Second Edition This version: August 29, 2001

Transcription

1 MULTIVARIABLE FEEDBACK CONTROL Analysis and design Sigurd Skogestad Norwegian University of Science and Technology Ian Postlethwaite University of Leicester Second Edition This version: August 29, 2001 JOHN WILEY & SONS Chichester. New York. Brisbane. Toronto. Singapore

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3 BORGHEIM, an engineer: Herregud, en kan da ikke gjøre noe bedre enn leke i denne velsignede verden. Jeg synes hele livet er som en lek, jeg! Good heavens, one can t do anything better than play in this blessed world. The whole of life seems like playing to me! Act one, LITTLE EYOLF, Henrik Ibsen.

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5 CONTENTS... iii CONTENTS... v PREFACE... ix 1 INTRODUCTION The process of control system design The control problem Transfer functions Scaling Deriving linear models Notation CLASSICAL FEEDBACK CONTROL Frequency response Feedback control Closed-loop stability Evaluating closed-loop performance Controller design Loop shaping Shaping closed-loop transfer functions Conclusion INTRODUCTION TO MULTIVARIABLE CONTROL Introduction Transfer functions for MIMO systems Multivariable frequency response analysis Control of multivariable plants Introduction to multivariable RHP-zeros Condition number and RGA

6 vi MULTIVARIABLE FEEDBACK CONTROL 3.7 Introduction to MIMO robustness General control problem formulation Additional exercises Conclusion ELEMENTS OF LINEAR SYSTEM THEORY System descriptions State controllability and state observability Stability Poles Zeros Some remarks on poles and zeros Internal stability of feedback systems Stabilizing controllers Stability analysis in the frequency domain System norms Conclusion LIMITATIONS ON PERFORMANCE IN SISO SYSTEMS Input-Output Controllability Perfect control and plant inversion Constraints on S and T Ideal ISE optimal control Limitations imposed by time delays Limitations imposed by RHP-zeros RHP-zeros amd non-causal controllers Limitations imposed by unstable (RHP) poles Combined unstable (RHP) poles and zeros Performance requirements imposed by disturbances and commands Limitations imposed by input constraints Limitations imposed by phase lag Limitations imposed by uncertainty Summary: Controllability analysis with feedback control Summary: Controllability analysis with feedforward control Applications of controllability analysis Conclusion LIMITATIONS ON PERFORMANCE IN MIMO SYSTEMS Introduction Constraints on S and T Functional controllability Limitations imposed by time delays Limitations imposed by RHP-zeros

7 CONTENTS vii 6.6 Limitations imposed by unstable (RHP) poles RHP-poles combined with RHP-zeros Performance requirements imposed by disturbances Limitations imposed by input constraints Limitations imposed by uncertainty MIMO Input-output controllability Conclusion UNCERTAINTY AND ROBUSTNESS FOR SISO SYSTEMS Introduction to robustness Representing uncertainty Parametric uncertainty Representing uncertainty in the frequency domain SISO Robust stability SISO Robust performance Examples of parametric uncertainty Additional exercises Conclusion ROBUST STABILITY AND PERFORMANCE ANALYSIS General control configuration with uncertainty Representing uncertainty Obtaining P, N and M Definitions of robust stability and robust performance Robust stability of the M -structure RS for complex unstructured uncertainty RS with structured uncertainty: Motivation The structured singular value Robust stability with structured uncertainty Robust performance Application: RP with input uncertainty μ-synthesis and DK-iteration Further remarks on μ Conclusion CONTROLLER DESIGN Trade-offs in MIMO feedback design LQG control H 2 and H 1 control H 1 loop-shaping design Conclusion CONTROL STRUCTURE DESIGN

8 viii MULTIVARIABLE FEEDBACK CONTROL 10.1 Introduction Optimization and control Selection of controlled outputs Selection of manipulations and measurements RGA for non-square plant Control configuration elements Hierarchical and partial control Decentralized feedback control Conclusion MODEL REDUCTION Introduction Truncation and residualization Balanced realizations Balanced truncation and balanced residualization Optimal Hankel norm approximation Two practical examples Reduction of unstable models Model reduction using MATLAB Conclusion CASE STUDIES Introduction Helicopter control Aero-engine control Distillation process Conclusion A MATRIX THEORY AND NORMS A.1 Basics A.2 Eigenvalues and eigenvectors A.3 Singular Value Decomposition A.4 Relative Gain Array A.5 Norms A.6 Factorization of the sensitivity function A.7 Linear fractional transformations B PROJECT WORK and SAMPLE EXAM B.1 Project work B.2 Sample exam BIBLIOGRAPHY INDEX

9 PREFACE This is a book on practical feedback control and not on system theory generally. Feedback is used in control systems to change the dynamics of the system (usually to make the response stable and sufficiently fast), and to reduce the sensitivity of the system to signal uncertainty (disturbances) and model uncertainty. Important topics covered in the book, include ffl classical frequency-domain methods ffl analysis of directions in multivariable systems using the singular value decomposition ffl input-output controllability (inherent control limitations in the plant) ffl model uncertainty and robustness ffl performance requirements ffl methods for controller design and model reduction ffl control structure selection and decentralized control The treatment is for linear systems. The theory is then much simpler and more well developed, and a large amount of practical experience tells us that in many cases linear controllers designed using linear methods provide satisfactory performance when applied to real nonlinear plants. We have attempted to keep the mathematics at a reasonably simple level, and we emphasize results that enhance insight and intuition. The design methods currently available for linear systems are well developed, and with associated software it is relatively straightforward to design controllers for most multivariable plants. However, without insight and intuition it is difficult to judge a solution, and to know how to proceed (e.g. how to change weights) in order to improve a design. The book is appropriate for use as a text for an introductory graduate course in multivariable control or for an advanced undergraduate course. We also think it will be useful for engineers who want to understand multivariable control, its limitations, and how it can be applied in practice. There are numerous worked examples, exercises and case studies which make frequent use of MATLAB TM 1. 1 MATLAB is a registered trademark of The MathWorks, Inc.

10 x MULTIVARIABLE FEEDBACK CONTROL The prerequisites for reading this book are an introductory course in classical single-input single-output (SISO) control and some elementary knowledge of matrices and linear algebra. Parts of the book can be studied alone, and provide an appropriate background for a number of linear control courses at both undergraduate and graduate levels: classical loop-shaping control, an introduction to multivariable control, advanced multivariable control, robust control, controller design, control structure design and controllability analysis. The book is partly based on a graduate multivariable control course given by the first author in the Cybernetics Department at the Norwegian University of Science and Technology in Trondheim. About 10 students from Electrical, Chemical and Mechanical Engineering have taken the course each year since The course has usually consisted of 3 lectures a week for 12 weeks. In addition to regular assignments, the students have been required to complete a 50 hour design project using MATLAB. In Appendix B, a project outline is given together with a sample exam. Examples and internet Most of the numerical examples have been solved using MATLAB. Some sample files are included in the text to illustrate the steps involved. Most of these files use the μ-toolbox, and some the Robust Control toolbox, but in most cases the problems could have been solved easily using other software packages. The following are available over the internet: ffl MATLAB files for examples and figures ffl Solutions to selected exercises ffl Linear state-space models for plants used in the case studies ffl Corrections, comments to chapters, extra exercises and exam sets This information can be accessed from the authors home pages: ffl ffl Comments and questions Please send questions, errors and any comments you may have to the authors. Their addresses are: ffl Sigurd.Skogestad@chembio.ntnu.no ffl ixp@le.ac.uk

11 PREFACE xi Acknowledgements The contents of the book are strongly influenced by the ideas and courses of Professors John Doyle and Manfred Morari from the first author s time as a graduate student at Caltech during the period , and by the formative years, , the second author spent at Cambridge University with Professor Alistair MacFarlane. We thank the organizers of the 1993 European Control Conference for inviting us to present a short course on applied H 1 control, which was the starting point for our collaboration. The final manuscript began to take shape in during a stay the authors had at the University of California at Berkeley thanks to Andy Packard, Kameshwar Poolla, Masayoshi Tomizuka and others at the BCCI-lab, and to the stimulating coffee at Brewed Awakening. We are grateful for the numerous technical and editorial contributions of Yi Cao, Kjetil Havre, Ghassan Murad and Ying Zhao. The computations for Example 4.5 were performed by Roy S. Smith who shared an office with the authors at Berkeley. Helpful comments and corrections were provided by Richard Braatz, Jie Chen, Atle C. Christiansen, Wankyun Chung, Bjørn Glemmestad, John Morten Godhavn, Finn Are Michelsen and Per Johan Nicklasson. A number of people have assisted in editing and typing various versions of the manuscript, including Zi-Qin Wang, Yongjiang Yu, Greg Becker, Fen Wu, Regina Raag and Anneli Laur. We also acknowledge the contributions from our graduate students, notably Neale Foster, Morten Hovd, Elling W. Jacobsen, Petter Lundström, John Morud, Raza Samar and Erik A. Wolff. The aero-engine model (Chapters 11 and 12) and the helicopter model (Chapter 12) are provided with the kind permission of Rolls-Royce Military Aero Engines Ltd, and the UK Ministry of Defence, DRA Bedford, respectively. Finally, thanks to colleagues and former colleagues at Trondheim and Caltech from the first author, and at Leicester, Oxford and Cambridge from the second author. We have made use of material from several books. In particular, we recommend Zhou, Doyle and Glover (1996) as an excellent reference on system theory and H 1 control. Of the others we would like to acknowledge, and recommend for further reading, the following: Rosenbrock (1970), Rosenbrock (1974), Kwakernaak and Sivan (1972), Kailath (1980), Chen (1984), Francis (1987), Anderson and Moore (1989), Maciejowski (1989), Morari and Zafiriou (1989), Boyd and Barratt (1991), Doyle et al. (1992), Green and Limebeer (1995), Levine (1995), and the MATLAB toolbox manuals of Grace et al. (1992), Balas et al. (1993) and Chiang and Safonov (1992). Second edition In this second edition, we have corrected a number of minor mistakes and made numerous changes and additions to the text, partly arising from the many questions and comments we have received from interested readers. All corrections to the first

12 xii MULTIVARIABLE FEEDBACK CONTROL edition are available on the web. We have tried to minimize the changes in numbering of pages, figures, examples, exercises and equation, so there should be little problem in using the two editions in parallel.