Robust Flight Control: A Design Challenge (Lecture Notes in Control and Information Sciences)

Jean-François Magni, Samir Bennani and Jan Terlouw (Eds.)

Robust Flight Control: A Design Challenge

i

This book was rst printed by Springer-Verlag, 1997 Lexture Notes in Control and Information S ien es, 224.

Editors Jean-François Magni, Do teur ès S ien es ONERA CERT, Département d'Études et Re her hes en Automatique, BP 4025, F31055 Toulouse Cedex, Fran e. Samir Bennani, Ir. Delft University of Te hnology, Fa ulty of Aerospa e Engineering, Kluyverweg 1, 2629 HS Delft, The Netherlands. Jan Terlouw, Ir. National Aerospa e Laboratory NLR, Flight Me hani s Department, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands.

ii

ROBUST FLIGHT CONTROL: A DESIGN CHALLENGE

EDITORS J.-F. Magni, S. Bennani & J. Terlouw

GARTEUR ACTION GROUP FM(AG08)

Resear h Establishments: Centro Italiano Ri er he Aerospaziali (CIRA, Italy), Deuts he Fors hungsanstalt für Luft- und Raumfahrt (DLR, Germany), Defen e Resear h Agen y (DRA, United Kingdom), Instituto Na ional de Té ni a Aeroespa ial (INTA, Spain), Laboratoire d'Automatique et d'Analyse des Systèmes (LAAS, Fran e), National Aerospa e Laboratory (NLR, The Netherlands), O e National d'Etudes et de Re her hes Aérospatiales (ONERA, Fran e).

Industry: Alenia Aeronauti a (ALN, Italy), Avro International Aerospa e (AVRO, United Kingdom), British Aerospa e, Dynami s (BAe-D, United Kingdom), British Aerospa e, Military Air raft (BAe-MA, United Kingdom), Cambridge Control Ltd (CCL, United Kingdom), Daimler-Benz Aerospa e Airbus (DASA, Germany), Fokker Air raft Company (FAC, The Netherlands), Saab Military Air raft (SMA, Sweden).

Universities: Craneld University (CUN, United Kingdom), Delft University of Te hnology (DUT, The Netherlands), Linköping University (LiTH, Sweden), Loughborough University (LUT, United Kingdom), University of Cambridge (UCAM, United Kingdom), University of Lei ester (ULES, United Kingdom), Universitá di Napoli Frederi o II (UNAP, Italy), Universitad Na ional de Edu a ión a Distan ia (UNED, Spain).

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iv

Contents Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1

Introdu tion. Jan Terlouw and Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutorial part 2

Multi-Obje tive Parameter Synthesis (MOPS). Georg Grübel and Hans-Dieter Joos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3

Eigenstru ture Assignment. Lester Faleiro, Jean-François Magni, Jesús M. de la Cruz and Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4

Linear Quadrati Optimal Control. Fran es o Amato, Massimiliano Mattei and Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5

Robust Quadrati Stabilization. Germain Gar ia, Ja ques Bernussou, Jamal Daafouz and Denis Arzelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 7

H1 Mixed Sensitivity. Mark R. Tu ker and Daniel J. Walker . . . . . . . . . . . 52 H1 Loop Shaping. George Papageorgiou, Keith Glover, Alex Smerlas and Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 -Synthesis. 9

Samir Bennani, Gertjan Looye and Carsten S herer . . . . . . . 81

Nonlinear Dynami Inversion. Binh Dang Vu . . . . . . . . . . . . . . . . . . . . . . . . . 102

10

Robust Inverse Dynami s Estimation. Ewan Muir . . . . . . . . . . . . . . . . . . . 112

11

A Model Following Control Approa h. Holger Duda, Gerhard Bouwer, J.-Mi hael Baus hat and Klaus-Uwe Hahn . . . . . . . . . . . . . . . . . . . . . . . . . 116

12

Predi tive Control. Jan Ma iejowski and Mihai Huzmezan . . . . . . . . . . . 125

13

Fuzzy Logi Control.

Gerard S hram, Uzay Kaymak and Henk B. Ver-

bruggen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

RCAM part 14

The RCAM Design Challenge Problem Des ription.

Paul Lambre hts,

Samir Bennani, Gertjan Looye and Dieter Moormann . . . . . . . . . . . . . . 149

15

The Classi al Control Approa h. Jim E. Gautrey . . . . . . . . . . . . . . . . . . . . 180

16

Multi-Obje tive Parameter Synthesis (MOPS). Hans-Dieter Joos . . . . . 199

17

An Eigenstru ture Assignment Approa h (1).

Lester Faleiro and Roger

Pratt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

v

18

An Eigenstru ture Assignment Approa h (2). Jesús M. de la Cruz, Pablo Ruipérez and Joaquín Aranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

19

A Modal Multi-Model Approa h. Carsten Döll, Jean-François Magni and Yann Le Gorre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

20

The Lyapunov Approa h. Jamal Daafouz, Denis Arzelier, Germain Gar ia and Ja ques Bernussou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

21

An

H1 Approa h. Mark R. Tu ker and Daniel J. Walker

-Synthesis Approa h (1). 23 A -Synthesis Approa h (2). 22

24

A

. . . . . . . . . . . . 300

Samir Bennani and Gertjan Looye . . . . . . 321 Jan S huring and Rob M.P. Goverde . . . 341

Autopilot Design based on the Model Following Control Approa h. Holger Duda, Gerhard Bouwer, J.-Mi hael Baus hat and Klaus-Uwe Hahn

25

360

Flight Management using Predi tive Control. Mihai Huzmezan and Jan M. Ma iejowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

26

A Fuzzy Control Approa h. Gerard S hram and Henk B. Verbruggen . 398

HIRM part 27

The HIRM Design Challenge Problem Des ription. Ewan Muir . . . . . . 421

28

Design via LQ Methods. Fran es o Amato, Massimiliano Mattei, Stefano S ala and Leopoldo Verde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

29

The

H1 Loop Shaping Approa h. George Papageorgiou, Keith Glover and

Ri k A. Hyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

30

Design of Stability Augmentation System using

-Synthesis.

Karin Ståhl

Gunnarsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

31

Design of a Robust, S heduled Controller using

-Synthesis.

Johan An-

thonie Markerink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

32

Nonlinear Dynami Inversion and LQ Te hniques. Béatri e Es ande . . 525

33

The Robust Inverse Dynami s Estimation Approa h. Ewan Muir . . . . . 543

Con luding part 34

The Industrial View. Chris Fielding and Robert Lu kner . . . . . . . . . . . . 569

35

An Other View of the Design Challenge A hievements. Georg Grübel 605

36

Con luding Remarks. Samir Bennani, Jean-François Magni and Jan Terlouw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

Appendix A

Used Nomen lature. Anders Helmersson and Karin Ståhl Gunnarsson

Bibliography vi

614

Author Index

Fran es o Amato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 446 Joaquín Aranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Denis Arzelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 J.-Mi hael Baus hat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Samir Bennani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 149, 421, 321, 612 Ja ques Bernussou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Gerhard Bouwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Jesús M. de la Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 238 Jamal Daafouz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Binh Dang Vu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Carsten Döll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Holger Duda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Béatri e Es ande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Pierre Fabre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Lester Faleiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 218 Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 569 Germain Gar ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Jim E. Gautrey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Keith Glover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 466 Rob M.P. Goverde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Georg Grübel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 605 Klaus-Uwe Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Anders Helmersson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 614 Mihai Huzmezan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125, 379 Ri k A. Hyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421, 466 Jonathan Irving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Joseph Irvoas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Hans-Dieter Joos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 199 Uzay Kaymak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Paul Lambre hts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 421 Tony Lambre gts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Yann Le Gorre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Gertjan Looye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 149, 321 Robert Lu kner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Jan Ma iejowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125, 379 Jean-François Magni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 149, 258, 612 Johan Anthonie Markerink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Alberto Martínez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Massimiliano Mattei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 446 Philippe Ménard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Dieter Moormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 421 Ewan Muir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112, 421, 543

vii

George Papageorgiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 466 Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Roger Pratt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Pablo Ruipérez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 33, 149, 421, 446 Carsten S herer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Gerard S hram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135, 398 Jan S huring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 421 Phillip Sheen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Alex Smerlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Karin Ståhl Gunnarsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421, 486, 614 Jan Terlouw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 149, 421, 612 Mark R. Tu ker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 300 Hans van der Vaart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Henk B. Verbruggen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135, 398 Leopoldo Verde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Daniel J. Walker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 300

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1.

Introdu tion

Jan Terlouw and Chris Fielding 1

2

1.1 The Importan e of Advan ed Control Design Methods for the European Air raft Industry European manufa turers of military and ivil air raft have rea hed a high level of expertise in designing ight ontrol laws, to a point that they an solve virtually any realisti hallenge that might be foreseen in the near future. This

apability is a result of the lessons learned by generations of engineers who have extended and passed on their skills, always driven by the ultimate requirement - that one day their ight ontrol system (FCS) had to y.

However, the

large time and eort spent to solve all problems en ountered during the design pro ess poses the question whether improvements are possible. As the s ienti ommunity sometimes laims to have invented new methods to improve urrent ways of working, there is a natural interest from industry in what the resear hers have to oer. On the other hand, s ientists are interested in realisti appli ations to justify their work and to test new on epts. It is lear that there is a strong in entive for both worlds to work together, but a tually a hieving it an be di ult.

S ientists like to develop methods

whi h have general appli ability, and this is parti ularly true for ontrol theoreti ians. On the other hand, users of design methods are, from a professional point of view, mainly interested in dedi ated methods that solve their parti ular problems. The result is that many new ideas never really break through, be ause they are simply not spe ialised and elaborated enough, or be ause there is not enough liaison between the s ienti and industrial worlds. There are now a large number of ontroller design methods that have been developed over the past twenty-ve years (some have earlier origins). In this book twelve of them are treated:

   1

Multi-Obje tive Parameter Synthesis Eigenstru ture Assignment Linear Quadrati Optimal Control

National Aerospa e Laboratory NLR, Flight Me hani s Department, Anthony Fokker-

weg 2, 1059 CM Amsterdam, The Netherlands. 2

British Aerospa e Military Air raft, Aerodynami s Department, Warton Aerodrome,

Preston PR4 1AX, UK

1

 Lyapunov Te hniques  H1 Mixed Sensitivity  H1 Loop Shaping  -Synthesis  Nonlinear Dynami Inversion  Robust Inverse Dynami s Estimation  Model Following  Predi tive Control  Fuzzy Logi Control These methods have many dierent features. A ommon feature is that ea h of them is developed to a hieve advantages over lassi al te hniques. The laimed benets range from enhan ed performan e, resulting from multi-input multioutput ontrollers, to improved e ien y and simpli ation of the design pro ess. At the same time, the most important and obvious di ulty in adopting any new method is the la k of experien e of its use in pra ti e. This book is an attempt to redu e the gap between theory and prati e, with respe t to appli ation of modern ontrol design te hniques. It deals with ight ontrol of rigid body ivil and military air raft. The twelve te hniques mentioned above will be demonstrated on the basis of two ben hmark problems [145, 177℄. But rst, some general remarks will be made about ight ontrol laws as a part of FCS design.

Flight ontrol laws design The main fun tion of the ight ontrol system (FCS) of an air raft is to ontribute to its safe and e onomi operation, su h that the intended ight missions

an be a

omplished and unexpe ted events an be handled. The heart of a modern FCS onsists of the following omponents, arranged in a logi al way to benet from the prin iple of feedba k: sensors provide a ight ontrol omputer (FCC) information on air data, inertial data and o kpit data; an FCC in whi h ight ontrol laws are implemented to determine the ommands for the a tuation systems of the air raft ontrol surfa es and throttles for engines demands. For air raft, feedba k ontrol is used to provide tight pilot ommand tra king, to attenuate external disturban es su h as gusts and turbulen e and to provide robustness against modelling errors. In the early days of ight, safety was the main on ern for FCS designers. Pilots needed signi ant eort to maintain some ight onditions under all ir umstan es. Today, safety is even more important, be ause many more people are transported, higher osts are involved in establishing safety, and the reputation of airlines and air raft manufa turers is paramount, in an in reasingly ompetitive market. Fly by Wire allows the pilot to ontrol the

2

air raft states, as an alternative to the onventional dire t ontrol of the engines and ontrol surfa es.

It gives new opportunities to in rease the overall level

of safety through the exibility oered by the ontrol laws [78℄. For example, error-tolerant ontrol laws provide ight envelope prote tion, and help the pilot to re over from unusual attitudes and su

essfully a hieve riti al manoeuvres. The use of modern FCS an be bene ial from an e onomi point of view. For ertain types of air raft, fuel onsumption an be redu ed by allowing relaxed stati stability, ountera ted by the appli ation of a tive ontrol. Another advantage related to fuel onsumption is that for large air raft the weight of Fly by Wire systems is smaller than that of onventional systems. Furthermore, the so- alled family on ept an be introdu ed. Flying dierent air raft an be made almost the same for pilots, by making appropriate adjustments in the ight ontrol laws. As a result, dierent air raft feel almost the same, therefore helping to redu e pilot training osts. Most importantly, modern FCS have ontributed to improved dynami al behaviour. Certain military air raft annot be own without a stability augmentation system.

The open loop instability, whi h is related to agility of

the air raft, is utilised to obtain better performan e and manoeuvrability of the losed-loop system.

For ivil air raft, performan e an be in reased by

appli ation of a tive systems, for example to provide gust suppression and auto-trimming, in order to a hieve improved ride quality. The performan e benets a hieved, have the penalty of tremendous osts involved in the development of an advan ed FCS. In the past, the pilot sti k was typi ally onne ted with rods or ables to the ontrol surfa es. Sin e then, the in reased safety, and e onomi al and performan e demands have for ed air raft manufa turers to extend FCS to a high level of omplexity. The danger exists that the e onomi al benets des ribed above are nullied by higher design and maintenan e osts, while omplexity an potentially have a negative ee t on safety. The large number of fun tions and requirements have in reased the number of spe ialists areas needed for the FCS design pro ess. This makes the work

hallenging from a te hni al and management point of view. People who are responsible for mode logi , redundan y design, software and hardware development, design integration and erti ation have to work losely together. In the overall pro ess, ontrol laws designers assume a modest, but entral position. They have knowledge of ight me hani s, ontrol theory, handling qualities, airframe stru ture and FCS hardware.

Their task is inuen ed by the de-

sign requirements, the ight envelope, the air raft onguration omplexity, the stores arriage and weight distribution, the required autopilot modes, the air raft stability (or instability) levels and the aerodynami nonlinearity. The work of an industrial ight ontrol laws designer who uses lassi al design te hniques (see Chapter 15) may onsist of the following simplied sequen e of a tivities. The rst step is to derive a nonlinear dynami model of the air raft to be ontrolled. Getting familiar with the dynami al behaviour by means of trimming, stability and ontrol analysis and nonlinear simulations (for stable air raft) and understanding the inuen es of the modelling assump-

3

tions is most important at this stage. Linearisation and linear simulation of the model is also performed. The next step is to dene the ontroller ar hite ture and to make a rst design whi h in ludes gain s heduling to over the air raft's ight envelope. Implementation of the ontrol law in the nonlinear model, for o-line and piloted simulation, is arried out next. This pro ess might be repeated to optimize the design. In the design pro ess, nonlinearities and model un ertainties are important issues to understand and deserve mu h attention if a robust design is to be a hieved.

Robustness of ight ontrol systems Robustness investigation deals with the dis repan y between models and reality.

It is basi ally on erned with whether a ontrolled system will work

satifa torily under the ir umstan es it will meet in pra ti e. FCS designers have always used models to in rease their knowledge about ight ontrol, and have been invloved in robustness investigations in some form, sin e the very beginning of ight. At various stages of the ight ontrol laws design pro ess, model un ertainty

an be introdu ed, for example, when linearised versions of omplex models are derived. In this ase, the term un ertainty is a tualy a misnomer, be ause the deviation between linear and nonlinear behaviour an be quantied. The same is true if known variations in, for example, the position of the entre of gravity or a time-delay in the system, are negle ted. Depending on the design te hnique used, it may be ne essary to make su h modelling assumptions (temporarily) in order to obtain a model whi h is suitable for ontroller design. Model un ertainty an also be introdu ed unintentionally due to modelling errors, unknown hara teristi s of the air raft in relation to the environment, or ina

urate information about the signals owing through the system. For example, the pre ise value of aerodynami stability derivatives and air data may not exist. A feature of several ontrol design te hniques des ribed in this book is that they deal systemati ally and sometimes expli itly with robustness. Introdu ing these systemati s into the design y le may enhan e FCS design in terms of the ee ien y of the pro ess and the performan e of the resulting ontrol systems.

Potential ontribution of modern design te hniques It seems that the European aeronauti al industry is not in the rst pla e interested in modern te hniques purely to a hieve better air raft performan e. In fa t almost any te hnique, modern or lassi al, when used to solve realisti problems with enough knowledge of the method itself, with the ne essary tools available, and based on a thorough knowledge of ight me hani s, will eventually lead to the desired results. The real interest is in the systemati approa hes behind new methods, be ause this an simplify the design y le and make it more transparent. Global ompetition for es air raft manufa turers to

ontinuously improve the e ien y of their engineering a tivities. If it an be

4

demonstrated that advan ed design te hniques lead to a design y le with better tra ability of design de isions and simpliation of the overall pro ess, the

han e that modern ontrol te hniques will be used by industry will in rease. The omplexity of the design task and the related investment made in the past in human and non-human apital, explain the areful attitude from some air raft manufa turers to repla e their well-established lassi al te hniques. Moreover, lassi al te hniques have desirable features, for example the visibility of the resulting ontroller.

At the lowest level of detail of the ontrol

law, the fun tion of every gain and dynami element an be easily understood, whi h makes designs easy to modify and a

ept. On the other hand, the visibility after integration of subsystems is partly lost at a higher level. Another advantage is that gain and phase margins are open-loop measures with a lear link to robustness. This makes them very useful for synthesis. Even though it is true that superiority is often related to simpli ity and transparan y, whi h are typi al features of lassi al ontrol te hniques, the aeronauti al industry a knowledges some disadvantages as well.

Due to his-

tori reasons, the lassi al approa h in whi h ea h mode and ight ondition is treated as a separate problem has led to mode proliferation and the need for omplex algorithms. To avoid fun tional integration at the end of the FCS design, whi h is too late, an all en ompassing and onsistent design strategy is ne essary. Throughout the design pro ess a systems approa h strategy should be applied, supported by good requirements, design tools and design models. Appli ation of advan ed te hniques promises a signi ant redu tion of design time be ause it would remove the time- onsuming lassi al one-loop-at-a-time approa h and redu e the number of design points for whi h a ontroller has to be designed.

1.2

GARTEUR A tion Group on Robust Flight Control 3

In O tober 1994, GARTEUR

A tion Group FM(AG08) was established. For

the twenty-three member organisations of this group from seven European

ountries, GARTEUR proved to be an organisation oering ex ellent onditions and support for arrying out basi , pre ompetitive resear h. GARTEUR unites resear h establishments, the aeronauti al industry and universities in A tion Groups. In FM(AG08) the following organisations parti ipated:



Resear h Establishments

 3

Centro Italiano Ri er he Aerospaziali (CIRA, Italy, Capua)

The Group for Aeronauti al Resear h and Te hnology in EuRope (GARTEUR) was

formed in 1973 and has as member ountries: Fran e, Germany, The Netherlands, Spain, Sweden and the United Kingdom.

A

ording to its Memorandum of Understanding, the

mission of GARTEUR is to mobilize, for the mutual benet of the GARTEUR member

ountries, their s ienti and te hni al skills, human resour es and fa ilities in the eld of aeronauti al resear h and te hnology. More information about GARTEUR an be found in the GARTEUR Guide [4℄.

5



Deuts he Fors hungsanstalt für Luft- und Raumfahrt (DLR, Germany, Oberpfaenhofen)

  

Defen e Resear h Agen y (DRA, United Kingdom, Bedford) Instituto Na ional de Té ni a Aeroespa ial (INTA, Spain, Madrid) Laboratoire d'Automatique et d'Analyse des Systèmes (LAAS, Fran e, Toulouse)



National Aerospa e Laboratory (NLR, The Netherlands, Amsterdam)





O e National d'Etudes et de Re her hes Aérospatiales

 

CERT-ONERA, Fran e, Toulouse ONERA-Salon, Fran e, Salon de Proven e

Industry

   

Alenia Aeronauti a (ALN, Italy, Turin) Avro International Aerospa e (AVRO, United Kingdom, Woodford) British Aerospa e, Dynami s (BAe-D, United Kingdom, Filton) British Aerospa e, Military Air raft (BAe-MA, United Kingdom, Warton)

   



Cambridge Control (CCL, United Kingdom, Cambridge) Daimler Benz Aerospa e Airbus (DASA, Germany, Hamburg) Fokker Air raft Company (FAC, The Netherlands, Amsterdam) Saab Military Air raft (SMA, Sweden, Linköping)

Universities

       

Craneld University (CUN, United Kingdom, Craneld) Delft University of Te hnology (DUT, The Netherlands, Delft) Linköping University (LiTH, Sweden, Linköping) Loughborough University (LUT, United Kingdom, Loughborough) University of Cambridge (UCAM, United Kingdom, Cambridge) University of Lei ester (ULES, United Kingdom, Lei ester) Universitá di Napoli "Fediri o II" (UNAP, Italy, Naples) Universidad Na ional de Edu a ión a Distan ia (UNED, Spain, Madrid)

The A tion Group was haired by NLR (Terlouw); CERT-ONERA (Magni) provided the vi e- hairman.

In total eight meetings were held in Amster-

dam, Madrid, Cambridge, Capua, Toulouse, Linköping, Oberpfaenhofen and (again) Amsterdam, whi h gave an extra ultural dimension to the proje t. In view of the longer term obje tive of ontributing to e ien y improvement of the ight ontrol laws design pro ess, it was de ided to follow three main streams.

6

Design Challenge The rst stream was the Design Challenge des ribed in this book. Before the start of the A tion Group it was on eived that a thorough demonstration of modern design te hniques, applied to genuine ight ontrol problems, was required in order to get the desired feedba k from industry. The aim was to present the state-of-the-art with respe t to modern (robust) ontrol in su h a way that industry ould relate to it. At the same time it was the intention to

larify what is needed for a design method to be a

epted by an industrial design o e. To a hieve this, people from industry were asked to give inputs for two ben hmark problems, whi h were subsequently developed by people from the resear h establishments and universities. The rst one, the RCAM (Resear h Civil Air raft Model) problem [145℄, is based on the automati landing of a large, modern argo air raft. The se ond, the HIRM (High In iden e Resear h Model) problem [177℄, onsiders the ontrol of a military air raft a

ross a wide design envelope. Both ben hmarks are based on six degrees of freedom mathemati al air raft dynami s models, dened in Matlab/Simulink [121, 240℄. They in lude aerodynami , engine, atmosphere and gravity models.

In addition, a tuator and

sensor hara teristi s are taken into a

ount, together with models for wind, atmospheri turbulen e and windshear. An extensive set of design requirements is given, whi h an be tested with software for frequen y and time domain evaluations.

A standard nomen lature [237℄ and a standard report lay-out were

dened at an early stage, to avoid unne essary problems later on. In order to make the ben hmarks more realisti , parameter variations (time-delay, mass and entre of gravity variations for RCAM; variations in aerodynmi derivatives and measurement errors for HIRM) were dened. hardware implementation issues are onsidered.

Furthermore, some

This puts the ben hmarks

into the ategory of robust ight ontrol problems. At the start of the proje t it was de ided to limit the s ope of the demonstration of the te hniques to design and omputer simulations. Validation of the most promising ontrol laws and design te hniques might possibly be performed in a follow-on proje t, in whi h the use of a ight simulator and a ying testbed is re ommended. The Design Challenge was not aimed at giving the answer to the question whi h method is best?, but rather to show, step by step, how modern ontrol

an be applied. The design teams were asked to highlight four main points: 1. The eort to learn, to implement and to apply the method. 2. The appli ability of the design method to ight ontrol laws design. 3. The omplexity of the resulting ontroller, its implementation and erti ation issues. 4. The robustness and performan e of the designed ontroller. A large group of ontrol engineers primarily from the European aeronauti al

7

industry has evaluated the proposed designs. This book is a summary of the results of the Design Challenge.

Computer-Aided Control System Design The se ond stream of a tivities addressed the development of a framework for

omputer-aided ontrol system design.

Several industrial members of GAR-

TEUR A tion Group FM(AG08) indi ated a need for omputer support of the design pro ess and data. A prototype was developed by NLR, based on the

ommer ial software produ ts Matlab/Simulink, SiFrame and Informix. The prototype oers fa ilities for design pro ess denition and exe ution, in luding tool integration and a entral data repository. Most important is the automati

onsisten y ontrol of all (versions of ) design information in the framework. The  lassi al design pro ess of Craneld University, des ribed in Chapter 15, is implemented in the prototype, whi h was evaluated by several FM(AG08) organisations. The results of this eort are des ribed seperately in [224℄.

Robust Flight Control Tutorial and Literature Survey Database The third stream aimed at making available a literature overview of ontrol te hniques related to ight ontrol appli ations and at providing a tutorial do ument on advan ed ontrol te hniques. CIRA has established a Robust Flight Control Literature Survey Data Base, whi h an be a

essed via the Internet [206℄. From it, referen es and summaries of arti les on robust ight ontrol an be obtained. The aim of the database is to assist designers in lassifying their ontrol problems a

ording to similar problems already treated in the literature. As su h, it an help the designer to identify the most onvenient te hnique to be used. INTA has written a do ument [53℄ in whi h tutorials of all design te hniques that are des ribed in this book and several others are presented in detail.

1.3 Outline of the book The editors would like to point out that this book is the result of a group a tivity. With respe t to the ontents, it was onsidered to be important that as many FM(AG08) organisations as possible would get the opportunity to present their results, in order to over a wide variaty of design approa hes. The ontributions were not sele ted by the editors. The book onsists of four parts.

Part I ontains tutorials of all methods

that have been applied on either RCAM or HIRM or on both problems. Parts II and III over the RCAM and HIRM ben hmark denions and the proposed designs.

With a few ex eptions, ea h design hapter has basi ally

the same stru ture.

The designs are based on the twelve methods given in

se tion 1.2. Furthermore, one RCAM design is entirely based on lassi al te hniques.

8

In Part IV, three dierent views on the Design Challenge are given. Chapter 34 presents a view from industry. A questionnaire was designed by British Aerospa e and DASA to aid evaluators in their assessment of the Design Challenge entries. Chapter 35 dis usses the Design Challenge results from the s ienti resear her's point of view. An obje tive measure of stability robustness, namely the stru tured singular value, is given for ten RCAM designs. Finally, Chapter 36 ontains some on luding remarks of the editors. One of the onditions whi h made the Design Challenge possible was the fa t that all teams have used the same nomen lature, whi h is given in Appendix A.

A knowledgements Most of the work needed for writing this book was funded by the parti ipating organisations of GARTEUR A tion Group FM(AG08).

These organisations,

whi h are listed in se tion 1.2, are given thanks for their onden e in the group and their full support until the end of the proje t. In some ases national agen ies and other resear h funding bodies have given additional nan ial help, notably the Netherlands Agen y for Aerospa e Programs (NIVR). Without their support the Design Challenge would not have been possible. FM(AG08) also wishes to express its gratitude to Aérospatiale and DRA for making available the models on whi h the RCAM and HIRM ben hmark denitions are based. Another word of thanks is to the GARTEUR organisation, in parti ular the Flight Me hani s Group of Responsables and the Exe utive Committee, for making the publi ation of this book possible. The head of the NLR Flight Division, Jan van Doorn, who has a ted as the GARTEUR Monitoring Responsable of FM(AG08), has given essential ontributions behind the s enes. He was an indispensible link between the A tion Group and the GARTEUR organisation. The editors are grateful to Chris Fielding, Derek Laidlaw, Jim Gautrey, Lester Faleiro, Daniël Walker and Jonathan Irving for he king most hapters on the use of English and proposing many suggestions for improvements. Not all results of GARTEUR A tion Group FM(AG08) related to the Design Challenge ould be presented in this book.

Several design teams joined in

at a later stage or there were other reasons why their designs ould not be in luded. In this respe t Alex Smerlas (Univ. of Lei ester) [217℄, Aldo Tonon (ALN), Jürgen A kermann (DLR), Angel Perez de Madrid (UNED) and their

olleagues are a knowledged for their valuable ontributions. This book will be presented at a GARTEUR Spe ialists' Workshop on Robust Flight Control (CERT-ONERA, Toulouse, Fran e, April 14-15, 1997). Spe ial thanks is given to CERT-ONERA for organising and hosting this workshop.

9

10

Part I

Tutorial part

11

2.

Multi-Obje tive Parameter Synthesis

(MOPS)

Georg Grübel1 and Hans-Dieter Joos

1

2.1 Theoreti al Aspe ts 2.1.1 Global Goals Any ontrol law is parametrized in some way. For example, in a P-I-D ontrol stru ture with additional shaping lters there are the P-I-D gains and the lter parameters whi h are to be tuned for losed-loop performan e. Similarly, an LQR state- or output feedba k ontrol law is parametrized by the state- and

ontrol eort weights Q and R, an eigenstru ture state feedba k ontrol law is parametrized by the eigenvalues and some eigenstru ture parameters and an

H1 ontrol law is parametrized by its input/output weighting lter parameters. Control parameter tuning for a best possible robust performan e is a time onsuming task if performed manually. This is due to the multitude of different (nonlinear) design spe i ations whi h have to be dealt with.

This is

also true if one uses advan ed analyti al ontrol synthesis te hniques su h as



synthesis. Multi-obje tive parameter synthesis (MOPS) is a general te hnique whi h

omplements a hosen ontrol law synthesis te hnique.

Having hosen an

appli ation-spe i ontrol law stru ture with parametrization, or having hosen a general ontrol synthesis te hnique with its analyti ally given parameterization, the free design parameters (e.g.

the LQR-weights) are omputed

by a min-max parameter optimization set up.

The designer formulates this

set up by spe ifying the design goals as a set of well dened omputational

riteria, whi h an be a fun tion of stability parameters (e.g.

eigenvalues),

and time- and frequen y response hara teristi s (e.g. step-response overshoot and settling time, ontrol rates, bandwidth, stability margins et ).

By this

multi- riteria formulation all the various oni ting design goals are taken are of individually, but are ompromised on urrently by a weighted min-max parameter optimization. In parti ular, robust- ontrol requirements with respe t to variations in stru tured parameter sets and operating onditions an be taken are of by a multi-model formulation whi h en ompasses the worst- ase design onditions. 1

DLR German Aerospa e Resear h Establishment, Institute for Roboti s and System

Dynami s, Control Design Engineering Group (Prof. G. Grübel), D-82234 Wessling E-mail: dieter.joosdlr.de

13

For oni ting design riteria the te hnique provides a best-possible paretooptimal ontrol tuning. Sin e the multi- riteria in ludes performan e measures whi h are dire tly used as design drivers, they provide dire t quantitative information about the design oni ts and performan e onstraints. This yields all the ne essary information on how to improve the balan e of a design tradeo within a given ontroller stru ture or a hosen ontrol synthesis method. The method allows simple (linear) ontrollers to be optimized based on omplex (nonlinear) system evaluation models, thereby linking together the steps of ontrol design and of (nonlinear) design assessment. Our experien e shows that with the same engineering eort, a dedi ated ontrol performan e improvement of about 30% an be a hieved by numeri al multi-obje tive optimization as ompared to manual design parameter tuning in a sequential manner.

2.1.2 System Model Des ription Both linear and nonlinear design models an be taken into a

ount. In a multimodel approa h linear models together with nonlinear models an be used simultaneously. There is no restri tion on the representation of system disturban es. Robustness against stru tured parameter deviations or, for example, sensor failures is a hieved by applying a ommon ontroller to a set of xed worst- ase parameter models. This model set hara terizes the worst dynami s deviations within the range of operation, e.g. the ight envelope, or a part of it. For ea h su h model the appropriate set of riteria has to be spe ied. Hen e, the multimodel problem is transformed into a multi- riteria optimization problem. In general, there exists no theory that guarantees stability or performan e robustness a ross the range of operation, if only a nite number of operating points is onsidered simultaneously. It depends on the physi al properties of the system to be designed, whether runaways an exist. If they exist, they have to be added to the set of worst- ase operating points and treated simultaneously by the multi-model approa h. Worst- ase operating points an be omputed by a dual appli ation of the parameter optimization approa h:

Find those

parameter ombinations within a given un ertainty domain whi h yield the worst performan e for the hosen multi- riteria set up [20℄. Of ourse, robustness of the ontroller about an operating point an be enfor ed in the multi-obje tive approa h by adding suitable robustness riteria (e.g. gain/phase margins) to the set of otherwise spe ied performan e riteria.

2.1.3 Controller Stru ture Both linear ontrollers and nonlinear ontrollers (e.g. fuzzy ontrollers or adaptive ontrollers) an be used. If a spe i analyti al synthesis te hnique is applied within this framework, the ontroller stru ture is bound to this synthesis te hnique.

14

2.1.4 Design Spe i ations Ea h design obje tive may be mathemati ally des ribed by a well-dened riterion

i

whose value is the smaller, the better the obje tive is a

omplished.

Examples taken from the RCAM design hallenge spe i ations are: - Steady state error, settling time and rise time for demanded steady state value

ys :

=

Z tend

t1

(y(t) ys )2 dt

- Overshoot over demanded steady state value

ys :

= max (y(t)=ys ) t - Rise time dened as the time the unit step response

y(t1 ) = 0:10 to y(t2 ) = 0:90:

= t2 - Minimal damping of the eigenvalues

y(t)

takes from

t1 evi :

= 1 min ( Re(evi )=jevi j) : i In order to get smooth riteria as a fun tion of the tuning parameters, the min- or max-fun tions involved are smoothed by an exponential approximation; see also se tion 2.5. The above eigenvalue riterion minimal damping is reformulated in su h a manner that better damping results in a smaller riterion value.

2.1.5 Analysis Information To evaluate the hosen design riteria have to be performed.

i , the respe tive analysis omputations

This usually requires eigenvalue omputations, time

response simulations and frequen y response omputations. This analysis information is used to judge the quality of a design in addition to the riteria values whi h quantify the balan e of the a hieved optimum, and whi h provide further ontroller synthesis information (see se tion 2.1.6).

2.1.6 Controller Synthesis Information To ea h riterion,

i

di is T = [Tk ℄ are omputed

an upper-bound demand value or driver value

dened by the designer. Then the tuning parameters

by solving the min-max parameter optimization problem

min max f i =di g T

i

15

subje t to performan e and tuning onstraints:

gj (T )  0; Tkmin  Tk  Tkmax: This is the MOPS synthesis formula. By iterating the demand values as a fun tion of the a hieved riteria values

i

1,

di

the resulting ompromise

trade-o solution an be driven in a desired dire tion.

2.1.7 Pra ti al Implementation Aspe ts The overall omputing time for the synthesis mainly depends on the time for

riteria evaluations. Hen e fast algorithms and software implementations [101℄ are required for the orresponding analysis omputations. It is good pra ti e to use  heap riteria where possible. The number of riteria evaluations also depends on the number of models used in a multi-model set-up. Therefore it is also bene ial to minimize the number of models by a areful sele tion of worst- ase operating points or parameter deviations.

2.1.8 Relation with other Methods Multi-obje tive parameter synthesis loses the parametri design loop with modelling-, synthesis- and analysis methods a

ording to Figure 2.1.

synthesis model

D

T

C

synthesis

plant model

P

performance/cost criteria

controller model

closed-loop model

M

simulation/ analysis

I

Figure 2.1: Design loop losed by multi-obje tive parameter synthesis. It serves to automate ontrol tuning to given performan e spe i ations. It is neutral as far as the design steps modelling, synthesis and analysis are

on erned.

2.2 Example of Appli ation The approa h has been applied for robust ight ontrol [138, 102, 100℄, a tive antenna-beam ontrol [19℄, PWM-satellite attitude ontrol [98℄, maglev vehi le

16

ontrol [190℄, (semi-)a tive ar suspension and air raft landing gear ontrol [81, 209, 254℄, robot ontrol [153℄, and others. An example appli ation is the MOPSsolution for the RCAM design hallenge [130℄.

There, for the longitudinal

ontrol, LQR PI-output feedba k is used, whereas for lateral ontrol a lassi al

ontrol stru tue [35℄ is used, thereby demonstrating the appli ation for two dierent ontroller stru tures. A nonlinear worst- ase plant analysis, also using MOPS, was performed to he k robustness within the multi-model set-up.

2.3 Computational Aspe ts The method requires the set-up of a omputation loop a

ording to Fig. 2.1 and the availability of a suitable min-max parameter optimization software. For an engineering-e ient appli ation of this te hnique it is very bene ial to have a software framework whi h supports intera tive modular problem setup and demand spe i ation as well as automated performan e evaluation (su h as ANDECS_MOPS [99℄). Multi-model/multi-obje tive performan e evaluation an be fun tionally parallelized, e.g.

by using the PVM (Parallel Virtual Ma hine) lient-server

network on ept. Thereby the omputation time an be redu ed. The multi-obje tive optimization problem an be solved by any nonlinear programming tool, sin e minimizing a set of riteria an be transformed into a

onventional s alar nonlinear programming problem; see Ÿ2.5. Using appli ation-spe i engineering riteria in pra ti e, typi ally leads to non onvex optimization problems. Thus lo al minima may exist. However, a lo al minimum solution is also a lo al best-possible pareto-optimal solution. If su h a solution is not satisfa tory, other solutions an be found by hanging the demand values, or by hanging the starting values for the tuning parameters. To avoid lo al minima, a global optimizer has to be used whi h may have the disadvantage of rather long omputing times.

2.4 Comparative Study Multi-obje tive parameter synthesis allows full exploitation of a given ontroller stru ture, as a fun tion of the ontroller parametrization.

In parti -

ular, it allows the exploitation of the a hievable trade-os between ontrol performan e and required ontrol eort. This is possible in a most detailed, appli ation-spe i way and hen e, no matter what ontroller stru ture or ontroller synthesis method is used, this te hnique, in prin iple, always yields the best possible performan e in the hosen ontext. A potential benet of multi-obje tive tuning the design parameters of an analyti synthesis method (e.g. LQR, eigenstru ture synthesis, et .) instead of dire tly tuning the parameters of a given ontroller stru ture (i.e. state- or dynami output feedba k), is primarily that usually a smaller number of parameters is to be tuned. This parti ularly holds for multi-input/multi-output

17

systems. Also, built-in performan e and robustness features of the hosen synthesis method are automati ally guaranteed. On the other hand, dire tly tuning the parameters of a spe ied ontroller stru ture allows the designer to use appli ation-proven ontroller stru tures for whi h no analyti synthesis te hniques exist, and it allows him to extend and to adapt su h stru tures during the design pro ess. If an appropriate software framework is available whi h provides a predened omputation loop and a set of standard riteria to hoose from (e.g. ANDECS), the level of required training is moderate. In this ase, no spe i mathemati al theory is required. Design spe i ations are expli itely formulated in their most natural mathemati al form and a trans ription of design spe i ations into a synthesis-spe i weighting form is not required. In omplex design-de ision problems with, say, more than 5 riteria to be handled simultaneously, an integrated data system is mandatory, to keep tra k of the de ision iterations made during the design pro ess (this holds for any design-iteration logi ). The ANDECS software provides su h an integrated data system, whi h is spe i ally-designed for multi-obje tive/multi-model design iterations.

2.5 Mathemati al Appendix 2.5.1 Preferen e order, goal oni ts and satisfa tory ompromise sets for riteria ve tors The main advantage of a multi-obje tive design is the possibility to formulate an individual riterion for ea h spe ied demand, while treating all riteria during optimization simultaneously. Here, some terms are laried by introdu ing the related on epts [128℄: - better solution in the ontext of a preferen e order for ve tor-valued

riteria, - goal oni t and  ompromise in the ontext of pareto-optimality and - satisfa tory ompromise in the ontext of demand level. (i) The individual riteria

i (T ) are ombined to give a riteria ve tor (T ).

The following preferen e order allows one to partially ompare su h ve tors:

A set of tuning parameters

T1

is said to be better than

the orresponding riteria ve tor

(T 1)

is smaller than

where smaller means

(T 1) < (T 2 ) , i (T 1) < i (T 2 ) for all i 18

T 2 , if

(T 2),

Smaller equal is dened as

(T 1)  (T 2 ) , i (T 1)  i (T 2 ) for all i 1 2 and i (T ) < i (T ) for at least one i. (ii) Trying to improve several riteria simultaneously normally leads to a goal

oni t in the sense that no riterion an be improved further without worsening another one. More pre isely:

A set of tuning parameters

T  is alled a ompromise solution,

or pareto-optimal solution, if there is no T with

(T ) < (T  ).

(iii) Usually, ompromise solutions are not unique. There exists a whole set of pareto-optimal solutions and it is up to the design engineer to de ide what trade-o is a best satisfa tory ompromise in his design ontext. The term satisfa tory an be made more pre ise by introdu ing the demand level

d referring to a riteria ve tor :

A set of tuning parameters

if T 

T  forms a satisfa tory ompromise,

belongs to the set of ompromise solutions and if

(T )  d ; where in

ve tor d

d

the demands of the designer are quantied. The

is alled the demand level.

Fig. 2.2 illustrates the above denitions for the ase of 2 riteria

2 .

Assume that that

(fT g)

1

and

fT g denotes the set of all feasible tuning parameters T and

(fT g)

is the orresponding value set. The thi k border part of

in Fig. 2.2 is the set of ompromise solutions and

Cs

marks the subset of a

satisfa tory ompromise. Note, that all solutions with riteria values smaller than the demand level

d

are satisfa tory solutions.

2.5.2 Finding a satisfa tory ompromise set by means of min-max optimization A parti ular, satisfa tory ompromise an be found by means of parameter optimization. From the riteria ve tor

(T ) and the demand level d one an form

a s alar fun tion

 = max f i (T )=di g : i

19

c2

c({T}) d* Cs c1 Figure 2.2: Demand level and satisfa tory ompromise set in two-dimensional

riteria spa e

Of ourse, we have solution if solution of

  1. 

(T )  d and therefore we have a satisfa tory

Moreover, it an be shown [205℄ that a minimum

 = min max f i (T )=di g i T

is a ompromise solution. Again, if



is less than or equal 1, the

ompromise solution is satisfa tory. Hen e the problem of nding a satisfa tory ompromise solution is redu ed to a s alar min-max optimization problem.

This is also known as goal at-

tainment with a zero ideal point [90℄. Fig. 2.3 illustrates what best possible solution is a hieved by min-max optimization in a two dimensional riteria spa e. The min-max optimization problem an be solved using standard nonlinear

 as obje tive fun tion. However, the non due to the maximum fun tion may ause problems if gradient

programming methods applied to smoothness of

based solvers are applied.

In this ase, it is preferable to reformulate the

optimization problem in one of two ways: 1. The un onstrained min-max optimization problem with smooth riteria

i (T ) is equivalent to the onstrained problem [181℄ min max f i (T )=dg , minftg ; s:t: i (T )  t : T

i

i

T;t

Solving the min-max problem in this way yields exa t solutions.

20

c2

c({T}) d* c* c1 Figure 2.3: Satisfa tory ompromise found by min-max optimization

2. Approximate solutions are found if the fun tion

 is approximated by a

smooth fun tion, as proposed in [138℄:

X  max f i (T )=di g = lim !1 1= ln ( exp( i (T )=di )) i

i

=  + lim !1 1= ln (

X

i

(( i (T )=di

exp

))) :

This approximation formulation is well suited for numeri al omputation, sin e the argument of the exponential is always less than or equal to zero. The approximated min-max problem an be solved as an un onstrained parameter optimization problem. Of ourse, the fun tion

 an be minimized dire tly if optimization methods

su h as dire t sear h methods are used, whi h do not require smooth obje tive fun tions.

21

3.

Eigenstru ture Assignment

Lester Faleiro , Jean-François Magni , Jesús M. de la Cruz and Stefano S ala 1

2

3

4

3.1 Introdu tion The theory presented here on erns the design hapters 17, 18, 19 and some aspe ts of hapter 28.

The main on epts of eigenstru ture assignment as a

design te hnique will be explored, in orporating a short explanation of how to

hoose a desired eigenstru ture based on design spe i ations. The mathemati al methods used will also be summarised, and some omments given on the use of eigenstru ture assignment.

3.2 Eigenstru ture Analysis The equations that des ribe an air raft and their relation to the time response of that air raft an be grouped together in matrix form:

x_ = Ax + Bu (3.1) y = Cx + Du where the most important of these matri es, A, des ribes the internal dynami s of the air raft. The B matrix des ribes the distribution of the a tuator inputs to the states of the air raft, and the C matrix denes how the states an be observed as outputs of the system. D is usually zero for an air raft, though non-zero matri es o

ur when air raft a

elerations are in luded in the outputs.

x

is the state ve tor,

u

y is the output measurement n states, m inputs and p outputs.

is the input ve tor and

ve tor. It will be assumed that the system has

A an be further de omposed into its onstituent eigenvalues and eigenve -

tors. The derivation of these an be found in any standard text on linear matrix algebra. Let the

n eigenvalues and eigenve tors of the system be dened by:

 = [1 : : : i : : : n ℄ and V = [v1 : : : vi : : : vn ℄ 1

(3.2)

Department of Aeronauti al and Automotive Engineering and Transport Studies, Lough-

borough University, Loughborough, Lei estershire LE11 3TU, United Kingdom. 2

CERT ONERA, Département d'études et Re her hes en Automatique, BP 4025, F31055

Toulouse Cedex, Fran e. 3

Dep. Informáti a y Atomati á. Fa ultad de Cien ias Físi as. Universidad Computense.

28040 Madrid, Spain. 4

Flight Control and Me hani s department, Centro Italiano Ri er he Aerospaziali, 81043

Capua, Italy.

22

where

AV = V 

The eigenve tor set

V

(3.3)

is a basis set for the state spa e

x; thus any ve tor in

the state spa e an be expressed as a linear ombination of the eigenve tors of the air raft system. These eigenve tors are also alled the right eigenve tors of the system. The left, or dual basis eigenve tors of the same system are given by

W , where

W T = [w 1 : : : w i : : : w n ℄ ; W A = W

(3.4)

Solving the state-spa e equations given in (3.1) yields an expression for the time response that an be found in most standard ontrol texts:

y(t) =

n X i=1

Cvi wTi ei t x0 +

n X i=1

Cv i wTi

Z t

0

ei (t  )Bu( )d

(3.5)

It is lear from this equation that there are two omponents to the time response. The rst is dependent on the initial onditions of the system, and is alled the homogeneous omponent; the se ond is dependent on an input to the system, and is alled the for ed omponent. The entire time response of a linear system thus depends on four variables: The eigenvalues of the system The eigenve tors of the system The initial onditions of the system The inputs to the system Ea h of these plays a part in the determination of the time response, and di tates the overall ee t that modes and inputs play in the output response of the system. The homogeneous omponent of equation (3.5) an be written as

y(t) = where

n X i=1

Ci ei t vi

i are the s alars wTi x0 , i = 1 : : : n.

(3.6)

This shows that the output response

is omposed of a linear ombination of eigenvalue-eigenve tor sets of the matrix

A.

Ea h of these sets is alled a mode. In every mode the eigenvalue deter-

mines the de ay/growth rate of the response and the eigenve tor determines the strength of the oupling of this mode with the outputs.

ith mode with the j th output is given by C j v i , where C j is the row of C . If C j v i = 0, then equation th mode does not ontribute to the j th output; they have (3.6) shows that the i From (3.6) we an see that the oupling of the

j th

been de oupled. As an example of how the information about the nature of eigenstru ture

an be used, let us examine a simple linear representation of the longitudinal

23

dynami s of the RCAM model, in terms of four varying states of the system.

Mathemati ally, we an determine the time response of the system to

an arbitrary initial ondition, but this does not ne essarily give us a omplete understanding of the system dynami s. Mode 1 2

Eigenvalue

0:830  1:107i 0:011  0:126i

Damping ratio

Frequen y (rad/s)

0.6

1.38

0.09

0.13

Table 3.1: Modes of the open-loop system The eigenvalues of this nominal system are shown in Table 3.1.

It an

be seen that although there are four states in the system, there are only two modes in its dynami behaviour. It is known that the Phugoid and the SPPO (Short Period Pi hing Os illation) are the two os illatory modes that hara terise air raft longitudinal motion, and that the Phugoid usually has a mu h lower frequen y and damping than the SPPO. However, if these modes were in any way un onventional, a knowledge of the eigenve tors alone would not be su ient to understand the air raft. States

q  u w

Mode 1

0:014 0:010 0:015 1

6 61:5 6 8:3 6 14:5 6 20:7

Mode 2

0:002 0:0132 0:99 0:142

6 50:2 6 34:7 6 39:5 6 41:8

Table 3.2: Eigenve tors of the open-loop system (magnitude and argument)

The only way to ensure that ea h of the modes an be attributed to parti ular air raft hara teristi s is by a subsequent examination of the right eigenve tors of the system. For this ase, these are shown in Table 3.2. The eigenve tors for a mode are read verti ally down the table. It an be seen that Mode 1 is

hara terised by a large intera tion with

w, the standard hara teristi of the u, and omparatively

SPPO. Mode 2 is hara terised by a large intera tion with little with

w,

typi al of the Phugoid. The two modes an thus be designated

as 1. SPPO and 2. Phugoid. In the time domain, the peak for ea h of these states will dier a

ording to the phase angles (arguments) of the elements of the eigenve tor, given in degrees in Table 3.2. Note that usually the magnitude, rather than the phase, in eigenstru ture assignment an be more easily visualised for the purposes of design and analysis, so only the magnitudes will be used in eigenve tor des ription from now on. Additional information about the system an be obtained by using the left eigenve tors to determine the ee t that ea h input has on ea h mode of the system.

These input oupling ve tors are given by the produ t of the left

eigenve tors and the input distribution matrix,

24

W B.

For the above example,

the input oupling is given below: Mode

Æt

Æth

SPPO

85.4

19.3

Phugoid

31.5

13.7

This shows that the SPPO will be ex ited by a taileron input to a mu h larger extent than a throttle input, and the Phugoid is the same. This qualitative eigenstru ture analysis is a tool that an be used to examine the nature of the modes of a system qui kly. Classi al te hniques usually assume that a knowledge of the system dynami s is readily available with the model.

This

is a fair assumption, but may be ome redundant if more omplex modes are involved in the open-loop system. Additionally, this te hnique of analysis is invaluable during the eigenstru ture assignment pro ess in examining the sour e of design problems.

3.3 Eigenstru ture Assignment It was shown in equation (3.6) that the output response of a air raft an be des ribed by a representation involving its eigenvalues and eigenve tors. Thus, if the eigenstru ture of the air raft an be manipulated somehow, we have a means of altering its time response. Various forms of dire t eigenstru ture assignment methodology exist, from the rst tentative steps in output feedba k by Kimura [135℄ to their further development by Andry, Shapiro and Chung [211℄ to urrent work su h as that done by Sobel, Lallman and Shapiro [219℄, [221℄ and [220℄. In essen e, all these methods are similar, and fun tion in mu h the same way. They all require the designer to spe ify a set of eigenvalues and eigenve tors for the design, and they all produ e a proportional gain matrix ontroller.

3.3.1 Determination of the desired eigenstru ture The philosophy behind dire t eigenstru ture assignment is that whilst the designer is able to spe ify a set of desired losed-loop eigenvalues

d ,

she/he

is also able to spe ify exa tly whi h elements of the desired eigenve tors

Vd

she/he would like to set to zero, where

d = [d1 : : : di : : : dp ℄ ; V d = [vd1 : : : vdi : : : vdp ℄

(3.7)

This an be illustrated by the set of eigenve tors shown in Table 3.3. We would perhaps like the SPPO mode of response to be unae ted by forward velo ity and pit h angle, and vi e-versa. We therefore spe ify these elements in the desired losed-loop eigenstru ture to be zero. We are un on erned with the values of the remaining elements, designated by an 'x'.

A similar situa-

tion o

urs with the Phugoid eigenve tor. This pro ess is ee tively assigning elements of

vi

in (3.6) to zero.

25

States

q  u w

SPPO

Phugoid

x

0

0

x

0

x

x

0

Table 3.3: Example of desired losed-loop eigenve tors

The ontrol design problem an thus be stated as follows: Given a set of

d and a orresponding set of desired eigenve tors V d , nd m  p matrix K su h that the eigenvalues of the losed-loop system matrix (A + BKC ), obtained when using the output feedba k ontrol equation desired eigenvalues

an

u = Ky; in lude

d

(3.8)

(A + BKC ) are

as a subset, and the orresponding eigenve tors of

as lose as possible to the respe tive members of the set

V d.

3.3.2 The a hievable ve tor spa e Now, from the eigenve tor equation of the losed-loop system:

(A + BKC )vi = i vi ; i = 1 : : : p

Avi 

(3.9)

i vi + BKCvi = 0

A i I B

where





(3.10)



vi = 0 zi

(3.11)

zi = KCvi

(3.12)

So, for a non-trivial solution,



vi zi





2 Ker A i I B 

n rows of the null spa e (Ker) of A i I B spa e, N i . A se ond method that an be used

and the rst able ve tor



(3.13)



form the a hiev-

to determine this

spa e an be derived from (3.10):

Dene

vi = (A i I ) 1 BKCvi

(3.14)

N i = (A i I ) 1 B

(3.15)

and now the losed-loop eigenve tors should omply with

26

vi = N i zi

(3.16)

in order to obtain the required eigenvalues. The a hievable eigenve tors lie in the subspa e spanned by the olumns of the matrix

vai must

N i . Expanding this N i is of dimension

example into more general terms, the subspa e des ribed by

m. Ni

On e the desired eigenvalues have been hosen, the range spa e of matri es

onstrains the sele tion of the losed-loop eigenve tors.

desired eigenve tors

v di

In general, the

will not reside in the a hievable eigenve tor spa e. In

order to have the resulting eigenve tor as lose as possible to the desired one, an optimum hoi e is made by proje ting the desired eigenve tor onto the a hievable spa e,

N i.

This is illustrated diagrammati ally in gure 3.1 for a simple three dimensional system. achievable vector v ia

desired vector v id

Dimension 3

This vector space, defined by the null vectors, describes the set of points over which the desired eigenvalues can be realised.

Null space vectors

Figure 3.1: Representation of de oupling in a 3-dimensional state spa e In this example, the desired ve tor an be hosen to de ouple a mode from a dimension.

As an example, say we want this mode to be de oupled from

Dimension 2. Thus, for this system, the only possible a hievable eigenve tor is given by the interse tion between the null spa e (whi h is the only pla e where the desired eigenvalue will be produ ed) and the Dimension 1/Dimension 3 plane (the lo us of points whi h does not ontain any omponent of Dimension 2).

Sin e the desired eigenve tor

vdi

ontains desired de oupling information

(i.e. a zero in the Dimension 2 row), it will lie on the Dimension 1/Dimension 3 plane. In real systems, this on ept an be expanded to de ouple modes from air raft outputs.

On e the desired eigenstru ture has been worked out, the

nal eigeve tors of the system an be produ ed.

3.3.3 Determination of the nal eigenve tors Ri an be dened su h that: A

ording to [12℄, a reordering operator f
g fvdi gRi



= dli i



and fN i 27

gRi



~i  N = D i

(3.17)

where

li

and

di

are the ve tors of spe ied and unspe ied omponents of

Ni

respe tively. The rows of the null spa e

vdi

have been reordered in the same

way. The nal eigenve tor is given by (see [12℄)

where

(
)y

y vi = N i N~i li

(3.18)

denotes the pseudo-inverse.

It is also possible to determine the nal eigenve tors without the use of proje tion. For ea h desired eigenve tor, the de oupled elements are integrated into a row ve tor

gi vi = 0,

gi

su h that if

vdi = [x x 0 x℄T , gi = [0 0 1 0℄T .

Thus,

sin e the nal eigenve tor should also have the relevant elements

de oupled. Thus, equation (3.11) an be rewritten as



A i I B gi 0





vi =0 zi

(3.19)

and for a non-trivial solution,



vi zi



2 Ker



A i I B gi 0

 (3.20)

This ve tor in the null spa e an now be suitable partitioned and its rst entries an be used to form

vai

n

3.3.4 Determination of the feedba k gain vi an now be grouped into the eigenve tor matrix V . The zi ( omputed together with vi using (3.20)) are grouped into the matrix

These eigenve tors ve tors

Z.

From (3.12) the feedba k gain satises

KCV = Z Usually, the number of olumns of

p, therefore

V

and

Z

(3.21) is equal to the number of outputs

K = Z (CV ) 1

If the number of olumns is larger than

p,

(3.22)

a dynami fedba k an be used as

detailed in Ÿ3.4.4. When

vi is omputed as in (3.18), the orresponding ve tors zi an be found

easily in order to solve (3.22). However the resulting stati feedba k gain matrix

an be determined dire tly by substituting rearranged to give:

V

into equation (3.9), whi h an be

K = B y (V  AV )(CV ) 1

(3.23)

Other ways of al ulating the gain matrix for numeri al e ien y and in the

ase of matrix non-invertibility have been des ribed in the literature ([12℄, [133℄), and an be used instead of equation (3.23) if desired.

28

3.4 Robustness to Parameter Variation Standard eigenstru ture assignment, as des ribed in previous se tions, takes performan e and de oupling into a

ount, but does not relate to any robustness requirements. Four dierent, and sometimes o-operative, ways of ta kling this problem have been pursued with the RCAM problem.

3.4.1 Open-loop ve tor proje tion It has been shown by Wilkinson in [256℄ and [160℄ that for a perturbation in the losed-loop matrix

(A + BKC ) given by
(A + BKC ), the orresponding

rst order perturbation in the relevant eigenvalue is given by:


i = wi
(A + BKC )vi where w i and v i are normalized su h that w i v i = 1.

(3.24) On the assumption that

the open-loop eigenvalues do not vary a lot with parameter variation, (3.24) shows that any variation an be related dire tly to the eigenve tors of the system. Thus, if the open-loop eigenve tors are used as the desired eigenve tors, eigenvalue sensitivity to perturbation should not be deteriorated by feedba k. This thesis is used in the RCAM design in hapter 19.

3.4.2 Iterative assignment Kautsy et al. [133℄ proposed using iterative eigenstru ture assignment to de rease the sensitivity of an eigenvalue in a state-feedba k ontrol system. An iteration is used in whi h the ve tor

vi

is repla ed by a new ve tor with maxi-

mum angle to the remainder of the urrent right eigenve tor spa e

i = 1; 2; : : : ; n in turn.

V i for ea h

The new ve tor is obtained, letting:

V i = [v1 : : : vi 1 vi+1 : : : vn ℄

(3.25)

wi (ith left eigenve tor) is orthogonal to V i , and the new v i is found by prowi (now ee tively the desired ve tor for the ith mode) into N i (whi h

je ting

ontains the a hievable right eigenve tor spa e):

vi =

N i N Ti wi k N Ti wi k2

(3.26)

thus giving a ve tor that is as orthogonal as possible to the urrent spa e whilst retaining the desired eigenvalues of the losed-loop system. This means that a perturbation in any of the elements of the remaining eigenve tors due to parameter variation will not ae t the urrent mode. The iteration is ontinued until the redu tion in the ondition number of the

V

matrix is less than some



toleran e. This is be ause the ondition number of the matrix ( ) is a measure of the overall sensitivity of the system. At the end of this iteration, a

V

matrix

for a minimum sensitivity solution remains. Ba k substitution of this matrix into equation (3.23) produ es a feedba k gain matrix.

29

Of ourse, altering the eigenve tors in this way does inevitable result in a loss of performan e.

The pro ess of de omposition and proje tion would

result in a loss of desired de oupling. However, using the null spa e des ribed in equation (3.20) an help to over ome this problem, as the null spa e itself

ontains the de oupling required. A further des ription of the use of this pro ess is given in [77℄.

3.4.3 Stability margin improvement A se ond riterion in use is a measure of loop robustness in terms of gain and phase margins. If the air raft is represented by

G(s), a variety of loop transfer

fun tions an be used to determine losed-loop system robustness. The singular values of the sensitivity fun tion plementary sensitivity fun tion fun tion

T = L(I + L) 1

S = (I + L) 1 ,

the om-

and the balan ed sensitivity

S + T , where L is the open loop gain matrix, an be used to measure

the stability margins for multiloop feedba k ontrol systems ([152℄, [50℄ and [28℄). The design pro edure in hapter 18 uses these measures, and the design in [77℄ uses similar ones. The fun tions

S

and

T

may be al ulated at the a tuator inputs or at the

sensor outputs. At the inputs,

L = KG and at the outputs, L = GK . The peak  S , T or S + T gives a robustness

value of the maximun singular value ( ) of

guarantee for all frequen ies. The formulae applied to omputing the stability margins using the sensitivity fun tion are the following:

a = 1=(S ) Gain Margin Phase Margin where gains

= [1=(1 + a); 1=(1 a)℄ = 2sin 1 (a=2)

(3.27)

a  1. The gains of the loops may thus be perturbed simultaneously by
satisfying 1=(1 + a) <
< 1=(1 a) without destabilising the losed

loop system. Similarly, the feedba k loops may be perturbed simultaneously

 satisfying j  j< 2sin 1 (a=2) without destabilising the losed loop system. The best possible gain and phase margins are obtained when  (S ) = 1, o in this ase the gain margin is [ 6 dB, +1 dB℄ and the phase margin is 60 . Similar margin equations an be devised for the T and S + T . These stability by phases

margins are known to be onservative, and a better approa h is obtained by repla ing the maximum singular value



with the stru tured singular value



[44℄, [28℄. The above des ription gives only a measure of robustness. In order to use this information in a design synthesis, an iterative loop whi h ontains the eigenstru ture assignment design pro ess, but updates the hoi e of eigenvalue and eigenve tor an be used. This pro ess produ es variable results, depending on the air raft and the initial design spe i ations, but has nonetheless been found to be useful. Previous examples of the use of these stability margins to improve robustness of air raft ontrol systems an be found in [178℄ and [76℄.

30

3.4.4 A multimodel approa h A fourth way of improving the robustness of an eigenstru ture assignment design is to use the multi-model approa h des ribed in [150℄.

The RCAM

design des ribed in Chapter 19 uses this method. It relies on produ ing a bank of linear air raft models at dierent operating points. These models are denoted

(Ai ; B i ; C i ) i = 1 : : : p.

Extra freedom to

improve robustness is introdu ed with the multi-model approa h. Instead of assigning all the available eigenstru ture to one linear model, a dierent model may be used for ea h assignment.

Thus, models with parti ularly sensitive

eigenvalues an be isolated, and the relevant eigenvalue-eigenve tor pair an be re-assigned to improve the robustness of a parti ular mode on a parti ular model. Thus, for ea h eigenvalue in turn, hoose solve for

v i , ti :



Ai

i I B i gi 0



i

and a model

(Ai ; B i ; C i ) then



vi zi = 0

(3.28)

First ase: the number of eigenve tors to be assigned is equal to the number of outputs, solve for

K

by using:

K [C 1 v 1 C 2 v 2 : : : C p v p ℄ = [z 1 z 2 : : : z p ℄

K = [z1 z2 : : : zp ℄[C 1 v1 C 2 v2 : : : C p vp ℄ 1

(3.29) (3.30)

Se ond ase: more ve tors need to be assigned. It is ne essary to use a dynami feedba k. Let

K (s) denote the transfer fun tion matrix of the feedba k. K (s)

In [150℄

is justied the fa t that we have to solve for

K (1 )C1 v1 = z1 ; K (2 )C2 v2 = z2 ; : : : Note that now, the assigned eigenvalue

i

(3.31)

appears in the equation. Finding a

solution to (3.31) is far more di ult than in the previous ase (see [150℄, [161℄ for details.)

3.5 Con lusions This hapter has shown that the main pro ess of eigenstru ture assignment an be broken up into two. The rst, and arguably most important, element is the spe i ation of eigenstru ture based on the designers requirements and experien e. The se ond is the mathemati al pro ess of eigenstru ture assignment itself. This latter pro ess onsists of nding an a hievable eigenve tor spa e whi h will produ e the desired losed-loop eigenvalues whi h have been spe ied for performan e. Ve tors an then be hosen from this spa e to give required de oupling. Additional manipulation to redu e eigenvalue sensitivity an also be employed. Robustness an best be a hieved by using eigenstru ture assignment

31

as a part of a large design strategy. Goal attainment, the use of singular values and multi-model design have been des ribed as used for the RCAM problem. Additionally, eigenstru ture an be further manipulated to give dynami

ontrollers, whi h have been des ribed for both the point design [77℄ and the multi-model design [55℄.

This is advisable in ases where additional design

freedom is required. Despite all the versatility and potential visibility of the method, eigenstru ture assignment is most useful as a tool within a fuller design environment, thus allowing the attainment of good performan e, de oupling and robustness in the resulting ontrol system.

32

4.

Linear Quadrati Optimal Control

Fran es o Amato 1, Massimiliano Mattei and Stefano S ala

1

2

4.1 Introdu tion Linear quadrati optimal ontrol is ertainly the most widely applied modern

ontrol te hnique. The fundamentals of this theory, whi h date ba k at least to the Fifties (see the germinal paper [131℄ and the bibliography therein) an be found in the Spe ial Issue on the LQG problem [1℄ whi h appeared as an IEEE Transa tion on Automati Control in 1971; sin e then, many books have been written on this subje t (see among others [10℄ and [140℄). This ontrol te hnique allows the designer to take into a

ount both requirements on the amplitude of the ontrol inputs and the settling time of the state variables; moreover, when onsidering innite horizon optimization and provided that the weighting matri es are suitably hosen, an important feature of LQ ontrol is that the resulting losed-loop system exhibits very good guaranteed multivariable stability margins.

Many appli ations of the LQ theory

have been performed in the aeronauti al eld. One of the most important is

ertainly the design of the ight ontrol system of the AFTI/F-16 air raft by General Dynami s (see [70℄). When the omplete state is not available for measurement and some or all of the measures are ae ted by noise, one an use the Kalman optimal ltering theory [1℄ (whi h turns out to be the dual of the LQ optimal ontrol theory) to design an observer of the state variables; however the robustness margins are no longer guaranteed in the presen e of an observer. If sensor noise is absent or one does not are about it, it is possible to use the degree of freedom on the design of the observer to re over the LQ robustness margins; this is the elebrated Loop

Transfer Re overy (LTR) te hnique (see [226℄), whi h, however, an be applied only when the plant under onsideration is minimum phase. Appli ations of the LTR in the aereonati al eld an be found in [64℄, [203℄, and [249℄. Finally in [231℄ some appli ations in aeronauti s of the linear quadrati optimal stati output feedba k ontrol, developed in [172℄, are provided. 1

Dipartimento di Informati a e Sistemisti a, Università degli Studi di Napoli Federi o II

via Claudio 21, 80125 Napoli, Italy, Tel.+39(81)7683172, Fax+39(81)7683686 2

Centro

Italiano

Ri er he

Aerospaziali

Via

Tel.+39(823)623949, Fax+39(823)623335

33

Maiorise,

81043

Capua

(CE),

Italy

4.2 Plant Model Requirements and Controller Stru ture Let us start by onsidering the linear time-invariant plant

x_ = Ax + Bu u x(t)

where, as usual,

x(0) = x0

(4.1)

2 IR n is the state and u(t) 2 IR m is the ontrol.

The

steady-state Linear Quadrati (LQ) optimal ontrol problem an be stated as follows:

Problem:

Q

given

[0; +1) ! IR m

 0 and R > 0, nd, if existing, the ontrol law u : t 2

whi h minimizes the ost fun tion:

J (u) = If the pair

(A; Bu )

Z

0

+1 



xT (t)Qx(t) + uT (t)Ru(t) dt :

(4.2)

is stabilizable the problem is solvable and the optimal

ontrol law turns out to be a state feedba k ontrol law in the form

u(t) = Kx(t)

(4.3)

therefore we often talk of Linear Quadrati State Feedba k (LQSF) optimal

ontrol law; the optimal gain matrix

K

is given by

K = R 1 BuT P where

P

(4.4)

is the unique positive semidenite solution of the algebrai Ri

ati

equation

AT P + P A + Q P Bu R 1 BuT P = 0 : Finally the value of J (
) orresponding to the minimum is

(4.5)

Jopt = xT0 P x0 :

(4.6)

Let us onsider the losed-loop system in Figure 4.1 given by the onne tion of (4.1) and (4.3). As shown in [152℄ and [202℄, if the weighting matrix

R

is

hosen in diagonal form, this system exhibits, at the plant input, guaranteed lower and upper multivariable gain margins of

1=2 and +1 respe tively; more60o and

over, the guaranteed lower and upper multivariable phase margins are

+60o respe tively.

weighting matrix

Therefore LQSF optimal ontrol systems, provided that the

R

is properly hosen, have good robustness properties; this

fa t has further en ouraged ontrol engineers in appli ation of this te hnique in several elds. Now we assume that not all states are available for measurement and that some or all of the measures are ae ted by white noise

x_ = Ax + Bu u + Bw w y = Cx + m 34

(4.7a) (4.7b)

u

Bu

(sI-A)

x

-1

-

K Figure 4.1: LQSF system

where

y(t) 2 IR r

and

ww ( ) = w Æ(t  ) mm ( ) = m Æ(t  )

(4.8a) (4.8b)

are the auto ovarian e fun tions of the sto hasti pro esses that

m

w and m; we assume

is stri tly positive denite.

The steady-state Linear Quadrati Estimator (LQE) problem an be stated as follows: Find a linear state estimator

x^ = L(u; y)

(4.9)

whi h minimizes the steady-state mean square re onstru tion error

where

If the pair

(A; 1w=2 )

 T  Ex (L) = t!lim E e ( t ) e ( t ) x x +1

(4.10)

ex (t) = x(t) x^(t) :

(4.11)

is stabilizable and the pair

(A; C )

is dete table, the

estimator problem is solvable; moreover the optimal estimator (whi h takes the name of Kalman Filter) is a dynami system whi h possesses a Luenberger observer stru ture

_ = A + Bu u + L(y C ) x^ =  where the optimal gain matrix

tion

(4.12b)

L is given by L = C T m1

and

(4.12a)

(4.13)

 is the unique positive semidenite solution of the algebrai Ri

ati equa-

A + AT + Bw w BwT 35

C T m1 C  = 0 :

(4.14)

Finally the value of the ost fun tion orresponding to the optimum is given by

Exopt = tr() :

(4.15)

It is readily seen that the LQ and the LQE problems are duals of ea h other. An immediate onsequen e is that, if we onsider the losed-loop system in Figure 4.2, this system exhibits at the output, the same robustness margins of the LQSF system.

(sI-A)

^ x

-1

C

-

L Figure 4.2: LQE System

u

Bu

-

(sI-A)

-1

y

x C

Bu

K ^ x

+

(sI-A)

-1

+

+

L -

C Figure 4.3: Controller-Observer Stru ture for Feedba k.

Now onsider the deterministi version of system (4.7)

x_ = Ax + Bu u y = Cx

(4.16a) (4.16b)

a well known result, the so- alled Separation Prin iple, states that, if one designs a state feedba k gain

K with A+Bu K Hurwitz, and a Luenberger observer 36

in the form (4.12) with

A + LC

Hurwitz, the losed-loop system depi ted in

Figure (4.3) and des ribed by the equations

x_ _ y u

= Ax + Bu u = A + Bu u + L(y = Cx = K

(4.17a)

C )

(4.17b) (4.17 ) (4.17d)

is asymptoti ally stable; moreover, the eigenvalues of (4.17) are those of

Bu K

and those of

A + LC .

Now assume that

K

and

L

A+

has been designed

following an LQ optimal ontrol and Kalman Filter estimator philosophy respe tively; we know from the above dis ussion that the LQ s heme without observer in Figure 4.1 is robust at the plant input and that the LQE s heme

without state feedba k in Figure 4.2 is robust at the plant output. What an we say about the robustness of the whole LQ-LQE s heme of Figure 4.3? The answer, as shown by a ounter-example in [57℄, is, in general, nothing. This last point introdu es the LTR robust ontrol te hnique, whi h is a methodology to re over, in a ontroller-observer framework, the LQ (or the LQE) robustness margins. Assume that the number of inputs is equal to the number of outputs, that is

m = r (if this hypothesis is not fullled and m < r we

an introdu e further  titious inputs), and that we are interested in obtaining good performan e in terms of amplitude of the ontrol inputs and settling time and, at the same time, good robustness properties at the plant input (a tuators) in the s heme of Figure 4.3. We pro eed in the following way: rst the matrix

K is designed following equations (4.4) and (4.5) (after suitable matri es Q and R have been hosen); then the observer gain L is hosen in su h a way that the desired LQ margins are obtained at the plant input; the last part of this se tion is devoted to detail the pro edure to design su h

L.

This pro edure is

known as LQG/LTR.

Assumption: Let

the plant (4.16) is minimum phase.

L be the solution of an optimal estimator problem with  titious input

disturban e matrix and auto ovarian e matri es given by

Bw = I w = Bu BuT m =  2 I : In this ase we have that

where

( ) is the solution of

1 L( ) = ( )C T 2 

A( ) + ( )AT + Bu BuT It is shown in [141℄ that

1 ( )C T C ( ) = 0 : 2

1 lim L( ) = Bu U 

 !0

37

(4.18a) (4.18b) (4.18 )

(4.19)

(4.20)

(4.21)

U is any orthonormal matrix. Using (4.21) and denoting G(s) = C (sI A) 1 Bu as the transfer matrix of the plant and K (s;  ) as the transfer matrix

where

of the ontroller-observer, it is readily seen that

lim K (s;  )G(s) = K (sI

 !0 Sin e

K (s;  )G(s)

A) 1 Bu :

(4.22)

is the open loop transfer matrix of the ontroller-observer

s heme in Figure 4.3, dened by opening the loop at the plant input, and

A) 1 Bu

K (sI

is the transfer matrix of the LQ ontroller in Figure 4.1, obtained

by opening the loop at the plant input, topologi al arguments lead to the

on lusion that the LQ robustness margins are asymptoti ally re overed at

 ! 0. From (4.21) follows that, when  ! 0, the observer gain L( ) goes to innity; therefore, in pra ti al situations one onsiders a given value of  , for example  = 1, and he k the degree of the plant input in Figure 4.3, when

satisfa tion of ondition (4.22) (this an be done by plotting and omparing the singular values of

K (s;  )G(s) and K (sI A) 1 Bu ).

Then the value of

 is

redu ed until the approximation of the limiting ondition (4.22) is satisfa tory and ompatible with the fa t that faster and faster observers be ome more and more transparent to sensor noise. If we desire to re over the robustness margins at the plant output we an set up the dual pro edure: rst design an optimal Kalman Filter and then design an optimal LQSF ontroller using the  titious weighting matri es

Q = CT C R = 2 I :

(4.23a) (4.23b)

In this ase, the dual relations of (4.21) and (4.22) hold

1 UC  lim G(s)K (s; ) = C (sI A) 1 L !0 lim K () =

!0

(4.24a) (4.24b)

whi h ensure the re overy of the Kalman Filter margins at the plant output. We remember, however, that this last pro edure an only be applied when

r = m or r < m (in this ase it is ne essary to introdu e  titious outputs).

When the plant (4.16) is nonminimum phase, the full re overy of the stability margins annot be obtained; however, a partial re overy may result from the modied LTR pro edure des ribed in [226℄.

4.3 Possible Design Obje tives and Design Cy le Des ription LQ optimal ontrol performs a trade-o between ontrol amplitudes and settling time; this trade-o is strongly inuen ed by the hoi e of the weighting matri es

Q

and

R.

Large values of

R

with respe t to

38

Q

will result in weak

ontrol amplitudes and a slow regulation of the state variables; onversely we have stronger ontrol amplitudes and a faster regulation. For a system in the form (4.16) with

m  r and robustness re overy at the

plant input (a tuators), the design y le is usually omposed of the following steps:

Step 1 Choose the weighting matri es

Q and R;

Step 2 Evaluate the time behaviour of states and ontrols; Step 3 If the time behaviour is satisfa tory, go to Step 4, otherwise go to Step 1; Step 4 Let

 = ;

Step 5 Evaluate

L( ) a

ording to (4.19) and (4.20);

K (s;  )G(s) and K (sI A) 1 Bu ; if the re overy is not satisfa tory, set  = = , where > 1, and go to Step 5.

Step 6 Plot the singular values of

4.4 A Simple Design Example We will now provide a numeri al example in whi h the LQ method has been used to design a Proportional plus Integral feedba k multivariable a tion. This stru ture resembles the one used in the HIRM ontrol s heme des ribed in Chapter 28. Let us onsider the linearized model of the longitudinal dynami s of the HIRM air raft in straight and level ight (Ma h=0.40, altitude=10000 feet) in the form

x_ = Ax + Bu u y = Cx where

x(0) = x0

(4.25a) (4.25b)

x = ( V  q  )T , u = ( ÆT S engineF )T

and

y = x;

we have the

following system matri es:

0

A =

B  0

Bu =

B 

9:150 10 2:717 10 3:458 10 0:00 2:482 5:855 10 1:203 0:00

2 3 3 2

6:553 6:136 10 1 9:806 1 1 1 1:166 10 9:859 10 6:091 10 7 C ; A 1 1 1:547 10 2:651 10 0:00 0:00 1:00 0:00 6:043 10 5 1 4:570 10 7 C ; C = I4 : 2:284 10 6 A 0:00

If we want to synthesize a ontroller whi h regulates velo ity and pit h rate, we have to dene an auxiliary matrix:



Cr = 01 00 01 00 39



su h that

yr = ( V q )T = Cr x :

(4.26)

We an now make referen e to the losed-loop s heme shown in Figure 4.4. Considering that the state-spa e realization of the integrator is

x_ i = e yi = xi ; where

e=r

(4.27a) (4.27b)

Cr x is the tra king error and r

is the referen e signal, we have

the following losed-loop system state equation





u Kp x^_ = A + B C r



where x ^ = xx i

 

Bu Ki x^ + 0 r 0 I

(4.28)

 ; equation (4.28) an be rewritten as





x^_ = A^ + B^ K^ x^ + B^2 r

(4.29)

where

A^ =







A 0 ; B^ = Bu 0 Cr 0



 

; B^2 = 0I

are the state-spa e matri es of an auxiliary  ti ious system and

K^ = ( Kp Ki )

(4.31)

is the state feedba k gain whi h we are going to design with the LQ method.

r

+ -

e

1 s

Ki

+

u

x

Linear plant

Cr

y

+

Kp

Figure 4.4: Closed Loop System

Now the problem is the hoi e of the weighting matri es

Q and R

appli ation of the LQ te hnique to design the PI gain matri es.

for the

Indeed our

e. This means that, ^ B^ ), the last two states, (A;

obje tive is to keep as low as possible, the tra king error in the quadrati ost fun tion dened by the system

whi h are related to the integrators, should be emphasized by in reasing the

40

relative entries of

Q.

In terms of the hoi e of

R,

a good trade-o between

performan e and ontrol energy must be found. In Figure 4.5 the time response of the system is shown under a demand

q. The results obtained by dierent hoi es of the R are ompared. Q = diag(( 10 8 10 8 10 8 10 8 10 6 10 1 )) ; 8 < R0 R0 = diag ( 25 10 5 10 13 ) : R = 100 R0 : 1000 R0

of 5deg/se on matrix

weighting

It is evident from the plots that, by in reasing the norm of the matrix

R, the

20

6

15

4

q (deg/s)

teta (deg)

ontrol energy, and onsequently the time response, des reases.

10 5 0 0

2 0

5

−2 0

10

5

30

dts (deg)

20 R=R0 R=100*R0 R=1000*R0

10 0 −10 −20 0

5

10 Figure 4.5: Simulation Results

41

10

5.

Robust Quadrati Stabilization

Germain Gar ia1;2 , Ja ques Bernussou11 , Jamal Daafouz1;2 and Denis Arzelier

5.1 Introdu tion A fundamental problem in ontrol theory is the robust stabilization problem [56℄. From a pra ti al point of view, it is ne essary to hara terize a lass of

ontrollers whi h ensures, at least, asymptoti stability for the ontrolled un ertain system.

A way to address this problem, is to extend the on ept of

Lyapunov stability to the ase of un ertain systems. The idea is to nd a single Lyapunov fun tion for the ontrolled system from whi h a single ontroller being dedu ed. When su h a Lyapunov fun tion exists, the system is said to be quadrati ally stabilizable this is why the orresponding on ept is alled quadrati stabilizability . Numerous papers deal with the quadrati stabilization problem. For norm bounded un ertain systems whi h are entral in this

hapter, a solution is given in [193℄, [86℄ and onne tions between quadrati stabilizability and

H1 ontrol are presented in [134℄.

Stability is a minimum requirement and is not su ient in pra ti e when a reasonable performan e level has to be obtained. A ommon and dire t way to a

ount for performan e is to put some onstraints on the losed-loop pole lo ations leading to robust pole lo ation design. performan e measure as

Another way is to dene a

H2 or H1 norms and, due to plant un ertainty, one

an at best minimize an upper bound on these norms. Su h approa hes are referred to as guaranteed ost designs [87℄, [91℄. It is also possible to ombine pole lo ation and guaranteed ost designs. The rst problem addressed in this hapter is to nd a linear ontrol law su h that the losed-loop system poles belong to the disk

 + j 0 and radius r.

D(r; )

with enter

The disk for pole lo ation an be hosen in su h a way

that a good ompromise between mode damping and speed is guaranteed. For

ontinuous systems, it su es to in lude it in a se tor lo ated in the left half

omplex plane. If

 is a omplex mode for the ontrolled system, !n = jj, its = Re[℄, its damping fa tor and z = !n 1 ,

undamped natural frequen y, its damping ratio, then

8 2 D(r; ) !n <  + r; < r ; z > r 1 1

LAAS-CNRS, 7 avenue du olonel Ro he, 31077 Toulouse Cedex 4

2

Also with INSA, Complexe S ientique de Rangueil, 31077 Toulouse Cedex 4

42

Another ontrol design problem whi h is dealt with in this hapter, onsiders the disk pole lo ation ombined with a guaranteed

H2 ost. When working in

the quadrati framework, two main approa hes are possible. The rst one (now very popular) is hara terized by the use of an LMI formulation (Linear Matrix Inequality) when writing the onditions for quadrati stabilizability, in luding or not performan e requirements. Being linear with respe t to the unknown matri es, the LMI formulation proposes a onvex parametrization of the robust

ontrollers. Among the good features asso iated with LMI, one an stress the fa t that there exist e ient numeri al tools (industrial pa kages) working on interior point methods.

Another interesting feature lies in the ability of the

LMIs to aggregate several onstraints, provided these are written in terms of LMIs (the ase for stru tural onstraints, integral quadrati onstraints, et . ). The se ond approa h relies on the use of Ri

ati type equations, a tool whi h it is not surprising to nd here, in the framework of linear systems with quadrati fun tions. E ient numeri al tools exist to solve parameter dependent Ri

ati equations. An advantage in expressing the onditions through Ri

ati equations is that ontrol interpretation is mu h easier. Usually in a Ri

ati equation, two weighting matri es, one for the states, and the other for the ontrol, appear. This is the ase for the Ri

ati equations arising in the quadrati stabilizability problem. Their role and ee ts on the derived ontrol are well understood and it is possible by a judi ious hoi e or by a trial and error method to sele t a

ontrol s heme satisfying some requirements. It should be noted that Ri

ati equations an be derived be ause the pole lo ation region is relatively simple (a ir le). This is the reason why in the following, the quadrati approa h will be illustrated by developing the results through the Ri

ati framework.

For

more omplex regions, no analyti al solutions in terms of matrix equations an be obtained. But for a large lass of regions named LMI regions, the problem

an be solved by LMI te hniques. For more details, see [46℄.

5.2 Preliminaries Throughout the hapter, the symbols

0; 1

respe tively denote the null matrix

and the identity matrix of appropriate dimension. of the matrix

M

M0

denotes the transpose

( omplex onjugate transpose for omplex matri es).

For

B , A < ()B means that the matrix A B is negative denite (semidenite). (M ) denotes the spe tral radius of M and  (M ) = (M 0 M )1=2 the maximum singular value. Let us onsider a ontinuous symmetri matri es

A

and

system des ribed by :

where IR

n

A

2

x_ (t) = (A +
A)x(t) + Bu(t) y(t) = Cx(t) nn , B 2 IR nm , C 2 IR pn , u(t) 2 IR m IR

(5.1)

is the input,

x(t)

2

is the state. In order to simplify the following developments, the ase of

un ertainty ae ting only the dynami matrix

A

is onsidered, noti ing that

most of the given results an be extended to un ertain

43

A and B matri es.

These

results an be found in the given referen es. There are several ways to model the un ertainty. One of the most popular is the following:

Norm bounded un ertainty where

D2

nr , IR

E

2

ln IR


A = DF E

(5.2)

dene the stru ture of the un ertainty and the

modelling parameter un ertainty

F

belongs to the set :

F = fF 2 IR rl : F 0 F  1g

(5.3)

In this way an ellipsoidal volume is dened as an un ertainty domain in the hyperspa e of the entries of of this hyperellipsoid.

A, the nominal model being dened in the enter

There exist some other ways to des ribe un ertainty.

We list below some examples whi h may be translated, after some elementary transformations, into a norm bounded un ertainty. Their pra ti al interest is dis ussed in some detail in the robust ontrol literature.

Bounded real un ertainty The un ertainty term is written as :


A = DF (1 D0 F ) 1 E nr , E 2 IR ln dene the stru ture of the un ertainty and the where D 2 IR modelling parameter un ertainty F belongs to F . D0 is a onstant matrix satisfying 1 D00 D0 > 0. Then, we have A +
A = A + D(1 D0 D0 ) 1 D0 E + D(1 D0 D0 ) 1=2
(1 D0 D0 ) 1=2 E with


0
 1.

0

0

0

0

Positive real un ertainty The un ertainty term is given by :


A = DF (1 + D0 F ) 1 E nr , E 2 IR rn dene the stru ture of the un ertainty and the where D 2 IR modelling parameter un ertainty F belongs to the set : Fp = fF 2 IR rr : F 0 + F  0g (5.4)

D0 is a onstant matrix of appropriate dimension satisfying D0 + D00 > 0. This ondition ensures that the matrix 1 + D0 F is invertible for all F 2 Fp . Then, we have A +
A = A D(D0 + D00 ) 1 E + D(D0 + D00 ) 1=2
(D0 + D00 ) 1=2 E 0 with

 1.

Moreover,

44

Stru tured un ertainty The above dened un ertainties are alled unstru tured un ertainties in the sense that they are dened through a single un ertainty matrix dened in a very global and general set.

F

whi h is

We an introdu e some stru tural

features on the un ertainty by dening multiblo k un ertainty terms, su h as


A = where


Ai

m X i=1


Ai

an be expressed by one of the following expressions


Ai = Di Fi Ei ; Fi 2 Fi

D00 i D0i > 0
Ai = Di Fi (1 + D0 i Fi ) 1 Ei ; Fi 2 Fpi ; D0 i + D0 0i > 0 with Di and Ei are onstant matri es of appropriate dimensions and the sets
Ai = Di Fi (1 D0 i Fi ) 1 Ei ; Fi 2 Fi ;

1

Fi and Fpi are dened respe tively like F and Fp .

In this way, one an take

into a

ount more pra ti al parametri un ertainty, but the onditions derived in the sequel are only su ient. In [85℄, the quadrati d stability on ept whi h is the ounterpart of quadrati stability in the ontext of pole pla ement in a disk was introdu ed. We re all below the denition this on ept.

Denition 5.1

The system

x_ (t) = (A + DF E )x(t)

is quadrati ally

if and only if there exists a positive denite symmetri matrix that :

for all

F

2 F with

(Ar + Dr F Er )0 P (Ar + Dr F Er ) P < 0

p

p

Ar = (A 1)=r; Dr = D= r; Er = E= r

P

d stable

2 IR nn su h (5.5)

(5.6)

This denition states that a system is quadrati ally d stable if there exists a single matrix

P

satisfying (5.5) for all the systems in the un ertainty domain.

Pole lo ation is meaningful in the ase of non time-varying un ertainty, i.e.

F

is a onstant matrix. It has been shown in [85℄ that equation (5.5) is in fa t a su ient ondition for quadrati stability, the matrix for the system (5.1), whatever

F

P

matrix in (5.5) is a Lyapunov

belongs to

F.

Furthermore, one may

expe t that for slowly varying un ertainty, the satisfa tion of (5.5) will ensure a good transient behaviour for the ontrolled system. It is to be noti ed that (5.6) is a dis rete Lyapunov inequality for the transformed system (5.6).

In

fa t, a system is quadrati ally d stable if and only if the transformed system is quadrati ally stable. This equivalen e allows to interpret the quadrati d stability as an

H1 norm onstraint as is done for quadrati stability in [134℄. The 45

ondition be omes: the system dened by

x_ (t) = (A +
A)x(t) is quadrati ally

d stable if and only if

kEr (s1 Ar ) 1 Dr )k1 < 1

(5.7)

In the light of this result, the quadrati d stability problem and in the sequel the quadrati d stabilization problem are equivalent to an

H1 synthesis problem

that an be solved using for example an LMI formulation or a Ri

ati equation

approa h. It is well known that the LMI te hniques are powerfull, parti ularly in the ases where multiple onstraints and obje tives have to be taken into a

ount. In the ases where analyti al solutions an be derived, for example a Ri

ati equation, the omplexity of LMI omputations remains higher than that of solving a Ri

ati equation [84℄.

5.3 Quadrati d Stabilizability by Output Feedba k In this se tion, we use the equivalen e between the quadrati d stabilization problem and a disk.

H1 ontrol synthesis to solve output feedba k pole lo ation in

The output feedba k quadrati d stabilizability is formalized in the

following denition.

Denition 5.2

The system is said to be quadrati ally d stabilizable via output

K (s) su h that the u = K (s)y) is quadrati ally d stable for all F 2 F .

feedba k if there exists a linear time-invariant ompensator

losed-loop system ( losed by

From (5.7) written for the losed-loop system, the system is quadrati ally d stabilizable via dynami linear output feedba k if and only if

kEr (s1 Ar Br K (s)Cr ) 1 Dr k1 < 1

(5.8)

As before, the problem an be solved using some standard te hniques.

The

Ri

ati equation approa h leads to

Theorem 5.3

The system (5.1) is quadrati ally d stabilizable by an output

R1 ; R2 ; Q positive denite symmetri ma > 0 and two positive denite nn ; Y 2 IR nn satisfying : symmetri matri es X 2 IR A0r (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar X + Er0 Er + Q = 0 (5.9) Ar (Y 1 + C 0 (R2 ) 1 Cr E 0 Er Q) 1A0 Y + Dr D0 = 0 (5.10)

ompensator if and only if, given

tri es of appropriate dimensions, there exist

r

with :

Y 1

r

r

1

Dr0 XDr > 0 Er0 Er > A0r (X 1 Dr Dr0 ) 1 Ar + Q 46

r

(5.11) (5.12)

Condition (5.12) implies that

(XY ) < 1.

ompensator is given by :

Under the previous onditions, a

p

_ = (A + BK + rDKd) + L(y C) u = K

(5.13)

where:

K = (R1 ) 1 Br0 (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar Kd = Dr0 (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar L = (1 Y X ) 1 Ar (Y 1 + Cr0 (R2 ) 1 Cr Er0 Er Q) 1 Cr0 (R2 ) 1

5.3.1 Output d stabilization algorithm The following algorithm to he k quadrati d stabilizabilty an be dedu ed from the monotoni behaviour of the solutions of the previous Ri

ati equations. Step 1 :

Step 2 :

Choose positive denite symmetri matri es

1; R2 = 1; Q = 1

and

>0

R1 ; R2 ; Q, for example R1 =

Solve the two Ri

ati equations of theorem 5.3. If the solutions are positive denite and satisfy (5.11) and (5.12), Stop. The system is quadrati ally d stabilizable by output feedba k.

Compute the ontroller with

formula (5.13). Otherwise go to step 3. Step 3 :

Take

 = =2.

If

 is less than some omputational a

ura y 0 , Stop.

The

system is not quadrati ally d stabilizable by output feedba k. Otherwise go to step 2. It is obvious that the above algorithm onverges for some number of steps.

0

0 > 0

in a nite

has to be hosen su iently small. To solve the Ri

ati

equations some standard algorithms an be used.

5.4 Quadrati d Stabilizabilty and Guaranteed Cost In this se tion, the results of robust pole lo ation in a disk are ombined with another spe i ation requirement expressed through an

H2 norm of a transfer

matrix from an external perturbation to a ontrolled output.

In fa t, this

problem an be seen as a robust pole lo ation problem with the minimization of an upper bound on a linear quadrati ost (multi obje tive ontrol design). Let the un ertain system be des ribed by :

x_ (t) = (A + DF E )x(t) + Bu(t) + B1 w(t) z (t) = C1 x(t) + D12 u(t) y(t) = C2 x(t) + D21 w(t) 47

(5.14)

where

w

is a disturban e,

2

z

IR

s

is a ontrolled output and

F

2 F.

All

matri es are onstant matri es of appropriate dimensions. We assume without

C10 D12 = 0 and B10 D21 = 0. Let us also dene : Co = f > 0 : the onditions of theorem 5.3 are satisedg

loss of generality that

and :

K = fK(s) given by (5.13)

The ontroller

:  2 Co g

K (s) an be written as : _ (t) = H (t) + Ly(t) u(t) = K(t)

where

H ; L; K are given by theorem 5.3.

(5.15)

The losed-loop system is obtained

by ombining (5.14) and (5.15).

2 

x_ _



=

z   6 6 A + 6 4 | {z } |

0 00

0B 1{z0

A

+

 |



0 {z

}

3


7 0 7 {z }5

|

E




}|

7

D F E

D z = C1 D12 K {z | Ce The transfer matrix from

B

Ae

}| 



}

x 



|

H L  K 0 {z H 

 }|

{

0 1 C2 0 {z }



x + B1 LD21  {z } | {z X B1

C



w

}

w(t) to z (t) is given by :

HF (s) = Ce [s1 Ae DF E ℄ 1 B1 If K (s) 2 K, the H2 norm of HF is expressed as : kHF k22 = Tra e(Ce L (F )Ce0 ) = Tra e(B10 Lo (F )B1 ) where L (F ) and Lo (F ) are respe tively the ontrollability and

(5.16)

(5.17) observability

gramians solutions of :



(Ae + DF E )L (F ) + L (F )(Ae + DF E )0 + B1 B10 = 0 (Ae + DF E )0 Lo (F ) + Lo (F )(Ae + DF E ) + Ce0 Ce = 0

(5.18)

The problem solved in this se tion is the following :

Find

K (s)

and

F , kHF k2 

,

0



<

1

su h that

K (s)

being as small as possible.

48

2K

and

8F 2

5.4.1 Guaranteed ost ontrollers To solve this problem, note that the existen e of a ontroller satisfying the

onditions of theorem 5.3 is independent of the hoi e of the weighting matri es

R1 ; R2

and

Q.

In fa t, an appropriate hoi e for these matri es allows us to

solve the pole lo ation with guaranteed ost design problem. Let :

0 D12 + Æ1 R1 = D12 where

0 + Æ1 R2 = D21 D21

Q = C10 C1 + Æ1

Æ > 0 is a small parameter whi h prevents singularities.

We have the

following results.

Theorem 5.4

Suppose that system (5.14) is quadrati ally d stabilizable by out-

put feedba k. Then: i)

Co 6= ;

1 > 0 and P = P 0 > 0 su h that : A0e P + PAe + 1 PDD0 P + 1 1E 0 E + Ce0 Ce + Æ1 = 0

ii) There exists

Eo1 = f1 > 0 : equation (5.19) has a solution P > 0g. Co , 1 2 Eo1 , and F 2 F , we have : P (1 )  Lo (F ); 8F 2 F

iii) Let

 2 Co , K(s) tra e (B1 B10 P (1 )).

iv) For all

(5.19)

For all

is a guaranteed ost ontroller with



2

2 (1 ) =

5.4.2 Optimization problem The previous lemma suggests solving the following optimisation problem to nd the best guaranteed ost ontroller in the sense dened above. Min

[B1B10 P (1 )℄

tra e

 2 Co 1 2 Eo1

(5.20)

We propose the following algorithm :

Algorithm.

For a representative sample of values of

Step 0 : Initialize Step 1 : Take



 (1020, for example)

2 Co

 2 Co , do

and ompute the orresponding ontroller using theorem

5.3. Step 2 : For the ontroller obtained in step 1 , solve:

2 =

rg

A

fMin tra e[B1 B10 P (1)℄g 1

2 E o1 49

(5.21)

< 

Step 3 : If

H

,

H; L

L; K 



K;

,

go to step 2.

Else go to step 1.

tra e[B1 B10 P (1 )℄ is a onvex fun tion with Eo1 and then this optimization problem an be solved by a

It an be shown that in step 2 ,

1

respe t to

over

one-line sear h algorithm.

5.5 Pra ti al Considerations  To apply the method presented in this hapter, the rst step is to derive an un ertain model for the system. Usually a nominal model is available (linearization) and the un ertainties result from parameter variations, high fre-


A an be obtained by an The D matrix disA and the E matrix over the rows

quen y phenomena or non-linear ee ts. The term

a priori knowledge of the range of parameter variations. tributes un ertainties over the olumns of

A. The size of the un ertainty is adjusted by an appropriate s aling on the D and E matri es. For the RCAM design problem, the parameters whi h vary

of

are the mass and the entre of gravity.

 The high frequen y and non linear phenomena an be minimized by appropri-

ately shaping the sensitivity fun tions. A way to do this is to sele t judi iously the weighting matri es

R1 ; R2

and

Q.

Although no systemati method to

x these matri es exists, a trial-and-error approa h allows us to adjust them. Theorem 1 hara terizes the lass of ontrollers whi h pla es the poles in a disk and the weighting matri es an be used to nd in this lass, a ontroller whi h satises other requirements. With no un ertainty, that is

D and E equal

to zero, the ontroller derived from theorem 1 is lose to an LQG ontroller dened on the triple

Q =  1,

if



(Ar ; Br ; Cr ).

Then onsidering

R1 = 1, R2 = 1

and

  and  , a ontroller with small gains K and L will be

sele ted in the lass of disk pole lo ation ontrollers. These gains have a dire t inuen e on the sensitivity fun tions. Then with shape the sensitivity fun tions.

If

D

and

E

;

and

, it is possible to

are not equal to zero, a similar

hoi e leads to the same ee ts. In pra ti e, a ompromise an be obtained by a trial-and-error approa h.

 Another degree of freedom on erns the hoi e of the parameters dening the ir le

(; r).

The values of these parameters are imposed by the settling

time and overshoot spe i ations, but there exists a ertain latitude on their sele tion.

If the radius of the ir le is too small, the problem is onstrained

and the lass of ontrollers too. In fa t, a trial-and-error approa h allows us to obtain a satisfa tory ompromise.

 The last point on erns the onservative nature of the approa h.

Consider

rst the unstru tured un ertainty. It is well known that the quadrati approa h leads to onservative results be ause a xed Lyapunov fun tion is used for the design. To alleviate this, it is possible to use parameter dependent Lyapunov fun tion approa hes developed over the last few years. If un ertainty is stru tured, the onservatism is more important. A way to redu e it is to ombine a

50

 synthesis approa h with multipliers.

5.6 Con lusion In this hapter, a robust ontrol design based on the quadrati approa h was presented. The performan e requirements are onsidered following two dierent paths. The rst one onsists of lo ating the losed-loop system poles in a disk, the parameters dening the disk ( entre and radius) being hosen in a way that ensured good transient behaviour. The se ond one onsists of dening a ost fun tion (quadrati ) and minimising a ost upper bound, leading to the well-known guaranteed ost design. In fa t, these two means to express performan e are onsidered simultaneously in this

hapter. The derived onditions expressed in terms of parameter dependent Ri

ati equations an be solved with available numeri al powerfull te hniques.

51

H1 Mixed Sensitivity

6.

Mark R. Tu ker and Daniel J. Walker 1

2

6.1 Introdu tion Classi al approa hes to feedba k design have for many years provided reliable methodologies for designing ontrollers that are robust, but these te hniques have not extended well to the multivariable ase. Modern te hniques have subsequently looked at methods for designing multivariable robust ontrollers.

H1

ontrol theory has been establishing itself

sin e the 1980's. The approa h is based on minimising over frequen y the peak values of ertain system transfer fun tions that an be hosen by the design engineer to represent design obje tives. The

H1

mixed sensitivity approa h allows the design engineer to meet

stability and performan e requirements in the presen e of modelling errors, un ertainty and perturbations arising from disturban es or noise. Input and output signals are shaped with frequen y dependent weights to meet robustness and performan e spe i ations.

H1 mixed H1 minimisation is des ribed, followed by a mixed sen-

This hapter is a tutorial hapter that will des ribe the theory of sensitivity methods.

sitivity one degree-of-freedom single input and single output design method.

Next a two degree-of-freedom multivariable mixed sensitivity design is onsidered that in ludes disturban e inputs and a mat hing model.

H1 te hniques have been applied in the design hapters 21, 22, 23, 29, 30, -synthesis methods -synthesis tutorials are given in hapters

and 31 where mixed sensitivity as well as loop shaping and have been used. Loop shaping and 7 and 8 respe tively. More extensive treatment of [215, 159, 266, 96, 61℄.

6.2

H1 theory and appli ations an be found in

H1 Minimisation

z are the output errors or r are the exogenous signals (referen e inputs and disturban es), e are the measurements and u are the ontrols. Consider the standard problem of Figure 6.1, where

osts,

1

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: mrtsun.engg.le.a .uk Tel: +44 116 252 2567/2874 Fax: +44 116 252 2619 2

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: wjdlei ester.a .uk Tel: +44 116 252 2529 Fax: +44 116 252 2619

52

r

z P

K

u

e

Figure 6.1: Standard Problem

A

ording to the signals, the open loop system



z e



=



P11 P12 P21 P22



P , of 6.1, is given as





r r =P u u

 (6.1)

r to z an be derived as Tzr = P11 + P12 K (I P22 K ) 1 P21

The losed loop relationship taken from

K

The standard

(6.2)

H1 optimisation problem is to nd a stabilising ontroller

whi h is proper and minimises the supremum (lowest upper bound) over

frequen y of the maximum singular value of

Tzr , the transfer fun tion from the

referen e inputs to the output errors or osts. That is, minimise

 [Tzr (s)℄ k Tzr k1 = Re(sup s) > 0

(6.3)

A stabilizing ontroller a hieving the minimum losed loop norm, k Tzr k1 =

opt , is said to be optimal. A stabilizing ontroller a hieving a losed loop norm

> opt is said to be sub-optimal. P an be represented in state spa e form as

2

3

x_ 4 z 5 = e

2

A B1 B2 4 C1 D11 D12 C2 D21 D22

32

3

x 54 r 5 u

(6.4)

It is worth noting that three spe ial ases of the standard plant A 1-blo k problem is when

D12

and

D21

P

exist.

are square and su h a problem is

mathemati ally easier to solve than a 2-blo k problem where only is square. A 4-blo k problem is when neither

D12

or

D21

D12

or

D21

is square and su h a

problem is the hardest to solve. Generally, all problems will require the solving of two algebrai Ri

ati equations, referred to as the ontrol and lter equations respe tively. In fa t the system of

P

needs to be onstru ted so that the following on-

ditions apply [92℄.

53

1.

(A; B2 ; C2 )

is stabilisable and dete table. This is required for the exis-

ten e of stabilising ontrollers. 2.

D12 has full olumn rank and D21 has full row rank.

This is su ient to

ensure that the ontroller is proper.

 3.

A jwI B2 C1 D12

 has full olumn rank for all



solution to the lter Ri

ati equation. row rank for all

w

Also

w,

enabling a stabilising

A jwI B1 C2 D21

 has full

enabling a stabilising solution to the ontrol Ri

ati

equation. The

H1 optimisation an be solved using fun tions su h as hinfopt whi h

iteratively sear hes for the optimum solution for a parti ular suboptimal

opt and using hinf  whi h produ es a

. These fun tions are available in the

Matlab Robust Control Toolbox [45℄. The ontroller produ ed will be of the same order as the system

P

used.

A high order ontroller an easily result, and so ontroller redu tion is often performed to eliminate unwanted or redundant states. A more spe i system stru ture is now onsidered.

6.3 Mixed Sensitivity - One Degree of Freedom d r

e

+

+

u G

K

y

+

-

Figure 6.2: Closed Loop Feedba k System

r,

Figure 6.2 shows a simple losed loop feedba k system with referen e input output

this

y,

output disturban e

d,

error signal

e

and ontrol signal

y=d = e=r = (I + GK ) 1 = So

u.

From

(6.5)

This is dened as the output sensitivity. To a hieve small tra king error, good transient behaviour and high bandwidth the output sensitivity needs to be small at low frequen ies whi h an be a hieved by designing

K

to have high

gain at these frequen ies. Also

u=r = u=d = K (I + GK ) 1 = KSo = (I + KG) 1 K = Si K 54

(6.6) (6.7)

Si = (I + KG) 1 is dened as the input sensitivity. (Note that in a single input single output system So = Si ). To a hieve robustness it is ne essary to where

a

ommodate disturban es and un ertainties and it is also required to limit high frequen y ontrol eort. For this

KSo

must be designed to be small at

K

high frequen ies whi h an be a hieved by designing

to have low gain at

these frequen ies. In order to meet the low and high frequen y onditions, the design will in orporate frequen y dependent weights.

z1 W1

z2

W2

d r

+

u

e

+

+

G

K

y

-

Figure 6.3: Closed Loop Feedba k System with Weights Figure 6.3 shows the system of Figure 6.2 with added weights. From this it an be written

2

z1 4 z2 e

3 5

3

2

W1 = 4 0 I

W1 G  r  W2 5 u G

whi h hen e denes the augmented plant be obtained using 6.8 in 6.2 and so the

ontroller that minimises

P.

H1

The transfer fun tion

where

> 0.

an

W1 So

k Tzr k1 W2 KSo 1 If there is a bound on the H1 norm su h that k Tzr k1 <

W1 So

W2 KSo 1

Tzr

problem is to nd a stabilising

=





(6.8)

<

(6.9)

then

(6.10)

This implies that

k W1 So k1 < k W2 KSo k1 <

(6.11)

so it an be shown that

 [So ℄ < =[W1 1 ℄  [KSo℄ < = [W2 1 ℄

W1 and W2 an be hosen to give the KSo required to a hieve the required high and In fa t W1 needs to be a low pass lter whilst W2 needs

Hen e the frequen y dependent weights bounds on the terms low frequen y gains.

(6.12)

So

and

to be a high pass lter.

55

Broadly speaking, W1 and W2 determine the performan e and robustness properties respe tively. For example, if the weights have been s aled so that gamma is about one, it follows that other words,

W1

W1 1

provides an upper bound on

So , the 1 Likewise, W2

should be hosen to mirror the desired

determined largely by performan e requirements. an upper bound on

KSo.

So .

In

latter being will provide

This an be interpreted in terms of the losed loop

KSo at s = jw, the smaller the additive model error that

system's robustness to unstru tured additive model error; the larger any given omplex frequen y

will be required to destabilize the system. (This follows from the small gain theorem). Conversely, knowledge of the likely size of the additive model error

KSo. KSo

di tates the safe upper bound on

an also be interpreted in terms

of the gain of the losed loop system from the output disturban es to a tuator useage. It should also be noted that

So

an be interpreted as determining the

system's robustness to output inverse multipli ative perturbation. larger

So

Thus the

at a given frequen y, the less robust the system is to output inverse

multipli ative perturbation at that frequen y. Given the state spa e representation of the plant and weights as

G=



AG BG CG DG



W1 =

then the state spa e form of

2 2 6 6 6 4

x_ z1 z2 e

3 7 7 7 5

=

6 6 6 6 6 6 6 4

P



A1 B1 C1 D1



W2 =



A 2 B2 C2 D2

 (6.13)

is onstru ted as

AG B1 CG 0 D1 CG 0 CG

0 A1 0 C1 0 0

0 0 0 B1 A2 0 0 D1 C2 0 0 I

BG B 1 DG B2 D1 DG D2 DG

3 72 7 7 76 74 7 7 5

3

x 7 r5 u

(6.14)

Assuming that the onstru tion of this augmented plant meets the requirements given in Se tion 6.2, then the

H1 minimisation an be performed to produ e

a robustly stabilising ontroller. Note that a 2-blo k problem is being solved as

D21

is square.

6.4 Design Pro edure The following is a simple pres riptive pro edure for designing a one degree-offreedom ontroller using mixed sensitivity. 1. Sele t the linearised plant model. 2. Sele t

W1 1

and

W2 1

to bound

So and KSo .

3. Augment the plant with the weights plant

P.

56

W1

and

W2

to form the augmented

4. Synthesise a sub-optimal ontroller or an optimal ontroller where the

H1 norm is minimised. The smaller indi ating a more robust design.

5. System analysis. The results an be tuned by adjustments to the weights and so iterations of the design loop an be performed as required. (Note: The pole-zero an ellation phenomenon an o

ur in this one degreeof-freedom mixed sensitivity te hnique. Steps to prevent this possibly undesirable situation o

urring an be found in [241℄. The subsequent two degree-offreedom approa h does not suer from this phenomenon).

6.5 Design Example The following plant model is onsidered to represent the transfer fun tion from the input voltage to the angular position of a simple motor.

G= W1

1 s(s + 1)

(6.15)

is sele ted to be an integrator, this will result in integral a tion in

the ontroller enabling good tra king and small steady state errors.

A true

integrator annot be used as this would not onform to the requirements of Se tion 6.2, so an approximate integrator is used.

The gain of this weight

determines the losed loop bandwidth. The sele ted weight is

W1 = Weight

W2

1 s + 10 6

is sele ted to be a high pass lter.

(6.16)

W2

must be proper to meet

the rank onsiderations of the augmented plant as required in Se tion 6.2. The gain and bandwidth of the weight are hosen to allow low frequen y ontrol eort but limit high frequen y ontrol eort. The sele ted weight is

W2 =

20s + 4 s + 80

(6.17)

The augmented plant is onstru ted as in 6.14 and a sub-optimal ontroller is synthesised realising a

H1 norm bound of = 1:1 opt = 1:2642. An optimal

ontroller an give better results over the whole frequen y range, but this may be a hieved through high frequen y or dire t terms in the ontroller. A suboptimal ontroller is generated without these possible unwanted terms at a slight ost to the robustness and performan e. The output sensitivity

So frequen y response of Figure 6.4 shows the desired

low gain over the operating bandwidth so reje ting low frequen y disturban es. At high frequen ies the gain is unity and around the bandwidth there is a peak in the response. The smaller this peak, the more robust the design. The magnitude of this peak determines the smallest unstru tured output inverse

57

multipli ative disturban e that will de-stabilise the system (see Appendix of [242℄). The fun tion

KSo is similarly analysed.

Figure 6.5 shows the frequen y response the step response for the losed loop system. Iterations of the design y le an now be performed to meet robustness and performan e spe i ations as required. 10

0

0

−5

−10

−10

−20 Gain − [dB]

Gain − [dB]

5

−15 −20

−30 −40

−25

−50

−30

−60

−35 −40 −2 10

−70

−1

10

0

1

2

10 10 Frequency − [rad/s]

3

10

10

−80 −2 10

−1

0

10

1

10 10 Frequency − [rad/s]

2

10

3

10

Figure 6.4: (a) Output Sensitivity Frequen y Response (b) Frequen y Response of

KSo

1.2

1

Units

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

Figure 6.5: Step Response of the Closed Loop System

6.6 Mixed Sensitivity - Two Degree of Freedom So far it has been shown how a one degree-of-freedom ontroller an be produ ed using mixed sensitivity te hniques. The method is now extended to a two degree-of-freedom ontroller design. The system of Figure 6.6 shows a pos-

r1 ), an output disturban e input (r2 )

sible onguration, with referen e input (

58

z2

W2

r2 +

e1

r1

u

e2

+

G

K

z1

+

W1 -

M

Figure 6.6: Two Degree of Freedom Closed Loop Feedba k System

z1

M is an ideal model to mat h K is to be designed.   K an be partitioned as K = K1 K2 , su h that

and two output osts (

and

z2).

The system

the losed loop system to. Controller



u = K1 K2





e1 e2



(6.18)

K1 is a pre-lter and K2 is a feedba k ontroller. There are two methods K . Firstly, K2 ould be synthesised to robustly stabilise the loop against disturban es and un ertainty, and then K1 synthesised where

for generating the ontroller

to shape the losed loop to meet the performan e requirements. Su h a two stage approa h an oer greater exibility and may produ e better results, but

the method is ompli ated to implement needing a two step design pro edure and the resulting ontrollers are independent of ea h other and so overall are

K by

of a high order. A simpler one stage method is to generate the ontroller synthesising the feedba k ontroller

K2

and pre-lter

K1

together. Only one

synthesis is required, and the resulting ontroller is of a lower order as

K2 share the same state spa e.

K1 and

Here the one stage approa h is onsidered.

For this system the standard problem of Figure 6.1 an be formed.

The

system is represented by

2

3

2

3

z1 W1 M W1 W1 G 2 3 r1 6 z2 7 6 0 0 W2 7 6 7 = 6 7 4 r2 5 4 e1 5 4 I 0 0 5 u 0 I G e2 whi h hen e denes P . It an also be shown that   Tzr = W1 (So GK1 M ) W1 So W2 Si K1 W2 K2 So On e again the aim is to produ e a stabilising ontroller the

H1

norm of

Tzr .

K

(6.19)

(6.20)

that minimises

The four weighted fun tions to be minimised are the

dieren e between ideal and a tual systems, the output sensitivity, the ontrol eort to the referen e inputs and the ontrol eort to the outputs.

59

W1

and

W2

are frequen y dependent weights sele ted as before. So using

this design enables robustness and performan e riteria to be met in orporating performan e spe i ations in the mat hing model

P

an be formed using the denitions of

G, W1

M.

and

The state spa e form of

W2

as given in 6.13 and

using the state spa e representation of the mat hing model

M= The state spa e form of

2

2

x_ 6 6 z1 6 6 z2 6 4 e1 e2

3 7 7 7 7 7 5

Ag 6 B 1 CG 6 6 0 6 6 0 6 =6 6 D1 CG 6 6 0 6 4 0 CG

P



AM BM CM DM

 (6.21)

is

0 A1 0 0 C1 0 0 0

0 0 A2 0 0 C2 0 0

0 B1 CM 0 AM D1 CM 0 0 0

0 B 1 DM 0 BM D1 DM 0 I 0

0 Bg B1 B1 DG 0 B2 0 0 D1 D1 DG 0 D2 0 0 I DG

3 7 72 7 7 76 76 76 74 7 7 7 5

x r1 r2 u

3 7 7 7 5

(6.22)

Assuming that the onstru tion of this augmented plant with plant model and weights meets the requirements given in Se tion 6.2, then the

H1 minimi-

sation an be done to produ e a robust stabilising ontroller. It is noted that a 2-blo k problem is being solved as

D21

is square.

6.7 Design Pro edure The pro edure for designing a two degree-of-freedom ontroller using mixed sensitivity follows a similar pro edure to the one degree-of-freedom ase (6.4). The two degree-of-freedom ase requires the additional sele tion of a desired

losed loop system response model.

6.8 Design Example The following plant model has been sele ted.

3

2

(s + 5) 0 1 4 0:1(s + 5) 10(s + 1) 5 G= s(s + 1)(s + 5) s(s + 5) 0

(6.23)

The rst and se ond outputs are to be mat hed to the desired model

 1 0 M = s2 +2s+1 1 0 2 s +2s+1 60

 (6.24)

The ross oupling terms here are zero, so dening the requirement for the

losed loop system to be de oupled. Next, the weights

W1 and W2 are sele ted. These will be multivariable and W1 is sele ted as an integrator to provide good

frequen y dependent. Weight

tra king and small steady state errors. The gain on this term will determine

the error bound on the dieren e between the a tual and the ideal system, and will also determine the bandwidth for output disturban e reje tion.

W1_1 = W1_2 =

2 s + 10 6

(6.25)

The third output is the rate of the rst output and will be fed ba k to enhan e the ontrol and robustness.

Low frequen y a tivity of the signal is

required to enable the tra king of the rst output.

The weight sele ted is a

bandpass lter, sele ted to reje t disturban es around the bandwidth frequen y.

s W1_3 = 2 s + 2s + 1 Weight

W2

(6.26)

is sele ted to be a high pass lter and must be proper to meet

the rank onsiderations of the augmented plant as required in Se tion 6.2. The gain and bandwidth of the weight are hosen to allow low frequen y ontrol eort and to minimise high frequen y ontrol eort.

20s+4 0 W2 = s+80 20s+4 0 s+80 



A suboptimal ontroller is now synthesised realising an

1:1 opt = 0:5818.

(6.27)

H1

norm of

=

Figure 6.7 shows the frequen y response of the singular values of error between the a tual and the ideal responses. At low and high frequen ies, the error is small. Small error at low frequen ies will give good mat hing to the model resulting in small steady state errors. Around the bandwidth, the error is largest, although it is less than unity. Redu ing the error around the bandwidth will improve the overshoot and rise times of the losed loop system. Above the operating bandwidth the desired response is for low gain, whi h is a hieved as the error is small. The output sensitivity

So response is also shown in Figure 6.7, where there is

small gain at low frequen y for the ontrolled hannels, whilst the rate feedba k has unity gain over these frequen ies. At high frequen ies the gain is unity and around the bandwidth there is peak in the response. The smaller this peak, the more robust the design (as dis ussed in 6.5. Fun tion similarly analysed.

K2 So

and

Si K1

are

For the losed loop system Figure 6.8 shows the response of the rst hannel, its rate and the ideal response of the mat hing model and the response of the se ond hannel and the ideal response of the mat hing model. Iterations of the design y le an now be performed to meet robustness and performan e spe i ations as required.

61

5

0

0 −20

−5 −10 Gain − [dB]

Gain − [dB]

−40

−60

−15 −20 −25

−80

−30 −100

−35 −120 −2 10

−1

0

10

1

2

10 10 Frequency − [rad/s]

−40 −2 10

3

10

10

−1

0

10

1

2

10 10 Frequency − [rad/s]

Figure 6.7: (a) Frequen y Responses of the Dieren e Between

3

10

M

and

(b) Output Sensitivity Frequen y Response

10

S0 GK1

1

1.2

0.9 Ideal Output 1

1

0.8 0.8

0.7 0.6 Units

Units

0.6 Output 1 0.4

Ideal Output 2

0.5 0.4 0.3

Output 3

0.2

Output 2

0.2 0

0.1 −0.2 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

0 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

Figure 6.8: (a) Step Response to the First Input of the Closed Loop System (b) Step Response to the Se ond Input of the Closed

6.9 Con lusions H1 ontrol theory and how it an H1 mixed sensitivity one and two degree-of-freedom design proThe H1 mixed sensitivity method lends itself well to systems whi h

This hapter has given a brief tutorial on be applied in

edures.

are required to meet stability and performan e requirements in the presen e of modelling errors, un ertainty and perturbations arising from disturban es or noise. Un ertainty and disturban e an be expli itly in orporated into the design and stability is guaranteed subje t to bounded perturbations, although robust performan e is not. The frequen y domain pro edures are demonstrated using simple examples.

Weight sele tion an be made to a

ount for model

un ertainty. If model un ertainty is unspe ied, then the weight sele tion is broadly dened by robustness and performan e requirements.

Additionally

in the two degree-of-freedom design, a model is in orporated that is dire tly translated from the performan e requirements that the losed loop system is

62

required to meet. Generally, ontrollers are produ ed by iterative design pro edures. The weights are sele ted and the robustness and performan e analysed. Large order ontrollers an sometimes be generated, but in pra ti e it is usually possible to a hieve signi ant order redu tion.

63

H1 Loop-Shaping

7.

George Papageorgiou and Keith Glover , Alex Smerlas and Ian Postlethwaite 1

1

2

2

7.1 Preliminaries

k2 m A 2 IR nm an be dened as kkAx xk2 where x 2 IR is an input ve tor and k:k2 denotes the Eu lidean 2-norm. It an easily be dedu ed, after a few al ulations, that the gain of A will depend on the dire tion of the input ve tor x. To see this we dene the singular value de omposition (SVD) of 22 a matrix (see pp. 32-35 in [266℄). For example the SVD of a matrix A 2 IR The gain of a matrix

is



A = u1 u2 where

u1; u2 ; v1 ; v2

2 IR 2

and







(A) 0 0  (A)

u1 u2



and





 v1  ; v2 

v1 v2



are unitary matri es

 (A) denotes the maximum singular value of A and  (A) the minimum singular value of A. Therefore, as      v1  v v  = 1 0 1 2 v2  0 1

(see p. 19 in [266℄).

if

x = v1

then the gain of

Similarly, for

x = v2

A

 (A) with Ax in the dire tion of u1 . (A) with Ax in the dire tion of u2 .

will be

the gain will be

Dene the ondition number of

A as

(A) =

(A) : (A)

Hen e an ill- onditioned matrix, i.e. a matrix with a high ondition number, has signi antly dierent gains in dierent dire tions. A round matrix is a matrix with a ondition number near unity. Assume that we are given a linear time invariant (LTI) stable system with

m

inputs and

p

outputs, i.e.

G(j!)

2 Cpm . I

All inputs

system are assumed to have nite energy, i.e. a nite

2-norm of a signal u(t) as

kuk2 = (

Z

+1 1

2-norm.

u(t)

We dene the

1 uudt) 2 :

1

Cambridge University Engineering Department, Cambridge CB2 1PZ, England

2

Lei ester University, Lei ester LE1 7RH, England

64

G

into the

It is easy to prove (see pp. 18-25 in [61℄) that if output

y(t) will be bounded by

kuk2  1 the energy of the

supfkyk2 : kuk2  1g = kGk1

where the

1-norm of G is dened as kGk1 = sup (j!)℄: !

Therefore

kGk1 denotes the maximum gain of G over all frequen ies and all

input dire tions.

Consequently, if for example we wanted good disturban e

1-norm of 1-norm an be a onservative measure if we are

reje tion at the output of a plant we would try to minimise the the output sensitivity. The

not interested in all input dire tions. This motivates the use of the stru tured singular value If

P

 (see pp.

271-300 in [266℄).

is a given plant model, then

~ fa torisation of P where M

~ and N

P = M~ 1 N~

is a normalised left oprime

are stable rational transfer matri es satis-

fying the normalisation onstraint

N~ N~  + M~ M~  = I: The motivation for using

H1

te hniques to design robust ontrollers is

provided by the small gain theorem (see pp. 217-221 in [266℄). From the small

1-norm of a stable 1-norm sense, of the unstru tured perturbation to whi h the system remains stable. Hen e a typi al H1 ontrol problem would be to minimise the 1-norm of a transfer matrix, alled the

gain theorem it an be dedu ed that by minimising the transfer matrix we maximise the size, in an

generalised plant, over all stabilising ontrollers.

This optimisation problem

has an exa t solution [63℄. The transfer matrix we hoose to look at depends on the type of un ertainty present in the plant and the performan e spe i ations. The rst step of a typi al

H1 design pro edure would be to de ide on the type

of un ertainty to be used (see table on p. 227 in [266℄). This is di ult and requires good knowledge of the plant. Normalised oprime fa tor un ertainty is the most general type of unstru tured un ertainty (see pp. 418-419 in [96℄). The se ond step would be to hoose frequen y dependent weights a

ording to performan e spe i ations and solve the optimisation problem (see pp. 213245 in [266℄).

Some well studied

H1

H1 -synthesis (see

ontroller design te hniques are

loop-shaping, the S/KS design pro edure (see Chapter 6) and Chapter 8).

7.2 Overview of the Design Pro edure The

H1 loop-shaping design pro edure [164℄ is des ribed below:

1. Shape

G

W1 and W2 a Gs = W2 GW1 , is

open loop with frequen y dependent weights

ording to losed loop obje tives. The weighted plant, depi ted in Figure 7.1.

65

w1

?

W2 

Figure 7.1: The

w2

to outputs





w1 w2



W1 

? 6

- K1

z2

2. Minimise the



G

w2 z1

H1 loop-shaping standard blo k diagram

1-norm of the transfer matrix from disturban es w1 and z1

and

 ! zz1

2

Che k the a hieved

z2





over all stabilising ontrollers.





=

K1 (I I 1

.

 Gs K1 ) 1 I Gs



1

= : (7.1)

This gives a measure of how robust the desired

loop shape is. 3. Choose the position of

K1 in the loop.

Model redu e the ontroller and

design the ommand pre-lters.

4. Che k the time simulations and frequen y responses of the resulting

losed loop system to verify robust performan e.

Reiterations may be

required.

7.3 Justi ation of the Set-Up H1 loop-shaping developed by M Farlane and Glover [164℄ is a very intuitive

method for designing robust ontrollers as the notions of lassi al loop-shaping readily arry through. The designer an spe ify losed loop requirements su h as disturban e and noise reje tion by simply shaping the loop gains. An important feature of

H1 loop-shaping is that it enables the designer to push for the

best a hievable losed loop performan e subje t to a required level of robustness. This is be ause the designer has ontrol over the ross-over frequen ies of the loop gain singular values. When setting up a robust ontrol problem a de ision has to be made about the type of un ertainty to be used. This an be di ult as it requires good knowledge of the plant. The value of

in Equation (7.1) provides a level of

robust stability to oprime fa tor un ertainty whi h requires no assumptions to be made about the open loop stability of the perturbed plant.

Coprime

fa tor un ertainty is a general type of un ertainty similar to the single-input single-output (SISO) gain and phase margins.

When there is little detailed

knowledge about the un ertainty present in a plant method for designing robust ontrollers.

66

H1 loop-shaping is a good

G with W1 and W2 , Figure 7.1. In [164℄ it is proved that if is not too large (say < 4) the ontroller K1 does not alter the desired loop shape Gs very mu h. Hen e shaping G orresponds to shaping the loop gains Gs K1 and K1 Gs . All losed loop transfer fun tions an be bounded in terms of Gs , W1 , W2 and (see pp. 493-494 in [266℄). In [210℄ it is proved that there are no left half plane pole/zero an ellations between K1 and Gs . This is be ause K1

an be written as an exa t plant observer plus state feedba k. Hen e H1 loopPerforman e is spe ied by shaping the singular values of the plant

weights

shaping ontrollers an be gain s heduled. This was done in [120℄. Left half

plane pole/zero an ellations are undesirable as they an limit the a hievable robust performan e. The ost fun tion minimised provides robust stability.

.

A measure of the

-iteration [164℄. Appli ation of the method to real plants has shown that a value of  = 1= between 0:2 0:3 is satisfa tory in the same way o that a gain margin of 6dB and a phase margin of 45 are for a SISO design. A good value of  should guarantee that the required gain and phase margins

robustness of the desired loop shape is given by

Cal ulating the best

requires no

are a hieved. In [247℄ the phase and gain margins of a stable SISO losed loop system,

PM

and

GM

respe tively, are related to the obtained

GM

 (1 + )=(1 ); P M  2 ar sin():

A lot of resear h has gone into

.

H1 ontrol design te hniques. As a result a

great deal of powerful analysis and model redu tion tools are available to help with the erti ation of su h ontrollers, e.g. approximation, the

 -gap and -analysis.

the gap metri , Hankel norm

7.4 Classi al Loop-Shaping Before being able to design ontrollers using the

H1 loop-shaping design pro-

edure (LSDP) the designer must be ome familiar with the notions of lassi al loop-shaping (see pp. 130-137 in [266℄). Loop-shaping allows the designer to spe ify losed loop obje tives by shaping the loop gains. Closed loop obje tives su h as disturban e reje tion at both input and output of the plant, noise reje tion, output de oupling and tra king an easily be spe ied when loop-shaping.

di

r

-

6

-

K

u-

?up-

d

G

- ? -y ? 6n

Figure 7.2: A typi al losed loop system

67

Simple algebrai manipulations on the losed loop in Figure 7.2 give:

y r y u up

To (r n) + So Gdi + So d (7.2) So (r d) + Ton So Gdi (7.3) KSo (r n) KSod Ti di (7.4) KSo (r n) KSod + Si di (7.5) 1 and T = I S . It is simple to show where Lo = GK , Li = KG, S = (I + L) that So P = P Si and Si K = KSo . Disturban e reje tion at the plant output y , Equation (7.2), an be a hieved 1 by making  (So ) = (I +GK ) small to reje t d and  (So G) =  (GSi ) small to reje t di . Similarly, disturban e reje tion at the plant input up , Equation 1 (7.5), an be a hieved by making  (Si ) =  (I +KG) small to reje t di and (Si K ) = (KSo ) small to reje t d. Sin e (pp. 31-33 in [96℄) = = = =

(KG) 1   (I + KG)  (KG) + 1 (GK ) 1   (I + GK )  (GK ) + 1; we an dedu e that if

(KG) > 1 and (GK ) > 1 1 1  (KG)+1   (Si )   (KG) 1 1  (So )  1 :  (GK )+1  (GK ) 1

Therefore for disturban e reje tion of

d at y and di

at

up

 (So )  1 , (GK )  1  (Si )  1 , (KG)  1: Also, if

 (GK )  1 or (KG)  1 and assuming that G and K

are invertible

(this assumption is made for the purpose of illustration) then

  (So G)  (1K ) . Therefore for disturban e reje tion of di at y the singular values of the ontroller should be high at low frequen ies.

  (KSo)  (1G) . This gives the input required to an el the inuen e of d on up . This will be small if  (G)  1 but an not be set by the designer and onstitutes a physi al limitation of the plant. As designers we are not only interested in disturban e reje tion. For noise reje tion

(To ) must be made small at high frequen y.

Typi ally noise is only

important at high frequen y. Note that noise reje tion at low frequen y oni ts with disturban e reje tion as bandwidth of

G

T + S = I.

dedu ed by examining Equation (7.4). equivalent to reje ting

S to zero for es T

Large loop gains outside the

an make ontrol a tivity quite una

eptable.

This an be

Output de oupling and tra king are

d at the plant output be ause T + S = I .

Hen e for ing

to the identity. Figure 7.3 illustrates the desired loop shapes.

68

 (L)

!l

 (L)

log !

!h

Figure 7.3: Loop gain boundaries

7.5 Choi e of Weights In [164℄ it is proved that

K1

does not modify the desired loop shape signi-

antly, i.e. the loop-shaping ontroller is well- onditioned. Hen e shaping a tually shapes both

Gs K1

and

K1 Gs .

onstitutes the theoreti al justi ation of

Gs

This is a very important result that

H1 loop-shaping. The fa t that K1

is well- onditioned is intuitive by examining the ost fun tion minimised and noting that this transfer matrix an be written in two ways (see p. 485 in [266℄). All transfer fun tions an be bounded in terms of mentioned in Se tion 7.3.

Gs , W1 , W2

and

as

For example the input and output sensitivity are

bounded by

(I + GK ) 1  minf (M~ s )(W2 ); 1 + (Ns )(W2 )g (I + KG) 1  minf1 + (N~s )(W1 );  (Ms )(W1 )g; where

K = W1 K1W2 .

ea h frequen y, large low frequen y and

di at up .

 (M~ s ) = (Ms ) = ( 1+(W12 GW1 ) ) 21 at gain of Gs results in reje tion of d at y

Therefore, as

Choosing ill- onditioned weights ould result in poor disturban e

reje tion. Hen e the bounds give the designer a feel for how the weights affe t the losed loop performan e. It be omes obvious, by examination of the bounds, that the notions of lassi al loop-shaping readily arry through. The designer an not usually augment one of the singular values of the open loop plant (as a fun tion of frequen y) with a diagonal pre- ompensator while leaving the other singular values un hanged.

To over ome this problem the

open loop plant an be augmented with a full blo k pre- ompensator. The singular value de omposition of the plant as a fun tion of frequen y

an be written as

G(j!) = U (j!)(j!)V (j!) .

If ea h element of

V (j!) is V^ then,

approximated with a stable minimum phase transfer fun tion to give

G(j!)V^ (j!) ' U (j!)(j!).

Hen e a diagonal weight an now be designed that

augments ea h singular value of

(j!) dire tly.

Note that the same an be done

when designing the post- ompensator. This method provides great exibility

69

to the designer in terms of understanding how the hoi e of weights ae ts the a hievable performan e. Sele ting the weights in su h a way, does not ae t the robustness of the design, as the plant is not inverted. The resulting ontroller is given by

V^ W1 K1 W2 .

As an be seen disturban e and noise reje tion, output de oupling and tra king an easily be in orporated in the loop-shaping methodology. What has not been dis ussed is translating the time response requirements into frequen y response requirements. Time response requirements are spe ied in terms of overshoot

Mp , settling time ts and rise time tr

with respe t to applying a step

to the referen e of the losed loop. These requirements are set by shaping the loop gain near ross-over and

hoosing the bandwidth of the losed loop.

What we must rst understand

is what kind of information we an extra t from the frequen y response of a stable system.

For example when looking at a singular value plot of the

output sensitivity of a system (stable transfer matrix) one an easily see, at a parti ular frequen y, what the maximum gain is. So requirements of the type that

0 (j!)℄

 0:1 for ! < 0:1 rad/s an easily be in luded in the design  (Lo)  11 at frequen ies smaller than 0:1 rad/s.

pro edure by for ing

The Fourier series of a square wave with period

u(t) =

2 !0

is given by:

N 1 4X sin(2n 1)!0 t:  n=1 2n 1

For a reasonably a

urate representation of a square wave it is su ient to take

N = 6.

!0 = 1 rad/s an be onsidered to 1 to 11 rad/s. In reality we an not a hieve a

Hen e a square wave of frequen y

ontain frequen ies ranging from

perfe t square wave as the high frequen y omponents will be ltered out. If we insert

y(t) =

u(t) at the input of the losed loop then the output y(t) will be

N 1 4X jT [j (2n 1)!0℄j sinf(2n 1)!0 t + 6 To[j (2n 1)!0 ℄g:  n=1 2n 1 o

To follow a square wave of frequen y

!0 (this frequen y is related to ts ) we must !0 ! 11!0, hen e

make the output sensitivity su iently small over frequen ies

ontrol the gain of

To and get it as lose to unity as possible.

This an be done

by in reasing the loop gain in this frequen y range. To a hieve this it might be ne essary to in rease the bandwidth of the system (the bandwidth is related to

tr ).

We must also make sure that the system is su iently well damped,

and therefore ontrol the phase of phase of

To).

Lo (the phase of Lo is related to Mp and the

This may mean de reasing the bandwidth due to a tuator and

sensor limitations. It ould also mean de reasing the phase lags introdu ed by the weights at ross-over. Even though it is not straightforward to translate time response requirements into the frequen y domain there are general trends that an be followed. The rise time and overshoot are related to the damping of the system. The less damped the system the smaller the rise time and the greater the overshoot.

70

For a desired damping ratio

< 1, whi h is usually the ase for air raft, rise time

depends very little on damping. A well damped losed loop is a hieved by making sure that the roll-o of the augmented plant singular values at rossover is dB . Bode's phase-gain relationship and its genertypi ally smaller than de alisation to the multivariable ase [61, 266℄ illustrates how the roll-o of

40

(L)

is related to the phase of the loop gain and hen e the overshoot. Rise time is

ontrolled by setting the bandwidth of the system. A fast system orresponds to a system with a small rise time and in most ases a small settling time as well (see pp. 126-131 in [82℄).

7.6 Design Cy le G is as follows:

The typi al design y le given a plant 1. S ale

G.

The open loop plant must be s aled a

ording to the desired

output de oupling and a tuator usage.

This is be ause the open loop

singular values an not be asso iated with any one input or output (see Chapter 1 in [215℄ and p. 42 in [120℄). A badly s aled plant is equivalent to a badly formulated problem. 2. Choose the weights

W1

and

W2 .

Integrators (or near integrators in the

ase of rate following) are pla ed in

W1

to boost the low frequen y gain.

This ensures a zero steady state error if we are tra king an attitude, disturban e reje tion and output de oupling/tra king.

To in rease ro-

bust stability, hen e de rease phase lag at ross-over (i.e. slope of augmented plant singular values), a proportional gain is added to the pre ompensator. The value of the gain (position of the resulting zero) is a trade-o between speed of response (moves the integrator open loop pole away from the origin in losed loop) and robustness. The bandwidth is made as high as possible within the a tuator and sensor apabilities, i.e. robust stability onsiderations. If the open loop is unstable are must be taken not to make the losed loop too fast so that disturban e reje tion leads to input saturation hen e loss of ontrollability. noise reje tion, hen e it ontains low pass lters.

W2

is hosen for

The over-all design strategy is to make the loop as fast as possible within the limitations of the plant to use the a tuators to their limits for disturban e reje tion.

Open loop pre-lters are then designed to satisfy

handling quality requirements. This is based on the fa t that the disturban e reje tion problem is entirely de oupled from the nominal tra king problem (see [247℄ and referen es therein). 3. Choose the position of the ontroller. The ontroller an be implemented in three ways. Pla ed in the forward path gives a faster response at the expense of overshoots be ause all the ontroller dynami s are ex ited dire tly by the referen es.

Also any right half plane (RHP) zeros of the

ontroller are also RHP zeros of

To .

71

An example of implementing the

ontroller in the forward path is given in Figure 7.11. Pla ing the ontroller in the feedba k path leads to a slower more damped response but any RHP poles of the ontroller lead to RHP zeros of

To.

The ontroller

ould also be implemented in the observer form as proposed in [247℄ (see pp. 72-78). This onstitutes the optimal way of introdu ing the ontroller into the loop. 4. Design the ommand pre-lters.

The pre-lters are designed to meet

the handling quality requirements. Performan e is limited by losed loop non-minimum phase zeros (RHP zeros). 5. Perform time simulations and analysis to prove robust performan e.

H1

loop-shaping readily provides robust stability. We an a hieve nominal performan e but must test for robust performan e.

7.7 Two Degrees-of-Freedom Design Pro edure The two degree-of-freedom (DOF) design pro edure as introdu ed in [117, 154℄ guarantees robust performan e with respe t to an ideal step response model. Figure 7.4 illustrates the blo k diagram of the two DOF setup.

The losed

loop response from the referen e signals to the plant outputs follows that of a spe ied model

K1

Tr .

The ontroller

is the pre-lter and

ontroller lter

K1

K2

K2

K

is partitioned as

is the feedba k ontroller.

K = [K1 K2 ℄ where

The inner feedba k

is used to meet the robust stability requirements while the pre-

optimises the overall system to the ommand input. The use of the

step response model (SRM) is to ensure that

(I where

Gs K2 ) 1 Gs K1 Tr 1   2;

(7.6)

 is the model-mat hing parameter. From Equation (7.6) it is obvious  in reases (I Gs K2 ) 1 Gs K1 ! To. By setting  equal to zero the

that as

two DOF setup redu es to the one DOF problem des ribed earlier in Se tion 7.2.

r

w2

- I - K1 -+ ?- W1 +6 z1

K2

-

w1

G



- ?- - I -z 6 y

Tr

Figure 7.4: Two degrees-of-freedom onguration

The design y le, given a plant

G with no dire t feed-through, is as follows: 72

1. Sele t a pre- ompensator

W1 a

ording to the guidelines given in Se tions W2 is usually a onstant

7.5 and 7.6. Note that in the two DOF setup matrix.

2. Sele t a desired losed-loop transfer fun tion

Tr

between the ommands

and ontrolled outputs. 3. Set the s alar parameter

 to a small value greater than 1; something in

the range 1 to 3 will usually su e.

P . In Figure 7.5 the signals, with respe t to u the ontrol variables (the input to the shaped plant GW1 ), v the measured variables (r; y ), w the exogenous signals (r; w1 ; w2 ) and z the error signals (u; y; z ).

4. Form the generalised plant those in Figure 7.4, are:

-

w

-z

P

u

v



K

Figure 7.5: General ontrol onguration The state spa e representation of

2

As 6 0 6 6 6 0 P =6 6 Cs 6 6 I 6 4 0 C In Equation (7.7)

H = ZCs 

P

is given by :

0 Ar 0 0 2 Cr 0 0

0 H Br 0 0 0 0 I 0 I I 0 0 I

where

Z

Bs 0 I 0 0 0 0

3 7 7 7 7 7 7 7 7 7 5

(7.7)

is the solution to the Gener-

alised Filtering Algebrai Ri

ati Equation. The reader an refer to [164, 215, 246℄ for more information on the Algebrai Ri

ati Equations in loop shaping synthesis.

(As ; Bs ; Cs ) and (Ar ; Br ; Cr ) are the state-spa e GW1 and referen e model respe tively.

realisations of the shaped plant 5. Solve the standard the ontroller

K.

H1

optimisation problem for the plant

P

to get

The ontroller may be written in an observer form

as in [250℄. 6. Partition the ontroller in to a pre-lter

K1 and a feedba k ontroller K2 Sf = K1 (0) 1 K2(0).

and al ulate the s aling fa tor of the pre-lter as

The nal two DOF ontroller is illustrated in Figure 7.6.

73

r

-

Sf

-

-+ +6

K1

K2

Controller

-

W1

G

-y



Figure 7.6: Two degrees-of-freedom loop-shaping ontroller

Note that the s aling fa tor

Sf

is lo ated in the ommand path.

This

has been found to improve the nominal tra king properties of the losed loop.

7.8 Design Example We are going to present a design example to illustrate all the above points and how the designer ould produ e a good or bad design.

7.8.1 Presentation of Model Used We are going to design a longitudinal ontroller for the RCAM [145℄.

G

The

ÆT ; ÆT H , the outputs z;_ V and the states q; ; uB ; wB . The linearised plant model G and the denitions of these variables inputs into the linear model

are

an be found in [145℄. All angles are in rad and velo ities in m/s.

G has poles at 0:011j 0:126, 0:830j 1:107 and zeros at 4:338, 4:390.

We an easily distinguish between the phugoid and short period modes. The phugoid is slower and lightly damped. The plant is open loop stable with a nonminimum phase zero. The RHP zero exists be ause the verti al a

elerometer is lo ated behind the entre of rotation of the air raft. The physi al meaning of a non-minimum phase zero is that the plant goes initially in the opposite dire tion to that desired, so when the air raft pit hes up

z_ is initially going to

be positive. Non-minimum phase zeros within the bandwidth of the plant limit the a hievable performan e (see pp. 90-104 in [61℄). The a tuator dynami s and loop delays are given below. The loop delay transfer fun tion is based on a rst order Padé approximation (MATLAB tools ommands are used).

>> tailplane = nd2sys(1,[0.15 1℄); >> engine = nd2sys(1,[1.5 1℄); >> delay = nd2sys([-0.06 1℄,[0.06 1℄);

7.8.2 Design Spe i ations The tra king requirements are as follows [145℄:

74

-

Vair

f tr < 12 se tr < 5 se ts < 45 se ts < 20 se Mp < 5% for h > 305m Mp < 30% for h < 305m Table 7.1: Time response requirements

z_ = V sin f , hen e for a small ight path angle f the limb rate z_  V f . For good disturban e reje tion a 13 m/s wind step should not indu e a deviation in airspeed greater than 2:6 m/s for more than 15 se . There are no ross- oupling requirements dened between V and z_ . Note that

be omes

7.8.3 S aling The open loop plant is s aled a

ording to output de oupling and a tuator

usage. For onvenien e the units of the a tuators are onverted to degrees (d2r: degrees to radians). It was thought that

0:5

of a degree of thrust.

1 degree of tailplane is analogous to

This is ompatible with the physi al limits of the

a tuators [145℄. In reasing for example

0:5 to 1 will in rease the usage of the

throttle by in reasing the bandwidth of the system.

Bs = diag([d2r 0.5*d2r℄); % input s aling Cs = diag([3 1℄); % output s aling Similarly, in reasing the se ond entry in

Cs from 1 to 2, will in rease the speed

z_ .

of response of the airspeed and the de oupling with open loop s aled RCAM

Cs *G*Bs .

3

Figure 7.7 shows the

4

10

10

2

10

2

10

1

10

0

10

singular values

singular values

0

10

−1

10

−2

10

−2

10

−4

10

−6

10

−3

10

−8

10

−4

10

−5

10

−10

−2

10

−1

10

0

10 frequency (radians/sec)

Figure 7.7:

1

10

10

2

10

−2

10

Cs *G*Bs

−1

10

Figure 7.8:

0

10 frequency (radians/sec)

Gs , Gs K1

Changing the dire tionality of the plant signi antly (i.e.

1

10

and

2

10

K1 Gs

in reasing the

loop gain too mu h in a dire tion of low plant gain) results in redu tion of the a hieved robustness. Hen e weighting the throttle with a big number would

75

result in in reasing the losed loop bandwidth even though the throttle a tuator is slow resulting in poor stability.

7.8.4 Choi e of Weights To ensure that a diagonal weight

W1

augments ea h of the singular values of

the s aled plant independently a weight

V~

7.5. The total pre- ompensator be omes dynami s. In the MATLAB ode

V~

was designed as des ribed in Se tion

V~ W1 . V~

is the variable

used in this design has no

preW_V.

preW_V = 9.8313e-01 -1.0565e-01 -1.0565e-01 -9.8313e-01

W1 to boost the low frequen y gain. Having just an 90Æ of phase is added at ross-over. Hen e a proportional matrix gain is added to W1 . The position of the zeroes is a trade-o between speed of response and robustness. The post- ompensator W2 Integrators are added in

integrator redu es the robustness as

is designed for noise reje tion. Again the loser to ross-over that the singular values are rolled-o at, the bigger the redu tion of the a hieved robustness. The weights hosen are:

w1 = nd2sys(0.25*[3 1℄,[3 0℄); w2 = nd2sys([3 1℄,[3 0℄); preW = daug(w1,w2); w = nd2sys(1,[0.2 1℄); postW = daug(w,w); The bandwidth in ea h loop is pushed up as high as possible subje t to a

Gs is shown in Figure 7.8 (solid K1 does not alter the desired loop shape too mu h. Hen e shaping G open loop is equivalent to shaping both Gs K1 (dotted) and K1 Gs (dash-dot). Note that the singular values of Gs K1 and K1 Gs are virtually the same. The augmented plant has 12 states. The ontroller was synthesised using desired level of robustness. The weighted plant line). Figure 7.8 also illustrates how

>> [sysK,emax℄ = n fsyn(sys,1.1); >> emax emax = 3.3307e-01 >> emargin(sys,sysK) ans = 3.0560e-01 Hen e the a hieved states.

 = 0:31.

The ontroller

Therefore the overall ontroller

the poles in

K1

sysK

V~ W1 K1 W2

was model redu ed to

are lo ated around the bandwidth.

the model redu ed ontroller

sysK2

has

11

states.

7

Most of

The singular values of

(dashed) and the original ontroller

sysK

(solid) ontroller are shown in Figure 7.9. The dieren e an hardly be seen.

76

sysK2

gives an

 = 0:30.

The equations in Se tion 7.3 give a feel for the

magnitude of the a hieved gain and phase margins. The ontroller was implemented in the forward path. The pre-lters were

hosen to be rst order lags.

The singular values of

ted) are shown in Figure 7.10.

-analysis

A

So

(solid) and

To

(dot-

ould be arried out, as in p.

3-36 [18℄, to he k robust performan e. Note that the large positive area under the

So urve is due to the RHP zero (see the waterbed ee t pp.

1

97-103 in [61℄).

1

10

10

0

10

−1

10 0

singular values

singular values

10

−2

10

−3

10

−1

10

−4

10

−5

10

−6

−2

10

−2

−1

10

10

0

1

10 frequency (radians/sec)

Figure 7.9:

10

2

10

−2

−1

10

10

0

10

1

10

10

frequency (radians/sec)

sysK and sysK2

Figure 7.10:

To and So

7.8.5 Time Responses The SIMULINK blo k diagram of the linear model is shown in Figure 7.11.

z_dot

1 3.5s+1

V

1 7.5s+1

u Csc

− +

sysK2

W1

Bsc

W2

delays

actuators

RCAM

y

Csc

Figure 7.11: SIMULINK blo k diagram of the linear model Figure 7.12 shows the response to a ommand on

z_ (solid) at t = 1 s.

Note

the initial undershoot due to the non-minimum phase zero. Figure 7.13 illustrates the response to an airspeed ommand (dash-dot). Figure 7.14 illustrates the reje tion of a wind-shear of

13 m/s.

7.8.6 Two Degrees-of-Freedom Design Having designed a weighting fun tion

W1 that provides good disturban e reje -

tion, the design spe i ations in Table 7.1 an be in luded dire tly in the design pro edure using a two DOF approa h. The user-dened step response model,

77

Step on z_dot

Actuator usage

−0.2

−0.4

−0.6

−0.8

−1

20

40

60

0.4

0.35

1

0.3

0.2

0.1

0

−0.1

0.8

0.6

0.4

0.2

0

−0.2

−0.3 0

20

time (s)

40

−0.2 0

60

20

time (s)

Figure 7.12: Step on

Actuator usage

1.2

tailplane (solid) and thrust (dash−dot) in degrees

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid) and V_air (dash−dot) in m/s

0

−1.2 0

Step on V_air

0.5

z_dot (solid) and V_air (dash−dot) in m/s

0.2

40

0.3

0.25

0.2

0.15

0.1

0.05

0

−0.05 0

60

20

time (s)

z_ ommand

40

60

time (s)

Figure 7.13: Step on

Vair ommand

Tr in Figure 7.4, is usually diagonal, emphasising maximum output de oupling and exhibiting ideal handling qualities.

>> z_dot_model = nd2sys(0.5^2,[1 2*0.7*0.5 0.5^2℄); >> V_air_model = nd2sys(0.3^2,[1 2*0.7*0.3 0.3^2℄); >> T_r = daug(z_dot_model,V_air_model); As des ribed in the design y le

 > 1.

The nal value was

 = 1:5.

A few

iterations were required (bearing in mind robust performan e) before arriving to this hoi e. The generalised plant was formed from Equation (7.7) and a slightly suboptimal ontroller was obtained using standard routines [18℄.

H1

The degradation of the stability margin ( ) as

optimisation



in reases is

shown in Table 7.2. It is evident that the better the model-mat hing the less robust is the design we an a hieve. Balan ed residualisation (see pp. 449-454



1.1

1.2

1.3

1.4

1.5

1.6

1.7

4.15

4.30

4.44

4.59

4.73

4.91

5.06

Table 7.2: Stability margin as a fun tion of



in [215℄) was used to redu e the ontroller to 8 states. The ontroller was implemented as in Figure 7.6. Figure 7.15 shows the output response to a unit step input on demand and a

z_ . Figures 7.16 and 7.17 illustrate the responses to an airspeed 13 m/s wind shear respe tively.

By omparing the output oupling in Figures 7.12, 7.13 and 7.15, 7.16 it is evident that the two DOF s heme gives good performan e without signi ant deterioration of the losed loop robustness properties. It an be dedu ed that all the requirements are met. The interested reader is en ouraged to go through the example and hange the s aling and weighting fun tions to obtain a feel of how the dierent parameters inuen e the design.

78

Disturbance on z_dot

Actuator usage

10

5

0

−5

−10 0

20

40

5

0

−5

−10

−15 0

60

20

time (s)

40

0

tailplane (solid) and thrust (dash−dot) in degrees

10

0.4

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid) and V_air (dash−dot) in m/s

Step on z_dot

Actuator usage

15

−0.2

−0.4

−0.6

−0.8

−1

0

60

20

40

0.3

0.2

0.1

0

−0.1

−0.2

−0.3 0

60

20

time (s)

time (s)

Figure 7.14: 1DOF wind-shear

40

60

time (s)

Figure 7.15: 2DOF

z_ demand

7.9 Limitations of the Method and Ideal Plant Some plants have features that restri t the a hievable performan e (see pp. 143-153 in [266℄ and Chapters 5, 6 in [215℄). Su h limitations are for example RHP poles outside the bandwidth, RHP zeros within the bandwidth and ill onditioning. These limitations are design method independent.

Step on V_air

Actuator usage

Disturbance on z_dot

0.6

0.4

0.2

0

−0.2 0

20

40

0.35

0.3

0.25

0.2

0.15

0.1

0.05

60

0 0

time (s)

Figure 7.16: 2DOF

20

40

10

5

0

−5

−10 0

60

time (s)

Vair

10

tailplane (solid) and thrust (dash−dot) in degrees

0.8

Actuator usage

15

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

1

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

0.4

20

40 time (s)

demand

60

5

0

−5

−10

−15 0

20

40

60

time (s)

Figure 7.17: 2DOF wind shear

H1 ontroller design te hniques are frequen y domain based methods. This

is be ause robustness issues are more easily addressed in the frequen y domain. The potential weakness of

H1 loop-shaping is that there is sometimes

di ulty in translating time response requirements to the desired loop shape. This di ulty an be over ome by pushing for the highest possible losed loop bandwidth. Hen e the designer aims for a bandwidth higher than that required to satisfy the handling quality requirements, subje t to obtaining reasonable robustness. The design problem be omes harder when the plant is open loop unstable in whi h ase a high bandwidth ould lead to input saturation and loss of ontrollability during disturban e reje tion. In su h ir umstan es the bandwidth must be lowered. De oupling be omes a more di ult task, parti ularly if the spe i ations for ea h loop vary signi antly.

79

The ideal plant for ontrolling with an

H1 loop-shaping design would be

a plant that has similar properties in all loops. By similar properties we mean equally fast and powerful a tuators and sensors and not too ill- onditioned. Hen e the ross-over frequen y for all singular values an be made the same. When using lassi al ontrol the designer designs the ontroller dire tly. This is not the ase when using

H1

design te hniques as the ontroller is

the produ t of an optimisation and hen e the designer has to set-up the ost fun tion to be minimised. There is an evident transfer of tasks. As

H1 loop-

shaping provides robustness to a very general lass of perturbed plants the

designer has only got to worry about translating the performan e spe i ations to the desired loop shape. Other examples, tutorials, of designing loop-shaping ontrollers an be found in [18, 120, 215, 247℄.

80

-Synthesis

8.

Samir Bennani1 , Gertjan Looye and Carsten S herer

1

2

8.1 Introdu tion This hapter gives some ba kground theory on the Stru tured Singular Value,

, and provides a ight ontrol design example motivating and demonstrating -synthesis design. The issue addressed in the example illustrates the inherent ontrol paradigm that -synthesis partially the ne essary steps to arry out a

solves. Fundamentally,

 addresses the problem of retaining a desired performan e

level in the fa e of un ertainties, whi h is alled the robust performan e problem. For SISO systems, this is automati ally a hieved when the system has guaranteed robust stability and nominal performan e. This does not hold in the MIMO ase and in this respe t, the

- on ept

is a tool to address the

multivariable robust performan e problem. An important by-produ t of the method is that it rises modeling issues in the most general sense, i.e. that we mean system modeling, spe i ation modeling, un ertainty modeling, open loop or losed loop modeling and their validations are all issues whi h appear on e a designer is fa ed with

.

An attempt in

ta kling and predi ting the real world an be done only by formal tools, and this is where

 is intended to be used.

The singular value loop shape paradigm

as presented in [64℄ was a great leap forward in formalizing robust multivariable

ontrol theory.

This resulted in progress towards

H1

optimal ontrol, for

whi h omputable ee tive solutions are presented in [62, 62, 63℄. The lassi al multivariable feedba k problem is illustrated in gure 8.1. Usually, the plant is an element of a set of plants given by in the set

G~

G~ .

G

We shall onsider that ea h system

is linear, nite dimensional and a time invariant system whi h

an be represented by a transfer fun tion

onsists of three subproblems. nd a ompensator

K

G(s) .

The overall design problem

The Robust Stabilization problem (RS) is to

whi h makes the feedba k loop in gure 8.1 internally

stable for all possible plants

G~ .

The se ond problem, the Robust Performan e

problem (RP), whi h is mu h harder to a hieve, requires the ompensator

K

to make the losed loop system respond well under various external signals. 1

Fa ulty of Aerospa e Engineering, Stability and Control Group, Delft University of Te h-

nology, Kluyverweg 1, 2629 HS Delft, The Netherlands E-mail: s.bennanilr.tudelft.nl 2

Me hani al Engineering Systems and Control Group, Delft University of Te hnology,

Mekelweg 2, 2628 CD Delft, The Netherlands.

81

n r

e -

d K

u

~ G

y

Figure 8.1: Classi al feedba k onguration

This means that for all plants in external ommands noise (

n(t) ).

G~ , the plant outputs y(t)

a

urately tra k

r(t) , even in presen e of disturban es ( d(t) ) and sensor

A third problem alled the Control Eort minimization problem

is a onstraint imposed on the ompensator su h that the ontrol signals

u(t)

and/or other ontrol dependent signals remain within appli able limits. It has been remarked in [64, 65, 58℄ that the singular value on ept leads to

onservative robust performan e predi tions. Therefore, the stru tured singular value

 has been proposed as a more rened robust performan e indi ator.

In Beyond Singular Values and Loop Shapes [227℄ by Stein and Doyle the singular value loop shaping as a paradigm for multivariable feedba k system design in the arrangement as shown in gure 8.1 has been revisited. The main

on lusion drawn was that singular values within the lassi al design framework are ee tive in addressing the performan e robustness problem whenever the problem's design spe i ations are spatially round, but that it an be arbitrarily onservative otherwise. The origin of the problem lies in that onditions for robust performan e based on singular values are not tight (su ient, but not ne essary) and an severely overstate a tual requirements. The onservatism of the singular value loop shape paradigm in the lassi al framework ame from a too narrow denition and representation for a system. Furthermore, a general tight performan e spe i ation pro edure is la king. Finally, the stability analysis and synthesis tools were not addressing the fa t that perturbations arising in the system are stru tured. The onservatism introdu ed when using singular values an be surmounted by using the Stru tured Singular Value (SSV)



as

a tighter multivariable generalization of the stability margin. It will be shown that

 naturally arises from the stability analysis of a general lass of systems

alled Linear Fra tional Transformations (LFT's). Naturally, in the sense that the existen e of LFT's automati ally leads to the formulation of the robust performan e problem. General, in the sense that LFT's are both suitable for the analysis and the synthesis problem. Using LFT's to model sets of systems and the ontrol obje tives in mind, the robust MIMO design problem is formalized by spe ifying, the plant set

G~

over whi h the obje tives must be a hieved and the pre ise mathemati al

statements for the performan e and ontrol eort obje tives. This will be illustrated on a simplied ight ontrol problem that we des ribe rst. The design plant is a linear model of the longitudinal short period dynami s of a Cessna Citation 500 in landing onguration. The model states

82

q and the angle of atta k .

are the pit h rate

The state spa e representation

of the model dynami s is given as:



q_ _



=



Mq M 1 Z

The input is the elevator dee tion



ÆE .

q 



+



MÆE ZÆE



ÆE

(8.1)

The ontrol obje tive is to design a

pit h rate ontroller, su h that the losed loop response mat hes the handling quality model

1:5 . Hid (s) = qq ((ss)) = s+1 :5

From robustness onsiderations we

have to ensure that the system works well in the fa e of un ertain state spa e entries, alled the stability and ontrol derivatives, for trim speed variations up to

 10

m/s. During a full pit h ommand manoeuvre the angle of atta k

jj < 20 deg and the (jÆE j; jÆ_E j) < (10; 30) [deg, deg/s℄. is limited to

elevator dee tion and dee tion rate to

The mathemati al formulation of the performan e spe i ations in the ontrol problem and the model set over whi h these spe i ations have to hold an be done by using linear fra tional transformations and norm bounds. The advantage of the LFT formulation is that it gives a ommon base for un ertainty modelling, stability and performan e analysis of perturbed systems ( alled the analysis problem) and nally for ontroller synthesis (our synthesis problem). Ea h of these three steps will be su

essively illustrated by an appli ation on the air raft example. To illustrate the pra ti allity of

 as mature design tool

we on lude the example with a trade-o study, where the performan e and the robustness in the problem are gradually hanged.

8.2 Linear Fra tional Transformations (LFT's) Denote

M

as a

2  2 blo k-stru tured matrix:

v1 = M11 r1 + M12 r2 v2 = M21 r1 + M22 r2 together with matrix

hannels of by


:

M

with







relating

h

v1 = M11 + M12
(I here

l

v2

to

r2

as

v2 =
r2 .

M

i M22
) 1 M21 r1 = Fl (M;
)r1

indi ates that the lower hannels of

M

have been losed with

the same way we an lose the upper hannels of dimensioned matrix

Closing the lower

gives a Lower Linear Fra tional Transformation of

M


.

In

with some appropriately

that relates r1 to v1 in the following manner: r1 = v1 .

The upper LFT is given by:

h

v2 = M22 + M21 (I

i M11 ) 1 M12 r2 = Fu (M; )r2

Many ontrol problems t within this representation. A well known example is the input-output mapping of a linear system,

83

y = G(s)u.

It an be expressed

in terms of state spa e data as an LFT system.

A) 1 + D an be rewritten as

C (sI

G(s) = Fu





A B ;I C D s

It easy to see that

G(s) =



As we shall see this framework is parti ulary suitable to arry out parametri un ertainty modelling. This is illustrated on the air raft problem where due to hanging operating

onditions the state spa e entries of the nominal model (equation 8.1) vary substantially.

In table 8.1 the nominal values of the elements and the max-

imum relative variations an be found. parameter

M MÆE Mq Z ZÆE

Drawing the system dynami s in a

value

mult. pert.

-1.4796

0.20

-6.7679

0.20

-1.5773

0.20

-0.7441

0.20

-0.0900

0.20

Table 8.1: State spa e elements and perturbations for the design example

. α

+

Zα

+

Zδ

w1

w2

δ1

. q

+ +

Mq

δ2

Mα

z2

w5

∆Z δ δE

z3

w3

w4

δ1

∆Z α α

δ5 Mδ

z1

δ4

z4

∆M δ

z5

δ5

z

∆M q

w

α q

q δ3

.. .

∆Mα

A/C

δE

Figure 8.2: Blo k diagram of exam-

Figure 8.3:

ple system with perturbations

ample system in LFT-form

Representation of ex-

blo k diagram, we obtain insight in how the perturbations ae t the model, see gure 8.2. The perturbations in the table are the maximum absolute hanges

Æi by introdu ing s alings. Æ's arbitrarily within the given bounds. The model

the parameters an undergo whi h are normalized to We may hange any of these parametrized in the

Æ's ree ts a set of models.

To derive an LFT representa-

tion, the invariant part of the model and the un ertain elements (the delta's) are separated. This pro ess is known as pulling out the deltas.

All un er-

Æ1 ... Æ5 are diagonally augmented in the perturbation matrix
= diag(Æ1 ... Æ5 ). In gure 8.2 the signals in and from the delta's have been

ut; the signals (z1 ...z5 ) be ome the outputs from the onstant part and inputs of
, while the opposite holds for the signals (w1 ...w5 ). We an now read o tain elements

84

all signal relations given by the mapping and build the following matrix:

 q w1 w2 w3 w4 w5 ÆE _ Z 1 Z ZÆE 0 0 0 ZÆE q_ M Mq 0 0 M Mq MÆE MÆE z1
Z 0 0 0 0 0 0 0 0 0 0 0 0 0 ÆZ
E z2 0 z3
M 0 0 0 0 0 0 0 0 0 z4 0 ÆMq 0 0 0 0 0 0 0 0 0 0 ÆM
E z5 0  1 0 0 0 0 0 0 0 q 0 1 0 0 0 0 0 0 The so obtained matrix losed with the blo k diagonal matrix




as shown

in gure 8.3 provides the required LFT formulation of the un ertain air raft dynami s overing all possible parameter variations.

Noti e that the para-

metri un ertainty modeling pro ess reveals that un ertainty that is unstru tured at parameter level ( omponent level) be omes stru tured at system level

A/C).

(

Another possible way to apture a set of models is given for the

a tuator. The elevator position is hanged via an a tuator having rst order dynami s to

20%

15 . Ga t (s) = s+15

The devi e is assumed to give position errors up

in a frequen y range up to

may be even more.

1

rad/s, while at higher frequen ies this

The variation in the position error along the frequen y

1+1 . The Wpert (s) = 0:20 s=s=40+1 ~ a t = model set overed by the un ertain a tuator dynami s is given by G fGa t (1 + Æ6 (s)Wpert ) : Æ6 (s) stable kÆ6 (s) k1  1g. The weighting fun tion is used to normalize the unknown perturbation Æ6 and at any frequen y ! the magnitude of Wpert represents the relative un ertainty level in the a tuator is represented by a rst order transfer fun tion

model.

This type of modeling is alled multipli ative un ertainty modeling.

It is unstru tured at omponent level (the a tuator).

On e again, we shall

see how unstru tured un ertainty at omponent level be omes stru tured at system level, when inter onne ting these omponents.

8.3 The Extended Design Framework The performan e robustness problem is addressed in a general framework for system design, whi h onsists of a general problem des ription in terms of LFT systems, some key analysis results, a suitable measure of the magnitude for matrix transfer fun tions, the stru tured singular value

, and some ontroller

synthesis results. An important remark is that all elements onsidered in this design framework have a pra ti al software implementation in order to be useful for the engineering world. This issue is provided by the ex ellent and reliable software of [18℄. Furthermore, the pra ti ing engineer needs good tutorials to keep tra k of the theoreti al advan es in the eld, these are nowadays ri hly provided in referen es as [18, 189, 266, 61℄ and many others. elements of that framework are des ribed next.

85

The various

8.3.1 General problem des ription Using the LFT representation the lassi al multivariable ontrol problem as shown in gure 8.1 an be transformed into a more versatile form. In this way any performan e obje tive from the a tual inter onne tion and its ee t within other system loops is derivable.

w d u

∆ P

z e y

K Figure 8.4: The general problem des ription We re ognize for any un ertain losed loop system the three basi omponents:

P (our problem data) the ontroller K (possibly still to be designed) the un ertain elements
(belonging to a pre-spe ied set).

1. general system 2. 3.

All un ertain elements have been pulled out of the system and pla ed in the


-blo k. For synthesis and analysis the only thing we have to know is that the
-blo k is stable and norm bounded: jj
jj1  1. The always returning subdivision for the general system P onsists of re ognizing three pairs of input-output variables. The rst one (u(t); y (t)) onsist of the ontrol and measurement variables. Then we have (d(t); e(t)), the disturban e and error signals whi h onstitute the generalized performan e variables and nally the third pair

(w(t); z (t))

for the perturbation signals whi h are onne ted ba k

into the system through a norm bounded perturbation The design problem is to nd a ompensator general system

e

P

K


.

internally stabilizing the

while keeping the matrix transfer fun tion between

appropriately small for the whole set of allowable perturbations


.

d

and

In the

transformation pro ess, from the lassi al setup into the more general setup, any un ertainty arising at system omponent (a tuator, plant sensors et .) level be omes automati ally stru tured at the level of the generalized system

P.

Furthermore, the so obtained generalized problem des ription as given in

gure 8.4 is suitable for the synthesis as well as for the analysis problem, and has potential for expansion due to its general stru ture.

8.3.2 Analysis results (Doyle 1984) From the general system representation as shown in gure 8.4, a non onservative, ne essary and su ient ondition for robust performan e an be derived.

86

∆

∆ w

0

M ( P, K )

d

Figure 8.5: Closed loop system

z

w

e

d

M

0

∆p z

M ( P, K )

e

Figure 8.6: Stability onguration for Robust Performan e

Suppose the stabilizing ontroller

K

an a

ept the feedba k loop as shown

M (P; K ) = Fl (P; K ), shown in g2  2 blo k-stru tured transfer M (P; K )(s) whi h together with the
operator in a feedba k

in gure 8.4 to get the losed loop system

ure 8.5. The generalized losed loop system is a fun tion matrix

arrangement, forms the basi obje ts on whi h the system analysis problem is based. Under the ondition the system

M (P; K ) is nominally stabilized by K

the

following results apply:

Theorem 8.1

General Analysis Theorem Doyle (1984)

1. Nominal performan e is satised if and only if

 (M22 (j!)) < 1

8!

(8.2)

8!

(8.3)

2. Stability is robust if and only if

 (M11 (j!)) < 1 3. Performan e is robust if and only if

(M (j!)) < 1

8!

(8.4)

The third result represents the MIMO extension of the robust performan e problem, providing with ne essary and su ient onditions. It is established by starting with the denition that performan e is robust if and only if the transfer fun tion matrix from

d

!e

given by

Fu (M;
), remains in an

norm sense bounded by unity - that is, if and only if

 M22 + M21(I



M11 ) 1
M12 < 1

8 !; k
k1  1

H1

(8.5)

This norm bound is also a ne essary and su ient ondition for the system

M

in gure 8.6 to remain stable if we onne t a se ond norm-bounded per-

turbation, say


p (s)

a ross the

e

! d 87

terminals

In this respe t, robust

performan e is equivalent to robust stability in the fa e of two perturbations,


and
p , onne ted around the system M

in the blo k-diagonal stru tured

arrangement shown in gure 8.6. The system

if the fun tion

det(I

(
;
p )M (j!))

diag

M

is robustly stable if and only

!.

remains non-zero for all

8.3.3 The Stru tured Singular Value The observation to view performan e robustness as a stability test brought the fun tion

(:)

 for omplex matri es M

dened in equation 8.4. The fun tion

is the following:

Denition 8.2

The Stru tured Singular Value

2


4min (M ) = In words,

I


M



8 <  :

det(I

93 = 5 m ;


M ) = 0

for some with




= diag[
1 ;
2 ; : : : ;
℄

 (
i )  1 8 i

is the re ipro al of the smallest value of s alar

singular for some






1 :

(8.6)

whi h makes the matrix

in a blo k-diagonal set. If no su h



exists,



is

taken to be zero. This denition redu es to the onventional maximal singular value in absen e of stru ture ( i.e. when the number of blo ks, reason,



m , in

has been alled the stru tured singular value.

The denition 8.6 is not limited to

2 2




is one ). For this

blo k stru tures.

It an be used

to test stability with respe t to any number of diagonal blo ks: in that ase

robust stability is satised if and only if


pert (M11 (j!)) < 1

8!

for a given blo k-stru tured un ertainty from the set


pert

(8.7) . In this way it is

possible to establish robust stability with respe t to plants ae ted by several stru tured perturbations while tting in the robust performan e paradigm. Denition 8.6 also extends to real-valued perturbations redu ing many parametri system analysis problems to

- al ulations.

More generally still, the

stru tured singular value on ept (not value) extends to time varying systems. The al ulations required for these extended ases expressed in Linear Matrix Inequalities, ontinue to impose substantial hallenges even with the tremendous evolution in the eld of onvex optimization.

8.3.4 Numeri s for the stru tured singular value In general exa t omputation of the stru tured singular value is not possible. Therefore, we work with approximations via the upper and lower bounds of For a omplete tutorial on the stru tured singular value numeri s we refer to [188, 18℄.



.

and the involved

The al ulation of the stru tured singular

value

()

dius.

Based on this generalization omputable bounds an be given and re-

relies always on the parti ular hoi e of the un ertainty stru ture


and generalizes matrix measures as the singular value and the spe tral raned. Dene the following perturbation stru tures

88


= fdiag[Æ1Im ; Æ2;
3 ℄ :

Æ1 ; Æ3 2 C;
3 2 Cnn ; g. The set B
is the sub-set of
for whi h holds B
= f
2
: (
)  1g. We an also dene Q
, Q
= f
:
2
;

= I g   It an be shown that Q
2
,
Q 2
,  (Q
) =  (
) =  (Q
) Asso iI

I




ated with the set

, dene the set of s aling matri es

D = diag[D1; d2 ; d3Ik ℄ :

D

given by:

D1 2 Cmm ; d1 ; d2 2 IR + ; D1 = D1 > 0; I




is full D is a s alar, and vi e versa. It D 2 D and
2
holds, D
=
D. In nn it is easy to see that  (M ) =  (M ). Sin e the perturbation the ase
2 C
is bounded we have 
(M )    (M ). However, this bound is not of pra ti al use sin e the gap between  and the   an be arbitrarily large. On the other hand when
= Æ1 Im , with Æ1 2 C then (M ) = (M ), the spe tral radius of M . Using the transformations D and Q on M the bounds an be rened to: max (QM )  (M )  inf  (DMD 1 ) (8.8) Q2Q
D 2D

Note that where the diagonal blo k of

an be seen dire tly that for ea h I

I

In fa t the left inequality is an equality, but not useful as su h, sin e the optimization over

Q is not onvex; it shows lots of lo al minima and maxima.

More useful is the right inequality (whi h in a limited number of ases is also an equality, but in nearly all ases very tight), sin e the optimization over

D

is onvex. Furthermore, it is an upper bound and therefore safe. The perturbations onsidered, were omplex matri es or s alars. However, in the ase of the parametri un ertainties in the air raft, the perturbations

2 IR ).

are real (

In a ase like this, we would like to know, given the system

with un ertain parameters, the smallest possible ombination of perturbations, that auses the system to be ome unstable. It is obvious that the stru tured singular value for this even more onstrained set of perturbations (un ertainties are only allowed to vary along the real axis) is more di ult to determine. For



al ulations with a mixed omplex/real perturbation set, there exist reasonably tight upper bounds by nding optimal s aling matri es (D-G s ales), for more details the reader is referred to [266, 18℄.

8.3.5 Setting up the design problem For the analysis of the ight ontrol example, we rst have to spe ify the overall

ontrol ar hite ture then translate the design spe i ations into mathemati al obje tives by weighting the signals of interest. To demonstrate the exibility of the proposed framework we shall address the simultaneous design of a feedba k and ommand shaping ompensator whi h is often referred to as two degrees of

freedom ontrol. Upon the hosen ontrol ar hite ture we pla e on the physi al lo ation in the system, namely at signal level our requirements. These requirements are made frequen y dependent and be ome our weighting fun tions. In doing so we end up with the situation depi ted in gure 8.7 whi h is what we

all the inter onne tion stru ture. It onsists of the air raft model parametri un ertainties, the un ertain a tuator

89

Ga t (s),

G(s) with its K (s)

the ontroller

to be designed, an ideal model

Hid (s)

whi h we want to mat h and the per-

forman e weighting lters that pla e emphasis on the frequen y ontent and amplitude on the signals of interest. The inter onne tion stru ture in gure 8.7 is the pi torial equivalent of the mathemati al statement of the plant set together with the ontrol spe i ations (depending on the norm we hoose). It ~.

~

δE

z ~

α

1

αmax

δ1.

α

..

w

δ5

G

δE 1

1

.

δ max

δ max 1 S

δE

.

δE

δ5 15

Wp

qc

u

K

-

q

Win

q nom

Gact

n ~

Wpert

w + 6

q

qe

z6

Wn

-

H id

Figure 8.7: Inter onne tion stru ture of the example system

is often advisable to s ale the systems units appropriately. The nominal pit h rate ommand signal is therefore normalized with respe t to the maximum expe ted ommands with the lter

Win .

The pit h rate ommand input

q

goes

through the ideal model. The dieren e between the ideal model response and the a tual pit h rate measurement

q

is the tra king error. To emphasize how

large and up to what frequen y the error redu tion should o

ur, a lter

Wp (s),

ree ting the tra king obje tive, is pla ed on the error signal. However, tra king should not be a hieved at the ost of ex essive ontrol a tivity. Therefore both the elevator dee tion and rate are penalized. The dee tion and rate are

WÆ_E = 1=Æ_Emax .

W = 1=Æ

Emax and ÆE When one of these weighted signals is larger than one, then

weighted by the inverse of their maximum allowed values the obje tive is violated.

To prevent stall during a full pit h rate ommand

we provide an angle of atta k limiting fun tion by introdu ing a performan e

, using the inverse of the maximum allowable value we get the W = 1=max. Finally, a noise lter Wn is depi ted in gure 8.7. This lter s ales the normalized measurement noise n as a fun tion of the frequen y.

spe i ation on weight

Dis onne ting in gure 8.7 the ontroller and the un ertainties we end up with the open loop inter onne tion stru ture

P

P

as shown gure 8.8. The re-

:~:-sign indi ates z w), the ~ ~ _ performan e hannel given by e = [~ qe ; ~; ÆE ; ÆE ℄, d = [qnom ; n℄ and the measurement/ ontrol hannel with y = [q ; q + Wn n℄, and u = [ÆE ℄. It ontains maining system

has three pairs of inputs and outputs (the

a weighted output) These orrespond to the un ertainty hannel (

all required problem data for design. But sin e the weighting fun tions are in most of the ases our design parameters it is worth to start with the analysis on basis of the hypothesis that we are in possession of a stabilizing ontroller

K (s).

90

3

Wp2 2

10

Wp1 1

Z6 (complex)

10

Hid

0

10

P

gain

~q e ~ α ~ δE ~

. δ

10

w 1 .... w5 (real)

Z 1.... Z 5 (real) Z6 (complex)

q nom

−1

10

Werr −2

10

n

E

qc

Wn

−3

10

δE

(noisy) q

−4

10

−2

−1

10

10

0

1

2

10 10 frequency (rad/s)

10

3

10

Figure 8.8: Open loop inter onne -

Figure 8.9: Weighting fun tions for

tion stru ture

design example (not

Win )

s aled with

8.3.6 Weighting fun tion sele tion onsiderations Up to now, we have dened a set of models (nominal air raft model with perturbations), de ided on the ontroller ar hite ture (two degrees of freedom, measurements et .), and whi h performan e quantities we wish to take into a

ount (tra king

q,

maximum ontrol dee tions et .).

The question now

arising is how do the weights have to look like if we want our losed loop system to a hieve robust performan e?

Mu h an be learned about the system by writing down the transfer fun tions in the inter onne tion stru ture. For simpli ity, we will forget about the parametri un ertainties in the air raft model and onsider only the un ertainty at the a tuator. Re all that the rst step is to obtain the transfer fun tions of the open loop inter onne tion stru ture

P

(gure 8.8). These fun tions an be dire tly read

from the inter onne tion stru ture in gure 8.7. Denote the transfer fun tion from

ÆE to q is Gq and from ÆE to  as G . P is given as:

The

3  3 open loop inter onne tion

stru ture

2

z 6 q~e 6 6  6 ~ 6 Æ~ 6 E 6~ 6 Æ_ E 6 4 q q

3

2

7 7 7 7 7 7 7 7 7 5

6 6 6 6 6 6 6 6 6 4

=

0 0 Wp Gq Ga t Wp Hid Win W G Ga t 0 0 0 0 0 0 Win Gq Ga t 0

0 Wpert 0 Wp Gq Ga t 0 W G Ga t 0 WÆE Ga t 0 WÆ_E sGa t 0 0 Wn Gq Ga t

3 72 3 7 w 7 76 7 6 qnom 7 7 74 5 n 7 7 7 Æ E 5

All expressions required for analysis of the losed loop system

M = Fl (P; K )

an be obtained via the short ut:

Mij (s) = Pij + Pi3 (s)[I

K (s)P33 (s)℄ 1 K (s)P3j (s) i; j = 1; 2 91

(8.9)

K (s) partitioned into [K Kf ℄, here K (s) represents the ommand part, q to u, while Kf (s) for the feedba k task stands for the transfer fun tion from q to u. The input sensitivity fun tion Si is given by: [I K (s)P33 (s)℄ 1 = [I [K Kf ℄ [0 Gq Ga t ℄℄ 1 = [I Kf Gq Ga t ℄℄ 1 = Si

with

given by the transfer fun tion from

so that with the omplementary sensitivity The omplete analysis system

2

Wpert 6 Wp 6 diag 6 W 6 4 WÆE WÆ_E

3T 2 7 7 7 7 5

6 6 6 6 4

M

Ti

we get

Si + Ti = I .

is:

Ti Si K Si Kf Gq Ga t Si Hid Gq Ga t Si K Gq Ga t Si Kf G Ga t Si K G Ga t Si Kf G Ga t Si Ga t Ti Ga t Si K Ga t Si Kf sGa t Si K sGa t Si Kf sGa t Ti

3 2 7 7 7 diag 4 7 5

I Win Wn

3T 5

 approximated by the peak value of the s aled  (DMD 1 ), a -upperbound), preferably to a value lower than

Sin e we try to minimize singular value (

1, the diagonal blo ks (not ae ted by the D-s ales) must have a norm smaller than 1. This leads to the robust stability and nominal performan e onditions:

jjWpert Ti jj1 < 1 and:

2

Wp 0 0 0 32 Hid Gq Ga tSi K

0 76 G Ga tSi K

6 0 W 0

4 0 0 WÆE 0 54 Ga tSi K

0 0 0 WÆ_E sGa t Si K From the rst expressions

(M11 )

3

Gq Ga t Si Kf  G Ga t Si Kf 7 Ga t Si Kf 5 sGa t Si Kf

0 0

it is lear that the weight

loop gain to roll o: at low frequen ies



Win 0 0 Wn

< 1

Wpert

1

for es the

jTi j  1 and Wpert will be taken 20%. At

jTi j  Kf G, while Wpert will in rease (more un ertainty to jWpert Kf Gj < 1 (SISO in this ase), we need at least: jKf Gj < 1=jWpert j. In this example, it will have little ee t,

higher frequen ies

a

ount for unmodeled dynami s et .). For

sin e the plant with its a tuator model have already su ient roll-o. However, the ross-over frequen y of bandwidth of

Ti .

Wpert is used as a design parameter to limit the

This is important to prevent ex itation of, for example, stru -

tural modes by ontrol signals ontaining high frequen ies. A more interesting

ase is the performan e blo k

(M22 ).

To a hieve the tra king obje tive, we need:

jWp (Hid Gq Ga t Si K )Win j < 1. At low frequen ies jHid j = 1 and we assume that K  Kf : jWp (I + Gq Ga t Si Kf )Win j = jWp (I Gq Ga t Kf ) 1 Win j < 1 1 Win j < 1, If the loop gain is high, we have approximately: jWp (Gq Ga t Kf ) so that: jGq Ga t Kf j > jWp Win j. We will hoose Win onstant. Gq Ga t have

onstant gain at low frequen ies, so that we mainly inuen e the low frequen y shape of the ontroller

K

via

Wp .

Observe that the tra king performan e

may be destroyed by the noise input via

Wn :

Wp Gq Ga t Si Kf Wn .

jGq Ga t Si Kf j  1 at low frequen ies, we need at least: jWp Wn j < 1. 92

Sin e But, if

Wp

we give

high gain at low frequen ies to in rease the gain of the ontroller,

this requirement is easily violated. There are two simple solutions: in the rst

Wp is high, so that Wn gets Wp ; in the se ond pla e we an feed the noise to the performan e lter, so that Hid q (q + Wn n) is weighted. This is very obvious: the steady state value (at ! = 0) of Wn a

ounts for example pla e we an make the noise input low where

approximately the inverse shape of

for a sensor bias. This bias may violate the performan e index, be ause this is based on the error between the referen e and the exa t output. By applying the se ond solution, the error is related to the same biased measurement the

ontroller re eives. (In a standard feedba k onguration the transfer fun tion of the noise to the output is hanged from a omplementary sensitivity fun tion to a sensitivity fun tion, whi h has low gain at low frequen ies). We an hoose here to lower the gain of

Wn

if ne essary; this has a desirable ee t when we

design a ontroller for the plant without un ertainties, as will be shown later.

W _ sG S K W

W _ sGa t Si Kf also play an important role, jSi j  1, sGa t = 15j!=j! + 15  15 jK j < 1=jWÆ_E 15Win j and jKf j < 1=jWÆ_E 15Wn j

ÆE a t i in and ÆE mainly at higher frequen ies. In that ase

The terms as

s ! 1,

so that at least:

respe tively. These weightings impose an upper-bound on the high frequen y

gain of the ontroller. In many ases the ontroller rolls o at higher frequen ies, so that the weights do not have a great ee t.

However, in the ase of

plant perturbations or severe disturban es it is very important to penalize the rates to prevent the ontroller from produ ing ontrol signals with rates beyond the physi al limits of the plant, ausing rate saturation. We will design

q ) up to Win = 10=57:3

for ommands (

rad/s. Next, two performan e weights

are hosen, to illustrate their ee t on the ontroller shape:

s=20+1 W = 1000 s=20+1 Wp1 = jW20in j s= 0:5+1 p2 jWin j s=0:01+1

(8.10)

Note, that the ross-over frequen ies are equal. This is an important onsideration. shape of

In the low frequen y range there are two major parameters for the

Wp :

the steady state gain and the ross-over frequen y. We must be

areful to hange one at a time.

If the steady state error appears to be too

large (in a simulation for example) simply in reasing the gain means that also the ross-over frequen y in reases, leading to unintended other ee ts.

Usu-

ally, if a good ross over point is found, one an try to extend the slope into the low frequen y range.

This an be seen for

Note that the weight attens at sient behaviour.

! = 10

Wp1

and

W p2

in gure 8.9.

rad/s. This is useful to limit tran-

The weights on the elevator dee tion and rate are hosen

= 5710:3 rad 1 Æ_E 1 = 5730:3 (rad/s) 1 To limit the angle of atmax 57:3 1 1 ta k we hoose max = 2jWin j = 20 . Finally, we dene the noise lter: s= 0 : 01+1 s= 0 0:01 :01+1 . The DC gain is hosen low taking Wn = 0:0005jWinj s=2+1 = 57 :3 s=2+1 into a

ount that we also have to satisfy the performan e index Wp2 . as:

1

ÆEmax

93

8.4 -Synthesis 8.4.1 Formulation of the synthesis problem The next step is the ontroller synthesis problem. The obje tive is to nd a

K a hieving the desired performan e requirements for the P~ .

stabilizing ontroller whole set of plants

P~ = fFu (P;
pert ) :
pert 2
pert ; k
pert k1  1g

The denition of the

(8.11)

-synthesis obje tive:

Denition 8.3 -synthesis :

K (s) the worst ase performan e, 
(M ) = 
(Fl (P; K )).

Minimize over all stabilizing ontrollers the peak value of

min K (s)

k[Fl (P; K )℄k
< 1

i.e.

(8.12)

stabilizing with the shorthand notation of the to the

-norm

of the operator

1-norm we have kGk
= max! 
(G(j!)).

G

and similarly

The stru tured singular

value does not satisfy the denition of a norm. This notation is adopted only to ree t the fa t that we want to measure the size of the worst ase performan e.


(
) by its upper bound  (D(
)D 1 ). Dene Dpert , the s aling set for the perturbation stru ture
pert . For Dpert 2 Dpert and
pert 2
pert it follows from the denition of the invarian e of  under s aling, Dpert
pert =
pert Dpert , that the s aling set D for the augmented In order to perform al ulations we repla e

perturbation set is dened as:

D=



Dpert 0 0 I



: where Dpert 2 Dpert

 (8.13)

D-s ale orresponding to the performan e blo k
p -blo k is set D 2 D 
an be obtained from the upper bound relation (applied to some onstant matrix M ): 
(M )  min (DMD 1 ) (8.14) D 2D When
pert onsists of F full blo ks, the set D looks like Note that the

to one. With respe t to the s aling stru ture

D=f

diag

[d1 I; : : : ; dF I; I ℄ dj > 0g

(8.15)

D an have any phase without ae ting the value of  (DMD 1 ). Therefore the optimization along the frequen y over D an be repla ed by an optimization over stable minimum-phase D (s). Considering real-rational, stable and minimum-phase s alings D (s) to the a tual optimization formulation The elements of

is given as:

min

K (s) stabilizing

min 2D

D(s) stable, min-phase

kD(j!)Fl (P; K )(j!)D 1 (j!)k1 94

(8.16)

In this way the optimization problem of minimizing the worst ase performan e has been t into the

H1 -synthesis framework. Optimizing over D and K si-

multaneously is in general not onvex. Therefore an indire t s heme is used in

D KK (s) while holding D(s) xed and then optimizes over stable minimum-phase D(s) while holding K (s) xed. More details on erning the pra ti al implementation of the syn-

the hope of nding a ontroller minimizing

.

The pro edure is alled

iteration sin e it iteratively optimizes over the stabilizing

thesis problem an be found in [18, 189℄. In most engineering situations the proposed s heme has been proven to be su

essful.

8.4.2 Controller synthesis and analysis To illustrate the ee ts of the weighting fun tion sele tion, un ertainty model sele tion, i.e the trade-o between performan e and robustness we shall study four design ases in our example: 1. Nominal Performan e Design: All un ertainties are set to zero we designate

Knom the resulting ontroller.

2. Complex Un ertainty Design:

assume plant with only the omplex per-

Æ6 2
at the a tuators. The resulting ontroller will be denoted as K2 . The augmented perturbation
related to the robust performan e index is given by
2 = diag(Æ6 ;
perf ). turbation

3. Real and Complex Un ertainty Design:

Taking all un ertainties into a -

ÆR = diag(Æ1 ; : : : ; Æ5 ) 2
related to robust performan e is denoted as
3 = diag(Æ1 ; : : : ; Æ5 ; Æ6 ;
perf ).

ount leads to ontroller


R

K3 .

The real un ertainty is

. The augmented perturbation

 (M ) (M ) (M ) order

Knom K2 (Wp1 ) K2 (Wp2 ) K3 (Wp1 ) K3(Wp2 ) 0.89

6

28.66

28.66

28.66

28.66

1.71

1.86

1.88

2.11

1.51

1.60

1.78

1.89

9

9

15

15

Table 8.2: A hieved robust performan e levels

In table 8.4.2 the a hieved robust performan e levels expressed in

 values

for all ongurations are summarized. Row # 1 shows the results of the pure

H1

optimization.

Rows # 2 and # 3 reveal the robust performan e levels

a hieved after the rst and se ond

D K iteration.

In the last row the order of

the resulting ontrollers is given. The table ree ts a well known fa t that the robust performan e level de reases as the un ertainty and performan e levels in rease.

95

Controller Shape Analysis In gure 8.10 the frequen y responses of the ontrollers are depi ted. As already noted in se tion 8.3.6, the ontroller will not ne essarily roll o at higher frequen ies, sin e the ombination of the a tuator and the plant model already shows this behaviour. We an see that ontroller shapes atten out at higher frequen ies and lower gains.

In the se tion 8.3.6 we have seen that

K at higher frequen ies has to satisfy: jK j < 1=jWÆ_E 15Win j = 1=(57:3=30  15  10=57:3) = 0:2. This is onrmed by gure 8.10. For Kf we have: jKf j < 1=jWÆ_ 15Wn j = 1=(57:3=30  15  0:1  10=57:3) = 2. This is also E the gain of

satised.

3

10

0

2

Wpert^−1

10

10

Ti

K3 (Wp2)

−1

K2 (Wp2)

10

1

10

K3 (Wp1)

mag

Controller gain

T_i and Wpert

1

10

−2

10

K2 (Wp1)

0

10

K_f −3

10

Knom c −1

10

K_c

−4

10

−2

10

−5

−2

−1

10

10

Figure 8.10:

0

10 frequency (rad/s)

1

10

10

2

10

−2

−1

10

Controller frequen y

10

0

1

10 10 frequency (rad/s)

2

10

Figure 8.11: Input ompl. sensitiv-

responses

ity fun tion with

1 Wpert

Due to the a tuator un ertainty the ontroller will limit its bandwidth at frequen ies where un ertainty starts to be ome important. This is illustrated by the fa t that the bandwidth of the input omplementary sensitivity fun tion

1 are Wpert . The frequen y response plots of Ti and Wpert Indeed Ti rolls o near the ross-over frequen y of Wpert .

is limited by the lter given in gure 8.11.

In this way we prevent unmodelled higher order dynami s from ex itation by keeping the ontrol a tions within the lower frequen y range. for the nominal plant in gure 8.10, and no feedba k

Kf = 0.

Knom

The ontroller

K K approximately inverts the

has only a feedforward a tion

The feedforward a tion

plant and the ideal model is built in as a feedforward lter. We know a priori that in absen e of un ertainty no feedba k is required. It is interesting to see that this out ome is a hieved automati ally by the method.

Wp . By Wp1 ! Wp2 ) the ontroller

Another interesting ee t is the inuen e of the performan e weight in reasing the slope into the low frequen y range ( does exa tly the same.

We an use this to for e the optimization algorithm

to build integration or even double integration.

As in lassi al Bode loop

shape te hniques, the internal model prin iple holds and as we know in reasing tra king error requirements requires in reasing low frequen y ontrol gain.

96

3

10

Nominal Performan e Beside the shape of the ontroller we are interested in the trade-os it makes. In gure 8.12 we have for all ontrollers plotted the nominal performan e level

 (M22 ).

The overall shape is, a high value of

 (M22 ) at low frequen y orre-

sponding to an ee tive tra king requirement at these frequen ies. At higher

 (M22 ) rolls o.  (M22 )  0:9 is a hieved

frequen ies there is no performan e requirement so that plot Given all the ontrollers the best nominal performan e

by the system without un ertainty. For the other ontrollers the level is higher

(worse), be ause there is a trade-o against robustness to the perturbations. We know that the omplex un ertainty is about 20 % in the low frequen y range. This is about of the performan e degradation level of the se ond system with respe t to the nominal system. It is surprising that the third ontroller for the most un ertain plant a hieves a better nominal performan e level than the se ond ontroller. The on lusion is that the real un ertainty at low frequen ies, sets o the ee t of omplex un ertainty with respe t to the nominal performan e and that this ee t is reversed at high frequen ies.

K2 designed with Wp2 is modest in the performan e level of K2 .

For omparison we add the nominal performan e plot for

Wp2 .

Note that the inuen e of

4 1.2 K2 (Wp2) 1

3.5

K2 (Wp1)

Knom 3

K3

2.5

mu(M)

sig1(M22)

0.8

Knom

0.6

K2

2

1.5

K3

0.4

1 0.2

0.5

0 −2 10

−1

0

10

1

2

10 10 frequency (rad/s)

0 −2 10

3

10

10

−1

0

10

1

2

10 10 frequency (rad/s)

Figure 8.12: Nominal performan e

Figure 8.13:

levels ontrollers

levels ontrollers

10

3

10

Robust performan e

Robust Performan e To ompare the robust performan e levels a a hieved ontrollers.

-test

for


3

is applied on all

The result an be found in gure 8.13.

None of the

ontroller a hieves robust performan e. One of the purposes of this omparison is to reveal the ee ts of o-design spe i ations for the ontrollers

Knom.

K2

and

The question we have in mind is how robust is a robustly designed

ontroller? The nominal ontroller

Knom performs worst with a 400 % degradation at

low frequen ies. At higher frequen ies it has a better robust performan e level than

K2

and

K3 .

The result is expe table, sin e it is purely an open loop

ontroller. The shown

-plots

for

K2

and

97

K3

ree t the design result in the

Wp1 . Using Wp2 would show a too Knom ae ting the s ale of the plots. The K2

medium performan e ase, i.e. with weight dramati performan e ollapse of

ontroller has about a 70 % performan e degradation due to real perturbations. Note, that by taking into a

ount the real perturbations in the design, ase

K3 ,

the total robust performan e level improves onsiderably in the low frequen y range at the ost of the level at higher frequen ies. There is an overall better balan e between the performan e and robustness obje tive, whi h improves the better we model the un ertainty in the plant.

Robust Stability

(M11 )
3 , have been

The robust stability properties are shown in gure 8.14, the values of are plotted along the frequen y axis. Again all perturbations, i.e. taken into a

ount. The ontroller

Knom a hieves the best robust stability level, Kf = 0), and

whi h is not surprising anymore sin e there is no loop losure ( apparently there is no perturbation with norm

 1 to destabilize the nominal

K2 (M11 ) < 1). Note that for K2 and K3 two bounds are visible at lower frequen ies; they arise from real  approximations by optimizing an upper bound and a lower bound: the exa t value of  lies in between these bounds. If for K2 only the omplex perturbation is taken into a

ount (not shown) plant (a system with no feedba k has no robust stability problems). For

robust stability is a hieved (

the plot moves approximately 0.1 downwards, whi h means that there is a 10 % stability robustness degradation to unmodelled spe i ations in the design. It is remarkable that the robust stability level for one of

K2 :

K3

is higher (worse) than the

in a small frequen y range it is even possible to nd a ombina-

 (
i )

tion of perturbations (

 1) that destabilizes the plant ((M11 ) > 1).

We must realize that we are optimizing the peak value of

(M (j!)).

Taking

the parametri un ertainties into a

ount in the design improves this value

ompared to the

-test

for

K2.

In this sense we su

eeded in the third de-

sign. However, the balan e between performan e and stability robustness has moved in the wrong dire tion: the rst improved, the se ond got worse, while overall robust performan e improved. The designer has to be areful and has always to nd a right balan e. Espe ially, sin e in pra ti e

(M ) < 1

8 !)

(

is seldomly a hieved. However, for a given situation robust stability should be

(M11 (j!)) < 1 8 !).

preferably the rst to be guaranteed, i.e. (

Then, slowly

and arefully, the designer an buy performan e from the robust stability until the overall requirements are in balan e. We would like to remark that in the multivariable ase this philosophy still holds but things be ome more omplex be ause of dire tionality issues.

Time Simulations, Performan e Sensitivity We shall next analyze the systems performan e via pit h rate step ommand simulations shown in gure 8.17. Three model ases are onsidered:



nom: simulation with the nominal model;

98

1.2

1

0.8

K3

mu(M11)

K2 0.6

0.4

Knom

0.2

0 −2 10

−1

0

10

1

2

10 10 frequency (rad/s)

3

10

10

Figure 8.14: Robust stability ontrollers



pert1: simulation with a perturbed model: all parameters in table 8.1 are perturbed with



Æ = 20%; only MÆE

pert2: identi al perturbation, now

with

Æ = +20%;


MÆE = 20% the others are 20%.

The rst plot shows the rst ontroller: the nominal response oin ides well with the ideal model response. The perturbations have a dramati ee t on the tra king performan e, sin e we are in fa t looking at an open loop simulation. In the responses for the other ontrollers the ee ts are less dramati . Note that for the se ond ontroller

K2

designed with

Wp2

the steady state error

indeed has be ome nearly zero, even under the inuen e of the perturbations. Finally, we an see that

K3

K2 (with Wp1 ) -analysis.

performs better than

perturbations, as ould be expe ted from the

under the

Time Simulations, Robust Stability Aspe t Finally, we are looking at the perturbations that ould destabilize the losed loop systems in the ase of

K2 and K3 .

!peak = 6:5

rad/s and

 plots peak = 0:918 at

We an see in gure 8.14 that the

(lower bounds) of the losed loop systems show peak values of

peak = 1:024 at !peak = 6:9

rad/s respe tively. This

means that we an nd the smallest destabilizing perturbation with appropriate


-stru ture:


= fdiag[Æ1;


; Æ6℄ : with

Æ1 ;




; Æ5 2 IR ; Æ6 2 Cg I

 (
) = 1=0:918 = 1:089 and  (
) = 1=1:024 = 0:977 respe tively. For
in mind,  (
)  1
2
. system with K2 is robustly stable, sin e peak < 1 and the norm of the

the robust stability test we have a norm bounded The

perturbation therefore needs to be larger than 1 to destabilize the system. This

99

is not the ase for

K3 .

Using available software we have found perturbations

that will just destabilize the systems. For

K2

we have (for example):


= diag[Æ1;


; Æ6 ℄ = diag[ 1:0892; 0:8389; 0:7893; 1:0892; 1:0892; 1:0573 0:2618i℄ with norm   (
) = 1:098. For K3 :
= diag[ 0:9768; 0:9768; 0:0073; 0:9768; 0:9768; 0:9624 0:1667i℄ In gure 8.15,8.16 we simulate the losed loop system, without and with perturbations. To see how sensitive results an be, we also implement the pertur-

98 %

bation s aled to

1:02 % of its riti al value. We an see that both K2 and K3 indeed are destabilized while in reasing

and

systems with ontrollers

the perturbation levels over their riti al values, whi h on ludes the example.

controller K2

controller K3

0.35

0.3

nominal

0.3

0.25

0.98*pert 1.02*pert 0.25

1*pert

q (deg/s)

q (deg/s)

0.2

0.2

0.15

0.15 0.1

0.1

nominal 0.98*pert

0.05

1.02*pert

0.05

0 0

1*pert 0 0

1

2

Figure 8.15:

3

4

5 time (t)

6

7

8

9

1

2

3

10

Destabilizing pertur-

Figure 8.16:

bations

4

5 time (t)

6

7

8

9

10

Destabilizing pertur-

bations

Although this example is very simple, it is lear that

-synthesis is a pow-

erful tool where many fa tors an be taken into a

ount: design requirements, un ertainties, disturban e models et . In absen e of un ertainty, the two degrees of freedom ontroller inverts the plant and pla es in the feedforward path almost no feedba k.

This is a desirable strategy only in the absen e of un-

ertainties. We saw that slight un ertainties aused huge performan e degradation.

The di ulty in designing a good ontroller is the to nd the right

trade-os between the many usually oni ting requirements. We believe that the approa h as shown here provides us with tools to make sensible (balan ed) design de isions to a hieve robust performan e. We on lude by saying that the method for es the designer to understand his model and the intimately related spe i ations on it. The method links the design work with the pra ti al world.

8.5 Con lusion We have reviewed a general framework for ontrol system analysis and synthesis. The stru tured singular value



100

arose from the stability analysis of a

more general type of systems, namely linear fra tional transformations. This permits us to ta kle formally the robust performan e paradigm. To over ome the often reported di ulties in the la k of guidan e in the weighting fun tion sele tion during the design we have provided a simple and illustrative example whi h ontains all ingredients and steps that should be arried out in analyzing su h a design problem. Hopefully, we have su

eeded in larifying that a good design is a matter of balan ing requirements.

We might say that



is

the tool to guide us in nding the required trade-os between performan e and robustness. It pla es the hallenge on the side of the pra ti ing engineer. To be su

essful in improving the behaviour of omplex systems he will have to quantify his spe i ations and he will have to rely ontinuously on a better and deeper system knowledge.

The paradigm is no longer ontroller design,

but spe i ation design.

Knom 14

K2 , with Wp1

12

12

pert1

pert1

10 nom

ideal 10

pert2

q (deg/s)

q (deg/s)

8

nom

ideal 8 pert2 6

6

4 4

2

2

0 0

5

10

0 0

15

5

10

time (s)

K2 , with Wp2

12

15

time (s)

12

K3 , with Wp1 nom, pert1

10

10 pert

1 pert2

8

8 q (deg/s)

q (deg/s)

pert2

6

6

4

4

2

2

0 0

5

10

15

time (s)

0 0

5

10 time (s)

Figure 8.17: Step responses for designed ontrollers

101

15

9.

Nonlinear Dynami Inversion Control

Binh Dang Vu

1

9.1 Introdu tion Among the spe i methodologies for the ontrol of systems des ribed by nonlinear mathemati al models, dynami inversion is ertainly the most widely investigated by ontrol resear hers in the last two de ades. A omplete theory is now available for the design of feedba k ontrol laws whi h render ertain outputs independent of ertain inputs (disturban e de oupling and nonintera ting ontrol) or whi h transform a nonlinear system into an equivalent linear system (feedba k linearization or dynami inversion). The theory of nonintera ting ontrol was initiated by the pioneering work on linear systems by Falb and Wolovi h [75℄. The extension to nonlinear systems is due to the work of Singh and Rugh [212℄, Freund [83℄, following an idea due originally to Porter [196℄.

Feedba k linearization is based on some

early work of Krener [139℄ and Bro kett [34℄ demonstrating that a large lass of nonlinear systems an be exa tly linearized by a ombination of a nonlinear transformation of state oordinates and a nonlinear state feedba k ontrol law. A major breakthrough o

urred at the beginning of the eighties with the appli ation of mathemati al on epts derived from the eld of dierential geometry, Isidori et al. [125℄, Byrnes and Isidori [42℄. A good survey of the theory an be found in re ent books : Isidori [124℄, Nijmeijer and Van Der S haft [184℄, Slotine and Li [216℄. The basi feature of feedba k linearization is the transformation of the original nonlinear ontrol system into a linear and ontrollable system via a nonlinear state spa e hange of oordinates and a nonlinear stati state feedba k

ontrol law. The solution of this problem relies on the nonsingularity of the so- alled de oupling matrix. When this ondition is not satised, a dynami state feedba k ontrol law an be investigated.

Su ient onditions for dy-

nami feedba k linearization have been given by Fliess [80℄ who introdu ed the dierential rank of a system. The dierential rank measures the degree of independen e of the system outputs and an be onsidered as the nonlinear equivalent of the rank of the transfer matrix for a linear system. When the ondition of nonsingularity is satised by the given system (stati feedba k) or by a suitable extension of the given system (dynami feedba k), the feedba k ontrol law an be omputed by solving a set of state independent 1

O e National d'Etudes et de Re her hes Aerospatiales (ONERA), BA701, 13661 Salon

de Proven e Air

102

algebrai linear equations.

This is a result of the stru ture of the dynami s

whi h is assumed to be ane in the ontrols. As the input-output behaviour of the resulting state-feedba k system resembles that of a linear time-invariant system, any linear ontrol design te hnique

an be applied to a hieve the design performan e. However, in order to guarantee the internal stability of the system, it is not su ient to look at input-output stability, sin e all internal unobservable modes of the system must be stable as well. The rst step in analysing the internal stability of the system is to look at the zero dynami s. The zero dynami s of a nonlinear system are the internal dynami s of the system subje t to the onstraint that the output, and therefore all the derivatives of the output, are set to zero for all time. There have been many appli ations of nonintera ting ontrol and feedba k linearization to air raft ight ontrol problems : Asseo [15℄, Singh and S hy [213℄, Meyer and Ci olani [170℄, Dang Vu and Mer ier [51℄, Menon et al. [168℄, Lane and Stengel [149℄, Bugajski et al.

[41℄, Adams et al.

[6℄....

The main

advantage of the feedba k linearization te hnique is that it does not require gain s heduling to ensure ight ontrol system stability over the entire operational envelope of the air raft.

Traditional air raft ontrol designs have to rely on

linearized models obtained throughout the ight envelope of the vehi le, with linear ontrollers synthesized for the set of resulting linearized models.

9.2 Plant Model Requirements and Controller Stru ture 9.2.1 SISO ase The essentials of the approa h are most easily understood in terms of the simple single-input single-output problem. The method of synthesis onsiders a lass of nonlinear systems ane in

ontrol

x_ = f (x) + g(x)u y = h(x) where

n IR

(9.1) (9.2)

f , g are smooth ve tor elds on IR n and h is a smooth fun tion mapping

! IR .

Su h a system is feedba k linearizable of relative degree

r if there exist state

and input transformations

z = (x) u = (x) + (x)v where

z 2 IR r v 2 IR

(9.3) (9.4)

(x) 6= 0 and  is a dieomorphism whi h transforms (9.1) into a on-

trollable linear system

z_ = Az + Bv 103

(9.5)

Indeed, we time-dierentiate (9.2) to obtain

h (f (x) + g(x)u) x

y_ = If the oe ient of

u

(9.6)

is zero, we dierentiate (9.6) and ontinue in this way

until a nonzero oe ient appears. This pro ess an be su

in tly des ribed by introdu ing some onventional notation of dierential geometry.

h with respe t to the ve tor eld f

derivative of the s alar fun tion as

Lf h(x) =

The Lie is dened

h f (x) x

(9.7)

Higher order derivatives may be su

essively dened

Lkf h(x) = Lf (Lfk 1 h(x))

(9.8)

With this notation, (9.6) an be written

y_ = Lf h(x) + Lg h(x)u If

Lg h(x) = 0, then dierentiate (9.9) to obtain y = L2 h(x) + Lg Lf h(x)u f

If

(9.9)

Lg Lfk 1 h(x) = 0 for k = 1; :::; r

ends with

The number

(9.10)

1, but Lg Lrf 1h(x) 6= 0, then the pro ess

y(r) = Lrf h(x) + Lg Lrf 1h(x)u

(9.11)

r is alled the relative degree of (9.1). z 2 IR r zk = k (x) = Lk 1 h(x) k = 1; :::; r

Now if we dene the oordinates

then we get the linear

f

(9.12)

r-dimensional

ompletely ontrollable and observable,

ompanion form system

0

0 B0 B z_ = B B :  : 0 where

1 0 : : 0

0 1 0 : :

: 0 1 : :

1

0

1

0 0 B0C :C C B C B C 0C C z + B : C v = Az + Bv A 0A 1 1 0

(9.13)

v = Lrf h(x) + Lg Lrf 1 h(x)u

(9.14)

Su h a system is alled a Brunovsky anoni al form. Exa t linearization is a hieved when the relative degree is equal to the system order

(r = n).

The ontrol law is obtained by transforming the above linear system state variables and ontrol into the original oordinates, with ontrol

u = (x) + (x)v 104

(9.15)

where

(x) =

Lrf h(x) Lg Lrf 1h(x)

(x) =

1 Lg Lrf 1 h(x)

The ontrol law v is hosen depending on the ontrol task. y is required to be stabilized around zero, we hoose v as r 1 X v=

k zk+1 k=0

(9.16) For instan e, if

(9.17)

in order to a hieve the design performan e for the output dynami whi h is given by

y(r) + r 1 y(r 1) + ::: + 1 y(1) + 0 y = 0

(9.18)

Stabilization of (9.18) annot guarantee stabilization of (9.1). A omplete

hara terization of the stability properties of (9.1) requires a view of the entire state spa e. The key result of Isidori [124℄ is that there exists a transformation of oordinates whi h provides a so- alled normal form for (9.1), from whi h a

omplete stability pi ture an be obtained

x

! (z; ) z 2 IR r  2 IR n

r

z_ = Az + Bv _ = q(z; )

(9.19)

(9.20) (9.21)

The zero dynami s of the system (9.1) are dened by the equation

_ = q(0; )

(9.22)

whi h orresponds to the internal behaviour of the system when the ontrol is

hosen to onstraint the output to be identi ally null. For tra king ontrol problems, for instan e if

hoose

v as

v = yd(r)

r 1 X k=0

y

is required to tra k

k (zk+1 yd(k) )

yd, we (9.23)

in order to a hieve the design performan e for the tra king error

e = y yd

(9.24)

e(r) + r 1 e(r 1) + ::: + 1 e(1) + 0 e = 0

(9.25)

whose dynami is given by

Again the internal behaviour must be bounded. It an be shown that for any

 > 0, there exists Æ so that je(k) (t0 )j < Æ k = 0; :::; r 1 =) je(k) (t)j < 

8t > t0 > 0

k(t0 ) R (t0 )k < Æ =) k(t) R (t)k <  8t > t0 > 0

where

_R = q(zR ; R ) and zR = (yd ; yd(1) ; :::; yd(r 1))T 105

(9.26) (9.27)

9.2.2 MIMO ase The multi-input multi-output ase is qualitatively similar to the single-input single-output ase. Consider a nonlinear dynami al system in the form

x_ = f (x) + g(x)u y = h(x)

(9.28) (9.29)

x 2 IR n , u 2 IR m , y 2 IR m , and f , g and h are smooth fun tions of x. The problem onsists of nding m transformations of oordinates and a ontrol where

law

z i 2 IR ri v 2 IR m

z i = i (x) u = (x) + (x)v where

ri

i = 1; :::; m

(9.30) (9.31)

is the relative degree asso iated to the output

yi ,

whi h transform

(9.28) into an equivalent ontrollable linear system

z_i = Ai z i + Bi v

i = 1; :::; m

(9.32)

from whi h the auxiliary ontrol synthesis is performed. Under the ondition of nonsingularity of the matrix


ij (x) = Lgj Lfri 1 hi (x)

i = 1; :::; m

j = 1; :::; m

(9.33)

the linearizing oordinates are given by

zki = Lfri 1 hi (x)

i = 1; :::; m k = 1; :::; ri

u is obtained from (x) =
1 b

and the ontrol law

with

bi = Lrfi hi (x)

The ontrol law

(x) =
1 i = 1; :::; m

v is hosen depending on the ontrol task. rX i 1 ( r i) vi = ydi

ik (zki +1 yd(ki ) ) k=0

(9.34)

(9.35)

(9.36) For instan e, if

(9.37)

then we obtain a nonintera ting ontrol system whi h performs a de oupled tra king of

yd

by

y,

omponent by omponent. In this ase, the matrix




is

alled the de oupling matrix. The input-output behaviour is dened by the diagonal transfer matrix

H (s) = diag(

1 ) di (s) 106

i = 1; :::; m

(9.38)

with

(9.39) di (s) = i0 + i1 s + ::: + iri 1 sri 1 + sri The stru ture of a simple ontrol system (ri = 1 i = 1; :::; m) is depi ted in Figure 9.1. As the output y is required to tra k the ommanded value yd , we hoose v as v = !(y yd) (9.40) where

! = diag( i0 )

i = 1; :::; m

(9.41)

The ontrol law is then given by

u= (

h h h g(x)) 1 !(y yd ) ( g(x)) 1 f (x) x x x h

(

-1

g)

x yd +

ω

v

(

h x

−

-1

g)

− +

u

h

(9.42)

f

x x

.

x=f(x)+g(x)u

y

h(x) ω=diag(c0i) i=1,...,m Figure 9.1: Controller stru ture

9.3 Possible Design Obje tives and Design Cy le Des ription A straightforward appli ation of the linearization te hniques might result in a system :



with unstable unobservable modes resulting in undesirable losed-loop system behaviour;



with large ontrol eort leading to ontrol saturation.

Preliminary physi al onsiderations are ne essary to obtain a good design. By negle ting ertain physi ally small variables, the approximate linearization might result in better performan e. Singular perturbation theory an also be

107

used to reformulate the original dynami model as two or more lower-order systems whi h are better onditioned for linearization; a ommon example is the time-s ale separation between the translation dynami s of an air raft and its rotational dynami s. Good zero-order dynami behaviour and redu ed ontrol a tivity rely on a good hoi e of the ontrolled variables and their dynami s (ve tor

v in the design).

The overall approa h for the ontrol design is as follows :



Step 1. Reformulate if ne essary the original dynami system to obtain an approximate nonlinear model for whi h a state-dependent nonlinear inverse an be easily onstru ted.



Step 2. Constru t the nonlinear inverse. The losed-loop system formed by the nonlinear inverse and the approximate nonlinear model redu es to a linear time invariant system.



Step 3. Use any suitable linear ontrol design te hnique to synthesize a

ontroller for the above linear system (e.g. eigenstru ture assignment).



Step 4.

Transform the linear system state variables and linear ontrol

into the original oordinates and ontrol.



Step 5. Iterate on linear dynami hara teristi s to obtain required performan e with redu ed ontrol a tivity.



Step 6. Eventually omplete the design by synthesizing a robust ontroller using adequate linear te hniques.

9.4 A Simple Design Example The following example on erns the ontrol problem of a simplied V/STOL air raft model and is taken from Meyer et al. [171℄. A simple air raft is used that has a minimum number of states and inputs, but retains many of the features that are onsidered when designing ontrol laws for a real air raft su h as the Harrier. Figure 9.2 shows the PVTOL (planar verti al takeo and landing) air raft, whi h is the natural restri tion of V/STOL air raft to jetborne operation (e.g. hover) in a verti al plane. The air raft state is simply the

y, z , of the air raft entre of mass, the angle  of the air raft relative y-axis, and the orresponding velo ities, y_ , z_ , _ . The ontrol inputs, u1 ,

position to the

u2 , are the thrust (positive downward) and the rolling moment.

The equations of motion for the PVTOL air raft are given by

where -1 is the

y = u1 sin  + u2 os  z = u1 os  + u2 sin  1  = u2 gravitational a

eleration and  is a

(9.43) (9.44) (9.45) small oe ient giving

the oupling between the rolling moment and the lateral a

eleration of the air raft.

108

Φ

z

y Figure 9.2: The planar verti al takeo and landing air raft

Choosing

y and z as the outputs to be ontrolled, we seek a state feedba k

law of the form

u = (x) + (x)v

(9.46)

r = (r1 ; r2 )T ,

su h that, for some

y(r1) = v1 z (r2) = v2 Here,

(9.47) (9.48)

v is our new input and x is used to denote the entire state of the system.

Pro eeding in the usual way, we dierentiate ea h output until at least one of the inputs appears. This o

urs after dierentiating twi e and is given by



y z



=



0 1



+



sin   os 

os   sin 



u1 u2

 (9.49)

Sin e the de oupling matrix is nonsingular (although almost singular as

), we an linearize the system by hoosing the stati state

its determinant is feedba k law



u1 u2



=



sin  os 

os sin  



109

 

0 1



+



v1 v2

 (9.50)

The resulting system is

y = v1 z = v2 1  = (sin  + v1 os  + v2 sin ) 

(9.51) (9.52) (9.53)

This feedba k law makes the input-output map linear, but has the unfortunate

 unobservable. Constraining the outputs v1 = v2 = 0, the zero dynami s are found to

side-ee t of making the dynami s of and derivatives to zero by setting be

1  = sin  

(9.54)

Equation (9.54) is simply the equation of an undamped pendulum. Nonlinear systems, su h as (9.51)-(9.53), with zero dynami s that are not asymptoti ally stable are alled non-minimum phase. From the above analysis, it is lear that exa t input-output linearization of a system an lead to undesirable results. The sour e of the problem lies in



trying to ontrol modes of the system using inputs that are weakly ( ) oupled rather than ontrolling the system in the way it was designed to be ontrolled



and a

epting a performan e penalty for the parasiti ( ) ee ts. For the simple PVTOL air raft, the linear a

eleration should be ontrolled by ve toring the thrust ve tor (using moments to ontrol this ve toring) and adjusting its magnitude using the throttle. The PVTOL air raft is now modelled as

ym = u1 sin  zm = u1 os  1  = u2

(9.55) (9.56) (9.57)

so that there is no oupling between rolling moments and lateral a

eleration. Dierentiating the model system outputs,



ym zm



=



0 1



+



ym and zm , we get   u1 sin  0

os  0 u2

(9.58)

Now, however, the de oupling matrix is singular whi h implies there is no

u2 enters the system  , we must dierentiate (9.55)-(9.56) at least two more times        sin  u1 os  u1 ym (4) = u1_ 2 sin  2u_ 1_ os  +

os  u1 sin  zm (4) u2 u1 _ 2 os  2u_ 1_ sin 

stati state feedba k that will linearize (9.55)-(9.57). Sin e through

The de oupling matrix is invertible as long as the thrust,

u1

(9.59)

is nonzero.

Physi ally, this result in agreement with the fa t that no amount of rolling will ae t the motion of the air raft if there is no thrust to ee t an a

eleration. Linearizing the above system using the dynami state feedba k law



u1 u2



=

u1 _ 2 2u_ 1 _ u1

!

+



110

sin  os 

os u1

sin  u1



v1 v2

 (9.60)

results in

ym(4) = v1 zm(4) = v2

(9.61) (9.62)

Unlike the previous ase, the linearized model does not ontain any unobservable zero dynami s. Thus, using a stable tra king law for

v, we an tra k

an arbitrary traje tory and guarantee that the model will be stable. Of ourse, the natural question that omes to mind is : will a ontrol law based on the model work well when applied to the true system? If

 is small

enough, then the system will have reasonable properties, su h as stability and bounded tra king. This example shows that preliminary physi al onsiderations are ne essary to obtain a good design. By negle ting ertain variables whi h are physi ally small, the approximate linearization results in better performan e.

9.5 Con lusion Feedba k linearization or dynami inversion has drawn onsiderable attention over the last two de ades and oers a potentially powerful alternative ontrol design methodology. Dynami inversion is an attra tive te hnique as it avoids gain s hedules.

Instead, it uses dynami models and full-state feedba k to

globally linearize dynami s of sele ted ontrolled variables. Simple ontrollers

an then be designed to regulate these variables with desirable losed loop dynami s. Theory of feedba k linearization is still gradually developing. There are limitations and open problems. The main drawba k might be that modes be ome unobservable under the linearization or de oupling onstraints, whi h an be unsurmountable in ase they are unstable. The dimension of the unobservable manifold and the omplexity of the ontrol law an vary drasti ally a

ording to the assumptions made on the model used (e.g. small oupling terms negle ted or not).

Preliminary physi al onsiderations are then ne essary to obtain a

good design. The design method requires, more or less, a

urate knowledge of the state of the system, while no satisfa tory theory for the design of the nonlinear observers is available. A suitable nonlinear analogue of the separation prin iple still needs to be developed. One area of resear h, already initiated, is that of ombining the design te hnique developed so far, with appropriate robust te hniques whi h ould take into a

ount unknown parameters and unmodelled dynami s : LQ,

-synthesis,

ontrol, dierential games.

111

QFT, Lyapunov synthesis, adaptive

10. Robust Inverse Dynami s Estimation

Ewan Muir

1

10.1 Introdu tion Robust Inverse Dynami s Estimation (RIDE) [48℄, [33℄,[176℄ has developed from two other methods: the Salford Singular Perturbation Method [115℄ and Pseudo-Derivative Feedba k [194℄, [14℄. Both of these methods use the same type of multivariable Proportional plus Integral (PI) ontroller stru ture but use a high gain to provide the desired de oupling and losed-loop dynami s. RIDE is a development of both these methods whi h repla es the high gain with an estimate of the inverse dynami s of the air raft with respe t to the

ontrolled outputs. This inverse input gives RIDE strong similarities to Nonlinear Dynami Inversion [218℄ and is similar to the equivalent ontrol found in Variable Stru ture Control [244℄.

10.2 General Stru ture The RIDE ontroller onsists of 3 omponents: a model inverse input, a PI

ontroller and a feedforward, as shown in Fig. 10.1 below.

v _ T-1 +

yc

. v

Kudi

KV

^udi +

+ _

KI

+

. r

+

r

+

_

+

x u

Aircraft

y

KP

Figure 10.1: Stru ture of RIDE ontrol law 1

Defen e Resear h Agen y, Flight Dynami s and Simulation Department, Bedford, MK41

6AE, UK

112

- The model inverse provided by the dynami inverse input,

u^di

, a ts to

de ouple the outputs from ea h other and from the other air raft states by using moment an ellation. The inverse is uniquely for the outputs to be ontrolled and is therefore for a subset of the omplete air raft model only. - Having inverted the air raft model with respe t to the outputs and de oupled these, the PI ontroller then assigns to the outputs, the dynami s desired by the ontrol law designer. The integral a tion, with gain

KI , provides robust-

ness against errors in the estimate of the model inverse. The proportional gain matrix

KP

provides stability and is positioned su h that it provides pseudo-

derivative feedba k. - The feedforward omponent, onsisting of a washout lter on the demands and des ribed by equation 10.1, is used to tune the step response hara teristi s to give an appropriate onset of response. feedforward input = where

T

[(T s + I ) 1 KV s℄y

is a diagonal matrix of washout time onstants and

(10.1)

KV

is a matrix of

gains on the feedforward inputs. The stru ture provides de oupling between the outputs and assigns them a se ond order response whi h is spe ied by the designer. The transient response shape an be adjusted using the feedforward. The design method provides simple me hanisms for handling motivator position and rate limiting and it is anti ipated that motivator redundan y an be handled separately from the ontroller design.

10.3 Closed-loop System Chara teristi s and Gain Cal ulation (Output Feedba k Case) The poles of the losed-loop system with a RIDE ontrol law are determined by the following: - the open loop system transmission zeros, - the feedforward washout time onstants given in matrix

2I

- the eigenvalues of the matrix [s

+

CBKP s + CBKI ℄.

T;

As the rst set of poles oin ides with the transmission zeros of the open loop system, it is ne essary to ensure that the feedba k measurements sele ted give stable transmission zero lo ations. The se ond set is spe ied by the designer who sele ts the feedforward time onstants. The third set of eigenvalues an be assigned to the poles of a standard se ond order transfer fun tion of the form given in equation 10.2, through appropriate gain sele tion.

y = (s2 I + 2Zd n s + 2n ) 1 2n y

The proportional, integral and feedforward gain matri es,

(10.2)

KP , KI

and

KV

respe tively, are al ulated from the inverse of the motivator ee tiveness ma-

CB ) 1 , the matrix of desired losed system damping, Zd, natural frequen y, n , and feedforward gain, M , where Zd , n and M are diagonal matrix, (

tri es spe ied by the ontrol law designer.

113

KP = (CB ) 1 2Zd n

(10.3)

KI = (CB ) 1 2n

(10.4)

KV = (CB ) 1 M

(10.5)

For the output feedba k ase and using the gains al ulated in equations 10.3 to 10.5 above, the output equation for the losed-loop system is des ribed by equation 10.6.

y = (s2 I +2Zd n s + 2n) 1 [ 2n y + sM (T s + I ) 1y + s(CB )(^udi udi)℄ As

Zd, n , T

and

(10.6)

M are all diagonal matri es, ea h demand y will ae t y. Therefore, for the output feedba k ase, RIDE will

only one of the outputs

provide tra king of the demands with unity steady-state gain, the dynami s of the response an be spe ied and will be se ond order.

omponent, whose dynami s are spe ied by the matri es

T

The feedforward and

M , will shape

the initial response to any inputs. Any errors in the estimate of the dynami inverse input will be orre ted by the integral loop. happens will be dependent on the integral gain The role of the dynami inverse input, estimate,

u^di , is al ulated from

udi ,

KI .

The rate at whi h this

is to keep

y_ = 0

and thus an

u^di = (CB ) 1 CAx (10.7) 1 where the matrix ( (CB ) CA) is represented by the gain matrix Kudi in Fig. 1. Note that the state ve tor x need only ontain the rigid body states whi h dire tly ae t the outputs y .

10.4 Design Limitations RIDE does not take into a

ount expli itly any motivator or sensor dynami s during the design phase.

In many ases, the dynami s of the a tuators and

sensors will be su iently fast to maintain the desired performan e. Should this not be the ase, a areful hoi e of the design parameters will re tify the problem.

For example, the spe ied bandwidth of the losed-loop response

ould be redu ed and the feedforward used to maintain an adequate speed of response.

10.5 Controller Synthesis Aspe ts The simpli ity of RIDE naturally means that it does not provide the omprehensive solution promised by other more omplex methods.

114

RIDE does not

provide expli it guarantees in terms of either stability or performan e robustness. It is also limited in terms of the amount of spe i ation data whi h an be in orporated dire tly in the design stage.

Therefore separate analysis is

required on e the initial design has been done, to see if the ontroller meets the spe i ation. However, in pra ti e, RIDE has been found to produ e ontrollers with a

eptable time responses, even when performing highly dynami manoeuvres with non-linear air raft models [176℄, and it is possible for designers to a hieve satisfa tory gain and phase margins.

Also the integral a tion

provides robustness to errors in the dynami inverse input estimate. The simpli ity of RIDE, both in terms of the underlying mathemati s and the design pro ess, means that the learning urve is short and undemanding. Also, no spe ialist skills, design software or omputer hardware are required and the resulting ontroller is simple with a lear stru ture. A fuller understanding of the ontroller synthesis pro edure and of the design method an be obtained from the RIDE-HIRM ontrol law in hapter 33.

115

11.

A Model Following Control

Approa h

Holger Duda1 , Gerhard Bouwer1 , J.-Mi hael Baus hat1 and Klaus-Uwe Hahn 1

11.1 Introdu tion Design and development aspe ts for state of the art ontrol systems are based today on improved system models and omputer supported tools. One of the design aims for a ontrol system is a low feedba k authority.

High feedba k

gains, espe ially in multi input/multi output systems with un ertainties, may lead to stability problems, whi h are often di ult to predi t. A well-known re ipe to avoid this is:

Put all available information about the pro ess to be ontrolled into the feedforward bran h of your ontrol system. In view of ight ontrol system design it an be assumed that a detailed database of the air raft to be ontrolled is available, be ause it is usual to improve aerodynami databases of new air raft during ight testing using modern system identi ation te hniques [109℄. Therefore, it is highly re ommended to use this knowledge in the ight ontrol system design pro ess [108℄. One of the most promising approa hes, whi h takes the above mentioned aspe ts into a

ount, is the Model Following Control (MFC) te hnique. Even in the early stages of ight ontrol system resear h MFC on epts seemed to be promising [243℄. An improved theoreti al understanding of the identi ation of system dynami s promotes the appli ation of MFC systems [107℄. The design problem for the MFC on ept an be separated into three independent subtasks: First a ommand blo k has to be dened, whi h in ludes the desired dynami behaviour taking into a

ount the plant's performan e limits. Then a ontroller omplex onsisting of a feedforward and a feedba k ontroller has to be designed, whi h is independent of any ommand blo k hara teristi s, su h as manual ight ontrol laws or autopilot fun tions. The expression  ommand blo k has been introdu ed instead of model in order to avoid misunderstandings on erning the plant model. It predominantly denes the dynami behaviour of the overall MFC system.

The feedforward

ontroller ontains an inverse model of the plant to be ontrolled. Assuming no external disturban es, a perfe t knowledge and an ideal inversion of the plant, 1

DLR German Aerospa e Resear h Establishment, Institute of Flight Me hani s, D-38108

Brauns hweig

116

the omplete ontrol ould be performed by the feedforward ontroller without any feedba k a tivity.

In pra ti e, a feedba k ontroller is required, whi h

only has to manage the remaining ontrol part not overed by the feedforward bran h, whi h will always perform the majority of ontrol a tivity. Sin e the aim of the RCAM Design Challenge is to evaluate ontrol theories

on erning robust ight ontrol system design, one has to answer the rising question:

How an the MFC on ept ontaining high authority feedforward

ontrol enhan e robustness, whi h is mainly ae ted by feedba k

ontrol? The feedforward part of the MFC represents a kind of partly inherent robustness ompared with a pure feedba k system.

By an exa t denition of the

desired performan e and the limitations of the pro ess one omes to oni tfree ontrol a tions and, therefore, to minimum feedba k ontrol a tivity for manoeuvres. This leaves maximum authority to the feedba k ontroller to ope with un ertainties and disturban es reje tion. Besides these robustness aspe ts the MFC on ept provides several additional benets regarding pra ti al appli ations:



The ommand blo k denes predominantly the input ommand behaviour of the overall MFC system.

Therefore, tailored Flying Qualities or au-

topilot fun tions an be easily realised.



The feedforward and feedba k ontrollers are independent from the layout of the ommand blo k.

This separation allows a lear sharing of

responsibilities for the design tasks with well dened interfa es, whi h

an be performed by dierent teams. Therefore, design problems whi h may be observed during simulator or ight testing an be easily lo ated and solved.



The overall ontroller stru ture allows the denition of ertain ommand blo k modules for spe ial tasks, su h as manual ight ontrol laws or autopilot fun tions for an air raft family. A re-design for other (similar) air raft does not have to go through all the individual steps, but only the feedforward and feedba k ontrollers have to be adapted. The attainable

ommonality of ying hara teristi s for an entire ategory of air raft type is a protable element onsidering pilot training and erti ation aspe ts.

11.2 Typi al Appli ations MFC on epts have been utilised in ight systems sin e the sixties [243℄. The main appli ation of the MFC approa h is in-ight simulation.

The aim of

in-ight simulation is to impose the hara teristi s of a ight vehi le to be simulated on airborne simulators, su h as Calspan's TIFS (Total In-Flight Simu-

lator) [175℄, DLR's ATTAS (Advan ed Te hnologies Testing Air raft System)

117

and ATTHeS (Advan ed Te hnologies Testing Heli opter System) [40℄.

Fur-

thermore, MFC on epts have been realised in several experimental heli opter programs in the United States and were even hosen for the new operational Fly-by-Wire heli opter Coman he [93℄. The appli ation potential of MFC systems is demonstrated below by re ent resear h programs arried out at DLR Institute of Flight Me hani s.

They

have been hosen be ause all have been ight-demonstrated in a real-time and real-world environment.

11.2.1 In-ight simulation Various in-ight simulations have been arried out in the xed-wing and heli opter area, su h as the Airbus A3XX air raft, the Indonesian N250 air raft, the HERMES Spa eplane, and the Lynx heli opter. One of the most re ent appli ations of ATTAS has been the airborne simulation of an Airbus A3XX-type transport air raft.

The airframe model is

based on preliminary data of the unaugmented air raft without elasti modes. A typi al ight-test result is illustrated in gure 11.1.

Manually own turn

reversals learly show the ex itation of the low damped dut h roll of the implemented model. The time histories demonstrate a good mat hing between the A3XX model states (solid lines) and the measured ATTAS states (dashed lines) validating the model following me hanism.

Roll Command of the Pilot (deg) 10 0 -10 Bank Angle (deg) 20 0 -30 Angle of Sideslip (deg) 5 0

A3XX ATTAS

-6 0

100

200

300

Time (sec)

400

Figure 11.1: A3XX In-Flight Simulation (Flight-Test Results) Espe ially in the heli opter area the MFC on ept has been proven to be a very valuable tool due to the highly ompli ated ouplings of basi heli opter dynami s [104℄. The in-ight simulation of the Lynx heli opter shall serve as an example [31℄. This heli opter has ouplings opposite to the orresponding

ouplings of ATTHeS in its basi BO 105 mode. Flight tests have been arried out demonstrating, that all ATTHeS states mat h well the ommanded Lynx

118

model states. In general, the in-ight simulation was deemed by the pilots to be representative for the Lynx heli opter.

11.2.2 Flight ontrol system resear h Flight ontrol system resear h proje ts based on MFC on epts have been performed, whi h are summarised below:



LADICO (Lateral/Dire tional Control of an Air raft): This proje t was arried out within the framework of the GARTEUR A tion Group FM (AG 06) Low-Speed Lateral/Dire tional Handling Quali-

ties Design Guidelines. A lateral/dire tional ontrol system for a transport air raft was developed, featuring an MFC on ept [38℄.

Piloted

evaluations of the system in two ground based simulators demonstrated its performan e; all evaluation pilots rated the system as Level 1.



ARCORE (Arti ial Redundan y Con ept for Re onguration): Flight ontrol system re onguration on epts have been developed and ight tested [21℄. The investigated failure was an elevator stu k in the trim position, whi h has been ompensated by the re onguration ontroller using the stabiliser with its poor dynami s for altitude ontrol instead of the stu k elevator.



SCARLET (Saturated Command and Rate Limited Elevator time delay): Air raft-Pilot Coupling (A-PC) problems due to rate saturation have been investigated and ight tested.

An alternative ontrol s heme has

been developed, whi h ompensates for the additional time delay due to rate saturation. The ight test results were very promising [39℄.



ADS-33D riteria (Aeronauti al Design Standard): Flying Qualities Databases for modern Fly-by-Wire heli opters have been developed on ATTHeS [31℄.

11.3 Plant Model Requirements The MFC approa h is well tuned for the usually available pro ess knowledge in the ight ontrol area. Ideally, there should exist a nonlinear pro ess model within the whole ight envelope in luding a tuator systems, sensor systems, engine, time delays, elasti ity, et .:

x_ (t) = f (x(t); u(t)):

(11.1)

But the method also works, if there is only a linear state model of the rigid air raft for one referen e point

x_ (t) = A x(t) + B u(t); as it has been demonstrated in ight tests (se tion 11.2).

119

(11.2)

11.4 Controller Stru ture Ea h MFC system ontains the main three elements ommand blo k, feedfor-

ward and feedba k ontrollers, gure 11.2. The ommand blo k ontains the equations to ompute a sele ted state ve tor

x

and its time derivative

x_

depending on the input signals.

The feedforward ontroller omputes the ontrol inputs whi h are required for model following. It in ludes an inverse model of the plant. The use of the state derivative

x_

together with

x

generated by the ommand blo k (whi h

ontains dynami s) allows the use of pure stati gain matri es in the feedforward

ontroller [35℄. In pra ti e, no perfe t inversion of the plant to be ontrolled an be provided, therefore, the feedba k ontroller has to ope with these un ertainties and disturban e reje tion.

Disturbances . xC Command Block

Feedforward Controller

uFF

Plant to be controlled

+ uFB

xC

Feedback Controller

x

Figure 11.2: General layout of a MFC system

11.5 Possible Design Obje tives The main design obje tive for the omplete MFC system is to full the design requirements.

As already stated, the main three elements ommand blo k,

feedforward and feedba k ontrollers an be designed separately. The design obje tives for these subtasks an be summarised as follows: The ommand blo k must ontain the desired dynami behaviour of the overall MFC system regarding ontrol inputs. Nonlinear ee ts like a tuator rate and dee tion limitations have to be taken into a

ount within the ommand blo k. The feedforward ontroller has to provide an optimum inversion of the plant to be ontrolled. The feedba k ontroller has to ensure rapid and smoothly de aying error dynami s in the presen e of unknown external disturban es and model un ertainties in order to maintain the quality of model following. Nonlinear ee ts (a tuator rate and dee tion limitations) have to be onsidered for its design.

120

11.6 Design Cy le Des ription The design y le for the MFC approa h is separated into the subtasks for the

ommand blo k, the feedforward, and the feedba k ontrollers.

11.6.1 Command blo k A pra ti al and simple way to dene the ommand blo k is to use models of proven systems, su h as an air raft model with Level 1 manual ight ontrol laws (Fly-by-Wire) or optimum autopilot fun tions.

It is obvious that any

ommand hara teristi s implemented in the ommand blo k are limited by the dynami s of the plant to be ontrolled, mainly be ause of the nonlinear

onstraints of the a tuators. The main onstraint to be onsidered is that the dynami s of the air raft model implemented in the ommand blo k are not faster than those of the plant.

11.6.2 Feedforward ontroller For the design of the feedforward ontroller an inverse model of the plant is required. Assuming that the plant model an be represented by a linear state spa e system (equation (11.2)) the following feedforward ontrol law an be applied [35℄:

uF F (t) = B 1 [x_ C (t) A xC (t)℄ :

(11.3)

This equation indi ates that the inversion does not in lude dynami elements (whi h means zero order) if the state derivative

x_ together with x generated by

the ommand blo k are available. For this pro edure the ontrol input matrix of the plant

B

has to be inverted.

This leads to the fundamental problem,

that dierential equations des ribing typi al dynami systems to be ontrolled (air raft, heli opters, industrial robots, et .) often annot be inverted. In most of these ases the number of ontrol inputs is smaller than the number of states, therefore,

B is a non-square matrix.

One approa h to handle this problem is the appli ation of the Pseudo-

Inverse





By = BT Q B BT Q ; whi h strongly depends on the weighting matrix Q.

(11.4) Therefore, an alternative

method is used at DLR Institute of Flight Me hani s:

x is of the order n and the input ve tor u m (with n > m) equation (11.2) an be written as:

Assuming that the state ve tor is of the order









be sele ted properly providing that In this ase

B1







x1 + B1 u (11.5) B2 x2 has the order m. Its elements should the subsystem of x1 is fully ontrollable.

x_ 1 = A11 A12 A21 A22 x_ 2 The state ve tor to be ontrolled x1

is a square matrix and invertible.

121

In order to de ouple

x1 from x2 , the feedforward ontrol law equation (11.3) x2  x2C :

is extended to a de oupling term, with

uF F = B1 1 (x_ 1C |

{z

A11 x1C ) B1 1 A12 x2C : } | {z }

Inversion

(11.6)

De oupling

Dening the ontrol matrixes

M1 = B1 1 ; M2 = M1 A11 ; M3 = M1 A12 ;

(11.7)

equation (11.6) an be simplied to:

uF F = M1 x_ 1C + M2 x1C + M3 x2C :

(11.8)

This pure linear approa h an be extended to nonlinear elements, if required [22℄.

11.6.3 Feedba k Controller The feedba k ontroller has to ompensate for model un ertainties and disturban e reje tion, while the feedforward ontroller performs the majority of the ontrol a tivity. The main requirement for the feedba k ontroller an be dened by:

e(t) = xC (t) x(t)

!

Min.

:

(11.9)

Dierent methods an be utilised to design the feedba k ontroller, su h as ve tor performan e optimisation [113℄ or robust ontrol system design methods. It has been shown that for air raft appli ations a feedba k ontroller using all signi ant states with proportional and integral terms is su ient. Its stru ture is illustrated in gure 11.3.

uFB + 1/s

Command Block states xC

KP

KI

Plant states x

e -

Figure 11.3: Stru ture of the feedba k ontroller The feedba k ontroller is dened by:

uF B (t) = KP e(t) + KI 122

Z

e(t)dt:

(11.10)

The gains of the feedba k ontroller an be optimised independently from the layout of the feedforward ontroller and the ommand blo k. A proven pro edure used at DLR is based on a numeri al optimisation algorithm [126℄. A ve tor ost fun tion allows the formulation of ea h design obje tive separately and its ombination with individual weighting fa tors forman e and

l

for the ontroller a tivity.

k for the ontroller per-

The formal stru ture of the ost

fun tion is given by:

n X

m

Z te

Z t

e X u2l (t) dt + ::: e2k (t) dt + l J = k 0 0 l=1 k=1

(11.11)

This ost fun tion has to be tailored to the a tual design problem. For air raft appli ations mostly a number of about ten gains to be optimised is su ient. However, for highly elasti air raft this number may in rease.

11.7 A Simple Design Example In order to demonstrate the design pro ess a very simple example is dened, gure 11.4: The plant represents a simplied air raft model ontaining only the short period mode, whi h is represented by the following linear model with

  0:2):

a very poor damping ratio (



u





q_ = 0:24 w_ 80:6 . qc Command qc block

0:016 0:67



M1 M2



q + w

+ δt +





2:4 Æ : 6:5 t

(11.12)

q Plant

w

M3 Figure 11.4: Blo k diagram of the design example The ommand blo k in luding the model to be followed is dened as a rst order system, whi h provides a ommand for the pit h rate

q :

q_ = q u: (11.13) For the MFC design the matri es M1 to M3 have to be determined a

ording to equation (11.7). In this ase x1 = q and x2 = w is sele ted. In order to verify this design, the Bode plot from the ommand blo k output

q_C to the plant output q is al ulated, gure 11.5a.

The pure integral behaviour

demonstrates that the inversion works orre tly. The step responses larify the poor damping of the unaugmented plant and the realised rst order behaviour of the MFC system, gure 11.5b.

123

In this linear example, any desired dynami behaviour an be implemented in the ommand blo k, su h as an air raft model with Level 1 Flying Qualities. Under real onditions the nonlinear ee ts of the plant, su h as rate and dee tion limits should additionally be onsidered in the ommand blo k.

. a) Bode Plot of frequency response q/qc Amplitude (dB)

20 0 -20

Phase (deg)

-90

1

0.1

Frequency (rad/sec)

10

b) Step Responses 0 MFC system response q/u

-1 -2 pure aircraft system response q/δ t

-3 0

5

10

15 Time (sec) 20

Figure 11.5: Results in the frequen y and time domain of the design example

124

12.

Predi tive Control

Jan M. Ma iejowski1 and Mihai Huzmezan

1

12.1 Introdu tion Predi tive Control is very dierent from other ontrol te hniques, in that in its most powerful form it requires the on-line solution of a onstrained optimization problem. This makes it an unlikely andidate for ight ontrol. On the other hand, it oers some unique benets, sin e it expli itly allows for hard onstraints, and it an anti ipate pilot ommands if the ight traje tory is known in advan e. This makes it interesting for ight ontrol, parti ularly if higher-level ontrol fun tionality is onsidered. In this tutorial hapter we present the models used by predi tive ontrol, the ontrol problem formulation, dis uss solutions te hniques and ontroller properties, and omment on the problem of tuning the predi tive ontrol problem formulation so as to meet design spe i ations.

12.2 General Chara teristi s Predi tive Control, also known by several other names, su h as Model-Based Predi tive Control (MBPC), Re eding Horizon Control (RHC), Generalised Predi tive Control (GPC), Dynami Matrix Control (DMC), Sequential OpenLoop Optimizing ontrol (SOLO) et , is distinguished from other ontrol methodologies by the following three key ideas:



An expli it `internal model' is used to obtain predi tions of system behaviour over some future time interval, assuming some traje tory of ontrol variables.



The ontrol variable traje tory is hosen by optimizing some aspe t of system behaviour over this interval.



Only an initial segment of the optimized ontrol traje tory is implemented; the whole y le of predi tion and optimization is repeated, typi ally over an interval of the same length. The ne essary omputations are performed on-line.

The optimization problem solved an in lude onstraints, whi h an be used to represent equipment limits su h as slew rates and limited authority

ontrol surfa es, and operating/safety limits su h as limits on roll angle, des ent 1

Cambridge University Engineering Dept, Cambridge CB2 1PZ, England

125

rate, et . Predi tive ontrol has hitherto been applied mostly in the pro ess industries, where the expli it spe i ation of onstraints allows operation loser to onstraints than standard ontrollers would permit, and hen e operation at more protable onditions. The drawba k of this approa h for ight ontrol is of ourse the on-line omputational requirement. But this is a temporary problem, whi h will disappear within a few years as omputing te hnology advan es. If the internal model is linear, the onstraints are linear inequalities, and the performan e riterion being optimized is quadrati , then the optimization problem to be solved online is a onvex quadrati program, whi h is a relatively good situation. (See below for more details.) Most a ademi publi ations on predi tive ontrol deal with un onstrained problems. See [174, 27, 222℄ for some good examples. The usual formulations then be ome losely related to, or even variants of, the LQ te hnique treated in

hapter 5. In this ase ontrollers an often be pre omputed o-line, but mu h of the advantage of the predi tive ontrol formulation is lost. In this hapter we assume that onstraints are an essential part of the problem. The problem with onstraints is treated in some detail in [197℄.

12.3 System Models All ontrol methodologies assume some model of the system being ontrolled. An unusual feature of predi tive ontrol is that an expli it internal model is required as part of the ontroller; this internal model may be of the same kind as the assumed behaviour of the real system, but need not be. For the purposes of analysing behaviour of the omplete ontrolled system it is usual to assume the same kind of model, though not ne essarily with the same parameters. (So the situation is similar to that in Robust Control theory, for instan e  see

hapters 7,8.) The internal model is required in order to generate predi tions of future system behaviour, if a parti ular set of future ontrol a tions is assumed. For this purpose a nonlinear model ould be used, and there have been some studies of using neural network and other nonlinear models with predi tive ontrol. Su h models lead to non- onvex optimization problems, however, and to ontrol s hemes for whi h no analysis is possible. They have not yet been proven to be useful or ne essary in pra ti e [198℄. We will therefore assume that the internal model is linear. Mu h of the predi tive ontrol literature assumes that a linear time-invariant model is available in the form of a (multivariable) step or impulse response, and that predi tions are generated by onvolution: suppose that the multivariable

fgi : i = 0; 1; : : :g, that the ( ontrol) input u(k) and that the (to be ontrolled) output ve tor at time k is y(k). Also let
u(k) = u(k) u(k 1) be the hange in the input at time

step response sequen e is given by ve tor at time

k

is

126

k.

Then the output is given by

k X

y(k) =

i=

1

gk i
u(i) + d(k)

(12.1)

where it has to be assumed that the open-loop system is asymptoti ally stable for this to be valid, and

d(k)

is assumed to be a disturban e a ting on the

output. In this ase predi tions of the output are omputed by

y^(k + j ) = where

N

k+j X i=k+j N

gk+j i
u(i) + d^(k + j )

is a relatively large integer, and

d^(k)

is some estimate of

(12.2)

d(k + j ).

Frequently the disturban e is estimated as

d^(k) = y(k) y^(k)

(12.3)

and it is assumed that future disturban es are the same as the urrent one:

d^(k + j ) = d^(k):

(12.4)

The onvolution model is an ine ient one, sin e the same model an be represented mu h more ompa tly in either transfer fun tion or state-spa e form. Furthermore, representing the system by a model of this kind removes the restri tion to stable models.

The Generalised Predi tive Control (GPC)

form of predi tive ontrol uses a multivariable transfer fun tion form of model:

A(z 1 )y(k) = B (z 1 )u(k) + n(k): z 1

(12.5)

zA(z 1 ), B (z 1 ) are matri es of polynomials in this 1 )℄ 1 B (z 1 ) is the transfer fun tion matrix from the operator  so that [A(z input ve tor u to the output ve tor y . Although it is not ne essary to asso iate in whi h

is the one-step time delay operator (or the inverse of the

transform variable), and

ea h kind of system model with a spe i disturban e model, it is ommonly assumed [47℄ that the disturban e

n(k) in

this model is generated by passing

white noise through a lter whi h in ludes an integrator:

n(k) =

C (z 1 ) e(k) z 1

(12.6)

Inserting an integrator here leads to integral a tion in the ontroller, whi h is also obtained with the use of onvolution models by the assumption of onstant future disturban es. Generating a set of predi tions now involves solving a set of matrix Diophantine equations, but reasonable approximations an be obtained by using simpler pro edures [47℄.

127

The linear model an also be represented in state-spa e form:

x(k + 1) = Ax(k) + Bu(k) + w(k) y(k) = Cx(x) + Du(k) + v(k) where

x(k) is the state ve tor and w(k), v(k) are disturban es.

(12.7) (12.8) For ight on-

trol this model is usually the most appropriate, sin e linearised air raft models are available in this form, with the state variables representing physi ally meaningful quantities. If the disturban es are assumed to be sto hasti then predi tions of the states and outputs an be obtained by using a Kalman lter [151℄. If other assumptions are made then some other observer needs to be used to generate predi tions. To represent sto hasti disturban es with parti ular spe tra, the state of the model has to be augmented by arti ial states in order to use a Kalman lter, in just the same way as is done for LQG design [159℄. Integral a tion in the ontroller an be obtained by in luding integrators in the augmented model.

12.4 Problem Formulations Predi tive ontrol works by hoosing ontrol a tions to minimise some ost fun tion, su h as

J (k) =

N2 X i=N1

jjM x^(k + ijk) r(k + i)jj2Q

+

Nu X i=1

jj
u(k + i)jj2R

(12.9)

subje t to onstraints su h as

j
uj (k + i)j  Vj juj (k + i)j  Uj j(M x^)j (k + ijk)j  Xj where

x^(k + ijk)

is the predi tion of

matrix (for example,

M =C

x(k + i)

(12.11) (12.12)

k, M is some J (k)), and r(k) integers N1 , N2 and

made at time

if only outputs are to appear in

is some referen e (desired) traje tory for

Nu ,

(12.10)

as well as the weighting matri es

Q

Mx(k). and R,

The

are in prin iple hosen to

represent some real performan e obje tives (su h as prot maximisation in a pro ess appli ation [197℄), but in pra ti e they are tuning parameters for the

ontroller. It is assumed that the ontrol signals are onstant after the end of the optimisation horizon, namely that


u(k + i) = 0 for i > Nu .

The inequalities an be used to represent a tuator rate limits (12.10), a tuator authority limits (12.11), and operating/safety limits (12.12). In these inequalities

Uj , Xj

uj (k) denotes the j 'th omponent of the ve tor u(k), et , and Vj ,

are problem-dependent positive values.

The referen e traje tory

r(k) an either be the real pilot ommand ve tor

(set-point), or an be generated by passing the pilot ommand through some lter. In the latter ase the lter design is another tuning parameter. One of

128

the strengths of predi tive ontrol is that if future ommands are known  for example before the start of a turn or other manoeuvre  then these an be anti ipated by the ontroller, leading to smoother manoeuvres, fuel savings, et .


u(k) in the ontrol signals, u(k) themselves, sin e the required steady-state values of u(k ) are not known in advan e. Penalising non-zero u(k ) would `drag' The ost fun tion penalises non-zero hanges

rather than the ontrol signals

the ontrol signals away from the required steady-state values, thus preventing integral a tion, for instan e. The situation is summarised graphi ally in gure 12.1.

PAST

SET POINT

FUTURE

REFERENCE

PREDICTED OUTPUT

r(k+l)

y(k)=r(k) u(k+l) MANIPULATED INPUT

k-n

k-2 k-1 k k+1

k+l

CONSTANT INPUT

Nu

N1

N2

CONTROL HORIZON - Nu MINIMUM OUTPUT HORIZON - N1 MAXIMUM OUTPUT HORIZON - N2

Figure 12.1: Re eding Horizon Strategy

As was said earlier, ombining a quadrati ost su h as

J (k)

with linear

inequalities and a linear model leads to a Quadrati Programming (QP) prob-

 Let the solution of this problem be fu (k + i) : i = 1; : : : ; Nu g. Then  u (k + 1) is applied to the system being ontrolled, and the problem is solved again at time k + 1. (In general one an apply a longer initial segment, and lem.

re-solve the problem at longer intervals.) Other non-quadrati osts are also possible. For instan e, min-max osts are sometimes advo ated in order to obtain robust ontrol, while using absolute values or peak values instead of quadrati fun tions allows the use of Linear Programming, whi h redu es the on-line solution time [197, 8℄.

129

12.5 Solution Te hniques The basi solution te hnique for the onstrained predi tive ontrol problem is to use a standard QP solver (or LP solver if the ost fun tion is appropriate). It is important to appre iate that a solution of a QP problem is required online, and that this problem has to be solved in real time. In pro ess ontrol, where update rates are very low, this is not a big limitation with urrent omputing te hnology. (For example, large multivariable problems with a few tens of inputs, outputs and onstraints, take a few se onds to solve on 486-type

omputers.) But it learly is a problem for ight ontrol, for whi h a speed-up of something like

103 is required.

There are several possible alternatives to the use of standard QP solvers, whi h do not seem to have been investigated thoroughly for predi tive ontrol. The rst is obtained by noting that if there are no onstraints, or if none of the onstraints is a tive, then the solution an be obtained analyti ally, as the solution of a linear equation. (For details, see any of the referen es mentioned previously.)

The problem is that one does not know, before omputing the

solution, whether any onstraints are going to be a tive or not. Se ondly, it is also true that, if one knew exa tly whi h onstraints were a tive, then one

ould again obtain the solution analyti ally. So if one knows that the set of a tive onstraints at time

k is the same as that at time k 1, then one an nd

the solution very qui kly. Furthermore, it will often be the ase that the set of a tive onstraints an only hange in very limited ways from one step to the next; it is then feasible to obtain a small set of analyti solutions qui kly, and

he k whi h one is the a tual solution.

These approa hes exploit knowledge

and understanding of the parti ular optimisation problem being solved  ie ight ontrol of a parti ular air raft  whi h a general QP solver annot do. Another possibility, again not open to a general solver, is to guess that the solution at time

k

will be very similar to that at time

k

1,

and hen e

use that as an initial trial solution. This strategy has been employed in [119℄, for example.

Of ourse su h a strategy will o

asionally go wrong, when a

onstraint is approa hed losely, and a ba k-up pro edure is required for su h o

asions. One problem with standard QP solvers is that they give up if the optimisation problem being solved is infeasible, a situation whi h should not o

ur with proper spe i ation, but nevertheless might.

(Typi ally infeasibilities o

ur

`inside' the algorithm only, and are due to apparently unavoidable onstraint violations some time in the future; the feedba k a tion of the ontroller usually restores feasibility before the problem is en ountered by the system.) In [137℄ the use of Lawson's weighted least-squares algorithm is advo ated, in whi h the weight is iteratively adjusted to emphasise the most-violated onstraint. This algorithm solves the QP problem if it an, and gives a `reasonable' solution if the problem is infeasible.

130

12.6 Controller Properties When a linear model and quadrati ost is used, the resulting ontroller is linear time-invariant providing that either no onstraints are a tive, or that a xed set of onstraints is a tive. (For ea h su h set, a dierent linear ontrol law results.) Thus the ontrol law an be a linear law for long periods of time. However, when hard onstraints are approa hed the ontroller an behave in a very nonlinear way. In parti ular, it may rea t mildly to a disturban e whi h drives the system away from onstraints, but very sharply to a disturban e of similar magnitude but in the opposite dire tion, whi h drives the system towards onstraints. The ontroller stru ture is very dierent from more onventional ontrollers. It onsists of a predi tor, whi h an be ompared with onventional ontrollers, for example by omparing omplexity as measured by the number of state variables, and an optimiser, whi h annot be ompared in that way. Figure 12.2 shows the stru ture of a predi tive ontroller. Clearly a predi tive ontroller is more omplex, in terms of behaviour, in terms of algorithm stru ture, and in terms of omputation y le time, than a onventional ontroller. Veri ation and erti ation is a mu h more formidable task than for a onventional

ontroller.

REFERENCE

OPTIMISER

COMMAND

PLANT

OUTPUT

& PREDICTOR OBSERVER

using

INTERNAL MODEL

STATE ESTIMATE

Figure 12.2: Stru ture of a Predi tive Controller

It is easy to formulate the predi tive ontrol problem in su h a way that the

ontroller displays (multivariable) integral a tion, and reje ts onstant output disturban es.

This is a hieved by the ombination of a suitable disturban e

model and penalisation of non-zero non-zero

u(k).


u(k) in the ost fun tion rather than of

It is not lear, however, how to obtain double-integral (`type 2')

a tion if it is required. An appropriate disturban e model would be required, but it would also seem ne essary to penalise instead of


u(k) in the ost fun tion.


2 u(k) =
u(k)


u(k

1)

This means that steadily-in reasing on-

trol a tions ould result, whi h would not be a

eptable in most appli ations. Reje tion of persistent but bounded-amplitude output disturban es, su h as sinusoids, is easily a hieved by in luding a model of the disturban e (in a

ordan e with the `Internal Model Prin iple') and penalising

131


u(k).

12.7 Design Spe i ations The problem of translating ontrol system design spe i ations into spe i

N1 , N2 , Nu ), weighting maQ, R), predi tor, and possibly a referen e-generating lter, is a di ult and is still a subje t of urrent resear h. Choosing Q, R, and the pre-

hoi es of predi tion and optimisation horizons ( tri es ( one

di tor is losely related to the hoi e of weighting and ovarian e matri es in LQG ontrol; there again the relationship between these parameters and the design spe i ation is very indire t, but experien e gained over many years has led to some rules of thumb, at least. The problem is made onsiderably more

ompli ated by also having to hoose horizons [222, 173, 119℄. If it is assumed that tight ommand-following is attained by the ontroller, then the hoi e of referen e-generating lter approximately denes the behaviour in response to ommands  a kind of model-referen e approa h  at least for the ase of ina tive onstraints. However, the assumption of tight model-following may not be realisti . Time-domain ommand-following spe i ations are, in prin iple, easily a hieved by formulating appropriate inequality onstraints. For example, restri tions on overshoot or rise-time during step responses may be formulated as inequality

onstraints. In pra ti e, however, there are problems if too many onstraints are added, sin e the solution time in reases.

One should, however, be wary

of taking responses to parti ular ommands su h as steps to be representative of behaviour in response to other ommands, sin e the predi tive ontroller is nonlinear (if onstraints be ome a tive). Frequen y domain spe i ations an be he ked under the assumption that no onstraints are a tive, or that a parti ular set of onstraints is a tive. Frequen y response hara teristi s of the ontroller an be omputed (and some software is available to do this [118℄) under either assumption, sin e the ontroller is then linear and time-invariant (assuming a quadrati ost fun tion). No omplete systemati method is urrently known of modifying the optimisation problem parameters in su h a way as to a hieve parti ular frequen ydomain hara teristi s, but signi ant progress towards this is reported in [151℄. This is parti ularly relevant for a hieving stability and performan e robustness. Stability of the losed loop is always part of the design spe i ation, even if only impli itly.

In the absen e of a tive onstraints, it is known how to

enfor e stability. Essentially, either the predi tion horizon

N2

must be made

large enough, or `terminal' equality onstraints, whi h bind at time must be added to the problem formulation.

k + N2 ,

It has been shown that, from

the point of view of stability enfor ement, terminal equality onstraints an be ex hanged for an innite predi tion horizon [199℄.

Furthermore, several

methods are known of ensuring stability even in the presen e of onstraints, but under the assumption that the problem posed always remains feasible. This is a very strong and almost unveriable assumption, and some urrent resear h is aimed at removing it. Most stability proofs are based on proving the monotoni ity of the ost fun tion

J (k) with k, and hen e using the ost fun tion as a Lyapunov fun tion. 132

There have also been some attempts at exploiting the pie ewise-linear nature of the ontroller to prove stability. Whereas obtaining stability is not di ult in pra ti e for predi tive ontrol s hemes, there are not yet standard pro edures for obtaining spe ied stability margins. (This is essentially the same problem as the problem of obtaining spe ied frequen y response hara teristi s, whi h was dis ussed above.) Although tuning of predi tive ontrollers remains di ult, mu h progress is being made, and systemati pro edures, whi h tune only some of the free parameters, are be oming in reasingly lear [173, 151℄.

12.8 Appli ations Predi tive ontrol has mostly been applied in the pro ess industries, and parti ularly in the petro hemi al industries. In these appli ations there is strong motivation to exploit the apability of respe ting onstraints, sin e mu h money is to be made by operating as lose as possible to onstraints. Also time onstants are very big, so there is plenty of time to perform the ne essary omputations. It is important to stress that in these industries predi tive ontrol is a mature te hnology, whi h is used routinely. A few papers report the use of predi tive ontrol with high-bandwidth ele tro-me hani al systems su h as servos and automotive systems [23℄. Typi ally these either restri t the stru ture of the predi tive ontrol law a priori in order to obtain an easier optimisation problem [5℄, or pose a problem without

onstraints [68℄. Several studies of using predi tive ontrol in aerospa e appli ations have been reported, though only a minority of these have really addressed the online omputation problem [105, 223, 24, 214, 252℄.

12.9 Con lusions Constrained Predi tive Control is radi ally dierent from other ontrol approa hes whi h are onventionally used, or might be used, for ight ontrol. Not only is the design method rather dierent, but the implemented algorithm is quite dierent, be ause it works by repeatedly solving an optimisation problem on-line. As a straight repla ement for those approa hes, it is not urrently ompetitive, primarily be ause of the omputational load. Even when further advan es in omputing hardware bring the solution time down to a

eptable levels, whi h they will surely do, the predi tive ontrol approa h will give greater problems of erti ation than onventional ontrollers, be ause of the di ulty of analysing the ontroller behaviour. On the other hand, predi tive ontrol oers some unique benets: its very nonlinear behaviour when onstraints are approa hed, and its ability to anti ipate pilot ommands, instead of merely rea ting to errors propagating round the feedba k loop.

133

We believe that predi tive ontrol is worth investigating further for use in ight ontrol, if its unique benets are exploited to obtain higher-level fun tionality, in addition to routine stability augmentation. This is dis ussed further in Chapter 25.

A knowledgement We would like to thank Dr Angel Perez de Madrid, of UNED, for useful omments during the preparation of this hapter.

134

13.

Fuzzy Logi Control

Gerard S hram , Uzay Kaymak1 and Henk B. Verbruggen1 1

13.1 Introdu tion Designing ontrollers for everyday tasks su h as grasping a fragile obje t, driving a ar, or more ompli ated tasks su h as ying an air raft, ontinue to be a real hallenge, while human beings perform them relatively easily.

Unlike

most onventional ontrol systems, however, humans do not use mathemati al models nor do they use exa t traje tories for ontrolling su h pro esses. Moreover, many pro esses ontrolled by human operators in industry annot be automated using onventional, linear ontrol te hniques, sin e the performan e of these ontrollers is often inferior to that of the operators. One of the reasons is that linear ontrollers, whi h are ommonly used in onventional ontrol, may not be appropriate for nonlinear plants. Another reason is that humans use various kinds of information and a ombination of ontrol strategies, that

annot be easily integrated into an analyti ontrol law.

However, a lot of

experien e and knowledge is available from the experts (e.g. the pilot), whi h

an be made expli it and programmed as a ontrol strategy in a omputer. Knowledge-based (expert) ontrol tries to formalize the domain-spe i knowledge, and uses reasoning me hanisms for determining the ontrol a tion from the knowledge stored in the system and the available measurements. Knowledge-based ontrol systems try to enhan e the performan e, reliability and robustness of urrent ontrol systems by in orporating knowledge that

annot be a

ommodated in analyti models upon whi h onventional ontrol algorithms are based.

Knowledge-based systems an be used to realize the

losed-loop ontrol a tions dire tly, i.e. repla e onventional losed-loop ontrollers, or they an omplement and extend onventional ontrol algorithms via supervision, tuning or s heduling of lo al ontrollers. A ommon type of knowledge-based ontrol is rule-based ontrol, where the ontrol a tions orresponding to parti ular onditions of the system are des ribed in terms of

ifthen rules.

Fuzzy Logi Controllers (FLCs) are rule-

based systems, where fuzzy sets are used for spe ifying qualitative values of the

ontroller inputs and outputs.

Mu h of the expert's knowledge ontains lin-

guisti terms su h as small, large, et ., whi h an be appropriately represented by fuzzy sets. Using fuzzy logi , experts' (linguisti ) knowledge of the pro ess

ontrol an be implemented. The rst appli ation of fuzzy logi ontrol was in 1

Department of Ele tri al Engineering, Delft University of Te hnology.

2600 GA Delft, The Netherlands.

P.O.Box 5031,

{g.s hram}{u.kaymak}{h.b.verbruggen}et.tudelft.nl 135

ement kiln ontrol [114℄. The rules representing the ontroller a tions were derived from the ement kiln operator's handbook. Sin e then, fuzzy logi ontrol has been applied to various systems in the hemi al pro ess industry, onsumer ele troni s, automati train operation, and many other elds [66, 136, 236℄. In se tion 13.2, the basi prin iples of fuzzy sets and fuzzy logi are introdu ed. Next, the ontrollers are onsidered in detail, followed by a dis ussion on ontroller tuning in se tion 13.4. tools are des ribed.

In se tion 13.5, software and hardware

In se tion 13.6, the possibilities of fuzzy logi for ight

ontrol are dis ussed. The hapter ends with on lusions.

13.2 Basi Prin iples The basi idea of a fuzzy logi ontroller is to formalize the ontrol proto ol of a human operator, whi h an be represented as a olle tion of

ifthen rules, in

a way tra table for omputers and for mathemati al analysis. As an example,

onsider the ontrol of the F/A-18 during arrier landing [229℄.

Following a

three dimensional ight path, the task involves the ontrol of speed, sink rate, and angular attitudes in order to allow a safe ship-board landing. The ontrol strategy of the pilot onsists of several subtasks, e.g. roll angle ontrol. If the desired roll angle is positive large (roll angle error positive large), then the pilot imposes a positive lateral displa ement on the sti k:

If roll angle error is positive large Then lateral sti k position is positive

large

The rule des ribes a proportional relation between roll angle error and lateral sti k position. Usually, the rules are a ombination of proportional as well as derivative a tion in order to redu e rates. A typi al rule from the sink rate rule base is:

If sink rate error is near zero AND sink a

eleration is positive large Then longitudinal sti k position is negative medium The rst part of the rules, alled the ante edent, spe ies the onditions under whi h the rule holds, while the se ond part, alled the onsequent, pres ribes the orresponding ontrol a tion.

Both the ante edent and the on-

sequent ontain linguisti terms (large, small, near zero et .) that ree t the pilot's knowledge of the pro ess. The ante edent ondition is dened as a ombination of several individual onditions, using a onne tive, su h as the logi al AND operation.

It is possible that other rules may ombine the ante edent

onditions using dierent onne tives su h as the logi al OR or the omplement NOT. When the rules of the above mentioned type are to be represented in a form tra table for omputers, one needs to dene the linguisti terms and the onne tives that operate on the linguisti terms. In fuzzy ontrol, the linguisti terms are represented by fuzzy sets. Suppose that the pilot has a general idea of what a small or large value is, without a

136

sharp distin tion.

Su h a term an be des ribed by a fuzzy set, represented

by a so- alled membership fun tion [264℄, whi h is dened on the universe of dis ourse

X

as a fun tion:

 : X ! [0; 1℄: The position and shape (e.g.

triangular, trapezoidal or bell-shaped) of

the membership fun tion depend on the parti ular appli ation.

However, in

many ases triangular shapes are preferred be ause they are related to linear a tions. Consider for example the roll angle error. In Figure 13.1, a number of triangular-shaped membership fun tions are shown. Note that in this example the membership fun tions are pairwise overlapping and that their sum is always 1. A roll angle error of 15 degrees belongs for 50% to the set of a positive small error (PS) and for 50% to the set of a positive medium error (PM). In other words, the membership degrees

µ 1

P S (15) and P M (15) are both 0.5.

NL

NM NS ZE PS PM

-30

0

15

PL

30

Figure 13.1: Membership fun tions for roll angle error; negative (N), positive (P), large (L), medium (M), small (S), zero (ZE).

Fuzzy set operations are performed by logi al onne tives su h as AND ( onjun tion), and OR (disjun tion). For this purpose, the logi al onne tives from onvential Boolean logi have been extended to their fuzzy equivalents. The generalization of onjun tion to fuzzy sets is done by fun tions alled tnorms. Disjun tions are generalized by t- onorms. The most ommonly used

onjun tion operators are the minimum and produ t operators. Usually, the maximum or the probabilisti sum operator is used for the disjun tion.

In

Figure 13.2, the onjun tion and disjun tion operations on two fuzzy sets are shown when the minimum and maximum operators are used, respe tively.

conjunction (AND)

disjunction (OR)

Figure 13.2: Conjun tion and disjun tion of two fuzzy sets by minimum and maximum operator, respe tively.

137

13.3 Fuzzy Logi Control Using fuzzy sets and fuzzy set operations, it is possible to design a fuzzy reasoning system whi h an a t as a ontroller [162℄. In Figure 13.3, the stru ture of a typi al fuzzy logi ontroller (FLC) is shown. The ontrol strategy is stored

knowledge base scaling factors

membership functions

rule base

membership functions

fuzzification

reasoning mechanism

defuzzification

scaling factors

control actions

errors dynamic filter & scaling

dynamic filter & scaling

Figure 13.3: Blo k-s hemati representation of fuzzy logi ontroller. in the form of

ifthen rules

in the rule base. They represent an approximate

stati mapping from inputs (e.g. errors) to outputs ( ontrol a tions). The dynami lters are used to introdu e dynami s, e.g.

error and derivative of

error, and to introdu e an integration on the output. Moreover, s aling is performed to keep the signals between the input and output limits for whi h the fuzzy rules are dened. The membership fun tions provide a smooth interfa e from the linguisti knowledge to the numeri al pro ess variables. The fuzzi ation module determines the membership degree of the inputs to the ante edent fuzzy sets. The reasoning me hanism ombines this information with the rule base and determines the fuzzy output of the rule-based system.

In order to

obtain a risp signal, the fuzzy output is defuzzied and s aled. The omputational me hanism of the FLC an be explained on an example of a fuzzy variant of a PD (proportionalderivative) ontroller. Simple PD-like fuzzy ontrol rules an be dened as relations between the ontrol error error derivative


e and the ontrol a tion u.

e, the

As an example, assume that the

following two rules are a part of a fuzzy ontroller's rule base:

If e is small and
e is medium Then u is small If e is medium and
e is big Then u is medium Triangular membership fun tions are dened for the terms small, medium and big in the respe tive domains, see Figure 13.4. The omputational me hanism of the FLC pro eeds in ve steps: 1. Fuzzi ation: The membership degrees of the ante edent variables are

small (e), medium (e), medium (
e), big (
e)).

omputed (

2. Degree of fullment: The degree of fullment for the ante edent of ea h rule is omputed using fuzzy logi operators.

i

determines to whi h degree the

138

The degree of fullment

ith rule is valid.

In the example, the

produ t operator is used:

1 = small (e)
medium (
e) 2 = medium (e)
big (
e): 3. Impli ation: The degree of fullment is used to modify the onsequent of the orresponding rule a

ordingly. This operation represents the

then

if

impli ation dened as a t-norm, i.e. a onjun tion operator (e.g.

produ t). Hen e, the fuzzy outputs of the rules be ome:

0

1 (u) = 1
small (u) 0 2 (u) = 2
medium (u): 4. Aggregation:

the (s aled) onsequents of all rules are ombined into a

single fuzzy set.

The aggregation operator depends on the impli ation

fun tion used; for onjun tions, it is a disjun tion operator (e.g. max): FLC output

0

0

(u) = max(1 (u); 2 (u))

8 u 2 U:

5. Defuzzi ation: the resulting fuzzy set is defuzzied to yield a risp value. Defuzzi ation an be onsidered as an operator that repla es a fuzzy set by a representative value.

There exists a number of defuzzi ation

methods, su h as the entre of area method. In Figure 13.4, a small arrow marks the defuzzied value. The above type of FLC is alled a linguisti ontroller or a Mamdani type of ontroller.

A on eptually dierent type of FLC has been introdu ed by

Takagi and Sugeno [234℄.

In this type of ontroller, the onsequent part is

repla ed by a risp (non)linear fun tion.

The ontroller output is omputed

by taking a weighted average of the individual rule outputs.

Sin e the rule

outputs are risp, there is no need for defuzzi ation. The ontroller an be

ompared to a gain s heduling ontroller whi h has linear, lo al modules whi h are smoothly onne ted. In this way, the lo al linear models an be used for tuning and analysis (e.g. stability) of the FLC, while the global model aptures the nonlinearity of the system. However, the transparen y of the ontroller is de reased. In the RCAM design, the Mamdani type is used be ause this type of ontroller an implement the pilot knowledge most easily (Chapter 26).

Dire t and supervisory ontrol The motivation for many FLC appli ations is to mimi the ontrol behaviour of a human operator in a dire t ontrol onguration or in a supervisory ontrol environment. Many fuzzy logi ontrollers are implemented as dire t ontrollers in a feedba k loop. As the rule base represents a stati mapping between the ante edent and the onsequent variables, external dynami lters are used to introdu e the desired dynami behaviour of the ontroller (Figure 13.3).

139

2 µ

small

1

1 µ small (e)

0

µ

small

medium

µ medium(

1

1

e)

β1 0

µ

medium

1

µmedium(e)

µ 1

1

medium

β2

e)

0 e

u

µ

big µ big (

0

3 0

3 0

u

e

product µ

4

1

max

0 u

5 Figure 13.4: Computational me hanism of a FLC.

A fuzzy inferen e system an also be applied at a higher, supervisory level. A supervisory ontroller is a se ondary ontroller whi h augments the existing

ontroller for various onditions. Supervisory systems are hara terized by the additional exibility they bring to the ontrol system. A supervisory ontroller

an adjust the parameters of a low-level ontroller a

ording to the pro ess information, so that dynami behaviour whi h ould not be obtained by the low-level ontroller due to pro ess nonlinearities or hanges in the operating or environmental onditions an be a hieved. An advantage of a supervisory stru ture is that it an be added to already existing ontrol systems. Hen e, the original ontrollers an always be used as initial ontrollers for whi h the supervisory ontroller an be used for tuning the performan e. A supervisory stru ture an be used for implementing dierent ontrol strategies in one single ontroller (heterogeneous ontrol). These

on epts will be shown in Chapter 26, where separate ontrol strategies for low airspeed and engine failure are in luded.

13.4 Fuzzy Logi Control Design Two dierent methods an be used for designing fuzzy logi ontrollers: 1. Design the ontroller dire tly from the knowledge available from the domain experts.

140

2. Develop a fuzzy model of the plant from measurements, rst prin iples and expert knowledge, and use this model to design a ontroller or in orporate this model in a model-based ontrol s heme. The se ond, indire t method is des ribed in e.g. [17, 36, 127℄. In the rest of this se tion we will only on entrate on the dire t approa h, whi h will serve as a guideline for the design in Chapter 26. The design is hara terized by the following steps: 1. Determine the ontroller inputs and outputs.

For this step, one needs

basi knowledge about the hara ter of the plant dynami s (stable, unstable, stationary, time-varying, low order, high order, et .), the plant nonlinearities, the ontrol obje tives and the onstraints. The simplied plant dynami s together with the basi ontrol obje tives determine the dynami s of the ontroller, e.g. PI, PD or PID type fuzzy ontroller. In order to ompensate for the plant nonlinearities, non-stationarity or other undesired phenomena, variables other than error and its derivative or its integral may be used as the ontroller inputs. It is, however, important to realize that with an in reasing number of inputs, the omplexity of the fuzzy ontroller (i.e. the number of linguisti terms and the total number of rules) in reases onsiderably. In that ase, rule base simpli ation and redu tion te hniques need to be used for keeping the number of rules small [16℄. 2. Determine the rule base. The onstru tion of the rule base is a ru ial aspe t of the design, sin e the rule base en odes the ontrol proto ol of the fuzzy ontroller. Several methods of designing the rule base an be distinguished. One is based entirely on the expert's intuitive knowledge and experien e over all operating onditions. Sin e in pra ti e it may be di ult to extra t all knowledge from the operators, this method is often

ombined with a good understanding of the system's dynami s. Another method is based on using a fuzzy model of the pro ess from whi h the fuzzy ontrol rules are derived. 3. Dene the membership fun tions and the s aling fa tors.

The designer

must de ide, how many linguisti terms per input variable will be used. The number of rules needed for dening a omplete rule base in reases exponentially with the number of linguisti terms per input variable. On one hand, the number of terms per variable should be low in order to keep the rule base maintainable.

On the other hand, with few terms,

the exibility in the rule base is restri ted with respe t to the a hievable nonlinearity in the ontrol mapping. The membership fun tions may be a part of the expert's knowledge, for example the expert knows approximately what a large roll angle error means. If su h knowledge is not available, membership fun tions of the same shape, uniformly distributed over the domain, an be used as an initial setting and an be tuned later. For omputational reasons, triangular and trapezoidal membership fun tions are usually preferred to bell-shaped fun tions. Moreover, the latter

141

fun tions introdu e a nonlinear hara ter whi h may not be desirable in all ases. Generally, the input and output variables are dened on losed intervals. For simpli ation of the ontroller design, implementation and tuning, it is more onvenient to work with normalized domains, su h as the interval

[ 1; 1℄.

S aling fa tors are used to transform the values from the operat-

ing ranges to these normalized domains. However, one should be aware that su h s aling fa tors also s ale the nonlinearity in the ontroller whi h may not always be desirable. 4. Inferen e options. The hoi e of the inferen e operators also inuen es the shape of the mapping between inputs and outputs. The most used inferen e method is the max-min method, where the minimum operator is used for determining the degree of fullment and the impli ation, and the maximum operator for rule aggregation. Another method is the sumprodu t inferen e. The latter ombination is useful for an initial, linear setting of the FLC. This will be explained below. 5. Fine-tuning the ontroller.

The implementation of human heuristi s is

formalized by fuzzy logi in a systemati way. Altough ne-tuning the performan e of the ontroller is essentially a matter of trial-and-error, an understanding of the inuen e of various parameters an guide the pro ess. The s aling fa tors, whi h determine the overall gain of the fuzzy

ontroller and also the relative gains of the individual ontroller inputs, have mainly a global ee t. The ee t of a modi ation of membership fun tions and rules is more lo alized, for example hanging the onsequent of an individual rule. The ee t of the hange of the rule onsequent is the most lo alized and inuen es only that region where the rule's ante edent holds. 6. Stability analysis. The analysis of the ontroller is mainly based on time responses. A stability analysis of the nonlinear FLC is in general di ult. However, results an be obtained by using te hniques from nonlinear systems theory if a model of the pro ess under ontrol is available [66, 235, 251℄. The stability is only proven for the parti ular, simplied model. Re ently, the stability results have also been extended to more general

lasses of systems [43℄. The resulting ontrollers are usually onservative be ause of the onservative nature of the stability riteria. In order to simplify the design, it is possible to initialize the FLC as a linear fun tion between the input and output bounds. This limits the hoi e of membership fun tions and operators, and the ontroller be omes easier to analyse. One way of a hieving linear initialisation is using pairwise overlapping, triangular membership fun tions where the sum of the memberhsip fun tions equals 1. The defuzzied onsequents must be dened su h that the total mapping of the FLC is a linear fun tion. The defuzzied onsequents are the numeri al values after defuzzi ation of ea h individual rule onsequent. Se ondly,

142

produ t operators must be used for determining the degree of fullment and impli ation. The aggregation and defuzzi ation phase are then ombined in one step by the so- alled fuzzy-mean method, whereby the FLC output

y

is

determined as a weighted sum of defuzzied onsequents:

y= with

i

and

i

Nr X i=1

i i

are the degree of fullment and the defuzzied onsequent of

the ith rule respe tively, and

Nr

the number of rules. Note that defuzzi a-

tion is performed for ea h individual rule before aggregation takes pla e.

In

Chapter 26, the FLCs are initialized in this way.

13.5 Available Hardware and Software Tools Sin e the development of fuzzy ontrollers relies on intensive intera tion with the designer, spe ial software tools have been introdu ed by various software (SW) and hardware (HW) suppliers su h as Omron, Aptronix, Inform, Siemens, National Semi ondu tors, et . Most of the programs run on a PC under MSWindows, although some of them are also available for UNIX systems. The general stru ture of most software tools is depi ted in Figure 13.5. The

project editor

Figure 13.5: Generi stru ture of a software tool for fuzzy ontroller design. heart of the user interfa e is a graphi al proje t editor that allows the user to build a fuzzy ontrol system from basi blo ks. Input and output variables

an be dened and onne ted to the fuzzy reasoning unit.

If ne essary, one

an also use pre-pro essing or post-pro essing elements su h as dynami lters, integrators, dierentiators, et . The fun tions of these blo ks are dened by the user. The rule base and the related fuzzy sets are dened using the rule base

143

and membership fun tion editors. The rule base editor is a spreadsheet in whi h the rules an be entered or modied. The membership fun tions editor is used for dening the shape and position of the membership fun tions graphi ally. After the rules and membership fun tions are designed, the fun tion of the fuzzy ontroller an be tested using system analysis and simulation software (e.g. MATLAB/SIMULINK). On e the fuzzy ontroller is tested using various analysis tools, it an be used for ontrolling the plant either dire tly by the environment (via omputer ports or analog inputs/outputs), or through generating a run-time ode. Most of the programs generate a standard C- ode and also a ma hine ode for spe i hardware, su h as mi ro ontrollers or programmable logi ontrollers (PLCs). An alternative implementation is a multi-dimensional look-up table with a simple interpolation routine. This ould simplify validation and erti ation in

ase of ight riti al ontrol systems.

13.6 Fuzzy Logi for Flight Control Re ently, mu h attention has been paid to the appli ation of knowledge-based

ontrol te hniques for ight ontrol [228, 230℄. It shows that te hniques like neural networks and fuzzy systems an provide appropriate tools for nonlinear identi ation [156, 204℄, ontrol of high performan e air raft [183, 229℄ (inner loop as well as outer loop ontrol), heli opters [195, 233℄, spa e raft [26, 106℄, ight ontrol re onguration [142, 182, 263℄, and advisory systems [111, 232℄. In these appli ations, neural networks generally serve as nonlinear, sometimes adaptive, fun tion approximators, while fuzzy systems are used as supervisory, expert systems. An example of a fuzzy logi ontrol appli ation for ight ontrol is [229℄. The ne essary knowledge is extra ted from experien ed pilots. In Chapter 26, pilot heuristi s of ying an air raft are implemented in the design of a FLC as well. The FLC design onsists of longitudinal and lateral outer loop tra king

ontrollers ombined with lassi al inner loop attitude ontrollers. Additional, supervisory rules for low airspeed and engine failure are in luded whi h show how (gain) s heduling and ex eption handling an be readily in orporated.

13.7 Con lusions A fuzzy logi ontroller an be onsidered from the AI point of view as a real-time expert system implementing a part of a human operator's or pro ess engineer's expertise. From the ontrol engineering perspe tive, a FLC is a nonlinear ontroller. Re ently, a lot of resear h eort has been put into fuzzy logi

ontrol. The appli ations in industry are also in reasing. Major produ ers of

onsumer goods use fuzzy logi ontrollers in their designs for onsumer ele troni s, dishwashers, washing ma hines, automati ar transmission systems et . FLC appli ations are beginning to appear in the pro ess industry as well.

144

One of the main reasons put forward for using fuzzy logi is that an expli it mathemati al model des ription is not required for the design of a FLC. Instead the a tions of a human operator, who already has an internal representation of the plant, are modelled.

This an result in a more e ient ontroller

design, saving time and money. This is only true if expli it operator knowledge is available in a suitable form.

Also, for testing and ne-tuning the FLC, a

reasonable simulation model or the pro ess itself should be available. However, if little experien e or knowledge about the pro ess is present, and it is not possible to make eld tests for tuning the ontroller, fuzzy logi ontrol may not be suitable. One has to onsider espe ially the knowledge a quisition bottlene k if the experts' knowledge is not available expli itly.

An alternative is rst

building a fuzzy model of the nonlinear system from measurement data about the system, and then applying model-based ontrol te hniques. Many fuzzy logi ontrollers are implemented as dire t ontrollers in a feedba k loop. In situations where an existing ontroller needs to be extended for several operating onditions or when a more exible ontrol stru ture is required, supervisory fuzzy ontrol an provide an answer.

It is more di ult

to formulate an analyti ontrol law at this level, while a lot of linguisti information may be available, whi h an be used for designing the FLC. At this level, the ontrol problem starts to resemble more and more a de ision making problem, whi h an be solved by te hniques from fuzzy-de ision making. The implementation of human heuristi s is formalized by fuzzy logi in a systemati way. This fa t is also re ognized by the industry, and re ently efforts have in reased to dene a European industry standard for the development methodology of fuzzy logi systems, based on ISO-9000 general system development guidelines [248℄. However, ne-tuning the performan e of the ontroller is a matter of trial-and-error like in lassi al ontrol, but using the provided guidelines and an understanding of the inuen e of ontroller parameters, a satisfa tory ontroller an be obtained.

145

146

Part II

RCAM part

147

14.

The RCAM Design Challenge

Problem Des ription

Paul Lambre hts , Samir Bennani , Gertjan Looye and Dieter Moormann 1

2

2

3 4

Abstra t.

The RCAM design hallenge problem is dened in this

hapter using two main se tions.

The rst se tion dis usses the

basi ight dynami s model, the available inputs, outputs, parameters, et . and the modelling of a tuators, disturban es, et . After that the ontrol design spe i ations are given and the evaluation pro edure to be performed by all design teams is presented.

14.1 Introdu tion This hapter provides the RCAM design problem formulation. It is abstra ted from the GARTEUR FM(AG08) report: Robust Flight Control Design Challenge Problem Formulation and Manual: the Resear h Civil Air raft Model (RCAM) [145℄. This report formed the basis for the RCAM design hallenge, the results of whi h are given in the hapters 15 through 26.

Therefore its

ontents are given here with little modi ation, so that a lear pi ture of the information that was available to the design hallenge teams is given. However, sin e the design and evaluation software that was available to the teams is not supplied with this book, the des ription of this software has been ex luded. It is remarked that both the software itself and the des ription appeared helpful in

larifying the problem formulation, but was not intended to provide additional information. In se tion 14.2, a des ription of the model is given, in whi h analyti al expressions for all the parameters of interest, states, inputs and outputs of the system, are derived.

A detailed, oje t-oriented des ription of the model

omponents is also in luded (air raft, sensors, a tuators, engines, wind, et .). 1

Hoogovens Corporate Servi es B.V., HR&D-RSP-SDC 3G.16, P.O.box 10000, 1970 CA

IJmuiden, The Netherlands. 2

(Formerly: NLR, Amsterdam.)

Delft University of Te hnology (TUD), Fa ulty of Aerospa e Engineering, Kluyverweg 1,

2629 HS Delft, The Netherlands. 3

German Aerospa e Resear h Establishment (DLR), Institute for Roboti s and System

Dynami s Control Design Engineering, Oberpfaenhofen, D-82230 Wessling, Germany. 4

The following authors ontributed to the original RCAM design denition: Pierre Fabre,

Joseph Irvoas, Philippe Ménard (Aerospatiale), Anders Helmersson (LiTH), Jean-François Magni (CERT), Tony Lambre gts (DUT), Alberto Martínez (INTA), Stefano S ala (CIRA), Phillip Sheen (AVRO), Jan Terlouw (NLR) Hans van der Vaart (TU Delft).

149

In se tion 14.3 the design problem is formulated, and the riteria and pro edure adopted for evaluation of the proposed design are des ribed.

14.2 Des ription of the RCAM Model The purpose of this hapter is to dis uss the RCAM model in a general setting, su h that used nomen lature and terminology an be introdu ed, and some of the philosophy behind the stru ture and numeri al al ulations in the software

an be highlighted. The hapter is set up to have some tutorial value, but is by no means omplete in that sense. It is re ommended to onsult a standard referen e su h as [74℄ or [35℄ for more information on the derivation of equations of motion, et .

14.2.1 Blo k diagram of the system A six degrees of freedom nonlinear air raft model in luding nonlinear a tuators (position and rate limited) and a model of disturban es has been proposed by Aérospatiale.

A blo k diagram of this model is given in gure 14.1.

Ea h

box in this blo k diagram will be overed in more detail in following text. In subse tion 14.2.3, an analyti al des ription of the air raft dynami s is given. In subse tions 14.2.4 and 14.2.5, the sensor and a tuator dynami s are detailed. In subse tion 14.2.6, the analyti al models of wind disturban es are presented.

14.2.2 Nomen lature: inputs, states, outputs, parameters As far as appli able, nomen lature is used as dened in the Communi ation Handbook [237℄.

The following tables summarise this nomen lature, as it is

used both for the formulation of the algorithms and the naming of variables in the software. Additional information an be found in Appendix A of this do ument. The inputs to the model are given in table 14.1. In this table,

FE

denotes the

earth-xed referen e frame, whi h is dened as follows. The origin threshold.

OE XE

is lo ated on the runway longitudinal axis at the is positive pointing towards the north, and we as-

sume that the runway is also dire ted towards the north (runway 00), hen e

XE is positive along the runway in the landing dire tion. ZE is positive downward, and YE is in the appropriate

Furthermore,

dire tion for a right handed axis system (positive east).

FB

stands for the body-xed referen e frame, whi h is dened as follows.

OB ZB is

The origin

is at the vehi le entre of gravity.

forward,

positive downward and

(starboard side).

150

YB

XB

is positive

is positive to the right

ACTUATOR MODELS (including nonlinearities)

trim settings controls

uext

feedback path

sim Specific outputs for system analysis

U=[ DA DT DR THROTTLE1 THROTTLE2]

uc

lon

ACTUATORS

Measurements for longitudinal control laws wext WIND=[ WXE WYE WZE WXB WYB WZB ] WIND MODELS (constant wind, turbulence, windshear)

lat Measurements for lateral control laws

WIND

AIRCRAFT

RCAM MODEL ( 6 degrees of freedom, non linear, landing configuration)

Figure 14.1: Blo k diagram of the system

The three earth-xed wind inputs, u(6)u(8), are intended to be used for onstant wind velo ity omponents eg. headwinds, whereas the body-xed wind inputs, u(9)u(11), are intended to be used for gusts. The states used internally by the software are expressed in SI units and are dened in table 14.2. In this table, CoG denotes Centre of Gravity. The outputs from the model are given in SI units and are shown in table 14.3. In this table,

FV

denotes the vehi le- arried verti al frame, whi h is dened as

follows. The vehi le- arried verti al frame is parallel to the earth-xed referen e frame but moves with the vehi le. The origin at the vehi le's entre of gravity. the north,

ZV

XV

OV

is lo ated

is positive pointing towards

is positive downward, and

YV

is positive towards the

east. Only the model outputs labeled as measured an be assumed to be available as inputs to the ontroller that is to be designed.

The simulation outputs

are only intended to be used for evaluation and should not be used in the nal

ontroller. Note that there is some redundan y in the measured signals, e.g.

 an be determined from uV

and

vV :

depending on the ontrol strategy the

most onvenient signals may be used.

151

Symbol

ÆA ÆT ÆR

ÆT H1 ÆT H2 W xE W yE W zE W xB W yB W zB

Alphanumeri DA u(1) DT u(2) DR u(3) THROTTLE1 u(4) THROTTLE2 u(5) WXE u(6) WYE u(7) WZE u(8) WXB u(9) WYB u(10) WZB u(11)

= = = = = = = = = = =

Name aileron dee tion tailplane dee tion rudder dee tion throttle position of engine 1 throttle position of engine 2 Wind velo ity in the x-axis of FE Wind velo ity in the y-axis of FE Wind velo ity in the z-axis of FE Wind velo ity in the x-axis of FB Wind velo ity in the y-axis of FB Wind velo ity in the z-axis of FB

Unit rad rad rad rad rad m/s m/s m/s m/s m/s m/s

Table 14.1: Input denitions

Symbol

p q r  

uB vB wB x y z

Alphanumeri P x(1) Q x(2) R x(3) PHI x(4) THETA x(5) PSI x(6) UB x(7) VB x(8) WB x(9) X x(10) Y x(11) Z x(12)

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

Name roll rate (in FB ) pit h rate (in FB ) yaw rate (in FB ) roll angle (Euler angle) pit h angle (Euler angle) heading angle (Euler angle) x omponent of inertial velo ity in FB y omponent of inertial velo ity in FB z omponent of inertial velo ity in FB x position of air raft CoG in FE y position of air raft CoG in FE z position of air raft CoG in FE

Unit rad/s rad/s rad/s rad rad rad m/s m/s m/s m m m

Table 14.2: State denitions

Usually, it is possible to dene geometri air raft parameters within the bodyxed referen e frame. However, in the ase of RCAM this is not allowed, as the CoG is not a geometri ally xed point. referen e frame

FM

For this reason, a measurement

is dened.

The measurement referen e frame is geometri ally xed to the air raft.

The origin

OM

is lo ated at the leading edge of the mean

aerodynami hord, whi h is denoted as ba kwards,

ZM

YM

. XM

is positive pointing

is positive pointing to the right (starboard), and

is positive pointing up.

It is assumed that the aerodynami entre of the wing-body onguration (ACwb ) is also geometri ally xed: its o-ordinates in

FM

0:12 ; 0 ; 0).

are (

With these denitions, it is now possible to spe ify the parameters used in RCAM: they are given in table 14.4. Finally, RCAM provides the possibility to study the ee t of the parameter

hanges dened in table 14.5.

152

Symbol Measured

Alphanumeri

Name

Unit

q nx nz wV z VA V p r  uV vV y 

Q NX NZ WV Z VA V BETA P R PHI UV VV Y CHI

y(1) y(2) y(3) y(4) y(5) y(6) y(7) y(8) y(9) y(10) y(11) y(12) y(13) y(14) y(15)

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

pit h rate (in FB ) = x(2) Fx  horizontal load fa tor (in FB ) = mg Fz verti al load fa tor (in FB ) = mg 1 z omponent of inertial velo ity in FV z position of air raft CoG in FE = x(12) air speed total inertial velo ity angle of sideslip roll rate (in FB ) = x(1) yaw rate (in FB ) = x(3) roll angle (Euler angle) = x(4) x omponent of inertial velo ity in FV y omponent of inertial velo ity in FV y position of air raft CoG in FE = x(11) inertial tra k angle

rad/s m/s m m/s m/s rad rad/s rad/s rad m/s m/s m rad

 

x ny

PSI THETA ALPHA GAMMA X NY

y(16) y(17) y(18) y(19) y(20) y(21)

= = = = = =

heading angle (Euler angle) = x(6) pit h angle (Euler angle) = x(5) angle of atta k inertial ight path angle x position of air raft CoG in FE= x(10) lateral load fa tor (in FB )= Fy  see equations 14.1 and 14.5 mg

rad rad rad rad m -

Simulation

Table 14.3: Output denitions

14.2.3 Air raft dynami s model This subse tion des ribes the RCAM dynami s model orresponding to the

AIRCRAFT blo k in gure 14.1.

The dynami obje ts are depi ted in gure 14.2.

These obje ts are:



body des ribes the body dierential equations of motion (see subse tion 14.2.3);



two transformation obje ts des ribe the o-ordinate transformation between the body-xed o-ordinates of the body obje t and the geodeti oordinates of the gravity obje t, and between the body-xed o-ordinates of body and the geodeti o-ordinates of wind, respe tively (see subse tion 14.2.3);



al airspeed des ribes the relationship between the inertial movement, the wind, and the movement relative to the air (see subse tion 14.2.3);



engine_1 and engine_2 des ribe the relevant engine behaviour (see subse tion 14.2.3);

 

atmosphere des ribes the atmosphere model (see subse tion 14.2.3); aerodynami des ribes the aerodynami for es and moments (see subse tion 14.2.3);

153

Symbol

Alphanumeri MASS Aerodynami Parameters

 CBAR lt LTAIL

m

S St
x
y
z

Name air raft total mass

= = =

S STAIL DELX DELY DELZ Engine Parameters XAP T 1 XAPT1

= = = = =

YAP T 1

YAPT1

=

ZAP T 1

ZAPT1

=

XAP T 2

XAPT2

=

YAP T 2

YAPT2

=

ZAP T 2

ZAPT2

=

Default 120 000

Unit kg

6.6 24.8

m m

260.0 64.0 0.23  0 0

m2 m2 m m m

0.0

m

7:94 1:9

m m

0.0

m

7:94 1:9

m

mean aerodynami hord distan e between AC of the wing-body (ACwb ), and AC of the tail (ACt ) wing planform area tail planform area x position of the CoG in FM y position of the CoG in FM z position of the CoG in FM

x position of appli ation point of thrust of engine 1 in FM y position of appli ation point of thrust of engine 1 in FM z position of appli ation point of thrust of engine 1 in FM x position of appli ation point of thrust of engine 2 in FM y position of appli ation point of thrust of engine 2 in FM z position of appli ation point of thrust of engine 2 in FM

=

m

Table 14.4: Parameter denitions

Parameter MASS DELX DELY DELZ

m
x
y
z

: : : :

100 000 kg 0.15  0.03  0.0 

< < < <

Bounds

m


x
y
z

< < < <

150 000 kg 0.31  0.03  0.21 

Table 14.5: Possible parameter hoi es in RCAM



gravity des ribes the gravitational inuen e (see subse tion 14.2.3).

Body equations of motion The following two subse tions give the equations of motion for a rigid body with 6 degrees of freedom and other air raft motion relevant equations whi h as used within RCAM. For a more detailed derivation and explanation of these equations a referen e su h as [74℄ or [35℄ should be onsulted.

Translational motion. The equations for the translational movement in bodyxed o-ordinates are derived from the for e equation,

F = m ( aB + ! F

 VB )

(14.1)

is the sum of for es due to the engines, the aerodynami s and gravity,

is the mass of the air raft,

VB

is the inertial velo ity and

!

m

is the rotational

velo ity expressed in body-xed o-ordinates. The a

eleration (in body-xed

154

RCAM aero

u

sim RCAM

body6DOF equations

COG

engine1

airspeed kinetic

RCAM

air

long

wind

engine2

const. gravity

bodyfixed

bodyfixed

Trafo

Trafo

veh.carried

veh.carried

const.

lat Earth

atmosphere

wind

gust

wind gust

Figure 14.2: Dynami obje ts of RCAM air raft model inside the

AIRCRAFT

blo k of gure 14.1. Conne tion arrows between obje ts hara terise physi al intera tions

system) is the time derivative of velo ity:

2

dV d uB aB = B = 4 vB dt dt w

3 5

(14.2)

B

and the velo ity is the time derivative of the position ve tor expressed in the vehi le- arried verti al frame:

3

2

d x d XV = 4y5 VV = dt dt z

(14.3)

Additionally, some air raft spe i quantities are dened as follows:

h, whi h is the negative z - o-ordinate in the vehi le arried system h= z (14.4) The load fa tor n is dened as the relation of the external for es F (equation 14.1) to the gravity for e mg , with all quantities given in the body-xed The height

oordinate system. In order to have a zero load fa tor for horizontal ight, the z- omponent of

n is redu ed by one. 2 3 nx F n = 4 ny 5 = mg nz 155

2

3

0 405 1

(14.5)

The inertial ight path angle,

, is given as a fun tion of the speed omponents

in the vehi le- arried verti al referen e frame

tan = The tra k angle,

p 2wV 2 uV + vV

(14.6)

, is also dened as a fun tion of the the speed omponents

in the vehi le- arried verti al referen e frame

v tan  = V uV

(14.7)

Rotational motion. The equations of motion for the rotational movement of a rigid body in the body-xed axis system are derived from the moment equation,

M = I !_ + ! M

 I!

(14.8)

is the sum of moments about the entre of gravity due to the engines

and the aerodynami s,

!

is the inertial rotational velo ity, and

!_

is the iner-

tial rotational a

eleration in the body-xed axis system. Using the standard notation [74℄ we get:

2

2

3

3

p_ d 4p5 5 4 !_ = q_ = q dt r r_

(14.9)

Again using standard notation [74℄, the relation between the rotational velo ities and the Euler angles is;

2

d 4 = dt

_ _ _

3

32

2

1

For a normal air raft , the inertia tensor is;

3

p 1 sin  tan  os  tan  5=4 0

os  sin  5 4 q 5 r 0 sin = os  os = os 

2

Ix I = 4 0 Ixz

0 Ixz Iy 0 0 Iz

3

2

5

= m4

I

(14.10)

dened in the body-axis frame

40:07 0 0 64 2:0923 0

3

2:0923 0 5 99:92

(14.11)

2

where all numbers are expressed in square metres, m .

Co-ordinate transformation (Body-Fixed

, Vehi le-Carried)

The rotations between the body-xed and the vehi le- arried o-ordinate system are depi ted in gure 14.3.

To des ribe the angular orientation of the

air raft, a transformation using the three Euler angles

,  ,

and

is ne es-

sary. This transformation is a hieved by initially rotating the vehi le- arried

ZV -axis by the heading k2 -axis by the pit h angle ,

verti al system about the

angle

is rotated about the

and nally the body-xed

1

. Then, the result

Normal air raft are assumed symmetri about the OXZ body axis plane.

156

xB xV

θ

ψ

k1 yV

ψ φ

xV, yV

k2

xB , z v

yB

φ yB , z B zB

θ

k3

zV

Figure 14.3: Co-ordinate transformation body-xed

, vehi le- arried

referen e frame is obtained by rotating the result of that by the roll angle about the

XB -axis.



This results in the following transformation matrix from

the vehi le- arried verti al axis system to the body-xed axis system:

RBV = 32 32 2

os

os  0 sin  1 0 0 4 0 os  sin  5 4 0 1 0 5 4 sin 0 sin  0 os  0 sin  os  Note that

sin

os 0

3

0 05 1

(14.12)

RBV = RVT B .

For example, the transformation of velo ities from the vehi le- arried verti al frame

FV

to the body-xed referen e frame

FB

is given by:

VB = RBV VV with

2

uB VB = 4 vB wB

3 5

(14.13)

2

and

uV VV = 4 vV wV

3 5

(14.14)

Similarly, the a

elerations, rotational velo ities, positions, for es and moments

an be transformed between the o-ordinate systems.

157

Cal ulation of airspeed

Va is the dieren e between the inertial velo ity VB , and the wind velo ities, WB and WE (see table 14.1).

The ve tor airspeed, air raft,

of the

Expressed in the body-xed o-ordinate system this is al ulated as:

Va = VB Hen e, with

the airspeed

WB 2

ua Va = 4 va wa VA

RBV WE

(14.15)

3 5

(14.16)

is given as:

p

VA = (ua 2 + va 2 + wa 2 )

(14.17)

, and the angle of sideslip, , are dened as: w (14.18) tan  = a ua v sin = a (14.19) VA The derivatives of  and with respe t to time are: aax wa a u _ = az a2 (14.20) ua + wa 2 a ( u 2 + wa 2 ) va ( aax ua + aaz wa ) p _ = ay a (14.21) VA 2 ua 2 + wa 2 where aax , aay , and aaz are the x, y , and z -time derivatives of the airspeed in dua ). body-xed o-ordinates. (e.g. aax = dt Next, the angle of atta k,

Aerodynami equations The equations dening aerodynami for es and moments are determined by means of aerodynami oe ients.

Depending on the method of modelling

these oe ients may be dened in dierent referen e frames; e.g.

FB .

FW , FS , or

The referen e frame for aerodynami for es and moments that is used in

RCAM is the stability axis frame

FS .

Aerodynami for es. The aerodynami for es are determined by means of aero-

CD , CY , CL ), whi h are given , and the ontrol

dynami oe ients for drag, sidefor e and lift ( as fun tions of the angle of atta k,

,

the sideslip angle,

surfa e dee tions. The aerodynami lift oe ient,

CL , is dened as (see gure 14.4);

CL = CLwb + CLt 158

(14.22)

C

Lwb is the lift oe ient of the wing and body. It a ts on the aerodynami

entre of the wing and body and is parallel with the S axis. It is only a fun tion of angle of atta k

Z , and for  < 19 degree it is given by the following

equations:

CLwb = Here,

0



 rad 5:5 ( 0 )   14:5 180  rad 3 2 768:5 + 609:2 155:2 + 15:2  > 14:5 180

(14.23)

is the angle of atta k at whi h the wing/body lift is zero:

0 = 11:5

 180

(14.24)

The maximum lift oe ient is obtained at an angle of atta k of

 = 18 degree.

Negle ting the ee t of the tailplane, this is al ulated from equation 14.23 as:

CLmax = CLwb ( = 18

 ) = 2:75 180

(14.25)

CLt a ts on the aerodynami entre of ZS axis. It is given as: S (14.26) CLt = t 3:1 t S

The lift oe ient of the tailplane,

the tailplane and is also parallel with the

where

t denotes the angle of atta k of the tailplane and is al ulated from the

following equations:

t =  " + ÆT + 1:3

q lt VA

d " = ( 0 ) (14.27) d d = 0:25 d Here " is the downwash angle, ÆT is the tailplane dee tion, q is the air raft pit h rate, and lt is the longitudinal distan e between the aerodynami entre of the tailplane and the aerodynami entre of the wing and body. gure 14.4).

;

The aerodynami drag oe ient,

(See

CD , is a fun tion of the angle of atta k CD a ts on the

drag of the tailplane is negle ted and it is assumed that

aerodynami entre of wing and body:

CD = 0:13 + 0:07
(5:5 + 0:654)2 The aerodynami sidefor e oe ient,

(14.28)

CY , is also assumed to a t on the aero-

dynami entre of wing and body and is given by the following equation:

CY = 1:6 + 0:24ÆR where



is the angle of sideslip and

ÆR

(14.29)

is the rudder dee tion.

These non-dimensional oe ients an now be onverted to dimensional for es using the following relationships:

159

CL

wb

xB Va

CL

CoG

CD

α

ε

AC

δT

αt q=0

t

AC t

lt

Figure 14.4: Illustration of aerodynami for es



Aerodynami for e along

XS

X = D = CD



Aerodynami for e along

Aerodynami for e along

(14.30)

YS Y = CY



1 V 2S 2 A

1 V 2S 2 A

(14.31)

ZS

Z = L = CL

1  VA 2 S 2

(14.32)

To al ulate the translational motion of the air raft using equation 14.1, these for es need to be resolved into body axis for e omponents. The resolution from stability axes for es,

(D; Y; L), into the body-axes for es, (FxA , FyA , FzA ),

is given by the following expressions:

FxA = L sin  D os  FyA = Y FzA = L os  D sin  Aerodynami moments.

(14.33)

The moments due to the air raft aerodynami s are

Cl ; Cm ; Cn ),

determined by means of the moment oe ients, (

whi h are as-

sumed to a t about the aerodynami entre of the wing and body and are given

160

by the following equation:

2

Cl 4 Cm Cn

3 5

2

=

4 2

+

6 4 2

+

4

3

1:4 0:59 3:1 SSt lt ( ) 5 180 ) (1  15  3 2 3 11 0 5 p 2 7 0 4:03 SSt l2t 0 5 V A 4 q 5 1:7 0 11:53 2 3r 0:6 0 0:22 ÆA 0 3:1 SSt lt 0 5 4 ÆT 5 ÆR 0 0 0:63

(14.34)

where

p, q,and r are the rotational rates in body axes, ÆA is the aileron dee tion, ÆT is the tailplane dee tion, ÆR is the rudder dee tion. The moment oe ients about the entre of gravity are al ulated from these aerodynami entre based oe ients using the following equation:

2

with

3

2

3

1

0

0

2

31

Cl
x 0:12 CD 5 = 4 Cm 5 + 
y A  RBS
4 CY 5A C3n 02
z CL 31 2 32 Cl
x 0:12 0 CZ CY = 4 Cm 5 + 4 CZ 0
y 5A CX 5 4
z Cn CY CX 0

ClCG 4 CmCG CnCG

2

3

os  0 sin  RBS = 4 0 1 0 5 sin  0 os 

CX = CD os  + CL sin  CZ = CL os  CD sin 

(14.35)

(14.36)

The following expressions are used to onvert these non-dimensional moments oe ients into dimensional moments:



Rolling moment in body axes

LA = ClCG



(14.37)

Pit hing moment in body axes

MA = CmCG



1  V 2 S  2 A 1  VA 2 S  2

(14.38)

1  V 2 S  2 A

(14.39)

Yawing moment in body axes

NA = CnCG 161

These moments, in ombination with the moments due to thrust are then used to al ulate the rotational motion of the air raft from equation 14.8.

RCAM engine thrust al ulation The RCAM is a twin engined air raft model, and the thrust provided by ea h of the two engines is assumed to be aligned with the

x-body axis.

The thrust produ ed by a single engine is given by

Fi = ÆT Hi mg;

i = 1; 2 with m at the nominal mass of 120,000 kg and ÆT H1 the setting of the throttle handles. In equation 14.40,

(14.40) and

ÆT Hi

ÆT H2

determined by

should be expressed

in radians: this has no physi al meaning but appears to be onvenient in the

al ulations. The allowed value of

ÆT Hi

lies between

 0:5 180

and

 10 180

radians.

Note that the maximum thrust to weight ratio is about 0.35 (for both engines together). Hen e, the engine thrust ve tor at the enter of gravity is given in

FB

as:

2

F1 + F2 Fp = 4 0 0

3 5

(14.41)

Due to the geometri lo ation of the engines, see gure 14.5, the engine thrusts also ontribute to the moments a ting on the air raft. These moments

zM

6

6

front

F1

F2

6

6 O M - yM  P2 ACwb

ACwb



F1 ; F2

?xM P1

- xM

front

P1

P1 ; P 2

Figure 14.5: Appli ation points of thrusts. and

P2

are the points where the thrust is applied.

an be al ulated about the entre of gravity as follows:

2

3

2

3


x XAP T i Fi TEi = 4 YAP T i
y 5  4 0 5 (i = 1; 2) (14.42)
z ZAP T i 0 where
x,
y ,
z , XAP T i , YAP T i and ZAP T i are dened in table 14.4. 162

Atmosphere The atmosphere is onsidered to be onstant, irrespe tive of height and position, and onsequently we an dene the following:

kg

 = 1:225 3 m N

P = 101325:0 2 m T = 288:15 K where

 is the density of air, P

is the stati air pressure, and

(14.43)

T

is the absolute

temperature.

Gravity model Due to the restri ted altitude range to be used with this model, gravity is not

onsidered to be a fun tion of altitude. Hen e, gravity is assumed to have a

onstant value of:

g = 9:81 m/s2

(14.44)

14.2.4 Sensor models Models are not provided for the hara teristi s of the sensors:

they are all

assumed to be perfe t.

14.2.5 A tuator models and engine dynami s Both a tuators and engines are assumed to have rst order system dynami s with rate limits and saturations. The time onstants of the rst order system dynami s are:

  

engine models: 1.5 s, ailerons and tailplane a tuators: 0.15 s, and rudder a tuator: 0.3 s.

Numeri al values for rate limits and saturations are given as follows.



Rate limits for throttle movement are: rising slew rate



 = 1:6 180

 rad/s, = 1:6 180   rad  Æ 0:5 180 T Hi  10 180 rad.

rad/s, falling slew rate

throttle limits (saturations) are:

In ase of engine failure we an assume that the throttle setting for the

 rad ÆT Hi = 0:5 180 given by the transfer fun tion 1=(1 + 3:3s).

failed engine redu es to

163

with rst order system dynami s

  

  Æ_  25  rad/s; 25 180 A 180 rad,  saturations of aileron dee tion are: 25 180  ÆA  25 180   Æ_  15  rad/s; rate limits for tailplane dee tion are: 15 180 T 180  rad,  saturations of tailplane dee tion are: 25 180  ÆT  10 180   Æ_  25  rad/s; rate limits for rudder dee tion are: 25 180 R 180  rad.  saturations of rudder dee tion are: 30 180  ÆR  30 180 Rate limits for aileron dee tion are:

14.2.6 Wind turbulen e model Turbulen e is a sto hasti pro ess that an be dened by velo ity spe tra. Commonly used velo ity spe tra for turbulen e modelling are the Dryden spe tra.

V through a frozen turbulen e eld with a

rad/m, the ir ular frequen y of the turbulen e an be

For an air raft ying at a speed spatial frequen y of

al ulated as:

! =V


rad/s

(14.45)

With this, the spe tra an be des ribed as follows:

2Lug 1 V (1 + (Lug V! )2 ) L 1 + 3(Lvg V! )2 vg (!) = v2g vg V (1 + (Lvg V! )2 )2 L 1 + 3(Lwg V! )2 wg (!) = w2 g wg V (1 + (Lwg V! )2 )2 ug (!) = u2g

L

ug , vg , wg

L

L

ug , vg , wg and turbulen e standard deare dependent on altitude and atmospheri onditions.

The turbulen e s ale lengths viations

(14.46)

As an indi ator for the atmospheri onditions it is possible to take the wind

W20 ).

speed at 20 ft above the ground (

For moderat onditions,

m/s (30 kts) is sele ted. The turbulen e standard deviation

wg

W20 = 15:4

is then given

as follows:

wg = 0:1W20 ug and vg are assumed to be fun tions of wg and the altitude h. For h < 305 m (1000 ft): wg ug = vg = (0:177 + 0:0027h)0:4 and for

(14.47)

(14.48)

h > 305 m (1000 ft): ug = vg = wg

The turbulen e s ale lengths of altitude: for

3 < h < 305 m:

Lug , Lvg

Lug = Lvg =

and

Lwg

(14.49) are al ulated as a fun tion

h (0:177 + 0:0027h)1:2 164

(14.50)

and for

Lwg = h

(14.51)

Lug = Lvg = Lwg = 305 m

(14.52)

h > 305 m we take:

With this pro edure, the gust velo ities

ug , vg and wg are dened in the stability W xB , W yB

referen e frame. However, as an approximation the RCAM inputs and

W zB

are used.

To simulate turbulen e, white noise is ltered through forming lters. These lters an be derived from the Dryden spe tra given in equation 14.46. As an example, the transfer fun tion of the lter for simulating the gust velo ity will be onsidered. Given white noise

w, the spe trum of wg

an be obtained as:

wg = jHwg w (!)j2 w Where

wg

(14.53)

w = 1, and Hwg w (!) is the frequen y response fun tion of the forming

lter. Therefore,

L 1 + 3(Lwg V! )2 w2 g wg = jHwg w (!)j2 = Hwg w (!)Hwg w ( !) V (1 + (Lwg V! )2 )2

(14.54)

To obtain a stable and minimum phase lter, the following frequen y response fun tion is sele ted:

r

Hwg w (!) = wg Repla ing the variable

p

Lwg 1 + 3 LVwg j! V (1 + LVwg j!)2

(14.55)

j! by s, the following transfer fun tion is obtained: r

Hwg w (s) = wg The transfer fun tion for generating

p

Lwg 1 + 3 LVwg s V (1 + LVwg s)2

vg

is equivalent.

The transfer fun tion for generating

r

Hug w (s) = ug

(14.56)

ug

an be found as:

2Lug 1 V 1 + Lug s

(14.57)

V

It is important to note that for orre t appli ation of these lters the white noise inputs need to be independent. For a more detailed dis ussion on turbulen e modelling, the reader is referred to for example [35℄.

165

14.3 Design Problem Formulation and Evaluation Criteria 14.3.1 Motivation design and evaluation riteria Within the aerospa e industry there is a large amount of experien e in the ight ontrol system design area. For this reason, the main obje tive of the

ontrol problem stated here is not so mu h to obtain a satisfa tory ontroller, but more spe i ally to exhibit approa hes whi h might redu e the omplexity of ontrol laws and the overall ontrol system design y le. Some of the main features addressed by modern ontrol design te hniques provide the possibility to take into a

ount:

    

the multivariable nature of the ontrol problem the non linear behaviour of the plant the time-varying nature of the plant robustness to parameter hanges and un ertainties simultaneous performan e and robustness spe i ations.

From the onsideration of these features it is expe ted that improvements ould be made in areas su h as:

   

ontrol system ar hite ture development

ontrol law design y le

ontrol design solution

ontrol system implementation

The RCAM design hallenge onsists of the synthesis of a ontrol law apable of fullling an approa h to landing under various external onditions eg. turbulen e and windshear, while being robust to parameter hanges. Furthermore, the air raft guidan e must not degrade under engine failure.

Details on the

design obje tives are given in subse tion 14.3.2. For the uniform omparison of all design entries from the design hallenge parti ipants, a set of evaluation riteria is formulated in subse tion 14.3.3. To evaluate proper ontrol system logi and to make the hallenge more realisti , an evaluation traje tory has been designed to ree t typi al phases during approa h to landing. The evaluation riteria given in this subse tion are based on sets of signals from whi h ertain hara teristi s will be al ulated.

All

designs should be able to tra k the given traje tory within the spe ied bounds. Note that the hoi e of a traje tory as an evaluation riterion is independent of the ontrol law and ontrol design methodology. An important subje t onsidered in this hapter is the translation of design obje tives into evaluation riteria: the evaluation riteria should be su iently

166

representative for the onsidered design obje tives, but will not be able to

over all aspe ts. It is asked that the ben hmark problem parti ipants onsider the design obje tives given in subse tion 14.3.2 and for them to use their own methods to illustrate to what extent these are met by their ontroller design. For instan e, we give robustness spe i ations in terms of real parameter variations, although they are often also onsidered in the frequen y domain or in terms of gain and phase margins. The evaluation pro edure is only aimed at obtaining an obje tive measure for omparison with other designs.

14.3.2 Design riteria Introdu tion The ontroller design problem for the RCAM model is hara terised by a number of fundamental trade-os between oni ting design spe i ations.

For

typi al air raft autopilot systems we re ognise ve lasses of riteria:



performan e riteria: these ree t tra king error and disturban e reje tion hara teristi s of ertain signals;



robustness riteria:

these ree t the stability bounds with respe t to

parameter variations;



ride quality riteria: these ree t the desire to obtain su ient passenger and pilot omfort in the form of bounds on ertain maximum allowable a

elerations and minimum damping levels;

 

safety riteria: these ree t envelope safeguards;

ontrol a tivity riteria: these are a measure of the power onsumed by the ontrols and also give an indi ation of fatigue ee ts.

Performan e riteria The performan e of the ontrolled system an be spe ied in terms of ommand response hara teristi s to normalised referen e signals, tra king error and disturban e reje tion features (see [132℄). The ommand response hara teristi s are dened in terms of rise time

tr ,

settling time

ts

Mp . Rise y(t) takes from y = 0:10

and overshoot

time is dened here as the time the unit step response

y = 0:90, i.e., tr = t(y90% ) t(y10%). Settling time is here dened as the time y(t) to a hieve 99 per ent of its nal value. Finally, overshoot is dened as (ypeak y(1))  100% (see [82℄). the relative peak of y (t), i.e., Mp = y(1) P1- Lateral deviation. The ontrolled air raft's lateral deviation, eyb (t), dened to

for

as the dieren e between the a tual and ommanded lateral air raft position,

y(t) y (t), should be redu ed to 10 per ent within 30 s.

There should be very little overshoot in the response to a unit step in

Mp < 5%. 30% in order to obtain higher tra king

lateral ommand signals at altitudes above 305 m (1000 ft), i.e., At lower altitudes

Mp

may in rease to

167

performan e.

There should be no steady state error due to onstant lateral

wind disturban es. In the nal phase of ight (landing approa h glide path) the lateral deviation from the desired ight path should not ex eed that given in gure 14.6.

6Maximum deviation from lo alizer path

6Maximum verti al deviation

20 m

6m

5m

1.5 m

100 ft

400 ft

-

Altitude

100 ft

-

Altitude

400 ft

Figure 14.6: Maximum lateral de-

Figure 14.7: Maximum verti al de-

viation

viation

P2- Altitude

ommands,

response. The ontrolled system should be able to tra k altitude

h ,

with rise time

tr < 12

s and settling time

ts < 45

s.

There

should be very little overshoot in the response to unit steps in altitude om-

mands at altitudes above 305 m (1000 ft), i.e., Mp < 5%. At lower altitudes Mp may in rease to 30% in order to obtain higher tra king performan e. In the nal phase of ight (landing approa h glide path) the verti al deviation from the desired ight path should not ex eed that given in gure 14.7

P3-

Heading angle response. The ommanded heading angle,

tra ked by the a tual heading angle, settling time

ts < 30 s.

, with a rise time

,

tr < 10

should be s and and

There should be very little overshoot in the response

to unit steps in heading ommands at altitudes above 305 m (1000 ft), i.e.,

Mp < 5%.

At lower altitudes

Mp

may in rease to

30%

in order to obtain

higher tra king performan e. For unit RMS intensity lateral Dryden gust, the RMS of the heading angle error in losed loop should be less than that in open loop.

P4-

, should tr < 5 s and

Flight path angle response. The ommanded ight path angle,

be tra ked by the a tual ight path angle, settling time

ts < 20 s.

,

with a rise time

There should be very little overshoot in the response

to unit steps in ight path angle ommands at altitudes above 305 m (1000 ft), i.e.,

Mp < 5%.

At lower altitudes

Mp

may in rease to

30% in order to obtain

higher tra king performan e.

P5-

,

Roll angle response. In ase of engine failure in still air, the roll angle,

should not ex eed 10 deg; its maximum steady state deviation should not

ex eed 5 deg. During engine failure, sideslip angle



should be minimised; the

steady state roll angle that is needed to a hieve this, should be redu ed to zero with an overshoot of less than 50 % when the failed engine is restarted (the failed engine's throttle setting steps ba k to that of the a tive engine). Under moderate turbulen e onditions (see subse tion 14.2.6)

168

 should remain smaller

than 5 deg.

P6-

Airspeed response. The ontrolled system's airspeed,

to tra k speed ommands,

ts < 45 s.

VA ,

with a rise time

VA ,

should be able

s and settling time

There should be very little overshoot in the step response to speed

ommands at altitudes above 305 m (1000 ft), i.e.,

Mp

tr < 12

may in rease to

30%

Mp < 5%.

At lower altitudes

in order to obtain higher tra king performan e. In

the presen e of a wind step with an amplitude of 13 m/s (25 kts) there should be no deviation in the airspeed larger than 2.6 m/s (5 kts) for more than 15 s. There should be no steady state error due to onstant wind disturban es.

P7-

Heading rate.

In ase of engine failure, the maximum heading rate,

_,

should be less than 3 deg/se .

P8-

Cross oupling between airspeed

manded altitude

h

VA

and altitude

h.

For a step in om-

of 30 m, the peak value of the transient of the absolute

VA and ommanded airspeed VA should be smaller than 0.5 m/s VA of 13 m/s (25 kts), the peak value of the transient of the absolute error between h and h should

error between

(1 kt). Conversely, for a step in ommanded airspeed

be smaller than 10 m.

Robustness riteria R1-

Centre of gravity variation. Stability and su ient performan e should

be maintained for horizontal entre of gravity variations between 15 and 31 % and verti al entre of gravity variations between 0 and 21 % of the mean aerodynami hord (see table 14.5; we will not onsider variations in lateral dire tion).

R2-

Mass variations.

Stability and su ient performan e should be main-

tained for air raft mass variations between 100000 to 150000 kg.

R3- Time delay.

Stability and su ient performan e should be maintained for

transport delays from 50 to 100 ms.

Ride quality riteria Ride quality riteria should ensure su ient passenger and pilot omfort. The following spe i ations are designed to obtain an a

eptable level.

Q1- Maximum verti al a

eleration.

Under normal onditions during manoeu-

vres (no turbulen e) the verti al a

eleration at the entre of gravity should be minimised; it should be less than

Q2-

 0.05 g1.

Maximum lateral a

eleration. Under normal onditions during manoeu-

vres (no turbulen e) the lateral a

eleration at the entre of gravity should be minimised; it should be less than

Q3-

 0.02 g.

Damping. Unless stated dierently, there should be no overshoot in any

step response of any ontrolled variable at altitudes above 305 ft (1000 ft). 1

This value is used in industry during the design phase, in fa t the verti al and lateral

a

eleration limits depend on frequen y. They are even lower at 2 Hz.

169

Below that altitude overshoot may in rease to 30 % in order to obtain higher tra king performan e.

Safety riteria S1-

Airspeed. The airspeed must always be larger than

Vstall denotes the stall speed, i.e. to maintain ight. relation:

1:05  V

, where stall the speed below whi h the air raft is unable

This speed an be found from the following equilibrium

1 2 CLmax mg = SVstall 2

(14.58)

Substituting the relevant values from hapter 14.2, and assuming a mass of

120000 kg, we obtain Vstall = 51:8 m/s. S2-

Angle of atta k. In subse tion 14.2.3 it was given that the maximum lift

oe ient is obtained at an angle of atta k of 18 degree.

Hen e, it an be

on luded from the previous requirement that the stall speed orresponds to:

stall = 18 deg.

A value of 12 deg is onsidered a

eptable.

S3- Roll angle. The maximum roll angle  should be limited to 30 deg. S4- Sideslip angle response. At all times, sideslip angle should be minimised. For unit RMS intensity lateral Dryden gust the RMS of the sideslip angle in

losed loop should be less than that in open loop.

Control a tivity riteria C1- A tuator eort minimisation.

Under moderate turbulen e onditions (see

subse tion 14.2.6), mean a tuator rates for aileron, tailplane and rudder should be less than 33 % of the maximum rates (see subse tion 14.2.5).

C2-

Engine eort minimisation. Under moderate turbulen e onditions (see

subse tion 14.2.6), mean throttle rate should be less than 15 % of the maximum rate (see subse tion 14.2.5).

14.3.3 Evaluation pro edure: RCAM mission and s enario To be able to evaluate all kinds of dierent ontrol design pro edures and resulting ontrollers it is ne essary to nd a uniform evaluation pro edure, independent of the design method.

An established pro edure to do this is to

dene a mission and a typi al landing approa h s enario (see [258, 37, 262℄). This mission onsists of manoeuvres that an be evaluated by means of nonlinear simulations. The performan e of the ontrol law depends on the mission phase, within whi h hard riteria or bounds on ertain signals should be met and/or error signals must be minimised. The mission and s enario to be own by the RCAM model onsists of a landing approa h divided into the following segments (see gure 14.8)

170

Trajectory for RCAM evaluation

altitude (−ZE) [m]

1500 2 e

d

f

1

1000 c

b

3 g

h4

a

500

Runway

0 Wind

0 0 −5 −10 −15 y−position (−YE) [km]

−20

−25

−10

−15

−20

−5

0

x−position (XE) [km]

Figure 14.8: the landing approa h for RCAM



Segment I (0 to 1). Starting at an altitude of 1000 m and with a tra k angle of

 = 90 deg, 1

level ight is to be maintained with a onstant airspeed of 80 m/s . During this segment, the lateral features of the autopilot will be investigated by simulating failure of the left engine (engine 1). This is indi ated in gure 14.8: the failure o

urs at point at point

b.

a, after whi h the engine is restarted

The transient and steady state behaviour of the system will

be analysed.



Segment II (1 to 2). This segment onsists of a ommanded o-ordinated turn from points

to

d

_ =3

with a heading rate of

deg/se .

The obje tives are to

maintain a onstant speed of 80 m/s, to keep the lateral a

eleration

lose to zero, and to restri t the bank angle to

 = 30 deg with onsistent

rudder/aileron dee tions.



Segment III (2 to 3). The des ent phase will be started a

ording to the so- alled Frankfurt des ent pro edure (see [35℄), whi h has been proposed for reasons of environmental noise redu tion. This des ent pro edure is engaged later and is steeper than the lassi al des ent, whi h has a onstant glide slope angle of

= 3

deg. The starting altitude is

h = 1000 m. After a short

= 6 deg at point

period of level ight, the ight path angle is set to

e, and to = 3 deg at point f.

1

The desired airspeed is 80 m/s.

The nominal airspeed during the landing phase depends on the air raft mass, it is taken

equal to 1.3 times m/s.

Vstall :

with a maximum landing weight of 150000 kg this results in

171

 80



Segment IV (3 to 4). The glide slope of

= 3 deg is to be maintained during a wind shear

between points

and

g

h.

The air raft has to maintain safe ight and

should not deviate too far from the desired glide path. The wind shear model used in the evaluation pro edure is a two dimensional model derived from [201℄ (also see subse tion 14.3.4 for more information). The desired airspeed is 80 m/s. Throughout the evaluation pro edure a Dryden turbulen e eld, of s ale length

L = 305 m and

amplitude

 = 0:08 m/s,

is assumed to be a tive. Note that

the amplitude is only 5% of the amplitude for moderate onditions as dened in Chapter 14.2: this is done to prevent that the ee t of turbulen e on lateral and longitudinal a

elerations overrules other ee ts that we are interested in. Superimposed on top of this turbulen e is a 10 m/s onstant wind with a xed heading. This onstant wind is a tive in full respe t during Segments I and II

d, and is slowly redu ed to zero between points d of Segment II g of Segment IV (at the start of the wind shear model). The wind has no

until point and

verti al omponent and is dire ted along the negative earth-xed

x-axis,

i.e.,

it is a ross-wind during Segment I and a headwind during Segment III. To he k robustness properties the entire approa h will be own with a most forward, a nominal and a most aft horizontal entre of gravity lo ation. Furthermore, one ight will be exe uted with a nominal entre of gravity lo ation and a time delay of 100 ms.

14.3.4 Translation of design riteria into evaluation

riteria It should be noted that it is not possible to he k all desired autopilot features by ying a single landing approa h traje tory.

Furthermore, the evaluation

pro edure should be relatively simple and straightforward: we want to be able to apply it to a great variety of ontrollers.

Hen e, the evaluation riteria

should be independent of the type of ontroller used: they should onsist of

al ulable indi ators that enable us to obtain an obje tive omparison between

ompletely dierent ontrollers. For these evaluation riteria we will use the same lassi ation as was given in the denition of the design riteria.

    

performan e robustness ride quality safety

ontrol a tivity

For ea h of these items and for ea h of the four traje tory segments a single number will be al ulated. This number should not be onsidered to be the

172

nal word on overall autopilot performan e: it is merely an indi ator for one or two important aspe ts. In most ases it is hosen su h that a value of smaller than one is a

eptable. To further evaluate the dynami behaviour of the autopilot, we will onsider several plots of key variables during ea h of the segments.

We will ompare

the shape of the a tual traje tory with the demanded traje tory and provide bounds that should be respe ted for good performan e. Similarly, we will plot the most important deviations from the desired traje tory.

Segment I For segment I we will plot a plan view of the referen e traje tory and the four traje tories dened in subse tion 14.3.3, and then superimpose the bounds

a

given in gure 14.9. The points

and

b orrespond to the

beginning and end

First segment: top view

x−deviation [m]

100

50

0

0

a

b

1

−50

−100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−2

Figure 14.9: Segment I: the ee t of engine failure with bounds of the engine failure segment.

Performan e. The bound of 20 m is given to a

ount for the ee t of turbulen e. During engine failure, we allow a maximum lateral deviation of 100 m that should be qui kly redu ed to less than 20 m at the end of the segment (when the air raft should be stabilised again). With

eyb denoting the lateral deviation

in body o-ordinates for the traje tory with nominal entre of gravity and time delay we will use

 j eyb (t)j jeyb (t1 )j =2 + max



t

100

20

(14.59)

as a measure that should be smaller than one for su ient performan e. Note that during the entire segment the maximum deviation of 100 m should not be

t1 orresponds with point 1) the

ex eeded, and that at the end of the segment (

maximum deviation of 20 m is taken into a

ount.

Performan e deviation. The maximum dieren es between the lateral deviation of the traje tories with nominal and perturbed entre of gravity and nominal and maximal time delays are onsidered:


eyb (t) := max (jeybmax(t) eyb (t)j; jeybmin (t) eyb (t)j) 173

(14.60)

We will allow dieren es of 10% of the maximal allowable lateral deviations:






eyb (t)
eyb (t1 ) max =2 + t 10 2

(14.61)

should be smaller than one.

Ride quality. The maximum lateral a

eleration will be limited by:

jny j < 1

(14.62)

0:2

i.e.

jny j should be smaller than 0.2g: under normal ight onditions this value

should be mu h lower (0.02g, see subse tion 14.3.2), but engine failure is an emergen y situation su h that an unusually large lateral a

eleration is a

eptable.

Safety. During the segment, the maximum angle of atta k

max t



 will be limited:

j(t)j 3 < 1 12

(14.63)

 = 12 deg; the power is taken to stress the fa t that  > 12 deg qui kly be omes una

eptable (stall situation).

This implies we a

ept

Control a tivity. The rudder a tuator eort will be onsidered that is needed to stabilise the air raft after engine failure is lifted: this is al ulated as:

Z t1

tb with

ÆR2 dt

(14.64)

tb denoting the end of engine failure ( orresponds to point b in gure 14.9).

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

Segment II For segment II we will plot a plan view of the referen e traje tory and the four traje tories dened in subse tion 14.3.3, and then superimpose the bounds given in gure 14.10. Furthermore, to obtain a better insight in the results, we will plot lateral deviations with bounds as given in gure 14.11.

Performan e. The maximum lateral deviation (due to the turn) and the lateral deviation at the end of the segment (when the air raft should be stabilised again) are onsidered:



 j eyb (t)j jeyb (t2 )j max =2 < 1 +

t

200

20

174

(14.65)

Second segment: top view

Second segment: lateral deviations

1

300

0.5 d

2

200 lateral deviation [m]

y−position (−YE) [km]

0 −0.5 −1 −1.5

c

−2 −2.5 −3

1

100

0 1

c

d

2

−100

−200

−3.5 −4 −24

−23

−22

−21 −20 −19 x−position (XE) [km]

−18

−17

−300 0

−16

1

2

3 4 5 6 along track distance from point 1 [km]

Figure 14.10: Segment II: plan view

Figure 14.11:

of the 90 degree turn with bounds

deviations

7

8

Segment II: lateral

during

the

90

degree

turn with bounds

Note that during the entire segment a maximum deviation of 200 m should not

t2 orresponds with point 2)

be ex eeded, and that at the end of the segment (

a maximum deviation of 20 m is taken into a

ount.

Performan e deviation.

As in segment I, the maximum dieren es between

the lateral deviation of the traje tories with nominal and perturbed entre of gravity lo ations and nominal and maximal time delays are onsidered. Again, we will allow dieren es of 10% of the maximal allowable lateral deviations:



Ride quality.




eyb (t)
eyb (t2 ) max =2 < 1 + t 20 2

As in segment I, the maximum lateral a

eleration

onsidered:

i.e.

(14.66)

ny

jny j < 1 0:02

will be

(14.67)

jny j should be smaller than 0.02 g.

Safety. As in segment I, the maximum angle of atta k during the segment is limited:

max t



j(t)j 3 < 1 12

(14.68)

Control a tivity. The rudder and aileron a tuator eort is al ulated as:

Z t2

t1


ÆR2 + ÆA2 dt

(14.69)

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

175

Segment III For segment III we will plot a side view of the four traje tories dened in subse tion 14.3.3. Figure 14.12 shows the referen e traje tory, the start and

2

end points of the segment (points

ommand a tions are labelled with

and

e

3) and the onsidered bounds; the f. We will also plot the verti al

and

deviation of the traje tories and overlay the bounds shown in gure 14.13. Third segment: side view

Third segment: altitude deviations

1100

1000

30 2

20 altitude deviation [m]

altitude (−ZE) [m]

e 900

800 f

700

600

500 −17

−15

Figure 14.12:

−14 −13 x−position (XE) [km]

−12

Segment

−11

III:

f

3

0 e

−10

−20

3

−16

10 2

−30

−10

side

−16

−15

Figure 14.13:

−14 −13 x−position (XE) [km]

−12

−11

Segment III: verti-

view of the -6 and -3 degree glides-

al deviations during the -6 and -

lope aptures with bounds

3 degree glideslope aptures with bounds

Performan e.

The maximum verti al deviation during the apture of the -6

degree glideslope and the verti al deviation at the end of the segment (when the air raft should be stabilised again) are onsidered.

Furthermore, speed

variations should be kept small in spite of the hange in required angle of atta k. With

ezb

denoting the verti al deviation in body o-ordinates for the

traje tory with nominal entre of gravity and time delay, we will demand



 j ezb (t)j jezb (t3 )j jVA VA j =3 < 1 max + +

20

t

for su ient performan e.

6

4

(14.70)

Note that during the entire segment a maximum

deviation of 20 m should not be ex eeded, and that at the end of the segment

t3

(

orresponds with point

3)

a maximum deviation of 6 m is taken into a -

ount. Speed variations should not ex eed 4 m/s, i.e. 5% of

VA = 80 m/s).

Performan e deviation. The maximum dieren es between the verti al deviation of the traje tories for the nominal and perturbed entre of gravity lo ations and nominal and maximal time delays are onsidered:


ezb (t) := max (jezbmax(t) ezb (t)j; jezbmin (t) ezb (t)j)

(14.71)

We will allow dieren es of 10% of the maximal allowable verti al deviations:






ezb (t)
ezb (t3 ) max =2 < 1 + t 2 0:6 176

(14.72)

Ride quality. The maximum verti al a

elera