In this text, the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimization problems that involve linear matrix inequalities.
Author(s): Laurent El Ghaoui, Eric Feron, Vendataramanan Balakrishnan
Series: Studies in Applied Mathematics 15
Publisher: Society for Industrial and Applied Mathematic
Year: 1994
Language: English
Pages: 206
Contents......Page 6
Preface......Page 8
Acknowledgments......Page 10
1.1 Overview......Page 14
1.2 A Brief History of LMIs in Control Theory......Page 15
1.3 Notes on the Style of the Book......Page 17
1.4 Origin of the Book......Page 18
2.1 Linear Matrix Inequalities......Page 20
2.2 Some Standard Problems......Page 22
2.3 Ellipsoid Algorithm......Page 25
2.4 Interior-Point Methods......Page 27
2.5 Strict and Nonstrict LMIs......Page 31
2.6 Miscellaneous Results on Matrix Inequalities......Page 35
2.7 Some LMI Problems with Analytic Solutions......Page 37
3.1 Minimizing Condition Number by Scaling......Page 50
3.3 Minimizing Norm by Scaling......Page 51
3.4 Rescaling a Matrix Positive-Definite......Page 52
3.5 Matrix Completion Problems......Page 53
3.6 Quadratic Approximation of a Polytopic Norm......Page 54
3.7 Ellipsoidal Approximation......Page 55
4.1 Di®erential Inclusions......Page 64
4.2 Some Specific LDIs......Page 65
4.3 Nonlinear System Analysis via LDIs......Page 67
5.1 Quadratic Stability......Page 74
5.2 Invariant Ellipsoids......Page 81
6.1 Input-to-State Properties......Page 90
6.2 State-to-Output Properties......Page 97
6.3 Input-to-Output Properties......Page 102
7.1 Static State-Feedback Controllers......Page 112
7.2 State Properties......Page 113
7.3 Input-to-State Properties......Page 117
7.4 State-to-Output Properties......Page 120
7.5 Input-to-Output Properties......Page 122
7.6 Observer-Based Controllers for Nonlinear Systems......Page 124
8.1 Analysis of Lur'e Systems......Page 132
8.2 Integral Quadratic Constraints......Page 135
8.3 Multipliers for Systems with Unknown Parameters......Page 137
9.1 Analysis of Systems with Multiplicative Noise......Page 144
9.2 State-Feedback Synthesis......Page 147
10.1 Optimization over an Affine Family of Linear Systems......Page 154
10.2 Analysis of Systems with LTI Perturbations......Page 156
10.4 Linear Systems with Delays......Page 157
10.5 Interpolation Problems......Page 158
10.6 The Inverse Problem of Optimal Control......Page 160
10.7 System Realization Problems......Page 161
10.8 Multi-Criterion LQG......Page 163
10.9 Nonconvex Multi-Criterion Quadratic Problems......Page 164
Notation......Page 170
Acronyms......Page 172
Bibliography......Page 174
Index......Page 200
