram of Fig. 12.3 shows the percentage distribution of the total number of simulations required over the 100 independent trials, to get the solution. The probability of success in attaining the true global solution is rather low, at only 65%. Tuning GA optimization parameters, such as the different GA-operator probabilities may, of course, improve the above results to a certain extent. However, there are few available guidelines as to how to do this tuning. Another possible approach would be to use alternate selection schemes and scaling and ranking procedures, such as those described in [82], Chapter 4. However, for the present problem the advantage to be gained from these techniques is not expected to be signiﬁcant. Finally, we note that in the context of the current ﬂight clearance process,

272

P.P. Menon, D.G. Bates, and I. Postlethwaite Table 12.3. Genetic algorithm result statistics Trial Avg. No. of Max. No. of Min. No. of Standard Success Simulations Simulations Simulations Deviation GA 100

4485

7500

2400

40

No. of Independent Trials [%]

100

35

αmax [deg]

30 25 20

15

10

5

0 0

828.364

65%

30

90 25

80

70

20

60

50

15

40 10

30 20

5

10

2000

4000

6000

8000

No. of simulations Fig. 12.2. No. of simulations vs. best ﬁtness

0 35.8

35.9

36

36.1

αmax [deg]

0 2000

4000

6000

8000

No. of simulations

Fig. 12.3. GA results histogram

the computational cost of the number of ﬁtness evaluations required by the above approach would be likely to prove prohibitive to its widespread adoption by industry, [68], Chapter 1. 12.4.3

Differential Evolution: Worst-Case Analysis Results

Random Initialization Like other evolutionary algorithms, DE works with a ﬁxed number, N p , of potential solution vectors, initially generated at random according to xi = xL + ρi (xU − xL ), i = 1, 2, ..., N p

(12.5)

where xU and xL are the upper and lower bounds of the parameters of the solution vector and ρi is a vector of random numbers in the range [0 1]. Based on initial experiments with different population sizes, N p was ﬁxed at 12 in the current study. Each xi consists of elements (x1i , x2i , ..., xdi ), which are the uncertain parameters deﬁned in Table 12.1. The dimension d of the optimization problem considered is therefore, 5. The ﬁtness of each of these N p solution vectors is evaluated using the cost function given in Eq. 12.3.

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Mutation The scaled difference vector Fm Di j between two random solution vectors xi and x j is added to another randomly selected solution vector xk to generate the new mutated as given in Eq. 12.6. Fm is the mutation scale factor, a real valued solution vector x¯ G+1 n number in the range [0, 1], (ﬁxed at 0.8 in this study). In the case of using a lower number for N p , it is advised by the algorithm’s creators to use a higher value for the weighting factor Fm , and vice versa. The superscript G represents the iteration number. x¯ G+1 = xk G + Fm Di j , Di j = xi G − x j G n

(12.6)

Fig. 12.4 shows a simple two dimensional example of the mutation operation used in the DE scheme. The difference vector Di j determines the search direction and Fm determines the step size in that direction from the point xk . x2 Global Solution

Fm Di j Di j xG k x¯ nG+1 Di j xGj xG i x1 Fig. 12.4. DE mutation strategy

Crossover During crossover, each element of the nth solution vector of the new iteration, xn G+1 , , and a chosen parent individual xn G as is reproduced from the mutant vector x¯ G+1 n given in Eq. 12.7, where j = 1, ..., dn and i = 1, ..., N p . Note that x¯ G+1 has elements n G+1 G+1 G has elements (x G , x G , ..., x G ). (x¯G+1 , x ¯ , ..., x ¯ ) and x n 1n 2n dn 1n 2n dn ! x ji

G+1

=

x ji G , if a generated random number > ρc x¯G+1 ji , otherwise;

(12.7)

ρc ∈ [0, 1] is the crossover factor, which is ﬁxed at 0.8 in the present study following the comment that a crossover factor of 0.8 often helps in convergence [198].

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Evaluation and Selection After crossover, the ﬁtness of the new candidate xn G+1 is evaluated using Eq. 12.3. If the new candidate xn G+1 has a better ﬁtness than the parent candidate xn G , then xn G+1 is selected to become part of the next iteration. Otherwise xn G is selected and subsequently identiﬁed as xn G+1 . Termination Criteria The same termination criterion as that chosen for the GA trials was used. Results Fig. 12.5 shows the best ﬁtness found over different numbers of simulations, for 100 trials of DE. 90 trials with the DE algorithm converged to the true global solution given in Eq. 12.4, giving the maximum AoA overshoot. Seven trials converged to solutions very close to the true global solution, and 3 trials gave different solutions. Compared to the GA, DE was seen to offer signiﬁcantly improved convergence properties, while the reduced number of initial random starting points (only 12 initial random points against 50 random initial points for the GA) meant that the total number of simulations required in each trial was also signiﬁcantly reduced. Table 12.4 provides the statistics of the results obtained from 100 independent trials of the DE algorithm, and also compares them to those from the GA. It veriﬁes the improvement obtained from the use of the DE algorithm. The average number of simulations required for DE, 3086 in this case, is 31% less than required by the GA. The probability of success of the DE algorithm is also high, at 90%. The left subplot in Fig. 12.6 shows the distribution of the maximum value of AoA achieved. The right subplot shows the distribution of the number of simulations over 100 independent trials of the DE algorithm, demonstrating the reliability and computational effectiveness of the algorithm. Note that, in addition to the improved results, another advantage of this method compared with GA’s is the reduced number of optimization parameters that must be adjusted by the user. Table 12.4. Comparison of global optimisation results statistics Trial

Avg.

Max.

Min.

Standard Success

Simulations Simulations Simulations Deviation

12.4.4

GA 100

4485

7500

2400

828.364

65%

DE 100

3086

4176

1152

567.57

90%

Hybrid Optimisation Schemes

Global optimization methods based on evolutionary principles are generally accepted as having a high probability of converging to the global or near global solution, if allowed to run for a long enough time with sufﬁcient initial candidates and reasonably appropriate probabilities for the evolutionary optimization parameters. As shown by the preceding results, however, the rate of convergence can be very slow, and moreover,

Optimisation-Based Flight Control Law Clearance 100

50

35

90

45

No. of Independent Trials [%]

40

αmax [deg]

30

25

20 15 10

5 0 0

2000

4000

6000

8000

No. of simulations Fig. 12.5. No. of simulations vs. best ﬁtness

80

40

70

35

60

30

50

25

40

20

30

15

20

10

10

5

0 35

35.5

αmax [deg]

36.1

0 0

2000

275

4000

No. of simulations

Fig. 12.6. DE results histogram

there is still no guarantee of convergence to the true global solution. Local optimization methods, on the other hand, can very rapidly ﬁnd optimal solutions, but the quality of those solutions entirely depends on the starting point chosen for the optimization routine. In order to try to extract the best from both schemes, several researchers have proposed combining the two approaches [46,135,241]. In such hybrid schemes, there is the possibility of incorporating domain knowledge, which gives them an advantage over a pure blind search based on evolutionary principles. In Ref. [165], a hybrid GA (HGA) scheme was developed using a switching strategy originally proposed in Ref. [135], and applied to a nonlinear ﬂight clearance problem. In the next section, we compare the performance of this HGA scheme with a novel hybrid DE (HDE) scheme developed for this study. 12.4.5

Hybrid Genetic Algorithm: Worst-Case Analysis Results

Hybrid Genetic Algorithm In Ref. [135], a decision making algorithm for hybrid optimisation is described. In the present study, a slightly modiﬁed version of this algorithm is employed. When a method - say local optimisation - is used, a reward or gain is obtained from that particular scheme. The reward associated with a method is the measure of how well the method helped in providing a solution which is better than the one found in the previous iteration. Similarly a reward can be assigned with the other method, say the GA. The reward associated with each optimisation scheme will determine the probability for that optimisation scheme to be chosen in the next iteration. The reward for each optimisation scheme keeps varying depending on how well it has performed in the previous iteration. A simple way to assign a reward, or weight, is with a weighted geometric average.

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The following equation is used to update the weighted reward for each optimisation scheme [135]: k+1 k k (12.8) WGA or Local = WGA or Local (1 − c) + cRGA or Local where W k , Rk are the weighted reward and the improvement in the solution at the iteration k, respectively, and c is a constant in [0, 1]. Rk is the improvement in the best cost function value obtained from an optimisation scheme over each iteration/generation. In case no improvement occurs, the value of Rk is set equal to zero. Obviously, if one knows at each time step which optimisation method is going to give an improvement towards the global solution, that particular optimisation algorithm can be chosen to accelerate the convergence. However, this prediction is almost impossible. When it is not known beforehand a decision has to be taken based on the previous reward and by calculating the associated probability. The decision making algorithm is summarized in Table 12.5. Due to the frequent occurrence of local optima in ﬂight clearance problems, Table 12.5. Hybrid genetic algorithm: decision algorithm 0 = 0.9, W 0 1. Initialize WGA Local = 0.1, c = 0.3, k = 1, set the calculation mode “Search”, the number of conﬁrmation zero, and generate initial population for GA 2. While the conﬁrmation number is less than a certain number (e.g. 15) k , (12.9) a) Calculate Rk , PGA b) (Flip Coin) = a random number between zero and one k then run GA and update W k , (12.8) c) If (Flip Coin) < PGA GA d) else choose the local algorithm with the following initial guess i. If the calculation mode is “Search”, choose one randomly from two best in the population, ii. else choose one randomly from the subset of population where the distance of each element from the current best is out of 1σ (standard deviation of the population from the current best) k , (12.8) iii. Update WLocal e) If the cost does not improve, 0 = 0.5, W 0 i. Initialize the following every ﬁve conﬁrmation: population, WGA Local = 0.5, c = 0.6 and set calculation mode equal to “Conﬁrm” ii. Increase the number of conﬁrmation f) else set the conﬁrmation number, equal to zero 3. end of While

it is desirable that, initially, the GA should have a higher probability of being chosen than the local algorithm. Hence, initially the weights for GA and the local algorithm are given as 0.9 and 0.1, respectively. The local algorithm used in the present study is the implementation of the SQP method [154], described in Section 12.4.1. Due to the improved convergence properties of the HGA algorithm (see below), it was possible to reduce the size of the initial population to 40 candidates. The initial guess for the local algorithm is taken from the population depending on the calculation mode. There are two modes in the algorithm, ‘search’ and ‘conﬁrm’. The normal starting mode of the algorithm is ‘search’ mode. Whenever there is no improvement in ﬁtness/cost the mode is changed to ‘conﬁrm’ mode. During the ‘search’ mode, the initial guess is chosen

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from the two best in the population. In the ‘conﬁrm’ mode, the initial guess is chosen from a subset of the population, chosen to be far away from the current best. From here onwards the decision-making is done based on probability matching depending on the rewards associated with each of the optimisation schemes. The probability of selecting the GA can be calculated from the following equation [135]: k k k k (12.9) = WGA / WGA + WLocal PGA A random number generator simulates a coin toss and depending on the result one of the optimisation schemes is chosen and proceeded with. If the scheme chosen is global optimisation, it proceeds with only one generation. If the local scheme is chosen, then the optimisation starts from the initial condition until it either converges or reaches a maximum number of cost function evaluations. At the end of a run of either of the optimisation schemes, the improvement achieved above the value of the best solution prior to the optimisation run is checked. The reward for a particular, local or global, optimisation is decided based on the improvement, the probabilities are updated and the sequence is repeated until no improvement occurs from either of the two methods. The algorithm stops when the conﬁrmation number equals a ﬁxed conﬁrmation number, ﬁxed at 15. The termination criterion is thus consistant with that used in the previous GA and DE analysis. Results To gain statistical conﬁdence about the performance of the hybrid genetic algorithm, the HGA algorithm was run 100 times. For 100 trials, the average number of cost function evaluations required was 2011, the maximum was 4468, the minimum was 1357, and the standard deviation was 547.42. Only eight out of the 100 trials failed to ﬁnd the global solution. Fig. 12.7 shows the number of ﬁtness evaluations versus the best ﬁtness for 100 trials of the HGA. Table 12.6 provides the statistics of the HGA results for 100 trials. The average number of cost function evaluations required was 2011, an improvement of 55% when compared with the standard GA. The success rate in ﬁnding the true global solution is also dramatically improved, from 65% to 92%. The success rate was seen to be inversely proportional to the number of cost function evaluations. The number of cost function evaluations can be reduced by decreasing the population size and/or the required number of conﬁrmations in the termination criteria (say 10 instead of 15), however, this will also reduce the resulting success rate. Table 12.6. HGA statistics Trial

Avg.

Max.

Min.

Std.

Success 92%

Simulations Simulations Simulations HGA 100

2011

4468

1357

547.42

GA 100

4485

7500

2400

828.364 65%

278

P.P. Menon, D.G. Bates, and I. Postlethwaite 100

35

90

No. of Independent Trials [%]

40

αmax [deg]

30 25 20 15

10

5

0 0

30

80

60

30

2000

1000

5

3000

4000

5000

0 35.8

35.9

36

0 0

36.1

αmax [deg]

2500

5000

No. of simulations

Fig. 12.8. HGA histogram 36.5

36

34

35.5 35 ∆Iyy /0.2

αmax [deg]

∆mass /0.1

32

αmax [deg]

10

20

36

34.5

30

34

∆Cm /0.05 δe

33.5

26

20 −1

15

40

10

38

22

20

50

Fig. 12.7. No. of simulations vs. best ﬁtness

24

25

70

No. of simulations

28

35

33

∆xcg /0.075

32.5 32

∆Cm /0.05 α

−0.5

0

0.5

Normalised uncertain parameter

1

31.5 −1

−0.5

0

0.5

1

Normalised uncertain parameter

Fig. 12.9. Sensitivity plots about the global solution

The global solution obtained is the same as given in Eq. 12.4. Note that four of the uncertain parameters in this case are on the upper bound and ∆∗Iyy is inside its bound. A sensitivity analysis is performed about the solution and is shown in Fig. 12.9, where the x-axis is normalized. As different allowable minimum and maximum bounds are deﬁned for each of the uncertain parameters, for the purposes of comparison the uncertain parameters are normalized to have a variation between -1 and +1 in the sensitivity plots. As shown in Fig. 12.9, the uncertain parameter ∆Iyy has many local maxima. Fig. 12.7, which gives the best cost over the number of simulations, shows the convergence property of the HGA algorithm over 100 trials. The upper subplot and lower subplot of Fig. 12.8 shows the histogram distribution of maximum AoA and the number of simulations taken respectively, over the 100 independent trials.

Optimisation-Based Flight Control Law Clearance

12.4.6

279

Hybrid Differential Evolution: Worst-Case Analysis Results

A hybrid differential evolution scheme was also developed and applied to the ﬂight clearance problem. The global optimisation scheme used is the same as that described earlier, classiﬁed as “DE/rand/1/bin”. The gradient based SQP local optimisation scheme “fmincon” is employed for the purposes of hybridisation. The algorithm begins with a set of random initial solution vectors in the search space. Like GA and many other evolutionary algorithms, DE also has the virtue of not getting trapped in local optima. To beneﬁt from this fact, the differential evolution is allowed to run for a deﬁned minimum number of initial iterations, ﬁxed at 12 for the present study. The improvement in best ﬁtness obtained is updated at every iteration of the algorithm. However, as the optimisation proceeds, DE may not provide an improvement in the best ﬁtness continuously. The local optimisation scheme is selected as and when there is no improvement in best ﬁtness from the DE. A random solution vector from the current solution set is chosen as the intial guess. In this study, it was observed that the choice of a random solution vector rather than the best solution vector (as adopted in Ref. [197]) as the initial guess for the local optimisation scheme provided the best improvement in convergence results. This is probobly explained by the choice of a random initial point ensuring sufﬁcient diversity in the search space for the algorithm to efﬁciently avoid local optima. If there is an improvement from the local optimisation, the solution obtained from the local optimisation is used to replace the initial guess point, the random solution from the current solution set. Otherwise, no replacement is done. When the local scheme is chosen, the optimization starts from the given initial condition and continues until it either converges or reaches a deﬁned maximum number of cost function evaluations. The switching algorithm is simple, and tries to search for the global optimum in a “greedy” way, demanding improvement in the achieved optimum value at each iteration. The termination criterion is the same as used earlier, i.e. the optimisation is terminated when the improvement in the best ﬁtness is less than 1e − 6 successively for a certain number of iterations of the hybrid algorithm, (ﬁxed at 15 in this study). A pseudocode for the hybrid DE algorithm is given in Table 12.7. Table 12.7. Hybrid differential evolution 1. 2. 3. 4.

Initialize random candidate solutions in search space Evaluate the ﬁtness of each solution and choose the best ﬁtness Apply DE for few initial iterations (e.g., 12); Update the best ﬁtness value in each iteration While the conﬁrmation number is less than a certain number (e.g, 20) a) Continue DE iteration (e.g., 2); Update the best ﬁtness b) If no Improvment in best ﬁtness i. Choose a random solution from the set, say X0 and apply local optimisation “fmincon” with X0 as initial point ii. Update the best ﬁtness and calculate the Improvement iii. If Improvement in best ﬁtness, then replace X0 in the set with the new solution else keep X0 in the set. c) If Improvment in best ﬁtness < 1e − 6 successively increment the number of conﬁrmations 5. end of While

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Results Again, 100 different trials were conducted as in the previous experiments. The results are summarized in Table 12.8 and compared with those obtained from the HGA algorithm. In 98 trials, the HDE scheme found the true global solution - 2 trials missed the global solution. The HDE scheme clearly outperforms the HGA scheme, with an average 45% reduction in computational overheads. The standard deviation of the required number of simulations for 100 trials of the HDE scheme is 84.5% less than that of HGA algorithm, as shown in Table 12.8. Fig. 12.10 demonstrates the fast convergence property of the HDE algorithm in its multiple trials. In a similar way, Fig. 12.11 shows the statistical distribution of maximum AoA and number of simulations taken in 100 independent trials of HDE algorithm. Note the distribution of the histograms compared to the results from the previous schemes. 40

No. of Independent Trials [%]

35

30

αmax [deg]

25

20

15 10 5 0 0

30

100

1000

2000

3000

4000

5000

No. of simulations Fig. 12.10. No.of simulations vs.best ﬁtness

90

25

80 70

20

60

15

50

40

10

30 20

5

10 0 35.8

35.9

36.1

36

αmax [deg]

0 0

500

1000

1500

No. of simulations

Fig. 12.11. HDE results histogram

Table 12.8. Hybrid optimisation statistics Trial

Avg.

Max.

Min.

Standard Success

Simulations Simulations Simulations Deviation

12.4.7

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2011

4468

1357

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92%

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1434

477

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98%

Deterministic Global Optimisation: Worst-Case Analysis Results

The (DIRECT) algorithm is a deterministic global optimisation algorithm developed by Jones et al in [112] in 1993. DIRECT is a modiﬁcation of the classical one dimensional Lipschitzian optimisation algorithm known as the Shubert algorithm [187, 211]. The DIRECT optimisation/search method does not require any derivative information to be

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supplied, and uses a center point sampling strategy. Without loss of generality, in the DIRECT algorithm, it is assumed that every variable has a lower bound of zero and an upperbound of one, since in our formulation, normalisation of the variables to this interval can always be done. Thus, the search space is an n-dimensional hypercube or box. It can be deﬁned as D = {x ∈ ℜn : 0 ≤ xi ≤ 1}. The algorithm works in the normalised parametric space, transforming to the actual search space as and when the cost function has to be evaluated. The main idea of the algorithm is as follows: As the algorithm proceeds, the search space will be partitioned into smaller hypercubes and each will be sampled at the centre point of the interval. Over iterations, the algorithm tries to ﬁnd all the potentially optimal hypercubes in the search space and then partition them, thereby obtaining the global solution. For full details of the lipschitzian optimisation method, the reader is referred to Ref. [211, 187]. Below, we summarise the most important stages of the DIRECT algorithm, as they are applied here in the context of ﬂight clearance. Center Point Sampling and Dividing Strategy The algorithm begins with the evaluation of the cost function about the center point, say c, of the normalised search space. The subsequent step is to divide the hypercube. The points c ± δei are sampled, where δ equals one-third of the side length of the cube (δ = 13 ε) and ei is the ith Euclidean base vector. wi is deﬁned as the min{ f (c − δei ), f (c + δei )}, 1 ≤ i ≤ N, and the division is done in the order given by wi , starting with the lowest wi . Therefore the hypercube is ﬁrst partitioned along the direction of lowest wi and then the remaining ﬁeld is divided along the direction of the second lowest wi and so on until the hypercube is partitioned in all directions. From this point onwards, the algorithm starts identifying the potentially optimal hyper boxes, dividing these hypercubes further and sampling at their centre points until the termination criteria is satisﬁed. Potentially Optimal Hypercubes Suppose the unit hypercube is divided into m smaller hypercubes. Let ci denote the centre point of the ith hypercube and εi the distance from the centre point to the vertices. One cube among these m hypercubes must be selected for further sampling. The deﬁnition of the potentially optimal hypercube is deﬁned as follows: Let ξ be a positive constant and fmin be the current lowest function value. A hypercube j is said to be potentially optimal if there exists some rate of change constant K > 0 such that f (c j ) − Kε j ≤ f (ci ) − Kεi , f or any i = 1, ..., m

(12.10)

f (c j ) − Kε j ≤ f (min) − ξ| fmin |

(12.11)

Fig. 12.12 illustrates the above deﬁnition further. The horizontal co-ordinate is the size of the hypercube, which is the distance from the centre to the vertices of the cube. This captures the goodness based on the amount of unexplored region in the search space. The vertical co-ordinate is the value of the cost function at the centre point of the particular hypercube. This captures the goodness of the interval with respect to the local search, that is the goodness based on the known function values. Each point on the graph

P.P. Menon, D.G. Bates, and I. Postlethwaite

Obj. func. value at centrepoint of hypercube f(c)

282

fmin fmin − ξ| fmin | Size of Hyperbox Fig. 12.12. Graphical interpretation of hypercube selection [112]

represents a hypercube. The ﬁrst condition in the deﬁnition forces the hypercube to be on the lower right of the convex hull of the asterisks. Hypercubes having low objective function values are inclined to fall on the convex hull of the set, as are (relatively) large hypercubes. The largest hypercube is chosen for division. The second condition insists that the lower bound for the interval, based on the rate of change constant K, exceeds the current best solution by a nontrivial amount. This condition prevents the algorithm from becoming too local in its orientation. In terms of Fig. 12.12, it implies that some of the smaller intervals might not be selected. In this way, the groups of hypercubes are larger, and consequently the iteration places a stronger emphasis on the value of the objective function at the center point of the hypercube, which biases the search locally. The parameter ξ was introduced to balance the local and global search [112]. In Fig. 12.12, the point (0, fmin − ξ| fmin |) changes the convex hull so that the hypercube with smallest objective function value need not be potentially optimal. By this approach, more sampling is done in larger, unexplored hypercubes. If ξ = 0.01, then the lower bound for the hypercube would have to exceed the current best solution by more than 1%. It has been suggested that best results are obtained for ξ with values ranging from 10−3 to 10−7, [112], and therefore a value of 10−4 was used in this study. The hypercubes represented by points on the lower right convex hull of this graph satisfy the above equations and thus are potentially optimal. The pseudo code for the DIRECT algorithm is given in the Table 12.9. For more details about the algorithm, and performance comparisons on standard test problems with other algorithms, the reader is referred to [69, 112]. Termination Criterion The same termination criterion as that chosen for the GA and DE trials was used.

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Table 12.9. Deterministic optimisation algorithm 1. Normalise the search space to unit hyperbox. 2. Sample the center point c1 of the hyperbox; Evaluate f (c1 ). Set fmin = f (c1 ), m = 1, t = 0 (iteration counter), and TC = 0 (Termination Counter) 3. While Termination criterion not satisﬁed (TC ≤ 15) do a) Identify the set S of ‘potentially optimal’ hyperboxes b) Select any rectangle/box j ∈ S c) Divide the box j as follows: i. Identify the set I of dimensions with the maximum side length ε. Let δ equal onethird of this maximum side length (δ = 13 ε). ii. Sample the function at the points c ± δei , ∀i ∈ I, where c is the center of the box and ei is the ith unit vector. iii. Divide the box j containing c into thirds along the dimension I, starting with the dimension with the lowest value of wi = min{ f (c ± δei )}, and continuing to the dimension with the highest wi . Update fmin , xmin and m. d) Set S = S − j. If S = GO TO STEP (b) e) Set t = t + 1. Calculate Improvement in f min obtained from previous iteration. TC = TC + 1 if improvement ≤ 1e−4 in subsequent iterations, if not set TC = 0. 4. END of While End of DIRECT Algorithm. BEGIN Hybridisation 5. Choose the solution xmin from STEP4, set xinitial = xmin and do ‘ f mincon‘ algorithm to reﬁne the global solution. End of H-DIRECT Algorithm.

Hybridisation of DIRECT The DIRECT algorithm is modiﬁed using a simple hybridisation strategy. In this study, a local optimisation method based on Sequential Quadratic Programming (SQP) is incorporated into the DIRECT algorithm. The implementation uses speciﬁcally the function “fmincon”. The solution obtained from the DIRECT algorithm is considered as the initial solution for the local optimisation method. The DIRECT algorithm including the local optimisation is referred to as H-DIRECT . The psuedocode of the modiﬁed algorithm is also given in Table 12.9. The hybridisation attempts to overcome one of the main known disadvantages of the DIRECT algorithm, i.e. its lack of precise convergence to solutions that are on the bounds of the uncertain parameter space, due to the center point sampling strategy. Results The global solution found for this example is the following:[∆∗mass, ∆∗xcg , ∆C∗ m , ∆∗Iyy , δe

∆C∗ mα , Max(α(t))] = [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908◦]. The maximum cost achieved by the DIRECT algorithm alone is 35 deg, at an expense of 550 simulations. The H-DIRECT algorithm was able to improve this result to 36.0908 deg, for an additional cost of 435 simulations. In a sense, it can be argued that the basic DIRECT algorithm performs relatively well. However, in order to track the true worstcase global solution, it is necessary to augment the method with a local scheme. The single parameter sensitivity analyses about the solution obtained from H-DIRECT is

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essentially the same as the one given in Fig. 12.9. The computational complexity of the H-DIRECT algorithm is comparable with, but slightly better than, that of the HDE (on average). Other signiﬁcant advantages of the H-DIRECT algorithm in the context of the industrial ﬂight clearance problem, however, are the repeatability of the results, and the guaranteed proofs of eventual convergence.

12.5 Clearance of Continuous Region of the Flight Envelope Due to limited time and resources, the current ﬂight clearance process employed by the European aerospace industry uses a gridding approach, whereby the various clearance criteria are evaluated over a gridding of points in the aircraft’s ﬂight envelope [68]. A major difﬁculty with this approach is the fact that there is no guarantee that the worstcase violations of a given clearance criterion have in fact been found, since in practice only a few selected points in the aircraft’s ﬂight envelope can be checked. All the previously published research on using optimisation methods for the ﬂight clearance problem have focused on clearing the control law at a single point in the ﬂight envelope, but for continuous variations in the uncertain parameters. Here, the ﬂexibility of global optimisation methods is exploited to show how continuous regions of the ﬂight envelope can be cleared in a single analysis. To extend the optimisation-based approach to clear a continuous region of the ﬂight envelope, the two most promising optimisation schemes from our study, namely the hybrid differential evolution (HDE) and hybrid DIRECT (H-DIRECT) algorithms, were selected. A continuous region of the ﬂight envelope, with Mach varying from 0.4 to 0.5 and altitude varying from 1000m to 4000m, is considered. In the general framework these parameters were then simply included in the vector of uncertain parameters which is returned by the optimisation algorithm to the simulation routine. Naturally, care must be taken to update the scope of each variable globally, when the different ﬂight conditions are considered. As simultaneous variations in uncertainties over a region of the ﬂight envelope are considered, the actual allowable bounds for parameters ∆Cmδe and ∆Cmα were scaled down by a factor of 62% (hence new bounds are [−0.031, 0.031]) and for parameter ∆Iyy by a factor of 50% (hence new bound is [−0.1, 0.1]), compared to the bounds given in Table 12.1, in accordance with current industrial practice. Other uncertain parameters have the same bounds as earlier. For the envelope clearance, a full pitch stick input of amplitude 80N was used. The hybrid differential evolution optimisation begins with 25 randomly generated ﬂight conditions and associated uncertain parameters, to give a candidate solution vector. The initial population size was doubled, due to the increased complexity contributed by the additional search variables, Mach and Altitude. About each ﬂight condition, the aircraft is trimmed straight and level. Once trim is achieved, the model is simulated for a 10 second simulation with the pull up maneuver as deﬁned in Section 12.2. The objective function was evaluated using Eq. 12.3. Apart from increasing N p , the number of initially generated solution vectors, from 12 to 25, no other variation was required in any of the other optimisation algorithm parameters. The basic differential optimisation algorithm is still the same as the one used so far, i.e. “DE/rand/1/bin”.

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To investigate the consistency and repeatability of the analysis, twelve different trials with the HDE algorithm were conducted. The performance of the optimisation algorithm over these trials is shown in Fig. 12.13. From Fig. 12.13 it is observed that 75 Mach

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the algorithm visits other potential local solution points during the course of tracking the global solution. Figs. (12.14, 12.15, 12.16) show the worst-case values of Mach, Altitude, α and ∆ computed in each of the 12 different trials, together with the required number of simulations. The statistics of the results from the different trials are given in Table 12.10. The trial which required the maximum number of simulations 0.03

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(the 3rd of the 12 trials) took 2 hours and 50 minutes on a Pentium 4 machine. The results of all trials are extremely consistent, providing strong conﬁdence that the true worst-case values of all parameters are being identiﬁed. Note also that in this analysis the worst-case values of the aerodynamic uncertainties ∆Cmδe and ∆Cmα are not on their bounds. For the uncertain aerodynamic parameters, the standard deviation value is slightly higher when compared to that of the other uncertain parameters. To analyse this further, a sensitivity analysis is performed about each of the twelve obtained solutions. The results are shown in Figs. (12.17 - 12.21). Note that the x-axis is normalised. 75

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Optimisation-Based Flight Control Law Clearance 75

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In a similar way, mach and altitude sensitivities over twelve different trials are given in Figs. (12.22-12.23). It can be seen from Fig. (12.17) that the dependence of αmax on ∆xcg is essentially linear between -0.075 to -0.02. Similarly from Fig. (12.18), the αmax dependence on ∆mass has a linear characteristic between -0.1 to 0.005. The sensitivity to ∆Cmδe variations is ﬂat between 0.02 to 0.04, and during this interval the αmax values are maximum as shown in Fig. 12.20. αmax has a maximum value for ∆Cmα variations between 0.0 to -0.02, as given in Fig. 12.19. These sensitivity analysis diagrams explain the reason for the slightly larger variation in the values of the worst-case aerodynamic uncertain parameters over the 12 different trials, since the worst-case response surface is nearly ﬂat over a certain region of the parametric space. The average computational complexity of clearing the continuous region of ﬂight envelope using the HDE algorithm, with the worst-case αmax solution accuracy having a standard deviation value of 0.0108, was 3984 simulations. When the same continuous region of the ﬂight envelope was analysed with HDIRECT, it took 3172 simulations, 20% fewer simulations compared to the HDE algorithm on average. The solution obtained from H-DIRECT is given in Table 12.11. The maximum cost obtained was 74.109 deg. Note that the solution is very close to the average value of the solutions obtained from the twelve different trials of HDE, given in Table 12.10. Moreover, since the H-DIRECT algorithm is deterministic, multiple trials are not required to validate results and formal proofs of convergence are available in literature. For the purposes of comparison, a maximum number of 5000 Monte Carlo simulations, a method currently widely used in industry, were performed and the solution is compared with the one obtained from the optimisation-based techniques. The worst-case AoA obtained over the same region of ﬂight envelope from the Monte Carlo simulation technique was only 59.23 deg. The solutions from the Monte Carlo simulations are also given in Table 12.11 for comparison. It is clear that the location of the worst-case uncertain parameter vector obtained from the Monte carlo

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simulations in the parametric space is entirely different from that of the actual worstcase parameter vector. Fig. 12.24 shows the α distribution obtained from the Monte Carlo simulations. Even though the Monte Carlo simulation used almost 2000 more simulations than than required by the H-DIRECT algorithm, the worst-case AoA obtained was only 59.23 deg, with less than 100 simulations attaining an AoA greater than 55 deg. This clearly demonstrates the signiﬁcant advantage of optimisation-based ﬂight clearance over sampling or gridding-based approaches, both in terms of efﬁciency (fewer simulations required) and accuracy (more reliable identiﬁcation of true worst-cases). Finally, Fig. 12.25 shows the ﬂight envelope, with the continuous

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Table 12.11. Flight envelope results α(max) [deg]

Solution Method

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region of ﬂight envelope considered for the present analysis shown as a rectangle. The worst-case ﬂight condition obtained from the analysis is also shown in Fig. 12.25. Note that the worst-case ﬂight condition obtained is not on the bounds of the considered region, hence there is a high chance of missing the true worst-case solution in the case of the standard gridding approach. To conﬁrm this further, a single ﬂight condition worstcase analysis was conducted at each of the vertices of the considered continuous region of ﬂight envelope. The results are given in Table 12.12. To evaluate the performance of the methodology still further, twelve trials of the local optimisation scheme “fmincon” 2000

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with a set of initial points generated by randomly sampling the parameter space were undertaken. To ensure a fair comparison, the local search with random initial guesses was allowed to run for three hours, slightly longer than the maximum time required for the HDE algorithm over the twelve trials. The maximum value of AoA obtained over the 12 trials of the local algorithm with random initial guesses was 65.0081 degrees, which is almost 10 degrees lower than the maximum value found by the proposed approach.

12.6 Computation of Worst-Case Pilot Inputs 12.6.1

Analysis I: Nominal Clonk Analysis

In this initial analysis, all parametric uncertainties in the ADMIRE simulation are set to zero, the pilot control inputs yREF (t) speciﬁed by the Clonk manoeuvre are applied over the ﬁnite time period τ, and the maximum value of the chosen cost function (AoA), given in Eqn. 12.3 is computed. According to the speciﬁcations for the Clonk manoeuvre [75, 72], the pilot’s pitch stick command switches, with a limited rate, between its maximum magnitude limits when the pitch attitude reaches its maximum or minimum. The roll stick command is simultaneously switched to the opposite extremum to that of the pitch stick command. However, once the roll stick reaches an extremum, it immediately starts moving in the opposite direction at a deﬁned rate, called the roll return rate. The pitch stick command, on the other hand, remains for some additional time on its magnitude limit - this time period is referred to as the pitch stick delay. At the next occurrence of a maximum or minimum of the pitch attitude the next switching for both the stick commands occurs, and this sequence is then repeated for a speciﬁed period of time.In this analysis, the Clonk model is used together with the full nonlinear 6-DOF ADMIRE simulation model consisting of the gain scheduled ﬂight control system, servo system having amplitude and rate saturation, sensor models and accumulated transport delays. The framework for Clonk analysis is illustrated in Fig. 12.26. The state, pitch angle θ(t), from the closed loop aircraft model is fed back to the Clonk generator block. The Clonk generator provides the pitch and roll reference input signals to the aircraft model according to the Clonk sequence generation logic discussed above. The aircraft model is initially trimmed straight and level at Mach 0.4 and Altitude 3000 meters. Results Fig. 12.28(a) shows the pilot input commands generated by the Clonk manoeuvre. It is observed that the occurance of maximum AoA is not a simple function of δe -delay and δa -dot, but may instead demonstrate a very irregular behaviour [201, 75]. For the Clonk analysis, a pitch and roll stick deﬂection rate of 720 N/sec and 500 N/sec, respectively, and a pitch stick delay and roll return rate of 1 sec and 128 N/sec, respectively, were recommended by the developers of the ADMIRE model, [73], and were used in this study. According to the speciﬁcations for the Clonk manoeuvre [75, 201], the pilot’s pitch stick command switches, with a limited rate, between its maximum magnitude limits when the pitch angle θ reaches its maximum of 23.19 degrees at 3.4 seconds. The roll stick command is simultaneously switched to the opposite extremum to that of the pitch stick command. Once the roll stick reaches an extremum, it immediately starts

Optimisation-Based Flight Control Law Clearance

AIRCRAFT − NONLINEAR MATHEMATICAL MODEL

AIRCRAFT − NONLINEAR MATHEMATICAL MODEL

. x(t)=f(x(t),u(t),∆ )

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moving in the opposite direction at a deﬁned rate, called the roll return rate. The pitch stick command, on the other hand, remains for some additional time on its magnitude limit - this time period is referred to as the pitch stick delay, and is ﬁxed at 1 second. At the next occurrence of a maximum or minimum of the pitch attitude the next switching for both the stick commands occurs, and this sequence is then repeated for the speciﬁed period of time. Fig. 12.28(a) shows the corresponding AoA and pitch angle time history - the maximum AoA achieved was 16.0038 degrees. Neither actuator magnitude saturation nor any signiﬁcant actuator rate limiting were observed for this sequence of pilot inputs. 12.6.2

Analysis II: Robust Clonk Analysis

In this section we show how GA-based global optimisation methods may be used to allow consideration of parametric uncertainty in the aircraft model during the Clonk Analysis. In this analysis, for the pilot control inputs yREF (t) speciﬁed by the Clonk manoeuvre over the ﬁnite time period τ, the combination of uncertain parameters, ∆ in the aircraft simulation model that maximises the chosen cost function (AoA), Eqn. 12.3, is computed. Fig. 12.27 provides the framework for performing the Robust Clonk Analysis. It consists of the closed loop ADMIRE aircraft model and Clonk generating logic as in analysis I, augmented with a global optimiser block. Based on the AoA response from the model, the optimisation algorithm (optimiser block) assigns the cost value, maximum AoA, as given in Eqn. 12.3 to each of the candidate uncertain parameter vectors, and over iterations tries to identify the uncertain parameter vector that maximises the cost. The Clonk generator logic adaptively generates the pitch and roll reference input depending on the pitch angle (θ) response from the closed loop model. Prior to the

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simulation of each different Clonk sequence, the aircraft model is trimmed straight and level at Mach 0.4 and altitude 3000 meters, to account for the effect of the uncertain parameters on the trimmed state of the aircraft.

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Results In this analysis, pilot control inputs determined by the Clonk manoeuvre were applied, while the GA-based optimisation software was used to compute the worst-case combination of uncertain parameters. The uncertain parameters deﬁne a hyperbox of dimension 5, and each uncertain parameter deﬁnes a gene for the problem considered. Table 12.1 shows the various uncertain parameters considered in the present study, and the bounds for each of these uncertain parameters. The maximum amplitude of pitch and roll input were limited to ±40N [73]. The chromosome length depends on the solution accuracy chosen, which for this study was 1e-6. With this accuracy, each chromosome consisting of 5 genes, is of total length 105 bits. A simple roulette wheel selection scheme is used as selection operator for the GA-based optimisation software. A single point crossover scheme and simple binary mutation are employed with probabilities 0.8 and 0.05 percentage, respectively, [166], [82]. The results of the Robust Clonk Analysis are shown in Fig. 12.28 (b) and Table 12.13. Fig. 12.28 (b) shows the control input sequence and AoA and θ time responses over the ﬁnite time period considered. Table 12.13 gives the uncertain parameter combination that gives the maximum overshoot in AoA. As expected, the effect of considering uncertainty in the aircraft simulation model has been to signiﬁcantly increase the maximum value of AoA achieved. Table 12.13. Analysis II results , ∆∗mass ∆∗xcg ∆C∗ m

δe

∆∗Iyy ∆C∗ m

α

max α(t)

[0.0413 0.0750 0.0499 -0.1218 0.0500 ] 28.6008

12.6.3

Optimisation-Based Worst-Case Pilot Input Computation

In the following sections, we show how optimisation methods may be used to compute worst-case pilot input signals for the ADMIRE model. The results should be compared with the two previous analyses using the Clonk maneouvre. The general framework for the proposed optimisation-based analysis is shown in Fig. 12.29. Three different types of analysis were performed, namely: 1. Analysis III: computation of worst-case pilot inputs for the nominal simulation model 2. Analysis IV: computation of worst-case uncertain parameters for worst-case pilot inputs computed on the nominal model 3. Analysis V: simultaneous computation of worst-case pilot inputs and uncertain parameters Depending on the type of analysis the search space is deﬁned by bounded pilot control inputs, bounded uncertain parameters, as given in Table 12.1 or bounded pilot control inputs along with the bounded uncertain parameters.

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AIRCRAFT − NONLINEAR MATHEMATICAL MODEL . x(t)=f(x(t),u(t),∆ ) y(t)=h(x(t),u(t))

u(t)=h(x(t),yREF (t))

Analysis III: Only Ref. Demand yREF (t) Analysis IV: Only ∆ Analysis V: Ref. Demand yREF (t) and ∆

GA Optimiser

Fig. 12.29. The framework for the optimisation-based analysis

12.6.4

Analysis III: Computation of Worst-Case Pilot Inputs for the Nominal Simulation Model

For the nominal simulation model, the pilot control inputs yREF (t) over the ﬁnite time period τ that maximise the chosen cost function (AoA), Eqn. 12.3, are computed. In the present study, pilot stick inputs only are considered. The throttle input is considered ﬁxed in these studies, because we are essentially dealing with the short-period time response. Also, to be consistent with the approach followed in the Clonk Analysis methodology, the rudder input is not applied. However, the inclusion of additional pilot inputs for future studies will be seen to be automatic. The optimisation problem is to search for a manoeuvre over a deﬁned time period, τ, which can induce large AoA values and thereby depart the aircraft. The search is deﬁned in a bounded 2-dimensional function space F = {Pstk (t), Rstk (t)}. The chromosome for the GA is deﬁned as an element in this function search space F consisting of two genes, each representing a ﬁnite time varying activity Pstk and Rstk respectively, for 0 ≤ t ≤ τ. In order to reduce the computational complexity of the optimisation problem, the continuous search space is discretised into four possible amplitude levels as shown in Table 12.14. The time axis is also discretised, in the sense that changes in the magnitude of pilot inputs are allowed to occur only at regular intervals of one second, and values of pilot inputs are held constant for at least one second. A one second input frequency is high enough to allow highly aggressive but still realistic pilot inputs, [202]. Since only four discretisation levels are used for the magnitude of the pilot inputs, in a binary representation, two bits can represent each pilot input signal over each one second interval of time. Note that the duration of typical test inputs used in industry for the validation of ﬂight

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control laws is between 5 and 7 seconds [202], [166]. Discussions with aircraft design experts and test pilots resulted in the choice of a 5 second sequence of control inputs followed by a 5 second period with the stick centered for this study. With this form of input, there are thus 1024 (45 ) possible different combinations for each gene in the GA, and a chromosome consists of two different genes corresponding to Pstk and Rstk . Table 12.14. Pilot control input discretisation levels and their binary representation Pitch Stick

Roll Stick

Binary Levels

Full MAX [+40N] Full MAX [+40N]

11

Half MAX [+20N] Half MAX [+20N]

10

Half MIN [-20N] Half MIN [-20N]

01

Full MIN [-40N]

00

Full MIN [-40N]

For analysis III, each chromosome is of length 20 bits - each gene consisting of 10 bits (5 times 2 bits per second) - representing an arbitrary manoeuvre of 5 seconds duration about the nominal ﬂight condition. At the end of the ﬁfth second, all the control inputs are brought to zero and the simulation is allowed to continue for another 5 seconds. Each simulation therefore lasts for a total of 10 seconds. The cost function depends only on AoA, which is determined by the short period dynamics of the aircraft. Usually, the time period of this mode is quite short, and hence, 5 seconds is a reasonable period of time for the simulation to effectively excite this mode. However, an additional 5 second time period is allowed to explore the possibility of any induced nonlinear effects in the near stall region of the aircraft. The maximum AoA value over the 10 second time history is the ﬁtness associated with each chromosome. In all subsequent analyses, the total simulation time is kept ﬁxed at 10 seconds. Each chromosome is interpreted as the corresponding sequence of control input deﬂections. The sequence of control input variables tagged with the time proﬁle are applied to the aircraft model, after initially trimming the ADMIRE model at straight and level ﬂight at Mach 0.4 and altitude equal to 3000 metres. Analysis III Results In analysis III, worst-case pilot control inputs are computed using the GA-based optimization software for the nominal simulation model. The maximum AoA obtained from the analysis is 50.0233 degrees. Fig. 12.30(a) shows the corresponding pilot control inputs while Fig. 12.30(b) shows the AoA time history, given by the continuous bold line. Note that the maximum AoA value is signiﬁcantly higher than the one obtained by using the pilot inputs determined by the Clonk manoeuvre in both Analyses I and II. Up until 3 seconds when the pitch stick is at its negative maximum -40N (push and nose down) the roll stick input is at its positive maximum. At the 3rd second the pitch stick is deﬂected to its positive maximum and the roll stick deﬂection is slaved

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50

70

No ∆

50

0

40

−50

Before stick shaping elements After stick shaping elements

0

2

6

4

8

10

30 α(t)

Pitch Stick (N)

∆ present

60

20

50 Roll Stick (N)

10

0

0

−50

−10

Before stick shaping elements After stick shaping elements

0

2

4

6

8

Time (Sec)

(a) Pitch and roll control Inputs

10

−20

−30

0

2

4

6

8

10

Time (sec)

(b) α(t) responses

Fig. 12.30. Analysis III and analysis IV results

to pitch stick, but in the opposite direction. This activity effectively provides a large demand on the elevons. Note that the present manoeuvre appears much less “agressive” than that employed in the Clonk Analysis, but in fact results in a much larger build-up of AoA. 12.6.5

Analysis IV: Computation of Worst-Case Uncertain Parameters for Worst-Case Pilot Inputs Computed on the Nominal Model

In this analysis, the combination of uncertain parameters, ∆, that maximises the chosen cost function (AoA), Eqn. 12.3, is computed for the worst-case pilot control inputs yREF (t) computed in Analysis III. In this case, the uncertain parameters deﬁne a hyperbox of dimension 5, and each uncertain parameter deﬁnes a gene for the problem considered. The chromosome length depends on the solution accuracy chosen, which for this study was 1e-6. With this accuracy, each chromosome is of length 105 bits, consisting of 5 genes each of 21 bits. Analysis IV: Results In this analysis, the worst-case pilot control inputs generated by analysis III are used, while the GA-based optimization searches for the worst-case combination of uncertain parameters ∆. This corresponds to searching for the worst-case pilot inputs and uncertain parameters separately. The uncertain parameter combination given in Table 12.15 gave the maximum value of AoA (61.7187 degrees) for this approach. The dotted line shown in Fig. 12.30(b) shows the AoA time history. Note that the worst-case values of two of the uncertain parameters are inside the hyperbox that deﬁnes the search space. Interestingly, if the value of ∆xcg is changed to be at its maximum allowable value (which would correspond to the “worst-case” value most likely suggested by an intuitive interpretation of ﬂight mechanics principles), the maximum AoA build up turns out to be only 48.7636 degrees. This result clearly shows the limitations of relying

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entirely on ﬂight mechanics intuition when analysing highly nonlinear ﬂight control problems. It also illustrates the inadequacy of current industrial approaches to identifying worst-case uncertain parameter combinations based on evaluating all combinations of minimum and maximum values of the parameters, [68]. Table 12.15. Analysis IV results , ∆∗mass ∆∗xcg ∆C∗ m

δe

∆∗Iyy ∆C∗ m

α

max α(t)

[0.1000 0.0104 -0.0500 -0.2000 0.0429 ] 61.7187

12.6.6

Analysis V: Simultaneous Computation of Worst-Case Pilot Inputs and Uncertain Parameters

For the ﬁnal analysis, the combination of uncertain parameters, ∆, and the pilot control inputs yREF (t) over the ﬁnite time period τ that maximizes the chosen cost function (AoA), Eqn. 12.3, are computed simultaneously. In this case, the chromosome is of length 125 bits, of which 20 bits consist of the pilot control input information as in Analysis III, and the remaining 105 bits consist of uncertainty parameter information as in Analysis IV. Analysis V: Results Analysis IV has already shown the signiﬁcant effect of the uncertain parameters on the maximum AoA value when a speciﬁc ﬁxed pilot input activity is considered. In this analysis both the pilot control inputs and uncertain parameters ∆ are optimised simultaneously. The solution from this analysis therefore has two parts, one the pilot control inputs and the other the value of the uncertain parameters. The results are shown in Fig. 12.31 and in Table 12.16. Fig. 12.31 shows the pilot control input combination and the corresponding AoA time history. Table 12.16. Analysis V results , ∆∗mass ∆∗xcg ∆C∗ m

δe

∆∗Iyy ∆C∗ m

α

max α(t)

[0.0767 0.0739 0.0400 0.1856 0.0500 ] 69.36189

By the sixth second the aircraft response in sideslip has gone outside of the range of the available aerodynamic database and hence the simulation stops. The worst-case combination of uncertainties is given in Table 12.16. The maximum AoA overshoot obtained is 69.3543, which is signiﬁcantly higher than that obtained in all of the previous analyses. Notice also that (a) the worst-case pilot control input is very different from the one computed in the previous analysis, and (b) the worst-case values of all the uncertain parameters are now located inside the search-space hyperbox.

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Pitch Stick Input (N)

50

70

Before stick shaping elements After stick shaping elements 60

0 50

Roll Stick Input (N)

2

6

4

8

10

α(t)

40

−50 0 50

30

20

0 10

−50

Before stick shaping elements After stick shaping elements

0

2

4

6

8

10

0

0

Time (Sec)

(a) Pitch and roll control Inputs

1

2

3

4

5

Time (Sec)

(b) α(t) responses

Fig. 12.31. Analysis V results

12.7 Conclusions This chapter has compared the suitability of a number of different optimisation algorithms for the problem of clearance of nonlinear ﬂight control laws for highly augmented aircraft. The study employed a typical nonlinear clearance criterion used by the European aerospace industry together with an industry standard simulation model of a high performance aircraft with a full authority ﬂight control law. The necessity of using global optimisation methods for ﬂight clearance was clearly demonstrated by the different results obtained by gradient-based optimisation when starting from different initial points in the parameter space. The GA method, on the other hand, converged to the exact global solution in 65 out of 100 different trials, while the DE algorithm found the true worst-case solution in 90 out of 100 trials. Particularly striking is the fact that DE achieves this improved accuracy in tracking the global solution with a reduced computational overhead - taking an average of 3086 simulations, 31% faster than the average of 4485 simulations required by the GA. Hybrid versions of both algorithms incorporating local gradient-based optimisation were shown to offer signiﬁcant advantages in terms of both reduced computational complexity and improved global convergence properties. The hybrid version of the GA (HGA) algorithm employing the SQP local optimisation scheme converged to the global solution in 92 out of 100 individual trials, requiring an average of 2011 simulations. The hybrid version of the DE (HDE) algorithm, again employing the SQP local optimisation scheme, converged to the exact global solution in 98 out of 100 independent trials while requiring an average of only 1106 simulations - 45% faster than the hybrid GA algorithm, and only about three times more computationally expensive than local gradient-based optimisation alone. These results, together with the fact the the DE method is particularly simple both to understand and implement, indicate that when appropriately augmented with local optimisation methods this approach has the potential

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to signiﬁcantly improve both the reliability and efﬁciency of the current industrial ﬂight clearance process. Also in this study, the hybrid global/local optimisation algorithm, HDE, was developed and applied to the problem of analysing the robustness of a nonlinear ﬂight control law over continuous regions of the ﬂight envelope. The ﬂexibility of global optimisation methods was exploited to allow for clearance of the ﬂight control law over continuous regions of the ﬂight envelope in the presence of multiple continuous variations in key aircraft parameters. The proposed approach avoids the principal limitations of the current industrial ﬂight clearance process, namely that only combinations of extreme variations in uncertain parameters are checked over a gridding of isolated points in the aircraft ﬂight envelope. It may therefore offer the potential to further improve both the reliability and efﬁciency of the industrial ﬂight clearance process for highly augmented aircraft. The hybrid differential evolution algorithm took an average of 3984 simulations to obtain the worst-case αmax solution accuracy with a standard deviation value of 0.0108. It was also demonstrated that, when clearing continuous regions of the ﬂight envelope, the worst-case need not necessarily occur at the vertices of the region, but may be well inside the continuous region of ﬂight envelope considered. Though these results are very promising and of signiﬁcant industrial interest, one signiﬁcant criticism voiced by the European aerospace industry of all of the probabilistic methods considered above was the lack of rigorous proofs of convergence to the global solution, with the resulting need for (expensive) multiple trials to provide statistical conﬁdence measures that the global solution has in fact been found. For ﬂight clearance in particular, the repeatability of the worst-case analysis is highly important in convincing industrial practitioners to adopt these new methods. To address this issue, a deterministic global optimisation algorithm, DIviding RECTangles (DIRECT), was also applied to the problem of nonlinear ﬂight clearance. Individual ﬂight conditions as well as a continuous region of the ﬂight envelope were considered in the analysis. The results were extremely promising and the algorithm achieved almost exactly the same quality of solution as that obtained from the best of the evolutionary methods, with a reduced effort in computational complexity. Incorporation of local gradient based optimisation into the DIRECT algorithm was shown to signiﬁcantly improve convergence to the global solution. Since the proposed algorithm, denoted H-DIRECT, is deterministic, multiple trials are not required to validate results and formal proofs of convergence are available. These properties make it a strong candidate for incorporation into the industrial ﬂight clearance process along with the proposed hybrid evolutionary candidate optimisation schemes. For the worst-case analysis at a single ﬂight condition, the H-DIRECT algorithm took 985 simulations, 121 simulations fewer than the average number of simulations required by HDE. To clear a continuous region of the ﬂight envelope, the number of simulations required by H-DIRECT was 3172, 20% less than that required by HDE on average. Finally, an analysis method based on the use of global optimisation and nonlinear simulation, is introduced in this chapter, which may be used as part of the process of clearing a ﬂight control law against departure susceptibility and/or violations of envelope protection limits. The ﬂexibility of global optimisation methods is shown to allow for the simultaneous computation of worst-case pilot inputs and worst-case

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combinations of uncertain parameters in the nonlinear aircraft simulation model. The results show that only such a simultaneous consideration of worst-case pilot inputs and uncertain parameters is likely to reveal the true worst-case behaviour of the aircraft. An important advantage of this type of analysis is the possibility of incorporating many other different ﬂight clearance criteria in the problem setup. Therefore, it is possible to identify the type of control input combination, in the presence or absence of uncertain parameters, that can give the worst-case violation of any deﬁned linear or nonlinear ﬂight clearance criteria. For example, it would be easy to exend the proposed approach to identify the control input combinations which can cause a Pilot Induced Oscillation (PIO). In this problem, any known PIO prediction criteria [68, 190] (for e.g., Gibbson criteria, Neal-Smith criteria, OLOP criteria etc.) can be used as the cost function for the problem, and the rest of the methodology will remain essentially the same. Also, alternaive cost functions, which focus completely on maximizing the actuator saturation could be considered. For a model like the ADMIRE, for example, the cost function could be deﬁned as follows: J = max YRLI/P −YRLO/P , where YRLI/P and YRLO/P are the input and output signals to the rate limiter scheme.

13 Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods Mikhail G. Goman1, Andrew V. Khramtsovsky1, and Evgeny N. Kolesnikov2 1 2

De Montfort University, Leicester, England, UK LE1 9BH [email protected] Bombardier A´eronautique, Montreal, Canada H4S 1Y9

Summary. In this chapter, the manoeuvring capabilities of the ADMIRE model are analyzed using qualitative methods via computation of attainable equilibrium sets, local stability maps and two-dimensional cross-sections of stability regions for stable equilibria considering the velocityvector roll manoeuvre. A nonlinear dynamic inversion control law is implemented as a prototype of a control and stability augmentation system. The analysis of the closed-loop dynamics allows one to assign the airplane manoeuvre limitations via speciﬁcation of ﬂight envelope critical boundaries. A functional interconnect between control inputs helps to avoid control surfaces saturation and to signiﬁcantly enhance the ADMIRE manoeuvrability at high angles of attack. A number of computational examples for attainable equilibrium sets and regions of attraction illustrate the applied investigation methodology. Keywords: qualitative methods, attainable equilibrium sets, local stability maps, manoeuvre limitations, nonlinear dynamic inversion, region of attraction, critical disturbances, aircraft departure.

Nomenclature G F Fx H, M h X,Y, Z L, M, N ny , nz n p, q, r pW , qW , rW V x yµ α, β θ, φ

= manoeuvre vector function = right-hand sides of aircraft rigid-body equations of motion = Jacobian matrix = ﬂight altitude and Mach number = unit vertical vector = longitudinal, side and normal aerodynamic force = rolling, pitching and yawing aerodynamic moment = side and normal load factor = load factor vector = body-axes angular rates = wind-body axes angular rates = ﬂight velocity = state vector = manoeuvre parameter vector = angle of attack and sideslip angle = body axis pitch and roll Euler angle

D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 301–324, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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ω = rigid body angular velocity vector Ω = velocity vector roll rate (Ω pW ) Subscripts ε = equilibrium Abbreviations ADMIRE = Aero Data Model In Research Environment AES = Attainable Equilibrium Set NDI = Nonlinear Dynamic Inversion RA = Region of Attraction

13.1 Introduction The ADMIRE model has been created using a realistic large-scale aerodynamic database for a generic ﬁghter aircraft with a delta-canard conﬁguration [115]. The aerodynamic functions are deﬁned over a wide ﬂight envelope with a signiﬁcant range of Mach number (0.3 ≤ M ≤ 1.2) and wide ranges of angle of attack (−30◦ ≤ α ≤ 90◦ ) and sideslip angle (−30◦ ≤ β ≤ 30◦ ) at low subsonic speeds. This ﬂight envelope provides a challenging case study for aircraft dynamics analysis and control law design. The objective of this case study is to test the potential of various nonlinear control law analysis and design methods, which can enhance a development cycle for aircraft control systems and facilitate the post design clearance process. An efﬁcient set of computational tools for aircraft ﬂight dynamics simulation, nonlinear dynamics analysis and control law design is necessary for this type of investigation. The tools should be capable of coping with inherent model nonlinearities and control constraints [68]. The application of qualitative theory and bifurcation analysis methods for the investigation of nonlinear aircraft dynamics is now a well established approach. Various kinds of aircraft instability and loss of control such as high incidence departures, wing rock behaviour, spin and roll-coupled manoeuvres have been effectively investigated using continuation techniques and bifurcation analysis methods. The advanced computational methods based on qualitative theory and bifurcation analysis, ﬁrst introduced in ﬂight dynamics in [33, 158], were further explored in [9, 49, 83, 91, 92, 108, 142, 188, 245]. Computation of aircraft global stability, parameter continuation and eigenstructure assignment techniques have been successfully applied to several control law design problems in [35, 84]. Bifurcation diagrams for high incidence ﬂight have been found useful for planning piloted simulations [143, 185] and the development of ﬂight clearance tools [87]. Early research results were discussed in detail in [88, 146]. The bifurcation analysis method has been recently extended for the evaluation of aircraft trimmed ﬂight conditions by the inclusion of manoeuvre kinematical constraints [4, 184]. This extension generalizes aircraft performance evaluation using stability and controllability analysis typically used for one-parameter continuation techniques. Although the computational methods based on continuation and bifurcation analysis have obvious advantages in solving ﬂight dynamics problems, they have so far not been widely adopted in engineering practice. One of the reasons for this is a lack of

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user-friendly software tools, which can be used by engineers who are not experts in qualitative theory, continuation and bifurcation analysis methods [85]. The main goal of this chapter is to demonstrate the potential of investigations based on the Attainable Equilibrium Sets (AES) computed for a variety of aircraft manoeuvres. The two-dimensional cross-sections of AES with local stability maps for the openloop system allow the designer to visualize airframe manoeuvring capabilities. The AES cross-sections for the closed-loop system are used for the assessment of a designed control law in terms of its non-local impact on aircraft stability and controllability. A reconstruction of two-dimensional cross-sections of regions of attraction for locally stable steady manoeuvres complements the investigation of the aircraft’s nonlinear dynamics, thus helping to deﬁne manoeuvre limitations. A brief outline of the ADMIRE model aerodynamic characteristics and the rigid body equations of motion used for qualitative investigation is given in section 2. The computation of attainable equilibrium sets and reconstruction of two-dimensional crosssections of regions of attraction are described in sections 3 and 4, respectively. The implementation of the Nonlinear Dynamic Inversion (NDI) control law is outlined in section 5. Section 6 contains a number of computational examples for the velocityvector roll manoeuvre at different ﬂight conditions. These results show how qualitative methods can contribute to the process of nonlinear dynamics analysis, control law design and validation.

13.2 Equations of Motion and ADMIRE Aerodynamic Characteristics The rigid aircraft equations of motion can be presented in autonomous form when equations for three aircraft coordinates specifying its position in space and for heading angle are omitted. The following autonomous 8th-order nonlinear system of ﬁrst order differential equations: V˙ = (1/m) [ ((X + T ) cos α + Z sin α ) + Y sin β ] +g(sin β cos θ sin φ − cosα cos β sin θ + sin α cos β cosθ cos φ) α˙ = q − (p cosα + r sin α) tan β + (1/mV)[Z cos α − (X + T ) sin α] +(g/V cos β)(sin α sin θ + cosα cos θ cos φ) β˙ = p sin α − r cosα + (1/mV) {Y cos β − [(X + T ) cos α + Z sin α ] sin β } +(g/V)(cos α sin β sin θ + cosβ cos θ sin φ − sin α sin β cos θ cos φ) p˙ = ((IYY − IZZ )/IXX )qr + L/IXX q˙ = ((IZZ − IXX )/IYY )pr + M/IYY r˙ = ((IXX − IYY )/IZZ )pq + N/IZZ θ˙ = q cos φ − r sin φ φ˙ = p + (q sin φ + r cosφ) tan θ

(13.1)

is normally considered for the investigation of aircraft dynamics using qualitative and bifurcation analysis methods [158, 33, 84]. System (13.1) is used for analysis of aircraft performance characteristics and also for spin dynamics investigation. Steady-state equilibrium solutions of system (13.1) correspond to a set of helical trajectories with

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rotation around a vertical axis. Note that this class of trajectories includes the straightand-level ﬂight (when angular velocity is zero), level turns, ascending and descending straight and helical trajectories and also all forms of aircraft spin. Manoeuvrability characterizes how fast an aircraft orientation in space can be changed in order to gain an advantageous attitude or to eventually divert the ﬂight trajectory to gain a better position. A manoeuvrability level is directly linked with maximum attainable values for the normal load factor, or angle of attack, and the angular velocity in roll. An aircraft motion with intensive rotation and arbitrary oriented aerodynamic force does not belong to the set of steady-state solutions of the system of equations (13.1). In this case, for manoeuvrability evaluation the effect of the gravitational force in system (13.1) is neglected and ﬂight speed is assumed to be constant. These simpliﬁcations allow the reduction of equations (13.1) to the 5th order autonomous system [207, 242]: α˙ = q − (p cosα + r sin α) tan β + (1/mV)[Z cos α − (X + T ) sin α] β˙ = p sin α − r cosα + (1/mV) {Y cos β − [(X + T ) cos α + Z sin α ] sin β } p˙ = ((IYY − IZZ )/IXX )qr + L/IXX q˙ = ((IZZ − IXX )/IYY )pr + M/IYY r˙ = ((IXX − IYY )/IZZ )pq + N/IZZ

(13.2)

The nondimensional aerodynamic coefﬁcients CX,Y,Z 1 and Cl,m,n 2 in the ADMIRE model are represented as nonlinear tabulated functions of Mach number M, motion parameters α, β, p, q, r and deﬂections of aerodynamic control surfaces δroe , δrie , δloe , δlie , δle f , δr , δc . The effect of the aeroelastic deformations (in quasistatic form) on aerodynamic stability and controllability characteristics is introduced through the dependence ˙ The on dynamic pressure q, ¯ the normal load factor nz and the pitch acceleration q. ranges of motion parameters and control surfaces deﬂections, for which the ADMIRE model is deﬁned, vary across the ﬂight envelope [115]. Some basic dependencies of the ADMIRE aerodynamic coefﬁcients for longitudinal and lateral-directional motions are presented in Figs. 13.1-13.14. The normal force and the pitching moment coefﬁcients at low subsonic speed (M = 0.4) are shown in Figs. 13.1 and 13.2, respectively, as functions of angle of attack for different elevon and canard deﬂections (the inboard and outboard elevons are used here as one integrated control surface). The canard deﬂections at high angles of attack generate signiﬁcant increments in the normal force coefﬁcient and in the pitching moment coefﬁcient as well. For example, in the angle of attack range 23◦ α ≤ 44◦ the elevons cannot produce the pitch-down moment; while canard deﬂection δc = −55◦ gives a substantial recovery margin Cm = −0.2. Fig. 13.3 shows the pitching moment coefﬁcient stability derivative Cmα (α, M) as a function of Mach number, and Fig. 13.4 shows the nonlinear dependence of the pitching moment coefﬁcient Cm (α) for several subsonic and supersonic Mach numbers. In the subsonic region the ADMIRE model is aerodynamically unstable in pitch with a static instability margin of approximately +7%. In the supersonic region the ADMIRE airframe gains static stability in pitch. 1 2

X = qSC ¯ X , Y = qSC ¯ Y , Z = qSC ¯ Z L = qSbC ¯ ¯ ¯ m , N = qSbC n l , M = qScC

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods CN vs AoA[deg] at M=0.4

305

C vs AoA[deg] at M=0.4 m

δe= −25°, δc=−55°

0.4

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25 −55 25 −55 25

0.3 0

1.5

0.2 25 0.1

m

−CZ

1 e

c

0.5 0 25

0

0

20

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C

δ =−25°, δ = −55°

60

25 −55 25 −55 25

0

−0.1 −0.2 −0.3 −0.4

80

Angle of attack, deg

Fig. 13.1. Normal force coefﬁcient CZ (α) for different elevon and canard deﬂections (M = 0.4)

0

20

40

60

Angle of attack, deg

80

Fig. 13.2. Pitching moment coefﬁcient Cm (α) for different elevon and canard deﬂections (M = 0.4)

−1

−3

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Cma [deg ] vs Mach

x 10

−3

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Cm vs AoA[deg]

x 10

M= 0.4 0.8 0.9 1.0 1.2

3

2 2

0 −1 −2

Cm

C

m

α

1 °

α=0

α=10° °

α=20

0 −1 −2

−3 −4 0

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1.5

Mach number, M

Fig. 13.3. Cmα (M) for different angles of attack

−3

−20

0

20

40

60

80

Angle of attack, deg

Fig. 13.4. Cm (α) for different Mach numbers (δe = δc = 0)

At low angles of attack the stability margin is ≈ −14%, however, it gradually decreases with angle of attack; for example, Cmα ≈ 0 at α = 10◦ when M = 1.1. Figs. 13.5-13.8 show the dependencies of the rolling moment Cl and yawing moment Cn coefﬁcients on angle of attack, sideslip and Mach number. The variation of these coefﬁcients with angle of attack and sideslip at M = 0.4 are shown in Figs. 13.5 and 13.6. At high angles of attack α > 20◦ the dependence on sideslip angle becomes very nonlinear. The dihedral stability derivative Clβ (α, M) and directional stability derivative Cnβ (α, M), presented in Fig. 13.7 and Fig. 13.8, respectively, can produce a destabilizing effect in the lateral-directional motion at high angles of attack and at high supersonic speeds. For example, the local departure criteria Cnβ at M = 0.4, shown in Fig. 13.14, dyn changes its sign at α = 24◦ indicating a potential onset of instability in the lateraldirectional modes at high angles of attack. The aerodynamic control derivatives Cl∆δe (differential deﬂection of the outboard elevon sections) and Cnδr (rudder deﬂection), shown in Figs. 13.9 and 13.10,

306

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov Cl vs α [deg]

α= −10 0 10 20 30 40

0.06 0.04

0.06 0.04 0.02

n

l

0.02

C

C

0

0

−0.02

−0.02

−0.04

−0.04

−0.06

−0.06

−0.08 −30

−20

−10

0

10

20

α = −10° 0 10 20 30 40

0.08

°

0.08

−0.08 −30

30

−20

−10

Sideslip, deg

Fig. 13.5. Rolling moment coefﬁcient Cl (α, β) at low subsonic speeds

3 M=0.4 0.8 1.0 1.2 1.5

1 0

x 10

1 0

C

Cl

n

β

30

2

β

2

−1

−2

−3

−3 0

10

20

M=0.4 0.8 1.0 1.2 1.5

−1

−2

−4 −10

30

Angle of attack, deg

0

10

20

30

Angle of attack, deg

Fig. 13.7. Dihedral stability derivative Clβ (α) for different Mach numbers

Fig. 13.8. Directional stability derivative Cnβ (α) for different Mach numbers

−3

1

20

−3

x 10

−4 −10

10

Fig. 13.6. Yawing moment coefﬁcient Cn (α, β) at low subsonic speeds

−3

3

0

Sideslip, deg

−3

x 10

°

0

α = −10 0 10 15 20 25

0.8

α= −10° °

−0.5

10

°

10

C

Cl

∆δ

nδ

r

e

0.6

x 10

−1

0.4 −1.5

0.2

0 0

0.5

1

1.5

Mach number, M

Fig. 13.9. Rolling moment control derivative Cl∆δe (α, M)

−2 0

0.5

1

1.5

Mach number, M

Fig. 13.10. Yawing moment control derivative Cnδr (α, M)

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

Pitching moment Cm

0.25

= − 25 deg

δe = −25°

0.08

−15 − 5 0 5 15 25

0.06

left

δe = − 25 deg

0.2 0.15 0.1 0.05 0 −0.05 −30

right δe =

−20

δleft= e

25 deg −10

0

10

°

δe= 25

°

15 0

0.04

°

−15

−25°

0.02 0 −0.02 −0.04 −0.06

25 deg

20

Differential elevons ∆δ , deg

l

right

δe

Rolling moment C

0.3

307

30

e

Fig. 13.11. Pitching moment coefﬁcient Cm vs differential elevon deﬂection ∆δe (α = 20◦ , δc = 0, M = 0.4)

−0.08 −30

−20

−10

0

10

20

Differential elevons ∆δ , deg

30

e

Fig. 13.12. Rolling moment coefﬁcient Cl vs differential elevon deﬂection ∆δe (α = 20◦ , δc = 0, M = 0.4)

respectively, have a high level of controllability in the subsonic region, but this is signiﬁcantly reduced in the supersonic region. The loss of control authority is inﬂuenced by compressibility effects and also by aeroelastic deformations at ﬂight regimes with high dynamic pressure. The left and right elevon sections are used for generating control moments both in pitch and roll channels. Due to physical deﬂection limits |δle | ≤ 25◦ , |δre | ≤ 25◦ , there exists an unfavorable coupling between control inputs in these two channels. The mean δle + δre value of the right and left elevon pairs δe = (it is supposed that the inboard and 2 outboard sections are deﬂected identically) generates the pitching moment, while the δre − δle differential elevon deﬂection ∆δe = generates mostly the rolling and yawing 2 moments. The pitching moment coefﬁcient for different mean elevon values (−25◦ ≤ δe ≤ 25◦ ) and various differential elevons ∆δe are shown in Fig. 13.11 for α = 20◦ at M = 0.4. Due to physical constraints of the elevons the available deﬂection amplitude for differential elevons ∆δe depends on the mean or symmetric elevons position δe . The maximum amplitude is reached when the mean elevons position is equal to zero δe = 0. The available amplitude for differential elevons shrinks to zero ∆δe → 0 when the mean elevons value approaches its maximum or minimum constraint limits δe = ±25◦. Fig. 13.12 shows the rolling moment coefﬁcient produced by differential elevons ∆δe for different δe values; one can see that the maximum aerodynamic control efﬁciency is reached when the elevons are close to their neutral position |δe | ≤ 5◦ . A canard deﬂection generates signiﬁcant pitching moments, which can be used for aircraft control along with elevons. The use of canard control can contribute to aircraft trimming at low angles of attack, but it is indispensable at high angles of attack for pitch-down recovery control (see Fig. 13.2). Different combinations of elevon and canard deﬂections can provide the same pitching moment. Therefore additional optimization criteria are required in order to determine the best control inputs. For example, a

0

2.

8

−0 .05

−0 .1

2.8

2.4

2.6

.15

−0

2.6

20

0.015

2.4

2.2

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov

0.01

0.05

3

0.005

0

0.1

05

0.

2.4

.05

−0

2.8

−0.1

dyn

10 2.2

−0.005

δ =10°

2. 2

−0.01

20 30 0

2

−0.015

3

3.2

5 0.0

0

C

2.6

2.8

3

0.15

1

0.

3

0

0.2

0

lef

5

0.1

−30

−20

5

−10

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8 2.

8 1.

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−20

2.6

−10

−0.05

nβ

0 2.4

Elevon deflection δe (deg)

308

0

10

Canard deflection δc (deg)

20

Fig. 13.13. Isoquants for pitching moment coefﬁcient Cm (solid lines) and lift to drag ratio CL /CD (grey concentric lines) at α = 20◦ and M = 0.4

−20

0

20

40

60

80

Angle of Attack, deg

Fig. 13.14. Lateral-directional stability parameter Cnβdyn (α) vs leading edge ﬂap deﬂection δle f at M = 0.4

control allocation scheme minimizing aerodynamic drag can be used [53]. To illustrate this approach Fig. 13.13 shows the isoquant curves in the plane of elevon and canard deﬂections for the pitching moments Cm = 0, ±0.05, ±0.1, ±0.15, etc. and for the lift to drag ratios CL /CD = 3.2, 3.0, 2.8, etc. The optimum canard deﬂection depends on angle of attack and Mach number. At ﬂight conditions with α = 20◦ and M = 0.4 the optimum canard deﬂection for zero pitching moment is equal to δc = −19◦ (the optimum point is marked by the ﬁlled circle in Fig. 13.13). In all computational examples presented hereafter the canard is deﬂected as a function of angle of attack and Mach number δc = fc (α, M), where fc is calculated to minimize the lift to drag ratio. Fig. 13.14 shows the dependence of the approximate departure criterion Cnβ > 0 dyn calculated for different leading edge ﬂap deﬂections. One can see that the wing leading edge ﬂaps improve the lateral-directional stability at high angles of attack. In order to use this effect the leading edge ﬂaps have been scheduled against angle of attack and Mach number: δle f = fle f (α, M).

13.3 Qualitative Methods and Attainable Equilibrium Sets For brevity, the equations of motion (13.2) in this section are represented in a compact vector form: dx = F(x, δ), x ∈ X ⊂ ℜn , δ ∈ U ⊂ ℜm , dt

(13.3)

where the state vector includes angle of attack, sideslip and rigid body angular rates – x = (α, β, p, q, r)T , the control vector includes the left and right elevons (or symmetric and differential elevons), and rudder deﬂections – δ = (δle , δre , δr )T (δe , ∆δe , δr )T . All equilibrium states in system (13.3) are deﬁned by solutions of the following system of nonlinear algebraic equations: F(x, δ) = 0, x ∈ X ⊂ ℜn , δ ∈ U ⊂ ℜm ,

(13.4)

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

309

∂F , calculated at an equi∂x librium point, are used for the evaluation of aircraft local stability. Any equilibrium solution of system (13.3) with constant-control input δ corresponds to a steady coordinated ﬂight along a helical trajectory with an arbitrary oriented axis. As the effect of the gravitational force is not included, these equilibria are considered as the pseudo steady state solutions in comparison with the equilibria of equations (13.1) [207, 242]. Critical ﬂight regimes such as spin, autorotation in roll or wing rock oscillations can coexist with the normal ﬂight conditions at the same control inputs. Entry to one of these critical regimes depends on the history of control actions and also on external disturbances. In addition to local stability analysis, these multiple-attractor dynamics require an investigation of regions of attraction (or domains of attraction) for each stable regime of motion. The identiﬁcation of manoeuvre limitations due to onset of instability or loss of control by means of direct numerical simulation only is an extremely difﬁcult and time-consuming task. A qualitative approach based on continuation and bifurcation analysis methods facilitates this task by providing estimates for critical control inputs and dangerous manoeuvre boundaries. Normally these estimates are further veriﬁed in numerical and piloted simulation using a full mathematical model. The continuation of equilibrium solutions of system (13.2) as a function of control inputs, or other parameters, is a basic computational procedure used in qualitative investigations of aircraft nonlinear dynamics. One-parameter continuation is normally performed for various trajectories in the parameter space along with simultaneous analysis of the eigenvalues of the linearized system (13.2). The bifurcational parameters when aircraft dynamics experience qualitative changes are connected with a number of equilibrium points and with the onset or disappearance of closed orbits. Local bifurcations can be analyzed entirely via migrations of equilibrium eigenvalues across the imaginary axis and periodic orbit multipliers through the unit circle. In this chapter we implement a more systematic approach for the evaluation of aircraft manoeuvrability boundaries. It is based on the computation of all attainable equilibrium states. The set of equilibrium states of system (13.2) is bounded due to natural physical constraints available for the control effectors. This attainable equilibrium set (AES) incorporates all possible trim points xε generated by the available bounded control from the allowable region U:

The eigenvalues and eigenvectors of the Jacobian matrix J =

AES = {xε : F(xε , δ) = 0, δ ∈ U ⊂ Rm }

(13.5)

The equilibrium xε of system (13.2) is characterized by only three parameters, namely, angle of attack α, sideslip angle β and wind-body axes roll rate pW Ω = p cosα cos β+ q cosα sin β + r sin α cos β. Two other wind axes projections qW and rW of angular velocity vector ω specify its inclination from the velocity vector V and deﬁne the radius of the aircraft’s helical trajectory. This equilibrium solution is a natural velocity-vector roll manoeuvre, which is speciﬁed by three kinematical parameters (α, β, Ω). The definition of the velocity-vector roll manoeuvre, for example given in [54], just restricts the velocity vector manoeuvre to a manoeuvre with zero sideslip and is incorrect in claiming that ω is parallel to V .

310

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov

13.3.1

Inverse Trimming Formulation

The inverse trimming procedure is used here for computation and visualization of attainable equilibrium manoeuvres deﬁned by system (13.2). Every sustained velocity-vector roll manoeuvre of interest is characterized by a vector of kinematical parameters yµ = (αµ , βµ , Ωµ )T or yµ = (nzµ , nyµ , Ωµ )T implicitly specifying the full state vector (α, β, p, q, r)T . To determine the required control inputs δε and equilibrium states xε for a manoeuvre, speciﬁed by parameter vector yµ , the following augmented system of trim equations is solved: 0 = F(xε , δε ), F ∈ ℜ5 (13.6) yµ = G(xε , δε ), G ∈ ℜ3 , where vector function G can be deﬁned as either G = (α, β, Ω)T or G = (nz , ny , Ω) (CN qS/mg,C ¯ ¯ Ω)T . Y qS/mg, The number of kinematical constraints in (13.6) is equal to the number of control parameters, thus system (13.6) consists of eight equations with ﬁve unknown states (α, β, p, q, r) plus three unknown controls (δe , ∆δe , δr ). This guarantees that for every manoeuvre speciﬁed by vector yµ , there is an isolated solution of system (13.6) represented by state vector xε (ﬁve parameters) and control vector δε (three parameters). In case of direct trimming of system (13.6), the control input δ is known a priori, while in the inverse trimming case it is deﬁned implicitly by system (13.6). Note that in both cases the local stability of the open-loop steady manoeuvre is deﬁned by the eigenvalues of the Jacobian matrix J = Fx (xε , δε ). Due to inertia and aerodynamic nonlinearities, system (13.3) at given control inputs may have several stable equilibrium solutions. In a one-parameter continuation of the equilibrium solution xε of system (13.3) there is a possibility of fold point bifurcations, which are speciﬁed by condition det [Fx (xε , δε )] = 0. Continuation of equilibrium solution xε and control input δε in system (13.6) in most cases is bifurcation-free (see Fig. 13.15). This guarantees a global convergence of the Newton-Raphson method to the unique equilibrium solution at every particular value of parameter vector yµ . An equilibrium path in manoeuvre parameter space according to (13.6) can follow any prescribed trajectory, which is identical to the so-called ”bifurcation tailoring” technique outlined in [35]. 13.3.2

Computational Procedure

The inverse trimming system (13.6) is solved numerically using the Newton-Raphson method. The ﬂight regime speciﬁed by vector yµ is accepted after successful convergence to solution (xε , δε ), where xε and δε belong to bounded regions X and U, respectively. The ﬂight regime is rejected if during the Newton-Raphson iterations the state vector x moves out of region X or further reduction of the error norm is not possible due to saturation of control inputs. To reconstruct multi-dimensional AES a number of its two-dimensional cross-sections are computed in the plane of two selected manoeuvre parameters on some predeﬁned grid of points. Each point in this grid belongs to AES if the inverse system (13.6) is solved successfully, i.e. the required manoeuvre satisﬁes the kinematical, state and control constraints. In this case the computed point is marked

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

311

Fig. 13.15. Continuation in direct and inverse trimming procedures (top and bottom plots, respectively)

Fig. 13.16. Computation of two-dimensional cross section of region of attraction (RA)

according to its eigenvalue spectrum. For each successful point the calculated results for state vector xε and control vector δε , combined with the eigenvalues and eigenvectors of the Jacobian matrix Fx , are stored in a special data structure for further use and visualization.

13.4 Regions of Attraction The region of attraction (RA) of a stable equilibrium xε is the ensemble of such state points x that the trajectory starting from x eventually approaches xε . For evaluation of multidimensional regions of attraction a method automating the computation and visualization of its two-dimensional cross-sections is implemented [83, 84]. Every locally

312

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov

stable equilibrium xε in the open- or closed-loop system may have a bounded region of attraction. A number of two-dimensional RA cross-sections are generated by direct integration of the dynamical system (13.3). Initial conditions are taken from a grid of points on the two-dimensional cross-section plane P2 (see Fig. 13.16). Final destinations of the integrated trajectories are identiﬁed by checking whether they enter into a close proximity of one of the available attractors stored in the database or leave the state space region of validity of the mathematical model [83, 88]. Grid points on the plane P2 are marked in accordance with the attractor which has been approached by the trajectory initiated in this point. The computed two-dimensional cross-sections allow one to determine critical disturbances leading to departure from the normal ﬂight conditions. A locally stable equilibrium with very small region of attraction should be considered as practically unstable.

13.5 NDI Control Law Control law design based on Nonlinear Dynamic Inversion (NDI) is an active area of research (see, for example, references [45, 58, 86, 128, 195, 212, 215]). In this study, this control law design method has been selected as the best candidate for the qualitative investigation of aircraft nonlinear dynamics and evaluation of its manoeuvring capabilities. The following vector form of equations (13.1) is used here for the NDI control law design: V˙ = g iVT (α, β)(n + h(θ, φ))

(13.7)

⎤ α˙ ⎣ β˙ ⎦ = Aω (α, β)ω + g E(α, β, θ, φ, n) V Ω ω ˙ = J−1 (−ω × Jω + M)

(13.8)

θ˙ = Bω (θ, φ)ω φ˙

(13.10)

⎡

(13.9)

where iVT , Aω , E, Bω (shown in the Appendix) are matrices with dimensions (1 × 3), (3 × 3), (3 × 1), (2 × 3), respectively3, J is the moment of inertia matrix, M is the vector of total moments including the aerodynamic, engine thrust and rotor gyroscopic moments. Measurements for load factor vector n allow direct inversion of the force equations [45]. The design objective is to track the outputs α∗ , β∗ , Ω∗ of a reference model. The reference model is introduced by three linear transfer functions in the channels of angle of attack, sideslip and velocity vector roll specifying the handling quality requirements [169]: ω2β ω2α β∗ Ω∗ k α∗ = 2 , = , = , 2 2 2 αcom s + kΩ s + 2ξαωα s + ωα βcom s + 2ξβωβ s + ωβ Ωcom 3

Note that matrix Aω is nonsingular.

(13.11)

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

313

parameters ωα , ξα , ωβ , ξβ , kΩ in (13.11) depend on ﬂight altitude H and Mach number M. The demanded control moment Mdem should transform the following vector equation ˙ Ω) in (13.8) by differentiation with respect to time: ˙ β, derived from the group (α, ⎡ ⎤ α¨ V˙ g ˙ wω + ⎣ β¨ ⎦ = A (13.12) E˙ − E + Aw · J−1 −ω × Jω + Mdem , V V ˙ Ω to the following decoupled group of linear equations guaranteeing asymptotic convergence of the closed-loop system to the outputs of the reference model (13.11) α∗ , β∗ , Ω∗ : ⎡ ⎤ ⎡ ∗ ⎤ α¨ − K1α (s)(α˙ − α˙ ∗ ) − K2α (s)(α − α∗ ) α¨ ⎣ β¨ ⎦ = ⎣ β¨ ∗ − K1 (s)(β˙ − β˙ ∗ ) − K2 (s)(β − β∗ ) ⎦ . (13.13) β β ˙ ˙ ∗ − K1 (s)(Ω − Ω∗ ) Ω Ω Ω Associating equations (13.12) and (13.13), the control demand Mdem can be explicitly expressed. As the next step the control vector δdem is obtained by the inversion of a function for physical moment depending on motion parameters and control deﬂections with the objective of generating the demanded control moment Mdem M(V, H, M, α, β, ω, δdem ) = ˙ ∗ ). ˙ ω, α, β, θ, φ, α∗ , α˙ ∗ , α¨ ∗ , β∗ , β˙ ∗ , β¨ ∗ , Ω∗ , Ω Mdem (V, n, n,

(13.14)

The transfer functions K{1,2}α,β (s) and K1Ω (s) in (13.13) specify the required stability and robustness properties in the closed-loop system. The NDI control law is perhaps the most adequate control strategy for aircraft stabilization and control, because it is designed using kinematical parameters α, β, Ω which are natural for the velocity-vector roll manoeuvre. In this study an NDI control law is used as a prototype of a control and stability augmentation system during the investigation of the ADMIRE manoeuvring capabilities. To achieve accurate dynamic inversion, the NDI method requires an exact knowledge about aircraft aerodynamic characteristics, inertia parameters, measurements of the normal load factor n and state variables α, β, p, q, r, θ, φ. The rate and deﬂection limits for control effectors were not considered during the NDI control law design. These constraints can dramatically change the closed-loop system behaviour and can lead to aircraft departure in case of an inherently unstable airframe. Examples of such dynamics are presented in the next section.

13.6 Computational Analysis The ADMIRE is a comprehensive nonlinear aerodynamic model covering a wide ﬂight envelope, which includes high angles of attack regimes at subsonic speeds, ﬂight regimes with transonic and supersonic speed regions. This section presents a number of computational examples illustrating the application of qualitative methods. A computational investigation of nonlinear dynamics using qualitative methods is performed with the objective to reveal the ultimate manoeuvring capabilities available for the ADMIRE in the open-loop airframe and in the closed-loop system with the NDI control law.

314

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov M = 0.4, β = 0 60

50

50

40

40

Angle of attack, deg

Angle of attack, deg

M = 0.4, Ω = 0 deg/s 60

30 20 10 0

30 20 10 0

−10

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−30

−20

0

−30 −300

20

−200

−100

0

100

200

300

Velocity roll angular rate, deg/s

Sideslip angle, deg

Fig. 13.17. AES cross-sections with local stability maps: (α, β) cross-section at Ω = 0 (left plot) and (α, Ω) cross-section at β = 0 (right plot). Flight regime M = 0.4 and H = 5 km. Stable

Saddle

Re λ

Im λ

Re λ

Im λ

Re λ

Re λ

Unstable saddle−focus

Unstable saddle−node

−2

Unstable node

Unstable focus Im λ

Im λ

0

Im λ

Re λ

Im λ

Re λ

Fig. 13.18. Colour codes for qualitatively different distributions of equilibrium eigenvalues

13.6.1

Open-Loop Dynamics

Fig. 13.17 shows two slices of the attainable equilibrium set in the plane (α, β) at Ω = 0 (left plot) and in the plane (α, Ω) at β = 0 (right plot) for the bare airframe at ﬂight regime with M = 0.4 and H = 5 km. Local stability of every equilibrium point is speciﬁed by different colours using the following classiﬁcation of eigenvalue spectrum (see Fig. 13.18): 1) ‘green’ area represents stable equilibria of stable focus type (all eigenvalues have negative real parts), 2) ‘red’ area represents aperiodically unstable equilibria of saddle-point type (one positive real eigenvalue), note that the ADMIRE is an aerodynamically unstable airframe at subsonic speeds at low angles of attack, 3) ‘yellow’ area represents oscillatory unstable equilibria of unstable focus type (a pair of complex eigenvalues with positive real part), 4) ‘cyan’ area represents aperiodically unstable equilibria of unstable node type (two positive real eigenvalues), 5) ‘magenta’ area

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods M = 0.4, H=5 km, α = 15 deg 300

200

200

Velocity roll angular rate, deg/s

Velocity roll angular rate, deg/s

M = 0.4, H=5km, α = 0 300

100

0

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20

100

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−100

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30

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200

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100

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−100

−200

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−10 0 10 Sideslip angle, deg

20

30

20

30

M = 0.4, H=5 km, α = 30 deg 300

Velocity roll angular rate, deg/s

Velocity roll angular rate, deg/s

M = 0.4, H=5 km, α = 25 deg 300

−300 −30

315

20

30

100

0

−100

−200

−300 −30

−20

−10 0 10 Sideslip angle, deg

Fig. 13.19. AES cross-sections with local stability maps in the plane (β, Ω) at α = 0, 15◦ , 25◦ , 30◦ . Flight regime M = 0.4 and H = 5 km.

represents unstable equilibria of saddle-focus type (one positive real and one unstable complex pair), 6) ‘blue’ area represents equilibria of saddle-node type (three unstable real eigenvalues). At low angles of attack |α| ≤ 15◦ in the absence of rotation, the equilibrium states of the ADMIRE model are aperiodically unstable (red region). Some green regions with stable equilibria appear at high positive and negative angles of attack due to aerodynamic nonlinearities (for example, see Cm (α) in Fig. 13.2 and Cl,n (α, β) in Figs. 13.5 and 13.6) and also at intensive velocity roll rotation due to the effect of nonlinear inertia moments (see equations (13.2)). Fig. 13.19 shows four AES slices in the plane (β, Ω) for the following values of angle of attack: α = 0, 15◦ , 25◦ , 30◦ (ﬂight regime is the same as in Fig. 13.17, M = 0.4 and H = 5 km). The local stability maps show that with variation of sideslip angle or velocity vector roll rate the equilibrium spectrum may change signiﬁcantly. For example, the regions with divergence instability (red area) adjoin with stable regions (green area), the stable regions adjoin with oscillatory instability regions (yellow area), etc. Nonlinear aerodynamics and inertia moments in equations (13.2) generated by intensive velocity-vector roll rotation can produce stabilizing and destabilizing effects. Note that on all slices the colour distribution is topologically consistent with possible types of eigenvalue bifurcations.

316

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov Delection of control surfaces

Equilibria eigenvalues λ=f(Ω,β=0)

δre (deg)

0

6

Saturation of right elevons

Im λ

−10 −20 −30

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200

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2

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10 5

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0 −5

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−2

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10

Oscillatory unstable

Stable

−4

r

δ (deg)

20

0

0

50

100

150

200

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250

300

−6

−1.5

−1

−0.5

0

Fig. 13.20. Deﬂections of control surfaces δre , δle , δr (left plot) and trajectories of equilibrium eigenvalues (right plot) for manoeuvre speciﬁed in Fig.13.19 by the vertical dashed arrow

The vertical dashed arrow in Fig. 13.19 (left top plot) represents a velocity-vector roll manoeuvre with maximum angular velocity Ωmax = 300 deg/s at zero sideslip (β = 0) and zero angle of attack (α = 0). Increase in angular velocity Ω stabilizes the ADMIRE dynamics when the equilibrium point crosses the green area and destabilizes when the equilibrium point enters the yellow area. Changes in colour correspond to equilibria bifurcations, the saddle-node bifurcation separates the red and green areas, the Hopf bifurcation separates the green and yellow areas. During computation of AES cross-sections the deﬂections of control surfaces and eigenvalues for equilibrium points are stored in a special data structure for further analysis. As an example, the control deﬂections and the locus of eigenvalues for the manoeuvre ”along the dashed arrow” (see Fig. 13.19) are presented in Fig. 13.20 on the left and right plots, respectively. With variation of the angular velocity Ω control deﬂections and equilibrium eigenvalues vary quite smoothly with two changes in equilibrium local stability. The right elevons saturate at δre = −25◦ at the boundary of AES when Ω → 300 deg/s. The ADMIRE manoeuvring capabilities vary with altitude and Mach number due to changes in aerodynamic stability and control efﬁciency characteristics. A number of AES slices with local stability maps in the plane (α, β) at Ω = 0 and in the plane (α, Ω) at β = 0 are presented for three Mach numbers M = 0.8, 1.0 and 1.2 in Figs. 13.21, 13.22 and 13.23, respectively. The aerodynamic instability at low angles of attack in the subsonic region changes to stability at transonic and supersonic speeds. The AES slices at transonic and supersonic speeds become mostly stable or ”green” and the size of the attainable equilibrium sets reduces with increasing Mach number. 13.6.2

Closed-Loop Dynamics

The ADMIRE airframe is aerodynamically unstable at low angles of attack in the subsonic speed region, however, the instability can also occur across the ﬂight envelope during intensive manoeuvring requiring signiﬁcant changes in the angle of attack and the angular velocity in roll. To stabilize the ADMIRE airframe the NDI control law (see

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods M = 0.8, H=5 km, β = 0 60

50

50

40

40

30

30

Angle of attack, deg

Angle of attack, deg

M = 0.8, H=5 km, Ω = 0 60

20 10 0

20 10 0

−10

−10

−20

−20

−30 −30

−20

−10 0 10 20 Sideslip angle, deg

317

−30 −300

30

−200

−100 0 100 Velocity roll angular rate, deg/s

200

300

Fig. 13.21. AES cross-sections with local stability maps in the plane (α, β) at Ω = 0 (left plot) and in the plane (α, Ω) at β = 0 (right plot). Flight regime M = 0.8 and H = 5km. M = 1, H=5 km, β = 0 60

50

50

40

40

30

30

Angle of attack, deg

Angle of attack, deg

M = 1.0, H=5 km, Ω = 0 60

20 10 0

20 10 0

−10

−10

−20

−20

−30 −30

−20

−10 0 10 20 Sideslip angle, deg

−30 −300

30

−200

−100 0 100 Velocity roll angular rate, deg\s

200

300

Fig. 13.22. AES cross-sections with local stability maps in the plane (α, β) at Ω = 0 (left plot) and in the plane (α, Ω) at β = 0 (right plot). Flight regime M = 1.0 and H = 5km. M = 1.2, H=5 km, β = 0 60

50

50

40

40

30

30

Angle of attack, deg

Angle of attack, deg

M = 1.2, H=5 km, Ω = 0 60

20 10 0

20 10 0

−10

−10

−20

−20

−30 −30

−20

−10 0 10 20 Sideslip angle, deg

30

−30 −300

−200

−100 0 100 Velocity roll angular rate, deg/s

200

300

Fig. 13.23. AES cross-sections with local stability maps in the plane (α, β) at Ω = 0 (left plot) and in the plane (α, Ω) at β = 0 (right plot). Flight regime M = 1.2 and H = 5km.

318

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov M = 0.4, Ω = 0 deg/s

M = 0.4, Ω = 0 deg/s 60

60

open−loop system

50

40

40

Angle of attack, deg

NDI closed−loop system

50

NDI control law

30

30

20

20

10

10

0

0

−10

−10

−20

−20

−30 −30

−20

−10

0

10

20

−30 −30

30

−20

−10

0

10

20

30

Sideslip angle, deg

Sideslip angle, deg

a) α

com

30

α

q q,Ω (deg/s)

α,β (deg)

40 20 0 β

com

β

10

1200

0

−10

10

1000

Ω

Ωcom 0

20

800

30

10

left

0

−10 0

200

0

0

−10

right 10 20 Time (sec)

600 400

10 δr (deg)

δle,δre (deg)

H, m

−20

20

30

b)

0

10 20 Time (sec)

30

100 0 −100 −200

500 0 −500 Xe, m

−Y , m e

c)

Fig. 13.24. Closed-loop dynamics with the NDI control law (M = 0.4 and H = 5 km): a) AES cross-sections for the open- and closed-loop systems in the plane (α, β) at Ω = 0; the NDI control law inputs for nose-pointing manoeuvre αcom = 20◦ + 10◦ cos(0.2πt), βcom = 10◦ sin(0.2πt), Ωcom = 0 (red circle); b) time histories for motion parameters and deﬂections of control surfaces; c) visualization of aircraft spatial trajectory

previous section) is considered as a prototype control and stability augmentation system. The attainable equilibrium sets in the open-loop and the closed-loop systems with the NDI control law totally coincide due to a natural conformity of the control law with aircraft dynamics in the velocity-vector roll manoeuvre. In case of conventional control laws which are normally designed separately for the longitudinal and lateral-directional motion modes, the resulting closed-loop system AES may be smaller then the AES for the open-loop system. The AES for the ADMIRE open-loop system may include both stable and unstable solutions (see previous examples with stability maps), but the ADMIRE closed-loop system with NDI control law is stable, or green, in all attainable equilibrium points. The closed-loop system operates perfectly if the control inputs αcom , βcom , Ωcom are within the bounds of the AES. In the absence of external disturbances this guarantees that there is no control saturation in the closed-loop system.

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

319

b,b_com (deg) vs time (s)

25.1

0.1

24.95

δle

Ω (deg/s)

δle,δre,δr

com

600

5

10

100

δr

0

β

−0.1 0

0 −20

0

−0.05

24.9

20

0.05

Ω

re

200

com

0

0

−50

δ

5 Time (sec)

50

400 H, m

α (deg)

com

25

β (deg)

α

25.05

0

a)

500

200 0 −200

−100 10

1000

5 Time (sec)

10

−Ye, m

0

Xe, m

b)

Fig. 13.25. Velocity-vector roll manoeuvre with NDI control inputs αcom = 25◦ , βcom = 0, Ωcom = 100sin(0.2πt) deg/s; ﬂight regime M = 0.4 and H = 5 km: a) time histories for motion parameters and deﬂections of control surfaces; b) visualization of aircraft spatial trajectory

Two examples of the ADMIRE dynamics with the NDI control law are shown in Figs. 13.24 and 13.25. The ﬁrst one is the nose pointing manoeuvre prescribing simultaneous variation of angle of attack and sideslip angle αdem = 20◦ + 10◦ cos(0.2πt), βdem = 10◦ sin(0.2πt) and maintaining zero projection of the angular velocity vector on the velocity vector, Ωdem = 0 (see the red circle in Fig. 13.24a). The time histories and visualization of the aircraft spatial trajectory for this manoeuvre are presented in Figs. 13.24b and 13.24c, respectively. The second example is the velocity-vector roll manoeuver with NDI control inputs αdem = 25◦ , βdem = 0 and Ωdem = 100sin(0.2πt) deg/s (see Fig. 13.25a and 13.25b). In both examples, the prescribed variations of motion parameters lie entirely within the AES and are not critically fast. They follow the demanded variations rather accurately and there is no saturation of control surfaces in performing these two manoeuvres. If control inputs αcom , βcom , Ωcom reach or exceed the AES boundary, or vary too fast, some of the control surfaces may reach their deﬂection and rate limits and the closed-loop system may lose stability properties speciﬁed by the NDI control law. The boundary of AES is critical if the open-loop system equilibria are unstable at the boundary. Saturation of control effectors in this case may destabilize the closed-loop system and lead to aircraft departure. Critical Disturbances The closed-loop system with the NDI control law (13.14) is locally stable at all attainable equilibrium states with predeﬁned closed-loop stability properties in the decoupled longitudinal, lateral-directional and velocity roll channels. However, due to bounded control, i.e. the control effectors’ amplitude and rate limits, the closed-loop system can lose stability at some critical disturbances and therefore is not globally stable. The size of the region of attraction (RA) shows the level of external disturbances, which can be tolerated by the closed-loop system. This critical level of external disturbances depends on the open-loop system eigenvalues and also on the closed-loop system poles

320

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov Cross−section of RA: α

=25°, β

com

=Ω

com

°

=0

Cross−section of RA: αcom=30 , βcom=Ωcom=0

com

36

40

34

38 36

30

Angle of attack (deg)

Angle of attack (deg)

32

28 26 24

Wgust=14 m/s

22 20

32 30 28 26

W

=8m/s

gust

24

18

22

16 14

34

20 −10

−5

0 Sideslip angle (deg)

a)

5

10

−10

−5

0 Sideslip angle (deg)

5

10

b)

Fig. 13.26. Cross-sections of region of attraction in the plane (α, β) for the NDI control law; ﬂight regime M = 0.4 and H = 5 km: a) αcom = 25◦ at βcom = Ωcom = 0; b) αcom = 30◦ at βcom = Ωcom = 0

assigned by the NDI control law. Fig. 13.26 shows two-dimensional cross-sections of region of attraction with perturbations in the plane of angle of attack α and sideslip β at ﬂight regime M = 0.4 and H = 5 km for two equilibrium points αcom = 25◦ , βcom = 0 and αcom = 30◦ , βcom = 0 when Ωcom = 0. Green squares deﬁne the initial angle of attack and sideslip (α0 , β0 ), when the closed-loop dynamics converges to the equilibrium point. Red points deﬁne initial values (α0 , β0 ), when the closed-loop dynamics do not converge to the equilibrium point during some time interval T and the state trajectory leaves the region X, where the aerodynamic model is speciﬁed. A radius of the circle inscribed inside the RA cross-section with a centre in equilibrium point xε can be used as a measure of critical wind gust disturbance, leading to aircraft departure. For example, when αcom = 25◦ the critical wind gust Wgust = 14 m/s and when αcom = 30◦ the critical wind gust Wgust = 8 m/s. Thus, the closer the equilibrium point to the AES boundary, the smaller the magnitude of critical wind gust disturbance leading to aircraft departure. Every green square and red point from the RA cross-section represents some convergence or departure dynamics, respectively. The time histories for departure and convergence processes are shown in Figs. 13.27a and 13.27b, respectively. There is no saturation of control deﬂections at all in the convergence process starting from the green pentagram shown in Fig. 13.26b. The process started from the red pentagram, which is out of the RA (see Fig. 13.26b), leads to aircraft departure after saturation both in amplitude and rate of deﬂections in the rudder and elevons (see Fig. 13.27b). Fig. 13.28 shows two RA cross-sections with perturbations in the plane (α, β) for two velocity-vector roll manoeuvres with Ωcom = 130 deg/s and Ωcom = 145 deg/s at αcom = 25◦ , βcom = 0. One can see that when angular velocity command input Ωcom approaches the AES boundary, the critical wind gust disturbance decreases signiﬁcantly. As was shown earlier, the AES slices of the ADMIRE model with the NDI control law at subsonic speeds have critical boundaries due to instability of the open-loop system and saturation of its control effectors. When control inputs αcom , βcom , Ωcom approach the boundary of the AES, the size of the RA becomes very small and the closed-loop system is highly susceptible to departure. A critical RA size ∆, reﬂecting

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

5

10

p

−10 0

15

5

10

0

δle,δre (deg)

δr (deg)

−5 left 5 10 Time (sec)

15

β 1

−5 0

5 10 Time (sec)

15

p

−150 2

−200

3

20

0

r

−100

0

15

right

−10 0

20

−20

5

5

0

p,q,r (deg/s)

q

0

q

0 −50

0

1

2

3

1

2

3

0

right

10

−10

0

left

−10

r

β

r

10

α

40

δ (deg)

α 10

60

α,β (deg)

20

0 0

δle,δre (deg)

50

20 p,q,r (deg/s)

α, β (deg)

30

321

−20

−20 −30

0

1

2

−30

3

0

Time (sec)

Time (sec)

a)

b)

Fig. 13.27. Time histories for region of attraction presented in Fig.13.26b (αcom = 30◦ , δ˙ max = 50 deg/s): a) stable dynamics (trajectory starts from green pentagram); b) departure due to saturation of control surfaces (trajectory starts from red pentagram) °

Cross−section of RA: α

24

24

22

22

20

20

18 16 14

Wgust=10 m/s

12 10 8

=0, Ω

com

=145 deg/s

com

18 16

W

=3 m/s

gust

14 12 10 8

6 4

=15°, β

com

26

Angle of attack (deg)

Angle of attack (deg)

Cross−section of RA: αcom=15 , βcom=0, Ωcom=130 deg/s 26

6 −10

−5

0 Sideslip angle (deg)

5

10

a)

4

−10

−5

0 5 Sideslip angle (deg)

10

b)

Fig. 13.28. Cross-sections of region of attraction in the plane (α, β) for the NDI control law; ﬂight regime M = 0.4 and H = 5 km: a) αcom = 15◦ , βcom = 0, Ωcom = 130 deg/s; b) αcom = 15◦ , βcom = 0, Ωcom = 145 deg/s

the level of available external disturbances, can be introduced as a safety margin for assigning the allowable region for control inputs αcom , βcom , Ωcom within a full AES. This safety margin sets realistic limitations on variations of angle of attack and roll angular velocity. These limitations can be used in the design of α- and Ω-limiters which guarantee stable manoeuvring even in the presence of external disturbances. In this way, a combined analysis of AES and RA allows the designer to set sensible limitations on aircraft manoeuvring. Interconnect Between NDI Control Inputs The AES cross-sections with local stability maps provide explicit qualitative and quantitative information about the open- and closed-loop system manoeuvring capabilities. The manoeuvre limitations naturally result from the shape of the AES and from the

322

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov Closed−loop system: M = 0.4, H= 5 km, β = 0 60

50

50

40

40

Angle of attack, deg

Angle of attack, deg

Open−loop system: M = 0.4, H=5 km, β = 0 60

30 20 10 0

departure due to control saturation

30

departure avoidance

20 10

NDI closed−loop AES

0 −10

−10

−20

−20 −30 −300

−200

−100

0

100

200

−30 −300

300

−200

−100

0

100

200

300

Velocity roll angular rate, deg/s

Velocity roll angular rate, deg/s

a)

b) a,a_com (deg) vs time (s) 300

β

0

−200 5

10

0

40

0 −20

β 0

40

40

20

20

Ω (deg/s)

0

−100

0

10

Ω

100

20

−20

5

α

com

5

Ωcom

−200

β

10

15

−300

0

5

10

15

DEL,DER,DR (r/g/b) (deg) vs time(s) 40

right

20 δr (deg)

δle, δre (deg)

right

Ω

200

com

0

−20

left

20

0

0

r

0 −20

0

α

40

δ (deg)

com

com

δle,δre (deg)

β

20

60

Ω

200

α, β (deg)

αcom

α

40

Ω (deg)

α, β (deg)

60

−20

−20

left −40 0

5 Time (sec)

10

−40 0

c)

5 Time (sec)

10

−40

−40 0

5

10

15

0

5

10

15

Time (sec)

Time (sec)

d)

Fig. 13.29. Variable attitude velocity-vector roll manoeuvre; ﬂight regime M = 0.4 and H = 5 km: a) AES cross-section and stability map in the plane (α, Ω) at β = 0 for the open-loop system; b) AES cross-section and stability map in the plane (α, Ω) at β = 0 for the closed-loop system with the NDI control law; c) time histories for motion parameters and deﬂections of control surfaces: αcom = 30◦ , βcom = 0, Ωcom = 100sin(0.2πt) deg/s (classical velocity-vector roll manoeuvre); d) pilot time histories for motion parameters and deﬂections of control surfaces: αcom = 30◦ , βcom = 0, Ωcom = 200sin(0.2πt) deg/s (variable attitude velocity-vector roll manoeuvre)

requirement for stability margin in the proximity of its boundary, i.e. minimum size of region of attraction (RA). Using this knowledge a control law designer can get maximum performance from the available airframe. Fig. 13.29 illustrates the efﬁciency of the AES in shaping the functional interconnect between the NDI control inputs αcom , βcom , Ωcom with the objective of increasing maximum angular velocity in roll manoeuvre. Velocity-vector roll manoeuvres at high angle of attack with αcom = 30◦ and βcom = 0 have a limit for roll rate Ωmax ≈ 90◦ due to saturation of control effectors and departure at the boundary of the AES (black dashed arrow in Fig. 13.29b and the time histories for motion parameters and deﬂections of control surfaces in Fig. 13.29c). However, the maximum roll rate can be signiﬁcantly increased if the velocity vector roll rate and the angle of attack simultaneously move along the AES boundary to lower angles of attack where the limit for roll angular velocity expands (bowed red arrow in Fig. 13.29b). Termination of a roll command stops rotation and returns the angle of attack back to its initial value αcom = 30◦ .

Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods

323

A coordinated change in the angle of attack and the velocity-vector roll rate may follow a safe path along the AES boundary, guaranteeing stability in the closed-loop system. pilot This effect can be achieved by a simple transformation of the pilot command input αcom in accordance with variation of the lateral control input Ωcom : pilot

αcom =

αcom . 1 + kα ∗ Ω2com

The manoeuvre shown in Fig. 13.29b and 13.29d can be called a variable attitude velocity-vector roll manoeuvre. Its implementation signiﬁcantly enhances manoeuvrability at high angles of attack. At supersonic ﬂight regimes a coordinated change of sideslip angle with variation of angular velocity in roll also has the potential to expand the manoeuvrability limit.

13.7 Conclusions The computational framework based on numerical implementation of qualitative methods described in this chapter has proved to be feasible and efﬁcient in the investigation of nonlinear aircraft dynamics. Computation of attainable equilibrium sets (AES), local stability maps and regions of attraction (RA) for the closed-loop system with NDI control law for the velocity-vector roll manoeuvre allow exhaustive analysis of aircraft manoeuvring capabilities and effective assignment of safe manoeuvring boundaries. The approach presented essentially complements the existing simulation methods which are extremely computationally-intensive and unreliable when applied for ﬂight safety evaluation. The qualitative methods can contribute to the control law design process through explicit assignment of manoeuvre limitations, bifurcation avoidance and comparative analysis of attainable equilibrium sets and local stability maps for open- and closedloop systems.

Appendix: Matrices from Equations in Section 5 The unit vectors for local vertical and ﬂight velocity are h(θ, φ) = [ − sin θ iV (α, β) = [

cos αcosβ

cosθ sin φ cos θ cos φ]T , sin β

sin αcos β]T ,

˙ Ω) triad is: ˙ β, The nonsingular matrix for the (α, ⎡ ⎤ − cos α tan β 1 − sin α tan β Aω (α, β) = ⎣ sin α 0 − cosα ⎦ , cos α cos β sin β sin α cos β

324

M.G. Goman, A.V. Khramtsovsky, and E.N. Kolesnikov

The gravitational force inﬂuence matrix is ⎡ ⎤ sin α cos α 0 ⎢ − cos β cos β ⎥ ⎥ E(α, β, θ, φ, n) = ⎢ ⎣ − cosα sin β cos β − sin α sin β ⎦ (n + h) 0 0 0 The Euler equations matrix is Bω (θ, φ)

=

0 1

cosφ tan θ sin φ

− sin φ , tan θ cos φ

Acknowledgements The presented results partially stem from the research projects in De Montfort University (1998-2002) funded by DERA/QinetiQ, Bedford UK. The authors are grateful to Yoge Patel for her fruitful cooperation. Special thanks go to Peter Messer and Jeff Knight at De Montfort University for the support given to GARTEUR FM(AG17) activities.

Part IV

Industrial Evaluation and Concluding Remarks

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14 Industrial Evaluation Matthieu Jeanneau1 , Fredrik Karlsson2 , and Udo Korte3 1 2 3

Airbus, France [email protected] Saab AB, Sweden [email protected] EADS Military Air Systems, Germany [email protected]

14.1 Introduction The Industrial Evaluation task is performed as part of the GARTEUR Action Group to show industry’s view of the work presented by the universities and the research establishments involved in the group. The evaluation also contributes to the process of knowledge transfer of new technologies into industry. The objective is not to ﬁnd the best method for aircraft control law design and analysis but to examine the potential of the presented methods. The purpose of the industrial evaluation is also to assess whether the presented methods can reduce the time and effort required to develop ﬂight control systems or help to develop ﬂight control systems with higher quality and reduced demands on on-board computer capacity.

14.2 Nonlinear On-Ground Control Law Design and Analysis 14.2.1

Reminder of the Objectives

This action group is dedicated to nonlinear design and analysis techniques for aircraft control. The ﬁrst proposed benchmark is a civil transport aircraft, provided by Airbus. The modelling and the requirements focus on the on-ground operation: taxiing, acceleration or deceleration on the runway. Nonlinear dynamics have a particularly strong effect on this phase of the operation of an airliner: examples include aerodynamic loads, ground forces on the tyres, engines, lateral-longitudinal motion couplings, actuators saturations, etc. 14.2.2

Overview of the Work Performed

The partners involved in research dedicated to the on-ground benchmark were ONERA, LAAS, DLR, University of Leicester and University of Bristol. The different studies performed cover an impressive and complementary range of activities: • Use, development and enhancement of modelling tools relevant for analysis and design purposes. D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 327–339, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

328

M. Jeanneau, F. Karlsson, and U. Korte

• Application to the benchmark of a complete modelling approach covering all phenomena impacting the motion and dynamics of the aircraft on-ground. • Development and application of control design methods. • Self-evaluation of the relevance of the proposed methods. • Rigorous evaluation of the achievable results. • Development and application of analysis tools offering stability and performance robustness guarantees. Two main directions were emphasized and discussed in Part II of this book. The ﬁrst one gathers works by ONERA, LAAS and the University of Leicester. It starts with the development of tools for handling LFT objects, as well as recommendations for the modelling of physical systems using LFT objects. The modelling of all the benchmark’s uncertainties, time-varying parameters and nonlinearities is then performed using such LFT objects. Anti windup control techniques are then investigated for the aircraft lateral control. These techniques prove to be relevant to cover nonlinear phenomena such as actuator saturations and ground tyres lateral slip phenomena. The extension of these techniques to produce gain-scheduled control laws is then proposed, in the form of an LPV-AW (Linear Parameter Varying-Anti Windup) approach. Finally analysis tools based on LFT modelling and LMI solvers are proposed. They provide formal robustness guarantees in the presence of uncertainties, time-varying parameters and nonlinearities, for both stability and performance assessment. A second direction was investigated by DLR. The modelling and simulation of the on-ground benchmark is fully based on object-oriented modelling. Each component of the benchmark is modelled through a speciﬁc object: aerodynamics, ground forces, ﬂight mechanics, actuators, etc. Interconnections between objects are also physically oriented. The ﬁnal model gathers all of the benchmark’s phenomena, making it an appropriate simulation environment for analysis or performance evaluation. This modelling can also easily be reduced to its most signiﬁcant components for design purposes for instance. A control law design procedure is then proposed, based on dynamics inversion techniques. This inversion is automatically and symbolically performed by the model compiler, allowing for fast design-evaluation loops. 14.2.3

Main Achievements

The ﬁrst impression following the works performed is a very good match between the needs expressed by the industrial benchmark and the developments and achievements provided by the partners. The research described in chapters 5 to 9 is deﬁnitely of great quality and rigour. These chapters offer perspectives for innovations in term of industrial process when dealing with highly nonlinear systems, which is the case when dealing with the on-ground motion of transport aircraft. The main achievements from the industrial point of view are the followings: • LFT modelling tools, including guidelines for LFT modelling, symbolic manipulations, model reduction and nonlinear symbolic LFT modelling. • On-ground benchmark modelling using LFT’s, including aircraft dynamics, actuators saturations, ground forces and time-varying dependencies such as ground velocity or nose-wheel deﬂection.

Industrial Evaluation

329

• LPV-AW design methods. This parameter varying anti-windup synthesis is coupled with an on-line estimator of the ground forces. Performance in the presence of saturations is guaranteed by design using LMI’s. • Stability and performance robustness assessment for LTI-uncertain and LTV nonlinear systems. The method developed in this action group extends the use of µanalysis to nonlinear systems. In addition, the method allows systems which are dependent on both uncertain and time-varying parameters to be considered. These results are based directly on the LFT modelling of nonlinearities, uncertainties and time-varying parameters, showing here again the complementarity of the works performed. • Object oriented modelling. This offers the possibility for interactive modelling with a reusable library and easy connection of objects, including the development of speciﬁc blocks for the benchmark. Automatic code generation then allows the integration of either a reduced or complete model into a design or simulation environment such as Matlab-Simulink, Dymola or any ﬂight simulation environment. This is of particular interest in a design process requiring iterative loops between synthesis and evaluation. • Symbolic inversion-based control design tools. These tools come with a proposed process for control laws design and evaluation based on object-oriented modelling and simulation. 14.2.4

Chapter Evaluations

Nonlinear Symbolic LFT Tools for Modelling, Analysis and Design In this chapter, the authors summarise the principle of LFT modelling and symbolic modelling support tools, such as LFT connections, concatenation, addition,multiplication,differentiation, etc. Mastering these tools requires a certain level of understanding. However, one of the main advantages of LFT modelling is the use of LFT objects, which make LFTbased systems very easy to handle. The recent development of LFT toolboxes now allows manipulation of these LFT objects with the most advanced manipulations techniques very easily. To use and master these tools, design engineers do not need to master the techniques behind the tools such as the ones presented in this chapter. This is probably a good point to spread LFTmodelling widely in industry.Thechapter also presentsthevariousphenomena that LFT modelling may cover: from uncertain parameter dependencies, to nonlinearities such as saturations or switches, including time-varying parameters or dynamics. Finally a guideline for LFT modelling is proposed. The main advantage of this guideline is that it follows closely the physics of the phenomena modelled, preventing the generation of models and results which are too abstract. Evaluation and understanding of LFT objects’ behaviour is thus simple, requiring only the understanding of the systems modelled. However sticking to the physics may lead to high order LFT objects. Fortunately, rules exposed in the guideline along with manipulation tools allow the complexity of LFT objects to be drastically reduced, without loss of clarity, as highlighted in the next chapter. The use of such objects is likely to expand in industry in the coming years. It offers a relatively simple way of handling nonlinearities. Also, numerous manipulation tools, design methods and analysis techniques based on LFT modelling have emerged in the past years - see chapters

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7 and 9 for instance. Results obtained in this action group are thus likely to help in closing the gap between industry and research developments in LFT-based modelling, design and analysis. Nonlinear LFT Modelling for On-Ground Transport Aircraft This chapter is a perfect illustration of the possibilities offered by LFT’s. A complete symbolic modelling of the on-ground benchmark is performed. This modelling directly and exactly expresses the physics and the equations of each component described in chapter 1. A concatenation of these simple LFT blocks is then performed. Manipulation tools are used to reduce the complexity of the full model, without loss of exactness. If complexity needs to be further decreased, reduction tools and physical simpliﬁcations can be considered. The ﬁnal LFT models, either the exact one or the simpliﬁed one, comprise all phenomena of interest in the benchmark. The nominal plant M(s) contains all dynamics coming from ODE or actuator transfer functions, plus the dependencies with respect to the elements of the ∆ block. The ∆ block allows the designer to consider various nonlinear phenomena: the trigonometric atan function, saturations of the ground-tyre contact (slip), uncertainties in the aerodynamic coefﬁcients, and timevariations of key parameters such as velocities or cornering gains. The illustration of the overall design process is performed on the lateral dynamics of the aircraft. However longitudinal LFT modelling was also performed, and the coupling of the two models is easily performed thanks to the LFT-object manipulations, producing a full longitudinal & lateral LFT model. Note that the identiﬁcation of lateral ground forces is performed using an interesting method based on NDI-based identiﬁcation. This appears suitable for both accurate identiﬁcation of the lateral forces and modelling of their uncertainties using LFT’s. Regarding the use of LFT’s in industry, this chapter highlights two points. Firstly there is a training requirement to become familiar with LFT objects and LFT manipulation tools. However, this training does not require particular skills and should be accessible to most design or modelling engineers. Powerful toolboxes also exist, such as the ones mentioned in chapter 6. They contain most of the necessary tools, along with tutorials and help documentation to facilitate their use. These toolboxes are developed using Matlab, which is the most common modelling and design tool in the ﬁeld of control in industry. Secondly the modelling stage itself using LFT’s is not an instantaneous process. However for control designs and analyses, modelling is a mandatory step. The best approach for industrial designers seeking to use LFT’s would be to start from the very beginning using LFT-objects, instead of their usual Matlab Control toolbox objects or Simulink block-based models for instance. When dealing with nonlinear systems, the switch should be easy to perform as the usual modellings are often either unsatisfactory or too complex. Besides, toolboxes such as the LFR toolbox allow the handling of LFT objects in Simulink. On-Ground Aircraft Control Law Synthesis Using an LPV Anti-Windup Approach The synthesis technique proposed in this chapter directly takes into account the saturations present in the system to avoid unexpected behaviours. The modelling of these saturations is based on LFT’s. The theoretical background of the proposed method

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is guaranteed using LMI’s and Lyapunov functions. The proposed design for the onground benchmark comprises 3 steps. A linear controller ensures the performances in the linear regime of the system. An anti-windup control loop is then added that minimises degraded performances in the presence of saturations. Finally an on-line estimation of the ground forces delivers to the anti-windup controller information about whether the saturations are active or not. One of the most attractive part of this chapter consists in the extension of the anti-windup approach to parameter-varying systems. The proposed LPV-AW approach permits a direct design of an AW controller which is gain-scheduled over the full range of admissible variations and whose performance is guaranteed. As previously mentioned, the method proposed is based on LMI’s and thus requires a LMI solver, which is not a common tool in the industry yet. Once again this requires training of designers who are not familiar with LMI’s and LMI solvers. However mastering the theory of LMI’s and all control applications based on LMI’s is not required. Handling LMI solvers is not difﬁcult in itself. When applied to aircraft, if the focus is on rigid systems only (neglecting structural dynamics), then the systems remain of low order and the LMI solvers usually work ﬁne. The proposed method is then a real innovation for industrial designers looking for methods adapted to nonlinear saturated systems. Problems may arise, however, when considering high order systems, such as aircraft with a many structural modes. The last point of interest for industry is the architecture of the proposed solution. It starts from a classical solution in the form of a linear feedback controller. The anti-windup controller is an additional loop acting to prevent phenomena induced by saturations. This approach therefore represents an improvement on current solutions, likely to preserve classical behaviour in the linear domain and improving the behaviour in the saturated domain, thus increasing the domain of use and the performance of the system. This continuity with classical solutions is a positive point for industrial applications, since industry is usually cautious about innovations that would completely modify their current approach. Rapid Prototyping Using Inversion-Based Control and Object-Oriented Modelling Object-oriented modelling is an increasingly mature technique which is being used more and more in various engineering ﬁelds. In this approach, physical objects and phenomena are modelled separately using class-objects described by their symbolic equations and stored in libraries. The main advantage is for multidisciplinary systems, whose sub-components really stick to the physics and are simply assembled using interconnectors. Model compilers then allow for simulation. Sets of tools also allow for analysis or evaluation of modelled systems. One is the automatic model inversion procedure, which eases the generation of inversion-based control laws. Based on their wide experience of aircraft applications, DLR were able to develop an aircraft ﬂight dynamics library, including objects modelling the behaviour and dynamics of components such as engines, ﬂight mechanics of the airframe, aerodynamics, kinematics, actuators, atmosphere, terrain description, and the necessary interconnectors between these objects. For the on-ground benchmark, a new class of object was deﬁned for the gears, based on the equations described in chapter 1. Modelling of the entire benchmark was then performed by gathering all necessary objects after deﬁning their characteristics

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as deﬁned in the benchmark. The resulting environment then offers many possibilities for post-process, simulation and analysis, including trimming of the aircraft and automatic generation of inversion-based control laws (with an automatic detection of the minimum number of times the outputs should be differentiated). When this approach was applied to the on-ground benchmark, interesting results were obtained. The ﬁrst was the poor behaviour of lateral control laws when the reference demand is the lateral load factor felt by the pilot. This generates strong oscillations on the aircraft yaw rate as well as on the steering nose-wheel. Physical explanations for this phenomenon are then provided, and a more classical control law based on yaw rate demands is designed. The full design process is very detailed and the results presented demonstrate the good behaviour of the controller from very low velocities up to high velocities. The automatic generation of inversion-based control laws is of particular interest to industry. This is ﬁrstly because such control laws are by nature parameter-dependent and therefore do not require aposteriori fastidious gain-scheduling. Secondly, this is a technique that tends to naturally emerge in industry for the control of nonlinear systems whose nonlinearities come from time-varying parameters that can be measured on-line. More and more control designers are thus aware of, even if they are not fully familiar with, these techniques. Another advantage is the usual good disturbance rejection properties of these techniques, as found by the author of this chapter when introducing cross-wind conditions. However, there is no explicit taking into account of robustness requirements. This requires a posteriori checks to guarantee stability and performance in the presence of unmeasured uncertainties or time-dependent parameters. Neither is there any explicit account taken of saturations, as such nonlinearities cannot be inverted. Robustness Analysis Versus LTI/LTV Uncertainties for On-Ground Aircraft This chapter presents an extension of the well-known µ-analysis approach, which is restricted to LTI uncertainties, in order to also handle LTV uncertainties. Nonlinearities modelled using LFT’s are then considered as LTV uncertainties. Thus robustness analysis can be conducted in the presence of LTI-uncertainties, time-varying parameters and nonlinearities. The method is applicable to both stability analysis and performance analysis. Domains of guaranteed stability are provided. Regarding performance, the proposed method deals only with L2-induced norms, which may be restrictive for aircraft. As with the AW techniques, the technique is based on LMI’s, which requires training. However as previously said the effort to handle these tools appears small compared to the potential beneﬁts. Analysis methods for nonlinear systems are often complex and numerically consuming. The method proposed seems simple and well suited for saturation nonlinearities. Regarding the possible conservatism of the method, it is inherent to the computation of an upper bound only for µ. Quantifying this conservatism would require the computation of the µ exactly, or at least a lower bound. The authors’ experience tends to show that this conservatism is small in practice. It would be interesting, however, to illustrate this on the application performed on the on-ground benchmark by ﬁnding a combination of δ parameters, close but outside the guaranteed domain of stability, for which the system is unstable.

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333

Open Issues and Perspectives

There remain some problems which have not been addressed in this project. No longitudinal control law was developed. This is probably less challenging than the lateral control problem. However the coupling between lateral and longitudinal dynamics is certainly of great interest. Longitudinal dynamics strongly impact the payload on each wheel, thus strongly modifying the lateral forces on the tyres. Lateral dynamics also affect the longitudinal forces by creating drag or frictions. Robustness of the lateral and longitudinal control laws to lateral-longitudinal couplings also needs to be analysed. Another open issue is the use of differential thrust and differential braking for enhancing the lateral control of the aircraft. This was not addressed in this project. In the future, the idea of bringing together the LFT-based techniques and the Modelica-based work should be promoted. Regarding the modelling, LFT’s are mathematical descriptions whose main advantage is the taking into account of dependencies. Object oriented modelling’s main beneﬁt is for system building and interconnections. Both allow for numerical treatments. The building of objects based on LFT descriptions allows to retain the beneﬁts of both techniques. Regarding design methods, there are even more beneﬁts to mixing the approaches. NDI techniques generate automatically parameter dependent solutions providing the parameters are measured. But they do not consider robustness requirements to either unmeasured or unmodelled parameters or saturations of the commanded inputs. Anti-windup LPV techniques on the contrary explicitly consider the saturations affecting the system and robustness constraints on the parameters. But the architecture proposed in chapter 7 requires a preliminary control loop for the system when saturations are not active, and this control often requires a gain-scheduling with respect to measured parameters. A mixed solution based on NDI techniques for the preliminary control loop (saturations inactive) enhanced by an LPVAW loop to cope with saturations, would gather the beneﬁts of both approaches. Regarding maturity, the techniques studied and developed in this Action Group have reached a promising R & D level. Promoted by research labs and universities, they were successfully applied on an industrial research benchmark. The next step now belongs to industry, which should appropriate these techniques, integrate them to their modelling, design and evaluation environments and validate them using their usual process, on realtime certiﬁed simulators with all system couplings and possibly up to the integration onboard aircraft for demonstration. With support from labs for tools, tutorials and training of designers, I am quite conﬁdent that most of these methods could set the standards for future techniques in the aeronautical industry when facing nonlinear problems.

14.3 Nonlinear Flight Control Law Design and Analysis 14.3.1

Evaluation Procedure

The evaluation is based on the information supplied in draft versions of the book chapters 10 to 13. Additionally, evaluation questionnaires were used for the design entries in chapters 10 and 11 to form a wider basis for these entries. The questionnaires were answered by a number of industrial engineers with good experience in ﬂight control law design and analysis at EADS and at Saab. The authors of the design and analysis entries

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also answered a ”self-evaluation” questionnaire to get a fair balance between the views of the authors and the evaluators. The analysis methods presented in chapter 12 are further developments of some of the techniques and thoughts presented in the work of GARTEUR Flight Mechanics Action Group 11 on ﬂight clearance. For this entry, the same questionnaire as for the design entries was used and hence several questions do not apply. The investigation presented in chapter 13 is not regarded as a speciﬁc design or analysis entry, but a supporting tool that can be used in both ﬂight control law design and analysis. 14.3.2

Evaluation of Entries

LPV Control Law Design for the ADMIRE Model The design and evaluation of Linear Parameter Varying (LPV) control laws presented in Chapter 10 covers several of the requirements and tasks speciﬁed in the design challenge described in Chapter 4. Using LPV control law design can eliminate or at least reduce the need for gain scheduling. Instead, however, an LPV model that approximates the nonlinear dynamics of the aircraft across the ﬂight envelope has to be developed. The coefﬁcients of this linear model are functions of measurable parameters such as Mach number, speed or dynamic pressure. In a wide ﬂight envelope it is likely that more than one model may be needed, which also is the case here. The book chapter was evaluated by four evaluators. The book chapter gives a description of the method at a high level and how it is used for the purpose intended here. A high level of understanding is needed to use this method for ﬂight control law design and the details of the principles of the method need to be well understood. This would certainly be easier for someone who is already familiar with H∞ design. The effort to learn the method is assessed as difﬁcult to acceptable. Once the method is learned, however, the effort to use it would be similar to the effort needed for methods such as the Linear Quadratic (LQ) design method which is widely used in industry today. The complexity of the design is acceptable and should be of similar complexity to the controllers currently used in industry. For the proposed design the LPV model is needed, but there are fewer data tables for gain scheduling. It is however possible that the complexity would increases for a real application as more considerations than are covered here need to be included. Such considerations would for example be command tuning for gross manoeuvring and ﬁne tracking or reduction/minimisation of overshoot for normal load factor with sufﬁciently small gain. The design is suitable for implementation in an aircraft’s on-board ﬂight control computer as long as the LPV models do not become too complex. It is assumed that the computer power needed for the LPV model is comparable with the computer power needed today for gain scheduling. When it comes to qualiﬁcation and certiﬁcation procedures this design is acceptable as the need to prove stability and performance is the same as for other methods used today. The robustness of the design is considered acceptable to high, as uncertainties or perturbations are considered in the design through weighting functions. The authors also indicate that the design is somewhat conservative. The performance in the presented

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design is acceptable but could probably be improved with additional tuning. For example, the coupling between angle of attack and angle of sideslip is too high at roll manoeuvres. The strengths of the method are that it provides automatic gain scheduling with guaranteed levels of performance and robustness. It also ﬁts well within the H∞ framework which already might be in use. The scheduling for a wider envelope can also be addressed once the tools are established. The weakness is that it is hard to understand how to implement requirements on performance into the design process and to tune it between different command regimes. It is hard to say when the LPV model is accurate enough to get the required performance. Special tuning of the control law structure for speciﬁc parts of the envelope or problem areas can only be done indirectly. Highly nonlinear aircraft dynamics would require LPV models of larger size and complexity and thus rapidely increase the demands on online computation. The inclusion of limiters might also pose some problems. The authors are probably optimistic on the time required to complete the design as compared to current methods. There is a considerable amount of work needed to get an adequate LPV model. Hoever, a re-design might be quicker and LPV probably has an advantage there. Once the model is done after a ﬁrst iteration, changes in basic aerodynamic data may be included quickly. It is not clear if changes in the model (mass and inertia, aerodynamics) will lead to large changes in the elements of the LPV controller which then would require full testing. It would be useful to have a kind of tutorial on building an LPV model, starting with a single coefﬁcient and scaling to a full model. The literature is typically heavier on the theoretical side and is not so much intended for the user that wants hands-on examples. Block Backstepping for Nonlinear Flight Control Law Design The method of block backstepping for ﬂight control law design is presented in Chapter 11. There is a good description of the basic principles of the method. As important topics for realistic controller design such as pilot command generation and blend over between different command variables is excluded, it is somewhat hard to assess the design made with this method. Comparisons with simulations with the current ADMIRE controller would have improved the evaluation. This book chapter was evaluated by three evaluators. The level of understanding needed to use this method is high, as detailed knowledge of the principles and the mathematical background is necessary. It is also considered difﬁcult to acceptable to learn the method. To take over the design and carry out a re-design would be hard as the method gives a lot of ﬂexibility. When the designer is familiar with the method, however, the effort is acceptable and similar to current methods, at least to make a ﬁrst quick design. It is, however, not completely clear if it would be possible to complete a design that covers all practical requirements. The complexity of the design is high to acceptable, but it is difﬁcult to give a general assessment. Implementation in an aircraft’s on-board ﬂight control computer would be suitable as the required computer power is about the same as for an LQ design. The design is acceptable for current qualiﬁcation and certiﬁcation procedures, which would be about the same as for the methods used today.

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From the presented simulations, the design seems to be acceptably robust. There is however no guarantee of robustness with respect to, for example, aerodynamic forces. The performance of the design seems to be high and even unrealistic for some cases. In the presented simulations of an extreme velocity vector roll (Fig. 5), the normal load factor is for example far beyond realistic limits. The control surface activity is unacceptably high for a the presented extreme velocity vector roll as the control surface rate is higher than 50◦ /s. The strength of the method is that it can treat nonlinearities and can cover a large part of the ﬂight envelope without gain scheduling. It should be easy to make a fast design once the method is understood. The method is ﬂexible, but also requires a lot from the designer. The weakness of the method is that it does not account for rate and position saturation explicitly. It can deﬁne desired dynamics only by ﬁrst order ﬁlters, while for the deﬁnition of the handling qualities of the short period mode and the dutch roll mode second order ﬁlters would be desirable. The method does not deliver guaranteed robustness for speciﬁc errors. It is probably hard to set the correct design parameters for the entire envelope. The book chapter leaves a number of questions open because of the limited number of design tasks addressed. Command generation and limiting is not provided and it would be interesting to see how a blend over from angle of attack command to normal load factor command would be realised. This is perhaps a method that is not suited for the entire ﬂight control task required for a ﬁghter aircraft. The method may be better suited for speciﬁc problem areas with strong nonlinearities. Optimisation-Based Flight Clearance of Nonlinear Control Laws In GARTEUR Flight Mechanics Action Group 11 on Advanced Techniques for Clearance of Flight Control Laws, optimisation-based clearance was pointed out as one of the most promising methods for worst-case search. The drawback noted was that its reliability depends on the optimisation method and high levels of conﬁdence also require high number of computations. The work presented here, in Chapter 12, covers several different optimisation methods, both local and global, which are applied on different ﬂight control law clearance problems using the full nonlinear simulation model ADMIRE implemented in Simulink. The book chapter gives a good overview of different optimisation methods that may be used in ﬂight control law clearance. The book chapter was evaluated by three evaluators. The same questionnaire as for the design entries was used and hence some of the questions were not relevant for this analysis method. One of the tasks in Action Group 11 was to ﬁnd the worst case of a combination of uncertainties at a number of single ﬂight condition points. This task has been studied in more detail here and results are provided for different optimisation methods. A full pitch stick pull command is used for the task and the maximum angle of attack was used as the cost function at the envelope point Mach 0.4 at 3000 meters. The local optimisation method based on gradient estimation shows a reasonable amount of simulations to ﬁnd the worst case. When using the genetic algorithm, GA, the number of simulations are ridiculously high, ten times more than the gradient based method. The optimisation could however have been terminated much earlier as there is probably no interest to

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look at simulations where the angle of attack exceeds 30-35◦, which is considered a dangerous or at least unacceptable case. The same holds for the differential evolution (DE) method, where the number of simulations is of similar order, even if the success rate is higher. Hybridisation of GA and DE shows an improvement in the success rate and a reduction in the number of simulations, although this is still high. The number of simulations would have been reduced with a termination criteria for unacceptably high angle of attack. The deterministic global optimisation method DIving RECTangles, DIRECT, and its hybridised version, shows the best result. From the numbers presented the hybridisation does not add much new information for the tested case, but increases the number of simulations by 80%. The number of simulations used with DIRECT are acceptable and it found dangerous cases. In a clearance process, DIRECT could be used ﬁrst to ﬁnd potentially dangerous cases and hybridisation of DIRECT then used for potentially dangerous cases if computer time and power is an issue for the user. Another task looked at is to clear continuous regions of the ﬂight envelope for a pitch stick pull command. This is done by simply including Mach number and altitude in the uncertainty vector used in the previous task. The problem with the investigation is that the used model ADMIRE does not have a properly working angle of attack limiter and the consequence is that there are many cases where the aircraft is actually departing and reaches angle of attack in the range 50-70◦, which is far outside any normal ﬂight envelope. This fact makes it hard to draw conclusions from the comparative results. Hybridisation of DE and DIRECT were used in this task as they were considered the most promising candidates from the ﬁrst task. The chosen envelope region is square in Mach number and altitude and stretches from Mach 0.4 to Mach 0.5 and from 1000 meters to 4000 meters. It seems like the use of hybrid differential evolution gives very high values (about 60◦ ) of angle of attack already at the ﬁrst few simulations. The use of hybrid DIRECT also found a very high angle of attack far outside the normal envelope. Comparison is also made with a Monte Carlo approach which also found a very high angle of attack, but at a different envelope point. The comparison of different maximum angles of attack found with DIRECT and Monte Carlo is hence not really relevant. The hybrid DIRECT does however use a reasonable amount of simulations when compared with Monte Carlo simulations. The third task investigated is to search for a worst case pilot input at the envelope point Mach 0.4 at 3000 meters. A so called Clonk manoeuvre where the pilot commands aim to work against (out of phase) with the aircraft response in pitch is ﬁrst considered. The purpose of the Clonk manoeuvre is primarily to drive the elevon actuators to rate saturation, but it is used here to ﬁnd the maximum angle of attack. The genetic algorithm optimisation routine is used to ﬁnd other pilot inputs that maximise the angle of attack. A diagonal (pitch and roll) pilot command is used with maximum or half of maximum pilot stick inputs. Each pilot input in a sequence is held for at least one multiple of one second and a pilot input sequence is ﬁve seconds. The optimisation shows that high angle of attack values are found, but it is not clear how the aircraft reaches that condition, e.g. in terms of air speed and attitudes. The search for worst pilot inputs is also used with a set of uncertainties. This then yields an even higher angle of attack, as expected. In all these cases, the angle of attack is far higher (50 -70◦ ) than a reasonable angle of attack limit so any relative comparisons are of less importance as all cases are

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dangerous and not acceptable. There is a need to try this search with a better aircraft simulation model in order to study the aircraft behaviour and hence draw conclusions on how useful the optimisation method is for this purpose. Knowledge of the principles of the different optimisation methods are needed in order to use them in ﬂight control law clearance and the user needs to be aware of the limitations of the different methods. Learning different optimisation methods could be done with acceptable effort. Once, the initial work is done to implement the chosen method it would take about the same time and effort as any conventional method it would replace. The strengths of optimisation methods are that continuous regions of the ﬂight envelope can be cleared. Optimisation methods are also relatively simple to understand. Most people involved in ﬂight clearance can probably use the method. The time and effort to clear the ﬂight and uncertainty envelope can potentially be reduced and the result of the clearance can be improved as worse cases can be found. The weakness would be that high computational load is required to clear the control laws for all evaluation criteria in the ﬂight envelope. The characteristics of a ﬁghter aircraft are complex, so it may be hard to draw conclusions from only worst case result. The presented optimisation methods could be used in industry and also improve some of the clearance tasks compared to using Monte Carlo simulations. In industry it is important to ﬁnd all dangerous cases and not only the worst case, so the setup would be slightly different compared to what is done here. Dangerous cases could be found faster through optimisation and possibly ﬁnd cases that are not found today. It would take a considerable amount of work to set up the optimisation method and a lot of different cost functions would be needed. This would require more simulations as it would need one set of simulations for each cost function while in Monte Carlo simulations, the same set of simulations can be used for different kind of criteria. There is still, of course, a lot of manual work to do in analyzing the results in terms of the actual aircraft behaviour. The methods need to be evaluated with models used in industry to see how feasible they are for real applications. Investigation of the ADMIRE Manoeuvring Capabilities Using Qualitative Methods The investigation in Chapter 13 based on attainable equilibrium sets, presents a method for visualisation of manoeuvring capabilities and limitations of an open loop airframe or a closed loop system with any control law. This kind of visualisation tool would be useful in the design process where the designer need to familiarise herself with the aircraft model and its potential. It is also a tool that can be used as a complement to the extensive simulations that are performed during ﬂight control laws clearance to evaluate ﬂight safety. Some of the aerodynamic characteristics of the ADMIRE airframe are presented in a classical way where certain coefﬁcients are plotted versus parameters such as Mach number or angle of attack. Looking at different combinations of aerodynamic coefﬁcients and derivatives is the classical way of familiarising oneself with the airframe’s characteristics. The book chapter then presents a qualitative method that can ﬁnd so called attainable equilibrium sets, AES. These sets can be used to reveal the ultimate

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manoeuvring capabilities of the open-loop airframe. Such information is very useful when deﬁning limiting functions as for example for angle of attack or roll rate. It also helps the ﬂight control law designer to maximise the performance of the aircraft. A velocity vector roll manoeuvre is used in the analysis and several very illustrative ﬁgures that show maximum attainable angle of attack, angle of sideslip and roll rate for various sections of the parameter space are presented. There is a great potential to save time and effort with this type of analysis. It gives a good overview of the airframe’s manoeuvring capabilities. There is still, however, a need to perform several simulations with the full nonlinear model for a ﬂight safety evaluation, but those simulations could be directed to critical areas using this method. This is a tool that certainly would be very useful both during design and ﬂight clearance. 14.3.3

Discussion

The scope of this GARTEUR action group was wide and the design and analysis entries are relatively few, so this is far from an complete picture of the available methods. The design and analysis challenge given in chapter 4 is quite ﬂexible and leaves a lot of considerations to the designers. A good understanding of ﬂight control and ﬂight mechanics is a requirement to produce a good design, but basic knowledge of handling qualities is also needed. Still, the challenge is fairly realistic with the ADMIRE model. Each of the presented methods has its own strengths and weaknesses but all could probably be used in a real industrial application. It is not obvious how great the beneﬁts would be, as there are very few presented comparisons with the benchmark, the old ADMIRE model. To carry out a complete design of a ﬂight control law is of course a very big task. The vision was to aim for the perfect control law with optimum performance and robustness, which is a conﬂict in itself. The intention with this was to drive the task to a level where the strength of the individual methods could be highlighted. There is always a need in industry to improve the process of making ﬂight control laws that are to be implemented in an aircraft. The presented optimisation method for analysis have the potential to improve the ﬂight clearance process. Still, the focus is on making this process faster and cheaper rather than improving the quality in terms of robustness and performance, as ﬂight control laws delivered today already are ”good enough”, i.e. all requirements are met. However, new ﬂight control problems will always arise and these may call for new methods. Industry needs to be one step ahead and to try out new methods with hands-on examples to be convinced of any new method’s beneﬁts. The GARTEUR framework is an excellent way to transfer knowledge of such methods from universities and research establishments to industry.

Acknowledgement The authors of this chapter would like to thank the additional evaluators Ralph Paul (EADS), Robert Hillgren (Saab), Stefan Larsson (Saab), Tor Andersson (Saab) and Henrik Hammarlund (Saab) for their contributions.

15 Concluding Remarks Declan G. Bates1 and Martin Hagstr¨om2 1 2

Department of Engineering, University of Leicester, Leicester LE1 7RH, UK [email protected] Department of Autonomous Systems, Swedish Defence Research institute, SE-164 90 Stockholm, Sweden [email protected]

15.1 Summary of Achievements The main objective of the Action Group was to investigate the potential beneﬁts of nonlinear design and analysis methods for control law development in aerospace vehicles. To guarantee the industrial relevance of the project, two highly realistic simulation models were developed, together with demanding design/analysis challenges. These benchmarks in themselves represent signiﬁcant achievements of the project, since there are still very few industrially relevent aircraft models, with realistic design and analysis speciﬁcations, available in the open literature on which control theoreticians can test and validate new techniques and algorithms. An additional beneﬁt of the on-ground transport aircraft benchmark developed by Airbus for the project is that it represents a non-standard control application (at least in the context of aerospace control!) and thus adds another new and challenging set of problems to those traditionally addressed by ﬂight control law designers. The process of undertaking this research has been beneﬁcial for all the participants, many of whom have taken part in previous related Action Groups and have by now developed effective working relationships. As usual, the information ﬂow has been in both directions, with industrial members providing valuable insight into the real problems and challenges faced by control law designers, and academic researchers highlighting the potential (and potential shortcomings) of the latest nonlinear design and analysis methods. The project has been particularly valuable for the doctoral students and post-doctoral researchers who participated, since it allowed them access to truly challenging problems which are also of particular interest to industry. In these respects, the Action Group has certainly demonstrated the value of GARTEUR research to the European aerospace industry - it has made a signiﬁcant contribution both to narrowing the “theory-practice gap” between academics and industry, and to educating the next generation of aerospace control engineers. How well have the design and analysis challenges formulated for the project been addressed by the nonlinear techniques described and applied in this book? Clearly, this is a difﬁcult question to answer deﬁnitively, or even objectively! The industrial viewpoint on the results of the different teams’ efforts is given in detail in Chapter 14. From a more research-oriented point of view, we believe that each of the design and analysis D. Bates et al. (Eds.): Nonlin. Anal. & Syn. Tech. for Aircraft Ctrl., LNCIS 365, pp. 341–342, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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methods discussed in this book have demonstrated the potential to make signiﬁcant improvements in different aspects of control law development for future aerospace vehicles. It is also striking that the application of even relatively well established methods (LPV, NDI etc) to control problems of the complexity of those described in this book has necessitated the development of interesting new theoretical extensions and variations on these methods - e.g. combining LPV with anti-windup techniques, nonlinear LFT modelling and analysis, combining NDI with object-oriented modelling, etc. Clearly, exposure of control theoreticians to “real-world” aerospace control problems in turn also provides a strong impetus for new theoretical developments, a nice demonstration of a (positive) feedback loop!

15.2 Future Research Almost all of the chapters in this book report research which has clear potential for further development and extensions which could allow the different design and analysis challenges to be addressed more comprehensively. The work on robust nonlinear modelling, analysis and design reported in Chapters 5-7 and 9-10 obviously just scratches the surface of the potential of this approach for extending the systematic and powerful techniques of robust control theory to highly nonlinear systems. The integration of LPV and anti-windup approaches for design in the presence of hard nonlinearities, as well as the use of LTV robustness analysis tools to analyse the effect of these nonlinearities, are both exciting approaches with the potential for signiﬁcant further development and application. The increasing availability of sophisticated software code, together with the ever more widespread use of linear robust control methods in the aerospace industry, should provide many opportunities for the further development of this approach over the coming years. The research on inversion-based approaches to nonlinear control design (Chapters 8, 11 and 13) provides extremely interesting developments of an approach which has gained widespread popularity in the aerospace industry, probobly due to its close links with ﬂight mechanics. The incorporation of object-oriented modelling, with its almost automatic inversion capabilities, into NDI approaches (chapter 8) seems particularly appropriate, and offers the potential for extremely rapid prototyping of control laws for new vehicles that are still in the very early stages of design. This capability should prove extremely interesting to uninhabited air-vehicle (UAV) designers, where highly novel airframe conﬁgurations and control strategies may be under consideration. The significant potential for further development of inversion-based approaches to ﬂight control law design is also illustrated in Chapter 11, where the advantages of block backstepping over more traditional NDI designs are readily apparent. Finally, nonlinear tools for control law analysis and clearance such as the optimisation-based methods discussed in Chapter 12 and the qualitative approach of Chapter 13 clearly offer the potential for further development in order to reduce the computational cost associated with the current simulation-intensive ﬂight clearance process. In particular, the potential of such methods to uncover problematic regions of the ﬂight envelope and pilot control sequences which can then be replicated in piloted simulation trials could prove extremely beneﬁcial.

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Index

α-demand system, 204 µ-analysis, 199 nz -demand system, 204 actuator dynamics, 241 adaptive controller, 138 ADMIRE actuators, 44 aerodata model, 41 description, 35 dynamic model, 38 engine model, 43 envelope, 37 ﬂight control system, 49 sensors, 46 aerodynamic angles, 234 forces, 234 Airbus benchmark architecture, 22 behaviour, 23 control objectives, 25 criteria, 27 design and analysis challenges, 24 manoeuvres, 29 on-ground transport aircraft, 3 aircraft attitude, 10 position, 10 pre-design phase, 148 aircraft-on-ground automatic inversion, 167 object oriented implementation, 151, 164 anti-windup design full-order, 127

parameter-varying, 133 reduced-order, 132 approximate departure criterion, 308 atmosphere model, 48 attainable equilibrium set, 309 automatic inversion, 148 backstepping block, 237 control law, 239 bicycle model approximation, 149, 156, 157, 160, 163 bifurcation analysis, 302 care free, 57 clonk analysis, 290 closed loop system, 240 command ﬁlter, 157, 159, 163 command generator, 239 command variables, 147 selection, 161 conical rotaton rate, 234, 237 control afﬁne form, 235 control allocation, 162, 166, 308 control law block backstepping, 239 NDI-TSS, 245 controller implementation, 246 coordinate system aircraft, 4 coordinate system aerodynamic, 5 body-ﬁxed, 4

358

Index

earth, 4 wheels, 5 corner speed, 57 critical disturbances, 319 D-K iterations, 199 design constraints, 59 design process ﬂight control laws, 162 rapid prototyping, 161 desired dynamics, 237, 241 desktop simulator, 171 differential algebraic equation, 156 differential algebraic index, 156 Differential evolution, 266 dihedral stability derivative, 305 DIRECT, 267, 280 directional stability derivative, 305 DIviding RECTangles, 280 dummy derivative, 157 Dymola, 151 dynamic inversion, 244 dynamics internal, 160 zero, 160 Earth-Centered Earth-Fixed frame, 151 Earth-Centred Inertial frame, 151 envelope design, 57 equation nonlinear solver, 154 equations of motion, 303 equilibrium points, 237 equilibrium states, 308 error dynamics, 240 Feedback Linearisation, 147, 160 feedback signal synthesis, 162 forces and moments, 10 aerodynamic loads, 10 brakes, 19 engines, 12 gravity, 12 nose wheel steering system, 21 rolling forces, 14 shock absorbers, 13 gain scheduling, 147, 246 GAM, 35

Generic Aerodata Model, 35 generic system, 235 Genetic Algorithms, 265 ground-forces estimation, 138 identiﬁcation, 105 linear approximation, 178, 187 modelling, 106 H-DIRECT, 283 handling qualities, 201 handling quality requirements, 312 handlinq qualities, 57 Hopf bifurcation, 316 Horner factorization, Afﬁne factorization, 74 Horner factorization, Sum decomposition, 73 Hybrid differential evolution, 279 Hybrid genetic algorithm, 275 Hybrid optimisation, 267, 274 inline integration, 167 interconnection diagram, 205 International Standard Atmosphere, 48 Inverse Feed Forward Compensation, 147 inverse trimming procedure, 310 invertible, 236 Jacobian matrix, 309 KYP lemma, 181 LFT, 76 controller design, 135 controller implementation, 138 design-oriented modelling, 102 exact modelling, 96 interconnection, 109 modelling, 92, 107, 120, 178 approach, 86 exact, 85, 86 nonlinear, 69, 85 symbolic, 69, 70, 85 nested substitution, 79, 82, 91 symbolic differentiation, 79, 90 symbolic modelling, 97 LFT,LHT, 69 LHT, 70 examples, 78 pseudocode, 89

Index routine factorization, 74 information management, 72 Sum decomposition, 75 LHT,ETD, 98 Lie derivatives, 159 Linear Matrix Inequalities, 199 LMI, 125, 127, 131, 184, 185 local stability, 309 local stability maps, 315 LPV, 95, 99, 102 LPV system, 199 LTV, 108 Lyapunov function, 238 manoeuvrability, 304 matched uncertainties, 244 model aircraft ﬂexible, 153 rigid-body, 153 causality, 149 compiler, 148 differential equations, 159 inverse stability, 160 inversion, 147, 159 automatic, 157, 167 object-oriented translation of, 155 Model Following Control, 147 Modelica, 151 across variables, 150 connectors, 150 data bus, 153 Flight Dynamics Library, 151 ﬂow variables, 150, 153 libraries, 151, 153 modelling block diagram, 149 multi-disciplinary, 149 object-oriented, 147, 148 signal ﬂow, 149 moment controlled, 234 multi-body system, 154 multiple-attractor dynamics, 309 NDI, 138, 241, 244 NDI identiﬁcation, 105 NDI,LPV, 95

359

Newton-Euler equations, 233 Nichols plot, 212 Nonlinear Dynamic Inversion, 147, 160 nonlinear dynamic inversion, 244, 312 normal acceleration, 234 normal force, 305 on-ground-aircraft, 92, 96 optimisation multi-objective, 167 ordinary differential equation, 155 orthogonal matrix, 236 Pad´e approximation, 208 parameter continuation, 302 parameter-dependent Lyapunov function, 199 piecewise linear ﬁtting, 204 pitching moment, 305 qualitative theory, 302 rapid prototyping, 148 reference model, 210, 312 reference signals, 236 region of attraction, 311 regions of attraction, 309 relative degree, 157, 159 rigid body motion, 233 robust performance, 209 robust stability, 214 robustness, 243 robustness analysis LTI/LTV uncertainties, 186 mixed-µ upper bound, 183 performance level, 184 stability margin, 183 structured singular value, 183 rolling moment, 305 saddle-node bifurcation, 316 saturated systems performance index, 125 stability analysis, 124 saturation modelling, 106 saturations, 122 sector conditions, 123 semi-positive realness, 180 Sequential Quadratic Progamming, 265 simulation

360

Index

batch, 162 desktop, 171 pilot-in-the-loop, 148, 162 real-time, 171 singular value decomposition, 236 state variables, 234 strict feedback form, 235 Sum decomposition, 75

trim points, 241 tuning parameters, 241

time varying, 236 time-scale separation, 244 tracking, 236 transonic, 57 trim computation, 164

weighting functions, 205 worst-case, 260 worst-case pilot inputs, 290

variable attitude velocity-vector roll manoeuvre, 323 velocity vector roll, 234, 241 velocity-vector roll manoeuvre, 309 virtual control gain matrix, 235 virtual control law, 237, 240

yawing moment, 305

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Vol. 340: Diehl, M.; Mombaur, K. (Eds.) Fast Motions in Biomechanics and Robotics 500 p. 2006 [978-3-540-36118-3] Vol. 339: Alamir, M. Stabilization of Nonlinear Systems Using Receding-horizon Control Schemes 325 p. 2006 [978-1-84628-470-0] Vol. 338: Tokarzewski, J. Finite Zeros in Discrete Time Control Systems 325 p. 2006 [978-3-540-33464-4] Vol. 337: Blom, H.; Lygeros, J. (Eds.) Stochastic Hybrid Systems 395 p. 2006 [978-3-540-33466-8] Vol. 336: Pettersen, K.Y.; Gravdahl, J.T.; Nijmeijer, H. (Eds.) Group Coordination and Cooperative Control 310 p. 2006 [978-3-540-33468-2] Vol. 335: Kozłowski, K. (Ed.) Robot Motion and Control 424 p. 2006 [978-1-84628-404-5] Vol. 334: Edwards, C.; Fossas Colet, E.; Fridman, L. (Eds.) Advances in Variable Structure and Sliding Mode Control 504 p. 2006 [978-3-540-32800-1] Vol. 333: Banavar, R.N.; Sankaranarayanan, V. Switched Finite Time Control of a Class of Underactuated Systems 99 p. 2006 [978-3-540-32799-8] Vol. 332: Xu, S.; Lam, J. Robust Control and Filtering of Singular Systems 234 p. 2006 [978-3-540-32797-4]

Vol. 325: Amato, F. Robust Control of Linear Systems Subject to Uncertain Time-Varying Parameters 180 p. 2006 [978-3-540-23950-5] Vol. 324: Christoﬁdes, P.; El-Farra, N. Control of Nonlinear and Hybrid Process Systems 446 p. 2005 [978-3-540-28456-7] Vol. 323: Bandyopadhyay, B.; Janardhanan, S. Discrete-time Sliding Mode Control 147 p. 2005 [978-3-540-28140-5] Vol. 322: Meurer, T.; Graichen, K.; Gilles, E.D. (Eds.) Control and Observer Design for Nonlinear Finite and Inﬁnite Dimensional Systems 422 p. 2005 [978-3-540-27938-9] Vol. 321: Dayawansa, W.P.; Lindquist, A.; Zhou, Y. (Eds.) New Directions and Applications in Control Theory 400 p. 2005 [978-3-540-23953-6] Vol. 320: Steffen, T. Control Reconﬁguration of Dynamical Systems 290 p. 2005 [978-3-540-25730-1] Vol. 319: Hofbaur, M.W. Hybrid Estimation of Complex Systems 148 p. 2005 [978-3-540-25727-1] Vol. 318: Gershon, E.; Shaked, U.; Yaesh, I. H∞ Control and Estimation of State-multiplicative Linear Systems 256 p. 2005 [978-1-85233-997-5]

Vol. 331: Antsaklis, P.J.; Tabuada, P. (Eds.) Networked Embedded Sensing and Control 367 p. 2006 [978-3-540-32794-3]

Vol. 317: Ma, C.; Wonham, M. Nonblocking Supervisory Control of State Tree Structures 208 p. 2005 [978-3-540-25069-2]

Vol. 330: Koumoutsakos, P.; Mezic, I. (Eds.) Control of Fluid Flow 200 p. 2006 [978-3-540-25140-8]

Vol. 316: Patel, R.V.; Shadpey, F. Control of Redundant Robot Manipulators 224 p. 2005 [978-3-540-25071-5]

Vol. 329: Francis, B.A.; Smith, M.C.; Willems, J.C. (Eds.) Control of Uncertain Systems: Modelling, Approximation, and Design 429 p. 2006 [978-3-540-31754-8]

Vol. 315: Herbordt, W. Sound Capture for Human/Machine Interfaces: Practical Aspects of Microphone Array Signal Processing 286 p. 2005 [978-3-540-23954-3]

Vol. 328: Loría, A.; Lamnabhi-Lagarrigue, F.; Panteley, E. (Eds.) Advanced Topics in Control Systems Theory 305 p. 2006 [978-1-84628-313-0]

Vol. 314: Gil’, M.I. Explicit Stability Conditions for Continuous Systems 193 p. 2005 [978-3-540-23984-0]

Vol. 327: Fournier, J.-D.; Grimm, J.; Leblond, J.; Partington, J.R. (Eds.) Harmonic Analysis and Rational Approximation 301 p. 2006 [978-3-540-30922-2]

Vol. 313: Li, Z.; Soh, Y.; Wen, C. Switched and Impulsive Systems 277 p. 2005 [978-3-540-23952-9]

Vol. 326: Wang, H.-S.; Yung, C.-F.; Chang, F.-R. H∞ Control for Nonlinear Descriptor Systems 164 p. 2006 [978-1-84628-289-8]

Vol. 312: Henrion, D.; Garulli, A. (Eds.) Positive Polynomials in Control 313 p. 2005 [978-3-540-23948-2]