Nonlinear Analysis and Synthesis Techniques for Aircraft Control (Lecture Notes in Control and Information Sciences)

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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

69

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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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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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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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.

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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

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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


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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

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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

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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


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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

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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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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.

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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.

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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

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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]

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0

1

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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)

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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)

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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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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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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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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].

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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.

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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

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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)

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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

ISV brk

F

θ

ISV NW

NW

Act

BT_MLG

EoM

F and M M

N1c

Tn

environment

1 s

aircraft states

environment aerodynamics

Fig. 6.1. On-ground aircraft model diagram

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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(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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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

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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

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10 time (sec)

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−0.4

20

5

0

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10 time (sec)

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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.

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

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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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135

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.

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

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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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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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)

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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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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

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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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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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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

Rapid Prototyping Using Inversion-Based Control

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.

156

G. Looye

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)

Rapid Prototyping Using Inversion-Based Control

157

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)

158

G. Looye

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

Rapid Prototyping Using Inversion-Based Control

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.

160

G. Looye

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

162

G. Looye

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

Rapid Prototyping Using Inversion-Based Control

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.

Rapid Prototyping Using Inversion-Based Control

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

Rapid Prototyping Using Inversion-Based Control

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

Rapid Prototyping Using Inversion-Based Control

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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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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181

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

Robustness Analysis Versus Mixed LTI/LTV Uncertainties

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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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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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

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dz

dz

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dz

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time (s)

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

60

heading (deg)

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40

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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)

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⎧ 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

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0.9

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

heading (deg)

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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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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

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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

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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

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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 =

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− 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

0.3s + 1 ; W = 0.005 Wnq (s) = 0.005 0.02s nz +1

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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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

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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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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.

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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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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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