Optimal and Robust Control: Advanced Topics with MATLAB«

by Fortuna; Luigi

• ISBN13: 9781466501911
• ISBN10: 146650191X
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-02-02
• Publisher: CRC Press

Summary

This book includes advanced topics in automatic control including the LQR and the H-Infinite approach. It covers advanced concepts of modeling analysis and presents the essential items of the new technique of LMI to show how it can be considered as an unifying tool for system analysis and controller design. The robustness property of the closed control loop is the guideline of this book.

Table of Contents

List of Figures	p. xi
Preface	p. xv
Symbol List	p. xvii
Modelling of uncertain systems and the robust control problem	p. 1
Uncertainty and robust control	p. 1
The essential chronology of major findings into robust control	p. 6
Fundamentals of stability	p. 9
Lyapunov criteria	p. 9
Positive definite matrices	p. 11
Lyapunov theory for linear time-invariant systems	p. 14
Lyapunov equations	p. 18
Stability with uncertainty	p. 21
Exercises	p. 24
Kalman canonical decomposition	p. 27
Introduction	p. 27
Controllability canonical partitioning	p. 29
Observability canonical partitioning	p. 31
General partitioning	p. 32
Remarks on Kalman decomposition	p. 39
Exercises	p. 40
Singular value decomposition	p. 43
Singular values of a matrix	p. 43
Spectral norm and condition number of a matrix	p. 45
Exercises	p. 50
Open-loop balanced realization	p. 51
Controllability and observability gramians	p. 51
Principal component analysis	p. 54
Principal component analysis applied to linear systems	p. 55
State transformations of gramians	p. 58
Singular values of linear time-invariant systems	p. 60
Computing the open-loop balanced realization	p. 61
Balanced realization for discrete-time linear systems	p. 65
Exercises	p. 67
Reduced order models	p. 69
Reduced order models based on the open-loop balanced realization	p. 70
Direct truncation method	p. 71
Singular perturbation method	p. 72
Reduced order model exercises	p. 74
Exercises	p. 77
Symmetrical systems	p. 79
Reduced order models for SISO systems	p. 79
Properties of symmetrical systems	p. 81
The cross-gramian matrix	p. 83
Relations between Wc2 and Woo	p. 83
Open-loop parameterization	p. 89
Relation between the Cauchy index and the Hankel matrix	p. 91
Singular values for a FIR filter	p. 92
Singular values of all-pass systems	p. 95
Exercises	p. 96
Linear quadratic optimal control	p. 99
LQR optimal control	p. 99
Hamiltonian matrices	p. 105
Resolving the Riccati equation by Hamiltonian matrix	p. 109
The Control Algebraic Riccati Equation	p. 110
Optimal control for SISO systems	p. Ill
Linear quadratic regulator with cross-weighted cost	p. 117
Finite-horizon linear quadratic regulator	p. 117
Optimal control for discrete-time linear systems	p. 118
Exercises	p. 119
Closed-loop balanced realization	p. 121
Filtering Algebraic Riccati Equation	p. 122
Computing the closed-loop balanced realization	p. 124
Procedure for closed-loop balanced realization	p. 125
Reduced order models based on closed-loop balanced realization	p. 127
Closed-loop balanced realization for symmetrical systems	p. 131
Exercises	p. 132
Passive and bounded-real systems	p. 135
Passive systems	p. 135
Passivity in the frequency domain	p. 136
Passivity in the time domain	p. 141
Factorizing positive-real functions	p. 143
Passification	p. 143
Passive reduced order models	p. 148
Energy considerations connected to the Positive-Real Lemma	p. 148
Closed-loop stability and positive-real systems	p. 149
Optimal gain for loss-less systems	p. 150
Circuit implementation of positive-real systems	p. 152
Bounded-real systems	p. 153
Properties of bounded-real systems	p. 156
Bounded-real reduced order models	p. 157
Relationship between passive and bounded-real systems	p. 157
Exercises	p. 158
H∞ linear control	p. 161
Introduction	p. 161
Solution of the H∞ linear control problem	p. 163
The H∞ linear control and the uncertainty problem	p. 168
Exercises	p. 171
Linear Matrix Inequalities for optimal and robust control	p. 173
Definition and properties of LMI	p. 173
LMI problems	p. 175
Feasibility problem	p. 175
Linear objective minimization problem	p. 176
Generalized eigenvalue minimization problem	p. 176
Formulation of control problems in LMI terms	p. 176
Stability	p. 177
Simultaneous stabilizability	p. 177
Positive real lemma	p. 178
A Bounded real lemma	p. 178
Calculating the H∞ norm through LMI	p. 178
Solving a LMI problem	p. 179
LMI problem for simultaneous stabilizability	p. 181
Solving algebraic Riccati equations through LMI	p. 184
Computation of gramians through LMI	p. 187
Computation of the Hankel norm through LMI	p. 188
H∞ control	p. 190
Multiobjective control	p. 192
Exercises	p. 198
The class of stabilizing controllers	p. 201
Parameterization of stabilizing controllers for stable processes	p. 201
Parameterization of stabilizing controllers for unstable processes	p. 203
Parameterization of stable controllers	p. 206
Simultaneous stabilizability of two systems	p. 209
Coprime facto rizations for MIMO systems and unitary factorization	p. 210
Parameterization in presence of uncertainty	p. 211
Exercises	p. 214
Recommended essential references	p. 217
Norms	p. 223
Algebraic Riccati Equations	p. 227
Index	p. 231
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