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MULTIVARIABLEFEEDBACK CONTROLAnalysis and designSigurd SkogestadNorwegian University of Science and TechnologyIan PostlethwaiteUniversity of LeicesterSecond EditionThis version: August 29, 2001JOHN WILEY & SONSChichester.New York.Brisbane.Toronto.Singapore

iiMULTIVARIABLE FEEDBACK CONTROL

BORGHEIM, an engineer:Herregud, en kan da ikke gjøre noe bedre enn leke i dennevelsignede verden. Jeg synes hele livet er som en lek, jeg!Good heavens, one can’t do anything better than play in thisblessed world. The whole of life seems like playing to me!Act one, L ITTLE E YOLF, Henrik Ibsen.

ivMULTIVARIABLE FEEDBACK CONTROL

CONTENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiiCONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vPREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix1 INTRODUCTION . . . . . . . . . . . .1.1 The process of control system design1.2 The control problem . . . . . . . .1.3 Transfer functions . . . . . . . . . .1.4 Scaling . . . . . . . . . . . . . . .1.5 Deriving linear models . . . . . . .1.6 Notation . . . . . . . . . . . . . . .112358112 CLASSICAL FEEDBACK CONTROL .2.1 Frequency response . . . . . . . . . .2.2 Feedback control . . . . . . . . . . .2.3 Closed-loop stability . . . . . . . . .2.4 Evaluating closed-loop performance .2.5 Controller design . . . . . . . . . . .2.6 Loop shaping . . . . . . . . . . . . .2.7 Shaping closed-loop transfer functions2.8 Conclusion . . . . . . . . . . . . . .1515212427394055623 INTRODUCTION TO MULTIVARIABLE CONTROL3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .3.2 Transfer functions for MIMO systems . . . . . . . .3.3 Multivariable frequency response analysis . . . . . .3.4 Control of multivariable plants . . . . . . . . . . . .3.5 Introduction to multivariable RHP-zeros . . . . . . .3.6 Condition number and RGA . . . . . . . . . . . . .63636468798487

viMULTIVARIABLE FEEDBACK CONTROL3.73.83.93.10Introduction to MIMO robustness .General control problem formulationAdditional exercises . . . . . . . . .Conclusion . . . . . . . . . . . . . 91. 98. 110. 1124 ELEMENTS OF LINEAR SYSTEM THEORY .4.1 System descriptions . . . . . . . . . . . . . .4.2 State controllability and state observability . .4.3 Stability . . . . . . . . . . . . . . . . . . . .4.4 Poles . . . . . . . . . . . . . . . . . . . . . .4.5 Zeros . . . . . . . . . . . . . . . . . . . . .4.6 Some remarks on poles and zeros . . . . . . .4.7 Internal stability of feedback systems . . . . .4.8 Stabilizing controllers . . . . . . . . . . . . .4.9 Stability analysis in the frequency domain . .4.10 System norms . . . . . . . . . . . . . . . . .4.11 Conclusion . . . . . . . . . . . . . . . . . .1131131221271281321351391431451521585 LIMITATIONS ON PERFORMANCE IN SISO SYSTEMS . . . . . .5.1 Input-Output Controllability . . . . . . . . . . . . . . . . . . . . .5.2 Perfect control and plant inversion . . . . . . . . . . . . . . . . . .5.3 Constraints on S and T . . . . . . . . . . . . . . . . . . . . . . . .5.4 Ideal ISE optimal control . . . . . . . . . . . . . . . . . . . . . . .5.5 Limitations imposed by time delays . . . . . . . . . . . . . . . . .5.6 Limitations imposed by RHP-zeros . . . . . . . . . . . . . . . . . .5.7 RHP-zeros amd non-causal controllers . . . . . . . . . . . . . . . .5.8 Limitations imposed by unstable (RHP) poles . . . . . . . . . . . .5.9 Combined unstable (RHP) poles and zeros . . . . . . . . . . . . . .5.10 Performance requirements imposed by disturbances and commands5.11 Limitations imposed by input constraints . . . . . . . . . . . . . . .5.12 Limitations imposed by phase lag . . . . . . . . . . . . . . . . . .5.13 Limitations imposed by uncertainty . . . . . . . . . . . . . . . . .5.14 Summary: Controllability analysis with feedback control . . . . . .5.15 Summary: Controllability analysis with feedforward control . . . .5.16 Applications of controllability analysis . . . . . . . . . . . . . . . .5.17 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 012106 LIMITATIONS ON PERFORMANCE IN MIMO SYSTEMS .6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .6.2 Constraints on S and T . . . . . . . . . . . . . . . . . . . .6.3 Functional controllability . . . . . . . . . . . . . . . . . . .6.4 Limitations imposed by time delays . . . . . . . . . . . . .6.5 Limitations imposed by RHP-zeros . . . . . . . . . . . . . .213213214218219220.

CONTENTS6.66.76.86.96.106.116.12viiLimitations imposed by unstable (RHP) poles . . .RHP-poles combined with RHP-zeros . . . . . . .Performance requirements imposed by disturbancesLimitations imposed by input constraints . . . . . .Limitations imposed by uncertainty . . . . . . . .MIMO Input-output controllability . . . . . . . . .Conclusion . . . . . . . . . . . . . . . . . . . . .2232242262282342462517 UNCERTAINTY AND ROBUSTNESS FOR SISO SYSTEMS7.1 Introduction to robustness . . . . . . . . . . . . . . . . . . .7.2 Representing uncertainty . . . . . . . . . . . . . . . . . . .7.3 Parametric uncertainty . . . . . . . . . . . . . . . . . . . .7.4 Representing uncertainty in the frequency domain . . . . . .7.5 SISO Robust stability . . . . . . . . . . . . . . . . . . . . .7.6 SISO Robust performance . . . . . . . . . . . . . . . . . .7.7 Examples of parametric uncertainty . . . . . . . . . . . . .7.8 Additional exercises . . . . . . . . . . . . . . . . . . . . . .7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .2532532552582592702772842892918 ROBUST STABILITY AND PERFORMANCE ANALYSIS .8.1 General control configuration with uncertainty . . . . . . .8.2 Representing uncertainty . . . . . . . . . . . . . . . . . .8.3 Obtaining P , N and M . . . . . . . . . . . . . . . . . . .8.4 Definitions of robust stability and robust performance . . .8.5 Robust stability of the M -structure . . . . . . . . . . . .8.6 RS for complex unstructured uncertainty . . . . . . . . . .8.7 RS with structured uncertainty: Motivation . . . . . . . . .8.8 The structured singular value . . . . . . . . . . . . . . . .8.9 Robust stability with structured uncertainty . . . . . . . .8.10 Robust performance . . . . . . . . . . . . . . . . . . . . .8.11 Application: RP with input uncertainty . . . . . . . . . . .8.12 -synthesis and DK -iteration . . . . . . . . . . . . . . .8.13 Further remarks on . . . . . . . . . . . . . . . . . . . .8.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .2932932963033053073093123143223263303393483519 CONTROLLER DESIGN . . . . . . . . .9.1 Trade-offs in MIMO feedback design9.2 LQG control . . . . . . . . . . . . . .9.3 H2 and H1 control . . . . . . . . . .9.4 H1 loop-shaping design . . . . . . .9.5 Conclusion . . . . . . . . . . . . . .355355359368382403.10 CONTROL STRUCTURE DESIGN . . . . . . . . . . . . . . . . . . . 405

viiiMULTIVARIABLE FEEDBACK ction . . . . . . . . . . . . . . . . . .Optimization and control . . . . . . . . . . .Selection of controlled outputs . . . . . . . .Selection of manipulations and measurementsRGA for non-square plant . . . . . . . . . .Control configuration elements . . . . . . . .Hierarchical and partial control . . . . . . . .Decentralized feedback control . . . . . . . .Conclusion . . . . . . . . . . . . . . . . . .40540741041641842042944145811 MODEL REDUCTION . . . . . . . . . . . . . . . .11.1 Introduction . . . . . . . . . . . . . . . . . . . .11.2 Truncation and residualization . . . . . . . . . .11.3 Balanced realizations . . . . . . . . . . . . . . .11.4 Balanced truncation and balanced residualization11.5 Optimal Hankel norm approximation . . . . . . .11.6 Two practical examples . . . . . . . . . . . . . .11.7 Reduction of unstable models . . . . . . . . . . .11.8 Model reduction using MATLAB . . . . . . . . .11.9 Conclusion . . . . . . . . . . . . . . . . . . . .45945946046246346446747647747812 CASE STUDIES . . . .12.1 Introduction . . . .12.2 Helicopter control .12.3 Aero-engine control12.4 Distillation process12.5 Conclusion . . . .479479480490500506A 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 functionA.7 Linear fractional transformations . . . .509509512515522526539541B PROJECT WORK and SAMPLE EXAM . . . . . . . . . . . . . . . . 545B.1 Project work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545B.2 Sample exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

PREFACEThis 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 (usuallyto make the response stable and sufficiently fast), and to reduce the sensitivity of thesystem to signal uncertainty (disturbances) and model uncertainty. Important topicscovered in the book, include classical frequency-domain methodsanalysis of directions in multivariable systems using the singular valuedecompositioninput-output controllability (inherent control limitations in the plant)model uncertainty and robustnessperformance requirementsmethods for controller design and model reductioncontrol structure selection and decentralized controlThe treatment is for linear systems. The theory is then much simpler and more welldeveloped, and a large amount of practical experience tells us that in many caseslinear controllers designed using linear methods provide satisfactory performancewhen applied to real nonlinear plants.We have attempted to keep the mathematics at a reasonably simple level, and weemphasize results that enhance insight and intuition. The design methods currentlyavailable for linear systems are well developed, and with associated software itis relatively straightforward to design controllers for most multivariable plants.However, without insight and intuition it is difficult to judge a solution, and to knowhow 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 coursein multivariable control or for an advanced undergraduate course. We also thinkit will be useful for engineers who want to understand multivariable control, itslimitations, and how it can be applied in practice. There are numerous workedexamples, exercises and case studies which make frequent use of MATLAB TM 1 .1 MATLAB is a registered trademark of The MathWorks, Inc.

xMULTIVARIABLE FEEDBACK CONTROLThe prerequisites for reading this book are an introductory course in classicalsingle-input single-output (SISO) control and some elementary knowledge ofmatrices and linear algebra. Parts of the book can be studied alone, and provide anappropriate background for a number of linear control courses at both undergraduateand graduate levels: classical loop-shaping control, an introduction to multivariablecontrol, advanced multivariable control, robust control, controller design, controlstructure design and controllability analysis.The book is partly based on a graduate multivariable control course given by thefirst author in the Cybernetics Department at the Norwegian University of Scienceand Technology in Trondheim. About 10 students from Electrical, Chemical andMechanical Engineering have taken the course each year since 1989. The coursehas usually consisted of 3 lectures a week for 12 weeks. In addition to regularassignments, the students have been required to complete a 50 hour design projectusing MATLAB. In Appendix B, a project outline is given together with a sampleexam.Examples and internetMost of the numerical examples have been solved using MATLAB. Some samplefiles are included in the text to illustrate the steps involved. Most of these files usethe -toolbox, and some the Robust Control toolbox, but in most cases the problemscould have been solved easily using other software packages.The following are available over the internet: MATLAB files for examples and figuresSolutions to selected exercisesLinear state-space models for plants used in the case studiesCorrections, comments to chapters, extra exercises and exam setsThis information can be accessed from the authors’ home pages: e.ac.uk/engineering/staff/PostlethwaiteComments and questionsPlease send questions, errors and any comments you may have to the authors. Theiremail addresses are: [email protected]@le.ac.uk

PREFACExiAcknowledgementsThe contents of the book are strongly influenced by the ideas and courses ofProfessors John Doyle and Manfred Morari from the first author’s time as a graduatestudent at Caltech during the period 1983-1986, and by the formative years, 19751981, the second author spent at Cambridge University with Professor AlistairMacFarlane. We thank the organizers of the 1993 European Control Conference forinviting us to present a short course on applied H 1 control, which was the startingpoint for our collaboration. The final manuscript began to take shape in 1994-95during a stay the authors had at the University of California at Berkeley – thanks toAndy 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 4.5were 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 assistedin editing and typing various versions of the manuscript, including Zi-Qin Wang,Yongjiang Yu, Greg Becker, Fen Wu, Regina Raag and Anneli Laur. We alsoacknowledge the contributions from our graduate students, notably Neale Foster,Morten Hovd, Elling W. Jacobsen, Petter Lundström, John Morud, Raza Samar andErik 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 AeroEngines Ltd, and the UK Ministry of Defence, DRA Bedford, respectively.Finally, thanks to colleagues and former colleagues at Trondheim and Caltechfrom 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 recommendZhou, Doyle and Glover (1996) as an excellent reference on system theory and H 1control. Of the others we would like to acknowledge, and recommend for furtherreading, the following: Rosenbrock (1970), Rosenbrock (1974), Kwakernaak andSivan (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 MATLABtoolbox manuals of Grace et al. (1992), Balas et al. (1993) and Chiang and Safonov(1992).Second editionIn this second edition, we have corrected a number of minor mistakes and madenumerous changes and additions to the text, partly arising from the many questionsand comments we have received from interested readers. All corrections to the first

xiiMULTIVARIABLE FEEDBACK CONTROLedition are available on the web. We have tried to minimize the changes in numberingof pages, figures, examples, exercises and equation, so there should be little problemin using the two editions in parallel.

PREFACEBlank pagexiii

1INTRODUCTIONIn this chapter, we begin with a brief outline of the design process for control systems. Wethen discuss linear models and transfer functions which are the basic building blocks for theanalysis and design techniques presented in this book. The scaling of variables is critical inapplications and so we provide a simple procedure for this. An example is given to show howto derive a linear model in terms of deviation variables for a practical application. Finally, wesummarize the most important notation used in the book.1.1 The process of control system designThe process of designing a control system usually makes many demands of theengineer or engineering team. These demands often emerge in a step by step designprocedure as follows:1. Study the system (plant) to be controlled and obtain initial information about thecontrol objectives.2. Model the system and simplify the model, if necessary.3. Scale the variables and analyze the resulting model; determine its properties.4. Decide which variables are to be controlled (controlled outputs).5. Decide on the measurements and manipulated variables: what sensors andactuators will be used and where will they be placed?6. Select the control configuration.7. Decide on the type of controller to be used.8. Decide on performance specifications, based on the overall control objectives.9. Design a controller.10. Analyze the resulting controlled system to see if the specifications are satisfied;and if they are not satisfied modify the specifications or the type of controller.11. Simulate the resulting controlled system, either on a computer or a pilot plant.12. Repeat from step 2, if necessary.13. Choose hardware and software and implement the controller.14. Test and validate the control system, and tune the controller on-line, if necessary.

2MULTIVARIABLE FEEDBACK CONTROLControl courses and text books usually focus on steps 9 and 10 in the aboveprocedure; that is, on methods for controller design and control system analysis.Interestingly, many real control systems are designed without any considerationof these two steps. For example, even for complex systems with many inputs andoutputs, it may be possible to design workable control systems, often based on ahierarchy of cascaded control loops, using only on-line tuning (involving steps 1, 45, 6, 7, 13 and 14). However, in this case a suitable control structure may not beknown at the outset, and there is a need for systematic tools and insights to assistthe designer with steps 4, 5 and 6. A special feature of this book is the provisionof tools for input-output controllability analysis (step 3) and for con

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