Max-plus methods for nonlinear control and estimation

Cover......Page 1
Editors......Page 2
Title......Page 3
e-ISBN 0-8176-4453-9......Page 4
Dedication......Page 5
Preface......Page 6
Contents......Page 10
Approximate Dependencies......Page 13
1 Introduction......Page 14
1.1 Some Control and Estimation Problems......Page 16
1.2 Concepts of Max-Plus Methods......Page 19
2 Max-Plus Analysis......Page 24
2.1 Spaces of Semiconvex Functions......Page 26
2.2 Bases......Page 28
2.3 Two-Parameter Families......Page 34
2.4 Dual Spaces and Reflexivity......Page 35
3 Dynamic Programming and Viscosity Solutions......Page 44
3.1 Dynamic Programming Principle......Page 45
3.2 Viscosity Solutions......Page 55
4 Max-Plus Eigenvector Method for the InfiniteTime-Horizon Problem......Page 70
4.1 Existence and Uniqueness......Page 71
4.2 Max-Plus Linearity of the Semigroup......Page 73
4.3 Semiconvexity and a Max-Plus Basis......Page 79
4.4 The Eigenvector Equation......Page 83
4.6 Computing B: Initial Notes......Page 96
4.8 A Control Problem Without Nominal Stability and a Game......Page 97
4.8.1 A Game Problem......Page 106
4.9 An Example......Page 108
5 Max-Plus Eigenvector Method Error Analysis......Page 110
5.1 Allowable Errors in Computation of B......Page 111
5.2 Convergence and Truncation Errors......Page 120
5.2.1 Convergence......Page 121
5.2.2 Truncation Error Estimate......Page 123
5.3 Errors in the Approximation of B......Page 132
5.3.1 A Method for Computing B......Page 135
5.4 Error Summary......Page 137
6 A Semigroup Construction Method......Page 142
6.1 Constituent Problems......Page 143
6.2 Operating on the Transformed Operators......Page 146
6.3 The HJB PDE Limit Problems......Page 147
6.4 A Simple Example......Page 151
7 Curse-of-Dimensionality-Free Method......Page 156
7.1 DP for the Constituent and Originating Problems......Page 159
7.2 Max-Plus Spaces and Dual Operators......Page 163
7.3 Discrete Time Approximation......Page 171
7.4 The Algorithm......Page 177
7.5.1 Pruning......Page 183
7.6 Examples......Page 184
7.7 More General Quadratic Constituents......Page 188
Wider Classes of Constituent Hamiltonians......Page 193
Other Nonlinearities......Page 194
8 Finite Time-Horizon Application: Nonlinear Filtering......Page 196
8.1 Semiconvexity......Page 200
8.2 Max-Plus Propagation......Page 205
9.1 Mixed L_inf /L2 Problem Formulation......Page 210
9.2.1 Dynamic Programming Principles......Page 213
9.2.2 Dynamic Programming Equations......Page 217
9.3 Max-Plus Representations and Semiconvexity......Page 218
9.4 Max-Plus Numerical Methods......Page 222
9.4.1 Nonuniqueness for the Max-Plus Affine Equation......Page 224
9.4.2 The Affine Power Method......Page 225
A.0.1 Sketch of Proof of Theorem 2.......Page 230
A.0.2 Proof of Theorem 3.13......Page 231
A.0.3 Proof of Lemma 3.15......Page 233
A.0.4 Sketch of Proof of Theorem 7.27......Page 235
A.0.5 Sketch of Proof of Lemma 7.31......Page 237
A.0.6 Existence of Robust/H
 Estimator and a Disturbance Bound......Page 241
References......Page 245
Index......Page 251
Series Editor......Page 254
Back Cover......Page 258