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Summary
Summary
While there are many books on advanced control for specialists, there are few that present these topics for nonspecialists. Assuming only a basic knowledge of automatic control and signals and systems, Optimal and Robust Control: Advanced Topics with MATLAB® offers a straightforward, self-contained handbook of advanced topics and tools in automatic control.
Techniques for Controlling System Performance in the Presence of Uncertainty
The book deals with advanced automatic control techniques, paying particular attention to robustness--the ability to guarantee stability in the presence of uncertainty. It explains advanced techniques for handling uncertainty and optimizing the control loop. It also details analytical strategies for obtaining reduced order models. The authors then propose using the Linear Matrix Inequalities (LMI) technique as a unifying tool to solve many types of advanced control problems.
Topics covered include:
LQR and H-infinity approaches Kalman and singular value decomposition Open-loop balancing and reduced order models Closed-loop balancing Passive systems and bounded-real systems Criteria for stability controlThis easy-to-read text presents the essential theoretical background and provides numerous examples and MATLAB exercises to help the reader efficiently acquire new skills. Written for electrical, electronic, computer science, space, and automation engineers interested in automatic control, this book can also be used for self-study or for a one-semester course in robust control.
Author Notes
Luigi Fortuna is a Full Professor of System Theory at the University of Catania, Italy. He was the coordinator of the courses in electronic engineering and Head of the Dipartimento di Ingegneria Elettrica Elettronica e dei Sistemi (DIEES). Since 2005, he has been the Dean of the Engineering Faculty. He currently teaches complex adaptive systems and robust control. He has published more than 450 technical papers and is the coauthor of ten scientific books. His scientific interests include robust control, nonlinear science and complexity, chaos, cellular neural networks, soft-computing strategies for control and robotics, micronanosensor and smart devices for control, and nanocellular neural networks modeling. Dr. Fortuna is an IEEE Fellow.
Mattia Frasca received his PhD in Electronics and Automation Engineering in 2003, at the University of Catania, Italy. Currently, he is a research associate at the University of Catania, where he also teaches systems theory. His scientific interests include robust control, nonlinear systems and chaos, cellular neural networks, complex systems, and bio-inspired robotics. He is involved in many research projects and collaborations with industries and academic centers. He is also a referee for many international journals and conferences. Dr. Frasca was on the organizing committee of the 10th Experimental Chaos Conference and was co-chair of the 4th International Conference on Physics and Control. He is the coauthor of three research monographs, has published more than 150 papers in refereed international journals and international conference proceedings, and is the coauthor of two international patents. He is also a Senior Member of IEEE.
For more information on Dr. Frasca's work, see his web page at the University of Catania.
Table of Contents
List of Figures | p. xi |
Preface | p. xv |
Symbol List | p. xvii |
1 Modelling of uncertain systems and the robust control problem | p. 1 |
1.1 Uncertainty and robust control | p. 1 |
1.2 The essential chronology of major findings into robust control | p. 6 |
2 Fundamentals of stability | p. 9 |
2.1 Lyapunov criteria | p. 9 |
2.2 Positive definite matrices | p. 11 |
2.3 Lyapunov theory for linear time-invariant systems | p. 14 |
2.4 Lyapunov equations | p. 18 |
2.5 Stability with uncertainty | p. 21 |
2.6 Exercises | p. 24 |
3 Kalman canonical decomposition | p. 27 |
3.1 Introduction | p. 27 |
3.2 Controllability canonical partitioning | p. 29 |
3.3 Observability canonical partitioning | p. 31 |
3.4 General partitioning | p. 32 |
3.5 Remarks on Kalman decomposition | p. 39 |
3.6 Exercises | p. 40 |
4 Singular value decomposition | p. 43 |
4.1 Singular values of a matrix | p. 43 |
4.2 Spectral norm and condition number of a matrix | p. 45 |
4.3 Exercises | p. 50 |
5 Open-loop balanced realization | p. 51 |
5.1 Controllability and observability gramians | p. 51 |
5.2 Principal component analysis | p. 54 |
5.3 Principal component analysis applied to linear systems | p. 55 |
5.4 State transformations of gramians | p. 58 |
5.5 Singular values of linear time-invariant systems | p. 60 |
5.6 Computing the open-loop balanced realization | p. 61 |
5.7 Balanced realization for discrete-time linear systems | p. 65 |
5.8 Exercises | p. 67 |
6 Reduced order models | p. 69 |
6.1 Reduced order models based on the open-loop balanced realization | p. 70 |
6.1.1 Direct truncation method | p. 71 |
6.1.2 Singular perturbation method | p. 72 |
6.2 Reduced order model exercises | p. 74 |
6.3 Exercises | p. 77 |
7 Symmetrical systems | p. 79 |
7.1 Reduced order models for SISO systems | p. 79 |
7.2 Properties of symmetrical systems | p. 81 |
7.3 The cross-gramian matrix | p. 83 |
7.4 Relations between W c 2 and W o o | p. 83 |
7.5 Open-loop parameterization | p. 89 |
7.6 Relation between the Cauchy index and the Hankel matrix | p. 91 |
7.7 Singular values for a FIR filter | p. 92 |
7.8 Singular values of all-pass systems | p. 95 |
7.9 Exercises | p. 96 |
8 Linear quadratic optimal control | p. 99 |
8.1 LQR optimal control | p. 99 |
8.2 Hamiltonian matrices | p. 105 |
8.3 Resolving the Riccati equation by Hamiltonian matrix | p. 109 |
8.4 The Control Algebraic Riccati Equation | p. 110 |
8.5 Optimal control for SISO systems | p. Ill |
8.6 Linear quadratic regulator with cross-weighted cost | p. 117 |
8.7 Finite-horizon linear quadratic regulator | p. 117 |
8.8 Optimal control for discrete-time linear systems | p. 118 |
8.9 Exercises | p. 119 |
9 Closed-loop balanced realization | p. 121 |
9.1 Filtering Algebraic Riccati Equation | p. 122 |
9.2 Computing the closed-loop balanced realization | p. 124 |
9.3 Procedure for closed-loop balanced realization | p. 125 |
9.4 Reduced order models based on closed-loop balanced realization | p. 127 |
9.5 Closed-loop balanced realization for symmetrical systems | p. 131 |
9.6 Exercises | p. 132 |
10 Passive and bounded-real systems | p. 135 |
10.1 Passive systems | p. 135 |
10.1.1 Passivity in the frequency domain | p. 136 |
10.1.2 Passivity in the time domain | p. 141 |
10.1.3 Factorizing positive-real functions | p. 143 |
10.1.4 Passification | p. 143 |
10.1.5 Passive reduced order models | p. 148 |
10.1.6 Energy considerations connected to the Positive-Real Lemma | p. 148 |
10.1.7 Closed-loop stability and positive-real systems | p. 149 |
10.1.8 Optimal gain for loss-less systems | p. 150 |
10.2 Circuit implementation of positive-real systems | p. 152 |
10.3 Bounded-real systems | p. 153 |
10.3.1 Properties of bounded-real systems | p. 156 |
10.3.2 Bounded-real reduced order models | p. 157 |
10.4 Relationship between passive and bounded-real systems | p. 157 |
10.5 Exercises | p. 158 |
11 H∞ linear control | p. 161 |
11.1 Introduction | p. 161 |
11.2 Solution of the H∞ linear control problem | p. 163 |
11.3 The H∞ linear control and the uncertainty problem | p. 168 |
11.4 Exercises | p. 171 |
12 Linear Matrix Inequalities for optimal and robust control | p. 173 |
12.1 Definition and properties of LMI | p. 173 |
12.2 LMI problems | p. 175 |
12.2.1 Feasibility problem | p. 175 |
12.2.2 Linear objective minimization problem | p. 176 |
12.2.3 Generalized eigenvalue minimization problem | p. 176 |
12.3 Formulation of control problems in LMI terms | p. 176 |
12.3.1 Stability | p. 177 |
12.3.2 Simultaneous stabilizability | p. 177 |
12.3.3 Positive real lemma | p. 178 |
12.3 A Bounded real lemma | p. 178 |
12.3.5 Calculating the H∞ norm through LMI | p. 178 |
12.4 Solving a LMI problem | p. 179 |
12.5 LMI problem for simultaneous stabilizability | p. 181 |
12.6 Solving algebraic Riccati equations through LMI | p. 184 |
12.7 Computation of gramians through LMI | p. 187 |
12.8 Computation of the Hankel norm through LMI | p. 188 |
12.9 H∞ control | p. 190 |
12.10 Multiobjective control | p. 192 |
12.11 Exercises | p. 198 |
13 The class of stabilizing controllers | p. 201 |
13.1 Parameterization of stabilizing controllers for stable processes | p. 201 |
13.2 Parameterization of stabilizing controllers for unstable processes | p. 203 |
13.3 Parameterization of stable controllers | p. 206 |
13.4 Simultaneous stabilizability of two systems | p. 209 |
13.5 Coprime facto rizations for MIMO systems and unitary factorization | p. 210 |
13.6 Parameterization in presence of uncertainty | p. 211 |
13.7 Exercises | p. 214 |
Recommended essential references | p. 217 |
Appendix A Norms | p. 223 |
Appendix B Algebraic Riccati Equations | p. 227 |
Index | p. 231 |