Linköping studies in science and technology. Dissertations No. 406

Transcription

1 Linköping studies in science and technology. Dissertations No. 406 Methods for Robust Gain Scheduling Anders Helmersson Department of Electrical Engineering Linkoping University S Linkoping, Sweden andersh@isy.liu.se November 7, 1995

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3 iii Abstract This thesis considers the analysis of systems with uncertainties and the design of controllers to such systems. Uncertainties are treated in a relatively broad sense covering gain-bounded elements that are not known a priori but could be available to the controller in real time. The uncertainties are in the most general case norm-bounded operators with a given block-diagonal structure. The structure includes parameters, linear timeinvariant and time-varying systems as well as nonlinearities. In some applications the controller may have access to the uncertainty, e.g. a parameter that depends on some known condition. There exist well-known methods for determining stability of systems subject to uncertainties. This thesis is within the framework for structured singular values also denoted by. Given a certain class of uncertainties, is the inverse of the size of the smallest uncertainty that causes the system to become unstable. Thus, is a measure of the system's \structured gain". In general it is not possible to compute exactly, but an upper bound can be determined using ecient numerical methods based on linear matrix inequalities. An essential contribution in this thesis is a new synthesis algorithm for nding controllers when parametric (real) uncertainties are present. This extends previous results on synthesis involving dynamic (complex) uncertainties. Specically, we can design gain scheduling controllers using the new synthesis theorem, with less conservativeness than previous methods. Also, algorithms for model reduction of uncertainty systems are given. A gain scheduling controller is a linear regulator whose parameters are changed as a function of the varying operating conditions. By treating nonlinearities as uncertainties, methods can be used in gain scheduling design. In the discussion, emphasis is put on how to take into consideration dierent characteristics of the time-varying properties of the system to be controlled. Also robustness and its relation with gain scheduling are treated. In order to handle systems with time-invariant uncertainties, both linear systems and constant parameters, a set of scalings and multipliers are introduced. These are matched to the properties of the uncertainties. Also, multipliers for treating uncertainties that are slowly varying, such that the rate of change is bounded, are introduced. Using these multipliers the applicability of the analysis and synthesis results are greatly extended.

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5 v Preface This thesis is a combined result of a personal ten-year experience as a control engineer and a couple of years of scientic research. After my graduation at Lund Institute of Technology in 1981, I moved to Linkoping where I worked with missile guidance during my rst three professional years. In 1984 I joined Saab Ericsson Space where I quite soon got involved in trajectory guidance and attitude control systems for sounding rockets, small satellites and shuttles [48, 49, 50, 47]. I also had the opportunity and privilege to participate in the launches of two record-breaking sounding rockets from Esrange, Sweden: Maxus Test reaching an apogee of 5 km and Maxus 1B reaching 717 km, both of which were guided by our control systems. During this time I used, what I believe is, the most common approach in aerospace industry to this kind of problem: to regard gain scheduling as a quasistationary problem. This means that the design in each ight condition, or operating point, is treated as a linear static problem, which is solved using traditional tools. With this I felt somewhat unsatised since no analytical tools or methods were available at that time to treat the time-varying aspects of the problems. Simulations were used for showing that the system behaved well when subjected to these changes. We must however bear in mind that this approach has been successful with very few failures, if any, due to this negligence. However, my hope with the formal approach is to be able to reduce the turn-around and cost for new control designs by reducing the simulation and test eorts. During my time as a control engineer I took some graduate courses at the department of Electrical Engineering at Linkoping University. In 199 I was convinced by Professor Lennart Ljung to join the group at Automatic Control where I got a part-time position as a Ph.D. student. Some of the applications in this thesis are inspired by my work at Saab Ericsson Space with guidance and control system for sounding rockets and small launchers. The problems are, however, not fully realistic since parameters and models have been modied to protect proprietary rights and the models have been simplied to better show the principles of the applied methods and algorithms.

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7 vii Acknowledgments I would like to express my sincere gratitude to the people who made this work possible by their inspiration and contributions. First of all I would like to thank my supervisor Professor Lennart Ljung for his excellent guidance and inspiring discussions throughout the elaboration of this thesis. He and Professor Torkel Glad have succeeded in creating a stimulating atmosphere in the group. Also, much of the nice spirit is due to our secretary Ulla Salaneck. I would also like to thank Professor Karl-Johan Astrom for drafting me for the automatic control team during my time as an undergraduate student at Lund Institute of Technology. Ever since the manuscript of this thesis started to grow as an embryo, it has been largely inuenced and improved by the many proof-readers: Karin Stahl Gunnarsson, Dr. Ke Wang Chen, Dr. Inger Klein, Dr. Tomas McKelvey, Dr. Fredrik Gustafsson, Dr. Henrik Jonson and Mats Jirstrand. Their many suggestions and constructive criticisms have greatly contributed to the nal touch of the thesis. I also would like to thank Peter Lindskog and Dr. Roger Germundsson for their valuable helps and hints on L A TEX and Magnus Sundstedt for keeping the computers running. The numerical solutions of the linear matrix inequalities have been performed using LMItool version.0, which was provided for free by Dr. Pascal Gahinet, INRIA, Le Chesnay, France. I am also grateful to Saab Ericsson Space for letting me be on leave for completing the thesis. I would also like to thank the sta at the Linkoping oce for creating a nice and friendly place to work at. This work was supported by the Swedish National Board for Industrial and Technical Development (NUTEK), which is gratefully acknowledged. Finally, I would like to thank my parents Anna and Torsten, as well as my brothers Lars, Karl Gustav and Bengt and their families for their love and support.

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9 Contents Abstract : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v Acknowledgments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii 1 Introduction Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1. Robustness and Structured Singular Values : : : : : : : : : : : : : : 1. Gain Scheduling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.4 Parameter Variations : : : : : : : : : : : : : : : : : : : : : : : : : : : Contributions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Outline : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 Preliminaries 9.1 Matrices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10. Linear Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Continuous-Time Linear Systems : : : : : : : : : : : : : : : : 10.. Discrete-Time Linear Systems : : : : : : : : : : : : : : : : : : 11.. Similarity Transformations : : : : : : : : : : : : : : : : : : : 1. Norms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1..1 Vector Norms : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.. Singular Value Decomposition : : : : : : : : : : : : : : : : : : 1.. Induced Matrix Norms : : : : : : : : : : : : : : : : : : : : : : 1..4 Rank and Pseudo-inverse : : : : : : : : : : : : : : : : : : : : 1.4 Signal Spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Lebesgue Spaces : : : : : : : : : : : : : : : : : : : : : : : : : Operators : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Induced Norms : : : : : : : : : : : : : : : : : : : : : : : : : : Hardy Spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : 15.5 Factorization of Transfer Matrices : : : : : : : : : : : : : : : : : : : 16 ix

10 x Contents.5.1 Inverse of Transfer Matrices : : : : : : : : : : : : : : : : : : : Factorization of Matrices : : : : : : : : : : : : : : : : : : : : The Riccati Equation and Its Solution : : : : : : : : : : : : : Spectral Factorization : : : : : : : : : : : : : : : : : : : : : : Canonical Factorization : : : : : : : : : : : : : : : : : : : : : The Kalman-Yakubovich-Popov Lemma : : : : : : : : : : : : 0.6 Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Small Gain Theorem : : : : : : : : : : : : : : : : : : : : : : : 1.6. Structured Uncertainties : : : : : : : : : : : : : : : : : : : : : 1.6. Passivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Positive Real Transfer Functions : : : : : : : : : : : : : : : :.7 Performance Bounds : : : : : : : : : : : : : : : : : : : : : : : : : : :.8 Matrix Inequalities : : : : : : : : : : : : : : : : : : : : : : : : : : : :.8.1 Continuous Time : : : : : : : : : : : : : : : : : : : : : : : : :.8. The Riccati Inequality : : : : : : : : : : : : : : : : : : : : : : 4.8. Linear Matrix Inequalities (LMIs) : : : : : : : : : : : : : : : 4.9 Structured Dynamic Uncertainties : : : : : : : : : : : : : : : : : : : Structured Nonlinear Dynamic Uncertainties : : : : : : : : : 8.9. Structured Parametric Uncertainties : : : : : : : : : : : : : : 0.10 Strictness of Quadratic Lyapunov Functions : : : : : : : : : : : : : : 1 Linear Matrix Inequalities.1 Some Standard LMI Problems : : : : : : : : : : : : : : : : : : : : : 4. Interior Point Methods : : : : : : : : : : : : : : : : : : : : : : : : : : 6..1 Analytic Center of an Ane Matrix Inequality : : : : : : : : 6.. The Path of Centers : : : : : : : : : : : : : : : : : : : : : : : 7.. Methods of Centers : : : : : : : : : : : : : : : : : : : : : : : 8..4 Primal and Dual Methods : : : : : : : : : : : : : : : : : : : : 8..5 Complexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9. Software Packages : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9.4 Nonstrict LMIs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40.5 Some Matrix Problems : : : : : : : : : : : : : : : : : : : : : : : : : : Minimizing Matrix Norms : : : : : : : : : : : : : : : : : : : : Minimizing Condition Number : : : : : : : : : : : : : : : : : Treating Complex-Valued LMIs : : : : : : : : : : : : : : : : : 4.6 Rank Conditions and Nonlinear Constraints : : : : : : : : : : : : : : Convexity and Complexity : : : : : : : : : : : : : : : : : : : 4 4 Model Parametrization Linearly Dependent Parametrization : : : : : : : : : : : : : : : : : : Scalar Uncertainties : : : : : : : : : : : : : : : : : : : : : : : Repeated Scalar Blocks : : : : : : : : : : : : : : : : : : : : : Linear Fractional Transformations (LFTs) : : : : : : : : : : : : : : : Upper and Lower LFTs : : : : : : : : : : : : : : : : : : : : : The Star Product : : : : : : : : : : : : : : : : : : : : : : : : : 51

11 Contents xi 4.. Conventions and Notations : : : : : : : : : : : : : : : : : : : 5 4. Nonlinear Parametrization : : : : : : : : : : : : : : : : : : : : : : : : Rational Parametrization : : : : : : : : : : : : : : : : : : : : Reducing Nonlinear LFT Models : : : : : : : : : : : : : : : : Block Uncertainties : : : : : : : : : : : : : : : : : : : : : : : : : : : : Dynamic Uncertainties : : : : : : : : : : : : : : : : : : : : : : : : : : Performance Requirements : : : : : : : : : : : : : : : : : : : : : : : Some Common Uncertainty Structures : : : : : : : : : : : : : : : : : Multiplicative and Additive Uncertainties : : : : : : : : : : : Coprime Factorization : : : : : : : : : : : : : : : : : : : : : : Treating the Frequency as an Uncertainty : : : : : : : : : : : : : : : Continuous-Time Systems : : : : : : : : : : : : : : : : : : : : Discrete-Time Systems : : : : : : : : : : : : : : : : : : : : : : Uncertainty Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 5 Structured Singular Values Rationale : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Structured Singular Values : : : : : : : : : : : : : : : : : : : : : : : Denitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : Upper and Lower Bounds : : : : : : : : : : : : : : : : : : : : Scaling and Multiplier Structures : : : : : : : : : : : : : : : : The Main Loop Theorem : : : : : : : : : : : : : : : : : : : : Strictness of Bounds : : : : : : : : : : : : : : : : : : : : : : : Connection with Bounded Real Lemma : : : : : : : : : : : : Contraction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Commuting and Convex Sets : : : : : : : : : : : : : : : : : : : : : : Commuting Matrices : : : : : : : : : : : : : : : : : : : : : : : Convexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : LMIs with Rank Constraints : : : : : : : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 6 Synthesis Problem Formulation : : : : : : : : : : : : : : : : : : : : : : : : : : : Solvability of LMIs : : : : : : : : : : : : : : : : : : : : : : : : : : : : Solvability of Strict LMIs : : : : : : : : : : : : : : : : : : : : Solvability of Nonstrict LMIs : : : : : : : : : : : : : : : : : : 9 6. Synthesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : An Ane Problem : : : : : : : : : : : : : : : : : : : : : : : : Complex Synthesis : : : : : : : : : : : : : : : : : : : : : : : Mixed Synthesis : : : : : : : : : : : : : : : : : : : : : : : : Shared Uncertainties : : : : : : : : : : : : : : : : : : : : : : : : : : : Structure of Shared Uncertainties : : : : : : : : : : : : : : : : Synthesis with Shared Uncertainties : : : : : : : : : : : : : : Complex Uncertainties : : : : : : : : : : : : : : : : : : : : : : 10

12 xii Contents Real Uncertainties : : : : : : : : : : : : : : : : : : : : : : : : The LFT Gain Scheduling Theorem : : : : : : : : : : : : : : : : : : Finding the Controller : : : : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Comparing Real and Complex Uncertainties : : : : : : : : : : LPV Synthesis : : : : : : : : : : : : : : : : : : : : : : : : : : Conservativeness : : : : : : : : : : : : : : : : : : : : : : : : : Rank Conditions and Convexity : : : : : : : : : : : : : : : : Nonconvex Problems : : : : : : : : : : : : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Model Reduction Model Reduction using LMIs : : : : : : : : : : : : : : : : : : : : : : Gramians and Internal Balancing : : : : : : : : : : : : : : : : Upper and Lower Bounds of Unweighted Approximations : : Optimal Hankel Norm Reduction : : : : : : : : : : : : : : : : LFT Model Reduction : : : : : : : : : : : : : : : : : : : : : : : : : : Solving for ^C and ^D : : : : : : : : : : : : : : : : : : : : : : : Solving for ^A and ^B : : : : : : : : : : : : : : : : : : : : : : : The Pure Complex Case : : : : : : : : : : : : : : : : : : : : : Some Algorithms : : : : : : : : : : : : : : : : : : : : : : : : : 1 7. Minimality : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Reducibility : : : : : : : : : : : : : : : : : : : : : : : : : : : : Verifying Minimality : : : : : : : : : : : : : : : : : : : : : : : Comparison with Classical Tests : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 8 Scalings and Multipliers Uncertainties and Multipliers : : : : : : : : : : : : : : : : : : : : : : Complex and Real Uncertainties : : : : : : : : : : : : : : : : Uncertainty Systems : : : : : : : : : : : : : : : : : : : : : : : Constant Scalings and Multipliers : : : : : : : : : : : : : : : : : : : : 1 8. Frequency Dependent Scalings : : : : : : : : : : : : : : : : : : : : : Scalings for Dynamic Uncertainties : : : : : : : : : : : : : : : Multipliers for Mixed Uncertainties : : : : : : : : : : : : : : : State-Space Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : Dynamic Uncertainties : : : : : : : : : : : : : : : : : : : : : : Mixed Uncertainties : : : : : : : : : : : : : : : : : : : : : : : State-Space Conditions : : : : : : : : : : : : : : : : : : : : : The D-K Iterations : : : : : : : : : : : : : : : : : : : : : : : : : : : Generalization of the D-K Algorithm : : : : : : : : : : : : : The Y -Z-K Iterations : : : : : : : : : : : : : : : : : : : : : : D-K Iterations for Gain Scheduling : : : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15

13 Contents xiii 9 Multipliers for Slowly Time-Varying Systems Parametrization of the Lyapunov Function : : : : : : : : : : : : : : : Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : Approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Connection with Time-Varying Lyapunov Functions : : : : : Structure of Uncertainty Augmentation : : : : : : : : : : : : An Example : : : : : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Frequency Dependent Scalings : : : : : : : : : : : : : : : : : : : : : Uncertainty Augmentation : : : : : : : : : : : : : : : : : : : Treating the Uncertainty as an Operator : : : : : : : : : : : : An Example : : : : : : : : : : : : : : : : : : : : : : : : : : : Structure of Uncertainty Augmentation : : : : : : : : : : : : Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Gain Scheduling and Robustness Gain Scheduling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Aerospace Applications : : : : : : : : : : : : : : : : : : : : : Parameters : : : : : : : : : : : : : : : : : : : : : : : : : : : : Model Representations : : : : : : : : : : : : : : : : : : : : : : Linearizing Nonlinear Models : : : : : : : : : : : : : : : : : : : : : : Dierential Inclusions : : : : : : : : : : : : : : : : : : : : : : Local Linearization : : : : : : : : : : : : : : : : : : : : : : : : Global Linearization with Fixed Equilibrium : : : : : : : : : Moving Reference Point : : : : : : : : : : : : : : : : : : : : : Robustness and Gain Scheduling : : : : : : : : : : : : : : : : : : : : Robustness Aspects of Gain Scheduling : : : : : : : : : : : : Examples of Applications A Rocket Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : Requirements : : : : : : : : : : : : : : : : : : : : : : : : : : : Complex- Design : : : : : : : : : : : : : : : : : : : : : : : : Mixed- Design : : : : : : : : : : : : : : : : : : : : : : : : : : Uncertain Resonant Modes : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : LFT Gain Scheduling : : : : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A Missile Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : The LPV Model : : : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 0 1 Conclusions 05 Bibliography 07

14 xiv Contents Glossary 17 Notations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 Acronyms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 0 Index 1

15 1 Introduction This thesis considers the problem of analysis and design of robust gain scheduling controllers. A robust controller maintains its stability and performance even if the plant to be controlled is uncertain and time-varying. A gain scheduling controller is parametrized by a function of the operating conditions. Uncertainties are treated in a relatively broad sense, covering parametric and dynamic uncertainties, constant or time-varying, which can be available or not to the controller in real time. The approach to analysis and synthesis as based on linear fractional transformations and linear matrix inequalities. This chapter introduces the concepts and gives an outline of the thesis. 1

16 Introduction 1.1 Background The analysis and design of robust controllers have attracted a large number of researchers for more than a decade. One fact that spurred this development was the fact that linear quadratic Gaussian (LQG) controllers can have arbitrarily bad robustness [5]. An important step towards a robust control theory was taken in 1981 when Zames introduced the H 1 control theory [97]. It was soon generalized to also include spatial structure. In 1981 Doyle and Safonov independently introduced similar concepts for this: Doyle called it structured singular values or [6], while Safonov called it stability margins K m of diagonally perturbed systems [81]. The H 1 synthesis, which is a fundamental tool for robust design, was a rather dicult problem until the advent of the two-riccati-equation method [8] in Then robust design tools became easier to use and found its way into many new applications. Relatively soon thereafter, linear matrix inequalities (LMIs) were found to be very well suited for formulating and solving control problems, including H 1 analysis and synthesis. Generalizations to more general control problems, such as gain scheduling synthesis, is possible [0]. In parallel with the theoretical results, numerical methods for solving LMIs eciently were developed and made available. 1. Robustness and Structured Singular Values The approach adopted in this thesis is based on structured singular values also denoted by. Consider a time invariant stable system G that is disturbed by an uncertainty element, which is block diagonal. The blocks in may correspond to disturbances, performance requirements, nonlinearities and parameter variations. - y G u In the analysis we pose the question: which is the \smallest" that can make the system G unstable? The size of is taken as the maximum singular value for parametric uncertainties, or as the L or ` induced norm for dynamic uncertainties. If constant uncertainties are considered we can employ the Nyquist criterion for determining stability. For single-input-single-output systems it states that the closed-loop system is stable if the Nyquist contour, i.e. the graph of 1 G(s) does not encircle the origin as the argument s traverses the imaginary axis. This can be generalized to multivariable systems, in which case the Nyquist contour is given by det(i G(s)).

17 1. Gain Scheduling The analysis involves the problem of nding the smallest such that det(i M) = 0. The structured singular value, denoted by (M), is dened as the inverse of the norm of such a. By letting M = G(j!) and making a sweep through all frequencies! we can provide an upper bound of that guarantees robust stability. Unfortunately the problem is sometimes too hard to compute and instead we must be content with upper and lower bounds. An upper bound can be determined using the small gain theorem on the scaled system, DMD 1. The scalings D are chosen from a set of matrices that commute with the uncertainties, i.e. D = D. By nding the minimum of kdmd 1 k we can nd an upper bound of (M), denoted by (M), and thus if kk is less than the inverse of this bound we have proved stability. The set of admissible scaling matrices depends on the class of uncertainties. If the uncertainties are time-invariant, frequency dependent scalings are allowed; if time-varying or nonlinear uncertainties are present, only constant scalings are admissible. The analysis can be further rened to also take into account that the uncertainties are parametric. This is described as real uncertainties since they do not cause phase shift, in contrast to the dynamic or complex uncertainties discussed so far. Real uncertainties are included in the set of complex uncertainties, but treating them as complex is sometimes too conservative. Real uncertainties appear naturally in models where they may enter as physical parameters. Dynamic or complex uncertainties are also relevant and can be used for taking unmodeled dynamics into account. Another important application of dynamic uncertainties is for specifying performance requirements. This can be performed by augmenting the model with disturbance inputs and performance outputs together with weighting functions. Thus, the robust performance problem, i.e. maintaining performance specications for an uncertainty model, can be included into the robust stability problem. The upper bounds can be computed eciently using linear matrix inequalities (LMIs). An LMI is an ane matrix function with positive or negative denite constraints. Many common control problems, such as H and H 1 synthesis, can be stated as LMIs. Also, the LMI method oers possibilities for analysis and design of the gain scheduling problem. 1. Gain Scheduling Gain scheduling is a nonlinear feedback of a special type. We will use the denition: Gain scheduling: a linear parameter varying (LPV) feedback regulator whose parameters are changed as a function of operating conditions. This thesis is devoted to how to analyze LPV systems and how to synthesize robust gain scheduling controllers.

18 4 Introduction In order to be able to analyze and synthesize gain scheduling systems we need a description of the plant. We assume that an explicit model is available that is linear in the states x but may have known nonlinear dependency on a parameter vector that aects the dynamics of the system: ( _x = A() x + B() u y = C() x + D() u; (1.1) where u is the plant's input, y its output and A, B, C, and D are parameterdependent matrices. Using linear matrix inequalities (LMIs) it is now possible to nd a controller that makes the controlled system quadratically stable for all combinations of parameters if such a controller exists. A system is quadratically stable if there exists a quadratic Lyapunov function, V (x) = x T P x where P is a symmetric matrix, such that _ V < 0 for all x 6= 0. We can extend the problem to also include measures for robustness and performance. In [1, 9] this problem is solved by nding a common solution to a set of LMIs. Three LMIs are in principle needed for every parameter combination. This is done by gridding the parameter space in a dense enough grid and including the grid points in the set of LMIs. It is sometimes possible to reduce the set of points to the vertices of the parameter space. In this thesis we solve a similar problem by rst nding a parametrization of the system (1.1) and then performing the analysis and synthesis on the parametrized system. The structure of the system is of a special type called a linear fractional transformation (LFT). The parameter dependency has been extracted from the original plant and has been placed in a feedback loop as an uncertainty block. The uncertainty block is parametrized by the original parameters and may also include model uncertainties and performance requirements. - () y ~G u The advantage with this parametrization approach is that only three coupled LMIs need to be solved in the synthesis step. Thus there is no need to grid the parameter space or even to check the vertices. The controller, which has the same structure as the LFT model, is more easily parametrized compared to the LPV description.

19 1.4 Parameter Variations 5 On the other hand, a parametrized model on LFT form is to be built from (1.1). During this stage the original model is augmented with inputs and outputs corresponding to the parameter uncertainties (). This process assumes that the model is suciently smooth with respect to the parameters and that they can be approximated with rational functions not necessarily by the parameters themselves but by bounded functions () thereof. Also, in the parametrization step we might introduce conservativeness in the design, since is an upper bound of. 1.4 Parameter Variations In the parametrization above we did not assume anything about how fast the parameters may change. By using this extra piece of information we can in some problems rene the performance of the system. We will here use three classes to describe how fast a parameter is allowed to change: a constant parameter, a slowly varying parameter (bounded _), a fast varying parameter. A constant parameter is virtually constant. For instance the mass of a vehicle can be considered constant during normal operation. The fuel consumption will reduce the mass at a rate that can be neglected compared to the dynamics of the vehicle. Typical examples of slowly varying parameters are velocity and altitude of an aircraft. These parameters have bounds on the rate of change set by e.g. engine performance and operational constraints. Fast varying parameters are not assumed to have any bounds on the rate of change. They are typically used for including nonlinear eects in the linear model (1.1). For showing robust stability we search for scalings that commute with the set of uncertainties. Depending on the characteristic of the uncertainty dierent sets of scalings are used. uncertainty scaling commuting property constant frequency dependent D(s) = D(s) slowly varying frequency dependent D(s) = D(s) + U(s) _V (s) fast varying constant D = D There exist various methods for analysis and design of systems with slowly varying parameters, see e.g. [9]. The approach proposed in this thesis is to augment the original uncertainty block with the time derivative of some blocks of. This yields a model of the same structure as before but with an augmented uncertainty structure.

20 6 Introduction 1.5 Contributions Chapters {5 give introductions to mathematical concepts and to structured singular values, and contain essentially nothing new. The main results are presented in the remaining chapters. The main contributions are listed below: Real- synthesis (chapter 6) and in particular theorem 6.1; Real- model reduction (chapter 7); D-K-like iterations for synthesis with real, constant uncertainties and gain scheduling (chapter 8); Uncertainty augmentation for analyzing systems with slowly varying uncertainties (chapter 9). The real- synthesis forms an important backbone for this thesis. It provides the main tool for gain scheduling synthesis in which the controller is parametrized using LFTs. This allows us to quite accurately approximate smooth functions over a relatively wide parameter range. Using real- synthesis we can not only design controllers but also apply model reduction on the parametrization for reducing the complexity of the model. The conservativeness in the analysis and design is reduced by introducing appropriate scaling or multiplier matrices complying to the temporal characteristics of the uncertainties. These matrices together with either the small gain or the passivity theorems are the main tools for the stability analysis and consequently the synthesis methods. There is an equivalence between the scaling matrices and quadratic Lyapunov functions, which is shown and discussed. Thus, the approach is nothing but a Lyapunov method using quadratic Lyapunov functions. 1.6 Outline The thesis is outlined as follows. In chapter, singular values, signal spaces and signal norms are discussed briey. It also covers factorization, stability and performance bounds. Chapter gives a short introduction to linear matrix inequalities and how they can be solved numerically. The uncertainty description of a parametrized model needed for the analysis and synthesis is discussed in chapter 4. The structured singular values are treated in three chapters. In chapter 5, the denition is given together with a computational method of the upper bound of (M). In chapter 6, new results on real and mixed synthesis are presented and compared with well-known complex synthesis. Results on model reduction based on the synthesis results are presented in chapter 7.

21 1.6 Outline 7 Scalings and multipliers are important tools for both analysis and synthesis. The well-known D-K iterations together with extensions for real- synthesis and gain scheduling are presented and discussed in chapter 8. In chapter 9 the concepts are extended to handle slowly time-varying systems, by using parametrized Lyapunov functions and frequency dependent scalings. In chapter 10, we discuss how to use the proposed methods in some applications, mainly from the aerospace area. Also, linearization of nonlinear models is discussed. This is exemplied in chapter 11, where a few applications are given. Finally the conclusions are given in chapter 1. A Short Tour of the Thesis The readers already familiar with uncertainty models, linear matrix inequalities and structured singular values can probable skip chapter {5. The main theoretical results are presented in chapters 6 and 7. The rst part of chapter 8 is probably also well-known while the last part, concerning D-K iterations with mixed-uncertainties and applications to gain scheduling problems, is new. Also the uncertainty augmentation concept for coping with time-varying uncertainty given in chapter 9 is novel. The examples of applications in chapter 11 would also illustrate some of the potentials and motivations for the techniques presented in this thesis.

22 8 Introduction

23 Preliminaries This chapter gives an introduction to some of the basic concepts for describing systems and their performance. The approach taken here is based on linear dynamic systems that are perturbed by some bounded elements. Three kinds of linear systems are dened: linear time-invariant (LTI), linear time-varying (LTV) and linear parameter-varying (LPV) systems. In order to quantify performance of systems a number of norms are used, such as L and ` norms for signals in the continuous and discrete time cases, respectively. Using these, induced norms for systems can be dened. For LTI systems the L and ` induced norms coincide with the H 1 -norm, which is the maximum gain of the system with respect to frequency. An upper bound of the L -induced gain of a system can be determined as a matrix inequality. Lyapunov and Riccati equations can also be modied into matrix inequalities, which provide criteria for stability and performance bounds. Using Schur complements the quadratic Riccati inequality can be rewritten into a linear matrix inequality (LMI). 9

24 10 Preliminaries.1 Matrices This thesis contains almost 600 matrices. The set of real-valued matrices with p rows and m columns is denoted by R pn and complex-valued matrices of the same size by C pn. The unit matrix of size n n is denoted by I n. Vectors are denoted by R n and C n and are assumed to be column vectors, i.e. R n1 and C n1. A square matrix Y is symmetric if Y = Y T where Y T denotes the transpose of Y. A matrix is Hermitian if Y = Y where Y denotes the complex conjugate transpose. A symmetric real matrix is also Hermitian. A unitary matrix U is square and satises U U = UU = I. A matrix Y is called positive denite, denoted by Y > 0, if it is Hermitian and if x Y x > 0 for all nonzero x; It is called positive semidenite, denoted by Y 0, if x Y x 0 for all x. The denitions are analogous for negative denite and semidenite matrices. We will also use Y > Z and Y Z to denote that Y Z > 0 and Y Z 0 respectively. There exist several methods to determine if a Hermitian matrix is positive denite. One possibility is to check if all its eigenvalues are positive. If they are non-negative the matrix is positive semidenite.. Linear Systems This thesis treats nite-dimensional linear dynamic systems with uncertainties. The main focus is on continuous-time systems...1 Continuous-Time Linear Systems Time-Invariant Systems A linear time-invariant (LTI) system is dened by its state-space representation _x = A x + B u y = C x + D u; (.1) where x R n is the state vector, u R m is the input vector and y R p is the output vector; _x = dt d x denotes the time-derivative of x. The system matrices A, B, C and D, of compatible sizes, are xed in time and describe the behavior of the system. Sometimes we will use the more compact notation, _x A B x = : (.) y C D u The notation G(s) = D + C(sI A) 1 B is also used for describing an LTI system. The symbol s can be interpreted both as the time-derivative operator dt d and as the argument of the Laplace transform of the system G.

25 . Linear Systems 11 A continuous-time LTI system is stable if A has all its eigenvalues, i, in the open left half plane, i.e. Re i < 0. A transfer function is proper if G(1) is bounded. A system is inversely stable, if G 1 (s) is proper and stable. Time-Varying Systems A linear time-varying (LTV) system has time-varying system matrices and is de- ned by _x = A(t) x + B(t) u y = C(t) x + D(t) u: (.) Parameter-Varying Systems A more general description is obtained by letting the system matrices depend on (time-varying) parameters (t). This model is called a linear parameter-varying (LPV) system: _x = A((t)) x + B((t)) u y = C((t)) x + D((t)) u: (.4) The parameters (t) S R s can either be an external input to the system or they can depend on the states of the system. In the latter case we can describe nonlinear eects by an LPV model... Discrete-Time Linear Systems Discrete-time systems are represented similarly by x(t + 1) = A x(t) + B u(t) y(t) = C x(t) + D u(t): (.5) Using the forward-step operator q dened by qx(t) = x(t + 1), this can be rewritten as qx = A x + B u y = C x + D u: (.6) A discrete-time LTI system is stable if A has all its eigenvalues strictly within the unit disc, i.e. j i j < 1. Time-varying and parameter-varying systems are dened analogously to the continuous-time case.

26 1 Preliminaries.. Similarity Transformations A linear system, either LTI, LTV or LPV, can be represented dierently in terms of the system matrices. Two representations, (A; B; C; D) and ( ^A; ^B; ^C; ^D), are similar if there exists a (constant) nonsingular transformation matrix T R nn such that ^A ^B T AT = 1 T B ^C ^D CT 1 : (.7) D The similarity transformation can be interpreted as a mapping from one base in the state space to another: ^x = T x. The input-output behaviors of two similar systems are identical.. Norms In the scalar case the gain of a system G at a given frequency! is given by the absolute value of G(j!). In this thesis we will treat systems with multiple inputs and outputs represented by a matrix of transfer functions. The gain of such a system G is not a single value but should rather be viewed as a range of gains. We will use matrix norms that are induced by Euclidian vector norms...1 Vector Norms If u C m denotes a vector, the Euclidian vector norm is dened by kuk = vu u t X m i=1 where u denotes the complex conjugate transpose of u... Singular Value Decomposition If M C pm, it can always be factored [58] into ju i j = p u u (.8) M = UV where = diag [ 1 ; ; : : : ; k ] are the singular values of M, k = min fp; mg, U C pk such that U U = I k and V C mk such that V V = I k. The maximum singular value 1 is denoted by and the smallest one k by. If M has not full rank then k = = 0. The singular values of M are related to the eigenvalues of M M and MM : M M = V U UV = V V :

27 . Norms 1 If p m then V is unitary and V 1 = V. Thus the eigenvalues of M M are given by i. When p m, analogous results can be derived, i.e. the eigenvalues of MM are given by i. Specically, the maximum eigenvalues of M M and MM are both equal to the square of the maximum singular value of M. The condition number of a nonsingular matrix is dened as the fraction between the maximum and minimum singular values:.. Induced Matrix Norms cond(m) = (M) (M) : (.9) Using the Euclidian vector norm, we can dene the induced norm of a matrix M C pm by kmuk kmk = sup u6=0 kuk = sup km uk: (.10) kuk=1 The Euclidian-induced norm of M is equal to the maximum singular value of M: kmk = (M): (.11) We will use both notations in the sequel. If we apply this to a frequency function G then the singular values of G(j!) are called the principal gains of G at!...4 Rank and Pseudo-inverse Due to numerical errors the rank of a matrix cannot in general be determined by counting the number of independent rows or columns of a matrix. A numerically more reliable method is to instead compute the singular value decomposition of the matrix M and counting the number of singular values that are signicantly larger than the numerical precision of the oating point operations. For instance in Matlab, which uses 64-bit oating point representation the numerical precision is about = If k is of the same order as 1 the rank of the matrix is set to be less than k. The pseudo-inverse M y of a matrix M that is not full rank can be determined using the singular value decomposition. If r 0 M = U V 0 0 (.1) then M y = V 1 r 0 U 0 0 (.1) where r = diag [ 1 ; : : : ; r ] such that 1 : : : r > 0.

28 14 Preliminaries.4 Signal Spaces For a more complete treatment on functional spaces related to control systems, see []..4.1 Lebesgue Spaces Consider a continuous-time signal x R! R n dened in the interval [0; 1). Restrict x to be square-lebesgue integrable Z 1 0 kx(t)k dt < 1: (.14) The set of all such signals is the Lebesgue space denoted by L n [0; 1) or just by L [0; 1). This space is a Hilbert space under the inner product hx; yi = Z 1 0 x(t) y(t)dt: The norm of x, denoted kxk, is dened as the square root of hx; xi. Similarly to the continuous-time Lebesgue space, L, we can dene the counterpart for discrete time signals x Z! R n in the interval [0; 1) by the inner product dened by hx; yi = 1X k=0 x(k) y(k): The set of signals such that hx; xi is bounded, that is hx; xi = 1X k=0 x(k) x(k) < 1; (.15) is the Lebesgue space denoted by `n [0; 1) or just by `[0; 1). The Extended Lebesgue Space The Lebesgue space L only includes signals with bounded energy. To also include unbounded signals, e.g. in order to discuss unstable systems, we need the extended space. Let P T denote the projection operator (P T x)(t) = x T (t) = ( x(t) t T 0 t > T (.16) The extended Lebesgue space, L e is dened as the space of continuous-time signals x R! R n such that x T L. The scalar product in L e is hx; yi T = hx T ; y T i = Z T 0 x (t)y(t)dt:

29 .4 Signal Spaces Operators An operator G is a function from one signal space to another. The operator is linear if G(u 1 + u ) = (Gu 1 ) + (Gu ) G(u) = (Gu) where R. For instance, linear systems are linear operators. An operator is causal if (Gu)(t) only depends on past values of u. Using the projection operator P T this can be written as P T G = P T GP T. An operator is parametric if (Gu)(t) only depends on u(t). Thus G is a timevarying function of u(t), i.e. (Gu)(t) = g(u(t); t). Linear parametric operators can be represented as linear systems with no states and only described by a D-matrix, which can be constant or time-varying..4. Induced Norms Based on the denition of the L and ` norms for signals, we can dene the induced norms or gains for operators, called the L -induced or `-induced norms. A continuous-time operator G is a function from one signal space u L m to another y L p : y = G u The L -induced norm is dened as kgk = sup ul u6=0 kguk kuk : The discrete-time case is dened analogously. A discrete-time operator G is a function from one signal space u `m to another y `p, and the `-induced norm is dened as kg uk kgk = sup : kuk u` u6=0.4.4 Hardy Spaces The Hardy space H 1 [] consists of all complex-valued scalar functions, G : C! C, of a complex variable s, that are analytical and bounded in the open right half plane, Re s > 0. This means that there exists a real number b such that jg(s)j b; Re s > 0: The smallest such bound b is the H 1 -norm of G, denoted kgk 1. The H 1 -norm is dened by kgk 1 = supfjg(s)j : Re s > 0g: (.17)

30 16 Preliminaries By the maximum modulus theorem we can replace the open right half-plane in (.17) by the imaginary axis: kgk 1 = supfjg(j!)j :! Rg: (.18) In the more general case of matrix transfer functions we have kgk 1 = supfkg(j!)k :! Rg: (.19) It can be shown that for LTI systems the H 1 -norm and the L -induced norm are equivalent, i.e. kgk 1 = kg(s)k. In this thesis we will focus on (nite-dimensional) real-rational functions, which are rational functions with real coecients. The subset of H 1 consisting of realrational functions is denoted by RH 1. If G is real-rational, then G RH 1 if and only if G is proper (jg(1)j exists and is nite) and stable (G has no poles in the closed right half plane, Re s 0). We denote by RH pm 1 multivariable functions in RH 1 with m inputs and p outputs..5 Factorization of Transfer Matrices In this section we will review some factorization methods for transfer matrices in state-space form. These factorization problems emerge in connection with statespace -analysis..5.1 Inverse of Transfer Matrices We start by stating the well-known matrix inversion lemma, see e.g. [58]: (D + CAB) 1 = D 1 D 1 C(BD 1 C + A 1 ) 1 BD 1 (.0) assuming that A and D are nonsingular. Using (.0) the inverse of a square transfer matrix dened by the realization G(s) = D + C(sI A) 1 B exists if D is nonsingular and is given by G 1 (s) = D 1 D 1 C(sI A + BD 1 C) 1 BD 1 : The eigenvalues of the closed-loop system is given by A BD 1 C..5. Factorization of Matrices A Hermitian or symmetric matrix P, positive denite or semidenite, can be decomposed into two factors P = D T D. Such a factorization is unique except for a unitary left factor on D. There exist a number of possibilities for doing this. Cholesky factorization nds an upper triangular factor, T such that P = T T T. Singular value decomposition of P = UV, then D = D = U 1= V. Note that if P is Hermitian then U = V.

31 .5 Factorization of Transfer Matrices The Riccati Equation and Its Solution Many factorization algorithms use the Riccati equation [, 8, 94]. The Riccati equation is a quadratic matrix equation XA + A T X XRX + Q = 0; (.1) where Q and R are symmetric matrices, and R is positive or negative semidenite. The equation has, in general, more than one solution but we will restrict our attention to the case when (A; R) is stabilizable, namely when there exists an X that makes A RX stable, i.e. all its eigenvalues have negative real parts. Such a solution of the Riccati equation does not always exist, but we can nd a sucient criterion for its existence by studying the Hamiltonian H associated to (.1) A R H = Q A T : (.) The Hamiltonian, if real, has symmetric eigenvalues both about the real and imaginary axes. If H has no eigenvalues on the imaginary axis and if (A; R) is stabilizable, then there exists a unique stabilizing, symmetric, semidenite solution to (.1), which is denoted by X = Ric H, see [, 8]. The solution is obtained by rst computing the modal subspace spanned by the generalized (real) eigenvectors of H corresponding to stable eigenvalues. It can be shown that this eigenspace can be written as the range space of [ I X ]..5.4 Spectral Factorization We will here consider the problem of factoring a square real-rational Hermitian transfer matrix W = W, where W (s) = W T ( s). We assume that W and W 1 are both proper and have no poles on the imaginary axis. This problem is described as the spectral factorization problem [] and the standard procedure is to write W = D 0 + G 1 + G 1 where D 0 is constant and G 1 is stable, inversely stable and strictly proper. We here conne ourselves to a particular structure of W, and assume that W = G G > 0 where G is stable but not necessarily inversely stable or even square. This particular structure emerges in state-space -analysis with complex uncertainties. Lemma.1 (Spectral Factorization []) Assume that G(s) = D + C(sI A) 1 B is stable and W = G G > 0. Then there exists a stable and inversely stable, ^G, such that W = ^G ^G. One such ^G is obtained by ^G(s) = ^D + ^C(sI A) 1 B with ^DT ^D = D T D and ^C = ^D T (D T C + B T X) where X = Ric H 0, A BD y C B(D T D) 1 B T H = ; C T C + C T D(D T D) 1 D T C A BD y C T D y = (D T D) 1 D T :

32 18 Preliminaries Proof A state-space realization of W is W = 6 4 A 0 B C T C A T C T D D T C B T D T D " # ~A 7 B 5 = ~ ~C D ~ A realization of the inverse of W has an A-matrix given by ~A = ~ A ~ B ~ D 1 ~ C = H Thus, H has no eigenvalues on the imaginary axis since W has no zeros there (W > 0). Also, ( ~ A; ~ B) is stabilizable, since A is stable. Thus, X = Ric H 0 exists and satises X(A BD y C) + (A BD y C) T X XB(D T D) 1 B T X + C T C C T D(D T D) 1 D T C = 0; (.) such that A BD y C B(D T D) 1 B T X is stable. Applying a similarity transformation to W with I 0 I 0 T = and T 1 = X I X I yields W ~ W = 6 4 A 0 B C T C XA A T X A T C T D XB D T C + B T X B T D T D Using (.) it is straightforward to show that and thus ^C T ^C = C T C + XA + A T X; A 0 B ~W = ^G ^G = 6 4 ^CT ^C A T ^CT ^D ^D T ^C B T ^DT ^D 7 5 : 7 5 : We can nd a (square) nonsingular ^D, e.g. by Cholesky, QR-factorization or singular value decomposition, such that ^DT ^D = D T D. It is evident that ^G is stable since G is such. Next, since ^D is nonsingular, ^G 1 exists and where ^G 1 (s) = ^D 1 ^D 1 ^C(sI ^A ) 1 B ^D 1 : ^A = A B ^D 1 ^C = A B ^D 1 ^D T (B T X + D T C) = A BD y C B(D T D) 1 B T X: Thus, ^A is stable and, hence, ^G is inversely stable.