LMIs in Control Systems: Analysis, Design and Applications

Introduction What are LMIs? General form Standard form Manipulations A few examples involving LMIs Eigenvalue minimization Matrix norm minimization A key step in mu-analysis Schur stabilization A brief history The seed planted (1890) The rooting period (1940-1970) The growing period (1970-2000) The Nourishing period (2000-present) Advantages About the book Structure Features Using it in courses Exercises PRELIMINARIES Technical Lemmas Generalized square inequalities The restriction-free inequality Inequalities with restriction The variable elimination lemma Schur complement lemma Schur complements Matrix inversion lemma Schur complement lemma Elimination of variables Variable elimination in a partitioned matrix The projection lemma The reciprocal projection lemma Some other useful results Trace of an LMI The maximum modulus principle The Parseval lemma Notes and references Exercises Review of Optimization Theory Convex sets Definitions and properties Hyperplanes, halfspaces, polyhedrons and polytopes Convex functions Definition and properties Criteria Mathematical optimization Least squares programming Linear programming Quadratic programming Convex optimization The problem Local and global optima The LMI problem Convexity The extreme result Standard problems Notes and references About this chapter The open source software CVX A counter example for numerical reliability Exercises CONTROL SYSTEMS ANALYSIS Stability Analysis Hurwitz and Schur stability Hurwitz stability Schur stability D-stability Special cases GeneralLMI regions Generalized Lyapunov theorem Quadratic stability Familyof systems Quadratic Hurwitz stability QuadraticSchur stability Quadratic D-stability Definition and main results Some special cases Time-delay systems The delay independent condition The delay dependent condition Notes and references Summary and references Affine quadratic stability Exercises H /H2 Performance H and H2 indices H index H2 index Equivalent definitions LMI conditions for H index Thebasic conditions Deduced conditions LMI conditions for H2 index Basic conditions Deduced conditions Notes and references Exercises Property Analysis Hurwitz stabilizability and detectability Hurwitz stabilizability Hurwitz detectability Schur stabilizability and detectability Schur stabilizability Schur detectability Dissipativity Definition Equivalent conditions Passivity and positive-realness Definitions The positive-real lemma The LMI condition Non expansivity and bounded-realness Definitions The bounded-real lemma The LMI conditions Notes and references Exercises CONTROL SYSTEMS DESIGN Feedback Stabilization State feedback stabilization Case of continuous-time systems Case of discrete-time systems D-stabilization H (a,B)-stabilization D(q,r)-stabilization General D-stabilization Quadratic stabilization Family of systems Quadratic Hurwitz stabilization Quadratic Schur stabilization Quadratic D-stabilization Problem formulation The solution Special cases Insensitive region design Sensitivity of matrix eigen values Insensitive strip region design Insensitive disk region design Robust stabilization of second-order systems Stabilization Robust Stabilization Stabilization of time-delay systems Case of delay independence Case of delay dependence Notes and references Exercises H /H2 Control H state feedback control The problem The solution Other conditions H2 state feedback control The problem The solution Other conditions Robust H /H2 state feedback control The problems Solution to the robust H control problem Solution to the robust H2 control problem LQ regulation via H2 control Problem description Relation with H2 performance The solution Notes and references Summary and references Dissipative, passive, and non-expansive control Exercises State Observation and Filtering Full- and reduced-order state observers Full-order state observers Reduced-order state observer design Full-order H /H2 state observers Problems formulation Solutions to problems Examples H filtering Problems formulation Solution to H filtering H2 filtering Problems formulation Solution to H2 filtering Notes and references Exercises Multiple Objective Designs Insensitive region designs with minimum gains Insensitive strip region designs with minimum gains Insensitive disk region designs with minimum gains Mixed H /H2 designs with desired LMI pole regions The problem Solutions to the problem Mixed robust H /H2 designs with desired LMI pole regions The problem Solutions to the problem Notes and references Summary of main results Further remarks Exercises APPLICATIONS Missile Attitude Control The dynamical model Models for non-rotating missiles Models for BTT missiles Attitude control of non-rotating missiles The problem The solution Attitude control of BTT missiles The problem Quadratic stabilization Numerical results and simulation Notes and references Exercises Satellite Control System modelling The second-order system form The state space form H2 and H feedback control H control H2 control Mixed H2/ H feedback control The problem Numerical and simulation results Notes and references Exercises APPENDICES Proofs of Theorems Proof of Theorem 4.1 Preliminaries Sufficiency Necessity Proof of Theorem 5.1 The first step The second step Proof of Theorem5.2 The first step The second step