Jean-François Magni, Samir Bennani and Jan Terlouw (Eds.)
Robust Flight Control: A Design Challenge
i
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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).
iii
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
viii
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 2.5. 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 fg 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 3.4.4. 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
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)
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
[S0 (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 ÆZE 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 ÆME 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: jjjj1 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 !; kk1 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 ; kpert 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
jju(k + i)jj2R
(12.9)
subje t to onstraints su h as
juj (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
< < <