Positive Polynomials in Control

front-matter......Page 1
1 Introduction......Page 11
2.1 Polynomial Definitions......Page 12
2.2 Positivstellensatz......Page 13
2.4 S -Procedure......Page 14
3.1 Reachable Set Bounds under Unit Energy Disturbances......Page 15
3.2 Set Invariance under Peak Bounded Disturbances......Page 19
3.3 Bounding the Induced L 2 → L2 Gain......Page 21
3.4 Disturbance Analysis Example......Page 22
4 Expanding a Region of Attraction with State Feedback......Page 23
4.1 Expanding the Region of Attraction for Systems with Input Saturation......Page 26
4.2 State Feedback Example......Page 27
References......Page 29
1 Introduction......Page 31
2 Background Material......Page 33
2.1 The Sum of Squares Decomposition......Page 34
2.2 Lyapunov Stability......Page 35
3.1 Recasting......Page 37
3.2 Analysis......Page 38
4 Examples......Page 41
4.1 Example 4.1: System with Saturation Nonlinearity......Page 42
4.2 Example 4.2: Whirling Pendulum......Page 44
4.3 Example 4.3: System with an Irrational Power Vector Field......Page 46
4.4 Example 4.4: Diabatic Continuous Stirred Tank Reactor......Page 47
5 Conclusions......Page 49
References......Page 50
1 Introduction......Page 52
2 Fixed-Order H ∞ Controller Synthesis......Page 55
3.1 Polynomial Semi-de.nite Programming......Page 56
3.2 Application to the H ∞ .xed order control problem......Page 60
4.1 Partial Dualization......Page 61
5 Robust Analysis by SOS-Decompositions......Page 64
5.2 Scalarization of Matrix-Valued Constraints......Page 65
5.4 Veri.cation of Matrix SOS Property......Page 66
5.5 Construction of LMI Relaxation Families......Page 68
5.6 Size of the Relaxation of LMI Problem......Page 69
6.1 Fourth Order System......Page 71
6.2 Active Suspension System......Page 72
7 Conclusions......Page 75
References......Page 76
1 Introduction......Page 79
2 Problem Statement......Page 81
3 H ∞ Design Technique......Page 82
4 Numerical Examples......Page 85
4.1 Optimal Robust Stability......Page 86
4.2 Flexible Beam......Page 87
5 Conclusion......Page 88
References......Page 89
1 Introduction......Page 92
2 Problem Formulation and Preliminaries......Page 93
3 Parameterization of Homogeneous Matricial Forms......Page 95
4.1 Parameterization of Positive De.nite HPD-QLF Matrices......Page 97
5.1 Example 1......Page 101
6 Conclusions......Page 104
References......Page 105
1 Introduction......Page 107
3 Notations and Preliminaries......Page 109
4 Constant Parameters......Page 111
5 Time-Varying Parameters with Bounded Variation......Page 114
6.1 Sketch of Proof of Theorem 1......Page 115
A Appendix on Polynomial Matrices......Page 117
A.2 Products of Polynomial Matrices......Page 118
A.3 Formulas Attached to the Inversion of the Map [theta]......Page 119
A.4 Di.erentiation of Polynomial Matrices......Page 120
References......Page 121
1.1 Representations of positive forms on the simplex......Page 122
2 Pólya’s and Putinar’s Theorems Give the Same Bounds for Optimization on the Simple......Page 126
3 A Negative Result......Page 129
4 Conclusion......Page 131
References......Page 132
1 Introduction......Page 134
2 Notation, De.nitions and Preliminary Results......Page 135
2.2 Localizing Matrices......Page 136
2.3 Multivariate Newton Sums......Page 137
3.1 The Associated Moment Matrix......Page 138
3.2 Construction of the Moment Matrix of μ .......Page 139
3.3 Conditions for a Localization of Zeros of G......Page 142
5.1 Proof of Proposition 1......Page 144
5.2 Proof of Theorem 6......Page 146
References......Page 150
1 Introduction and Notation......Page 152
2 The Plane, Half Plane, and Quarter Plane......Page 157
3 Non-compact Strips in the Plane......Page 158
References......Page 163
1 Introduction......Page 165
2 Stability Criterion......Page 166
3 Uncertain Polynomials......Page 169
4 Time–Delay Systems......Page 173
5 Conclusions......Page 175
References......Page 176
1 Introduction......Page 178
2 The Basic SOS/SDP Techniques......Page 179
3 Sparsity......Page 182
4 Equality Constraints......Page 183
5 Symmetries......Page 185
6 Combination of Techniques......Page 187
7.1 Domain of Attraction of a Lyapunov Function......Page 188
References......Page 189
1 Introduction......Page 192
2.1 Optimization Problems with Frequency-Domain Inequalities......Page 196
2.2 Linear-Quadratic Regulators......Page 198
2.3 Quadratic Lyapunov Function Search......Page 199
3.1 Semide.nite Programming......Page 200
3.3 General-Purpose Solvers......Page 201
4 General-Purpose SDP Solvers and KYP-SDPs......Page 202
4.1 Primal Formulation......Page 203
4.2 Dual Formulation......Page 204
5.1 Reduced Newton Equations......Page 207
5.2 Fast Construction of Reduced Newton Equations......Page 210
6 Numerical Examples......Page 214
7.1 Multiple Constraints......Page 216
8 Conclusion......Page 218
A.1 Optimality Conditions......Page 219
A.2 Algorithm......Page 220
A.3 Solving the Newton Equations......Page 225
B.1 Expression for H......Page 226
C Derivation of (53) and (54)......Page 227
C.1 Expression for H 1......Page 228
D Non-controllable (A, B )......Page 229
E.1 Change of Variables......Page 230
E.2 Linear Independence......Page 231
References......Page 232
1 Introduction......Page 236
2.1 Real Line......Page 238
2.2 Unit Circle......Page 240
3 The Optimization Problem......Page 241
3.1 Real Line......Page 242
3.2 Unit Circle......Page 243
4.1 Strict Feasibility......Page 244
4.2 One Interpolation Constraint......Page 247
4.3 Two Interpolation Constraints......Page 249
4.4 More Interpolation Constraints (k ≤ n + 1)......Page 252
4.5 Still More Interpolation Constraints (m > n + 1)......Page 257
4.6 Property of the Objective Function......Page 258
5 Matrix Polynomials......Page 259
5.2 Strict Feasibility......Page 260
5.4 More Interpolation Constraints......Page 261
6.1 Real Line......Page 263
6.2 Unit Circle......Page 264
7 Conclusion......Page 265
Appendix......Page 266
References......Page 267
1 Introduction......Page 269
2 SOSTOOLS Features......Page 272
2.1 Formulating Sum of Squares Programs......Page 273
2.2 Exploiting Sparsity......Page 275
2.3 Customized Functions......Page 276
3.1 Nonlinear Stability Analysis......Page 278
3.2 Parametric Robustness Analysis......Page 279
3.3 Stability Analysis of Time-Delay Systems......Page 280
3.4 Safety Veri.cation......Page 283
3.5 Nonlinear Controller Synthesis......Page 285
References......Page 286
1 Introduction......Page 289
2.2 Extraction Algorithm......Page 291
2.3 Example......Page 294
2.5 Number of Extracted Solutions......Page 295
3.1 Rank Condition Non Satis.ed but Global Optimum Reached......Page 296
3.2 Infeasible Extracted Solutions......Page 297
3.3 Reducing the Number of LMI Variables......Page 298
4 A Remark on the Numerical Behavior of GloptiPoly......Page 302
5 Conclusion......Page 304
References......Page 305
back-matter......Page 307