Cover
***
Series Info
Title
Book Info
Preface
vi
vii
viii
ix
***
Contents
Contents 2 - 4
Contents 5 - 7
Contents 8 - 9.5
Contents 9.6 - 11
***
Notations
xviii
1. Linear Control Systems
1.1 The Controlled System
p.3
p.4
1.2 Control and State Constraints
1.2.1 Constraints on Control and State
p.7
1.2.2 Open-Loop and Closed-Loop Control
p.9
1.3 Optimal Control with Norm-Minimal Cost
1.3.1 Minimum Energy Control
1.3.2 Minimum Magnitude Control
p.13
p.14
p.15
1.3.3 Controllability
1.3.4 The Finite-Dimensional Moment Problem
1.4 The Reachability Problem
1.4.1 Reachability
1.4.2 Calculating Controls for the Boundary of Reach Set
p.21
p.22
1.4.3 Time-Optimal Control
1.5 Optimal Open-Loop Control for Linear-Convex Systems
1.5.1 The Necessary Conditions of Optimality
p.26
p.27
p.28
1.5.2 Degenerate Cases
p.30
p.31
1.5.3 Sufficiency Conditions for Optimality
p.33
p.34
p.35
1.5.4 A Geometrical Interpretation
p.37
1.5.5 Strictly Convex Constraints on Control
p.39
1.6 Duality Methods in Problems of Optimal Control
1.6.1 The Primal and Dual Optimization Problems
p.42
p.43
p.44
p.45
1.6.2 General Remark: Feedforward Controls
2. The Dynamic Programming Approach
2.1 The Dynamic Programming Equation
p.49
p.50
p.51
p.52
p.53
2.2 The Linear-Quadratic Problem
p.55
p.56
p.57
2.3 Reachability through the HJB Equation: Hard Bounds
2.3.1 Forward Reachability
p.60
2.3.2 Backward Reachability or Solvability Problem
p.62
p.63
2.4 Reachability for Linear-Convex Systems
2.4.1 Forward Reachability: Calculating Value Functions
p.66
2.4.2 The Conjugate of the Value Function V_0
2.4.3 The Value Function (Backward)
p.69
2.5 Colliding Tubes: Calculating All Reachable Points
p.71
p.72
p.73
p.74
2.6 The Closed-Loop Target Control
p.76
p.77
2.7 Reachability within an Interval
p.79
p.80
p.81
p.82
2.8 Dynamic Programming: Time-Optimal Control
p.84
p.85
p.86
3. Ellipsoidal Techniques: Reachability & Synthesis
3.1 Linear Systems under Ellipsoidal Constraints
p.89
p.90
3.2 Ellipsoidal Approximation of Reach Sets
p.92
p.93
3.3 Recurrent Relations: External Approximations
p.95
p.96
p.97
p.98
p.99
p.100
p.101
3.4 The Evolution of Approximating Ellipsoids
p.103
p.104
p.105
p.106
3.5 The Ellipsoidal Maximum Principle and Reachability Tube
p.108
p.109
p.110
3.6 Example 3.6
p.112
p.113
3.7 Reachability Sets: Internal Approximations
p.115
p.116
p.117
3.8 Example 3.8
p.119
p.120
p.121
3.9 Recurrent Relations: Internal Approximations
p.123
p.124
p.125
p.126
p.127
3.10 Backward Reachability: Ellipsoidal Approximations
p.129
p.130
3.11 Control Synthesis: Solution through Internal Ellipsoids
p.132
p.133
p.134
p.135
p.136
p.137
p.138
p.139
p.140
3.12 Internal Approximations: The Second Scheme
p.142
p.143
p.144
p.145
4. Solution Examples on Ellipsoidal Methods
4.1 Multiple Integrator
p.149
p.150
p.151
4.2 A Planar Motion under Newton's Law
p.153
p.154
p.155
p.156
p.157
4.2.1 Computational Results
p.159
p.160
p.161
p.162
p.163
p.164
p.165
p.166
p.167
4.3 Damping Oscillations
4.3.1 Calming Down a Chain of Springs in Finite Time
p.170
p.171
p.172
p.173
p.174
p.175
p.176
p.177
p.178
p.179
p.180
p.181
4.4 Computation in High-Dimensional Systems
4.4.1 Computation: The Problem of Degeneracy
p.184
p.185
4.4.2 Regularizing the Ellipsoidal Estimates
p.187
p.188
4.4.3 Regularizing the Estimate for the Reachability Tube
p.190
p.191
p.192
4.4.4 Efficient Computation of Orthogonal Matrix S
p.194
4.4.5 Parallel Computation
p.196
5. The Comparison Principle: Nonlinearity & Nonconvexity
5.1 The Comparison Principle for HJB Equation
5.1.1 Principal Propositions for Comparison Principle
p.200
p.201
5.1.2 A Deductive Approach to Ellipsoidal Calculus
p.203
p.204
p.205
p.206
p.207
5.2 Calculation of Nonconvex Reachability Sets
p.209
p.210
p.211
p.212
5.3 Applications of Comparison Principle
5.3.1 Forward Reachability
p.215
p.216
p.217
p.218
p.219
p.220
p.221
5.3.4 External Ellipsoids for the Unicycle: Reachability
p.223
p.224
p.225
p.226
p.227
p.228
5.4 Ellipsoidal Methods for Non-Ellipsoidal Constraints
5.4.1 Degenerate Ellipsoids: Box-Valued Constraints
5.4.2 Integrals of Box-Valued Functions
p.232
p.233
p.234
5.4.3 Box-Valued Constraints: External Approximations
p.236
p.237
p.238
5.4.4 Box-Valued Constraints: Internal Approximations
p.240
5.5 Ellipsoidal Methods for Zonotopes
5.5.1 Zonotopes
5.5.2 Internal Ellipsoidal Tubes for a Zonotope
p.244
p.245
5.5.3 External Ellipsoidal Tubes for a Zonotope
p.247
p.248
p.249
p.250
p.251
6. Impulse Controls and Double Constraints
6.1 The Problem of Impulse Controls
6.1.1 Open-Loop Impulse Control: The Value Function
p.256
p.257
p.258
p.259
6.1.2 Closed-Loop Impulse Control:HJB Variational Inequality
p.261
p.262
p.263
p.264
p.265
6.2 Realizable Approximation of Impulse Controls
6.2.1 The Realistic Approximation Problem
p.268
6.2.2 The Approximating Motions
p.270
p.271
p.272
p.273
7. Dynamics and Control under State Constraints
7.1 State-Constrained Reachability and Feedback
7.1.1 The Reachability Problem under State Constraints
p.278
p.279
p.280
7.1.2 Comparison Principle under State Constraints
p.282
p.283
p.284
7.1.3 Linear-Convex Systems
p.286
p.287
p.288
7.2 State-Constrained Control: Computation
7.2.1 The Modified Maximum Principle
p.291
p.292
p.293
7.2.2 External Ellipsoids
p.295
p.296
p.297
p.298
p.299
p.300
7.2.3 Generalized Multipliers
7.2.4 An Example
p.303
p.304
p.305
p.306
7.2.5 Helpful Facts for State-Constrained Control Design
p.308
p.309
8. Trajectory Tubes State-Constrained Feedback Control
8.1 The Theory of Trajectory Tubes
8.1.1 Trajectory Tubes and the Generalized Dynamic System
8.1.2 Some Basic Assumptions
8.1.3 The Set-Valued Evolution Equation
8.1.4 The Funnel Equations: Specific Cases
p.317
8.1.5 Evolution Equation under Relaxed Conditions
8.2 Viability Tubes and Their Calculation
8.2.1 The Evolution Equation as a Generalized PDE
p.321
8.2.2 Viability through Parametrization
p.323
p.324
8.3 Control Synthesis under State Constraints
8.3.1 The Problem of State-Constrained Closed-Loop Control
p.327
8.3.2 State-Constrained Closed-Loop Control
p.329
p.330
8.3.3 Example
p.332
8.4 Obstacle Problems
8.4.1 Complementary Convex State Constraints
8.4.2 The Obstacle Problem and the Reach-Evasion Set
p.336
p.337
8.4.4 Closed-Loop Control for the Obstacle Problem
p.339
9. Guaranteed State Estimation
9.1 Set-Membership State Estimation
p.343
9.2 Hamiltonian Techniques for Set-Membership Estimati
9.2.1 Calculating Information Tubes
9.2.2 Comparison Principle for HJB Equations
p.347
9.2.3 Linear Systems
p.349
9.3 Ellipsoidal Approximations
9.3.1 Version-AE
p.352
9.3.2 Version-BE
9.3.3 Example: Information Set for Linear Systems
9.4 Discrete Observations
9.4.1 Continuous Dynamics under Discrete Observations
p.357
9.4.2 Discrete Dynamics and Observations
p.359
p.360
p.361
p.362
9.5 Viability Tubes: The Linear-Quadratic Approximation
p.364
p.365
p.366
9.6 Information Tubes vs Filtering Equations
9.6.1 Set-Valued Tubes through Stochastic Approximations
p.369
9.6.2 The Singular Perturbation Approach
10 Uncertain Systems: Output Feedback Control
10.1 The Problem of OFC
10.2 Generalized State and Rigorous Problem Formulation
p.374
10.3 The Overall Solution Scheme: General Case
p.376
10.4 Information Sets and Information States
p.378
p.379
p.380
10.5 Feedback Control in the Information Space
10.5.1 Control of Information Tubes
10.5.2 The Solution Scheme for Problem C
p.384
10.5.3 From Problem C to Problem C_opt
10.6 More Detailed Solution: Linear Systems
10.6.1 The "Linear-Convex" Solution
10.6.2 The Computable Solution: Ellipsoidal Approximation
p.389
p.390
10.7 Separation of Estimation and Control
10.8 Some Examples
p.393
p.394
11 Verification: Hybrid Systems
11.1 Verification Problems
11.1.1 The Problems and the Solution Scemes
11.1.2 Ellipsoidal Techniques for Verification Problems
p.399
p.400
11.2 Hybrid Dynamics and Control
11.2.1 The Hybrid System and the Reachability Problem
p.403
p.404
p.405
p.406
11.2.2 Value Functions: Ellipsoidal Approximations
p.408
p.409
p.410
p.411
p.412
p.413
p.414
p.415
p.416
11.3 Verification of Hybrid Systems
11.3.2 Ellipsoidal Methods for Verification
11.4 Impulse Controls in Hybrid System Models
11.4.1 Hybrid Systems with Resets
11.4.2 Two Simple Examples
p.422
11.4.3 Impulse Controls in Hybrid Systems
p.424
11.4.4 More Complicated Example: 3-D Bouncing Ball
p.426
p.427
p.428
p.429
References 1 - 19
References 20 - 49
References 50 - 76
References 77 - 102
References 103 - 128
References 129 - 153
References 154 - 176
References 177 - 202
References 203 - 228
References 229 - 256
References 257 - 275
Index B - E
Index E - P
Index R - V