## Optimal Control of Constrained Piecewise Affine SystemsOne of the most important and challenging problems in control is the derivation of systematic tools for the computation of controllers for constrained nonlinear systems that can guarantee closed-loop stability, feasibility, and optimality with respect to some performance index. This book focuses on the efficient and systematic computation of closed-form optimal controllers for the powerful class of fast-sampled constrained piecewise affine systems. These systems may exhibit rather complex behavior and are equivalent to many other hybrid system formalisms (combining continuous-valued dynamics with logic rules) reported in the literature. Furthermore, piecewise affine systems are a useful modeling tool that can capture general nonlinearities (e.g. by local approximation), constraints, saturations, switches, and other hybrid modeling phenomena. The first part of the book presents an introduction to the mathematical and control theoretical background material needed for the full understanding of the book. The second part provides an in depth look at the computational and control theoretic properties of the controllers and part three presents different analysis and post-processing techniques. |

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### Contents

3 | |

7 | |

9 | |

10 | |

12 | |

14 Constrained Optimization | 14 |

15 MultiParametric Programming | 17 |

SYSTEMS AND CONTROL THEORY | 20 |

LINEAR VECTOR NORMS AS LYAPUNOV FUNCTIONS | 93 |

82 Software Implementation | 97 |

84 Complex W for DiscreteTime Systems | 98 |

85 Construction of a Real W for DiscreteTime Systems | 99 |

86 Simple Algebraic Test to Substitute the Feasibility LP 88 | 106 |

87 Computational Complexity and Finite Termination | 107 |

872 A Complexity Comparison | 109 |

88 Polyhedral Positive Invariant Sets | 110 |

22 Invariance and Stability | 21 |

RECEDING HORIZON CONTROL | 26 |

31 Basic Idea | 27 |

32 Closedform Solution to Receding Horizon Control | 28 |

321 Timevarying Model and System | 30 |

322 Comments on the Closedform Solution | 31 |

33 Choice of the Cost Function | 32 |

34 Stability and Feasibility in Receding Horizon Control | 35 |

PIECEWISE AFFINE SYSTEMS | 39 |

41 Obtaining a PWA Model | 41 |

OPTIMAL CONTROL OF CONSTRAINED PIECEWISE AFFINE SYSTEMS | 43 |

INTRODUCTION | 44 |

51 Software Implementation | 47 |

CONSTRAINED FINITE TIME OPTIMAL CONTROL | 48 |

61 CFTOC Solution via mpMILP | 51 |

62 CFTOC Solution via Dynamic Programming | 53 |

63 An Efﬁcient Algorithm for the CFTOC Solution | 54 |

64 Comments on the Dynamic Programming Approach for the CFTOC Problem | 56 |

65 Receding Horizon Control | 58 |

66 Examples | 59 |

661 Constrained PWA SineCosine System | 60 |

662 Car on a PWA Hill | 61 |

CONSTRAINED INFINITE TIME OPTIMAL CONTROL | 64 |

72 Closedloop Stability of the CITOC Solution | 66 |

73 CITOC Solution via Dynamic Programming | 68 |

731 Initial Value Function and Rate of Convergence | 73 |

732 Stabilizing Suboptimal Control | 79 |

7A An Efficient Algorithm for the CITOC Solution | 80 |

741 Alternative Choices of the Initial Value Function | 84 |

75 Examples | 86 |

752 Constrained LTI System | 88 |

753 Car on a PWA Hill | 89 |

ANALYSIS AND POSTPROCESSING TECHNIQUES FOR PIECEWISE AFFINE SYSTEMS | 92 |

810 Examples | 112 |

STABILITY ANALYSIS | 114 |

92 Constrained Finite Time Optimal Control of Piecewise Afﬁne Systems | 115 |

93 ClosedLoop Stability and Feasibility | 116 |

931 Computation of the Maximal Positively Invariant Set | 118 |

932 Computation of the Lyapunov Stability Region | 123 |

94 Examples | 128 |

942 Constrained PWA SineCosine System | 129 |

943 Constrained PWA System | 131 |

944 3Dimensional Constrained PWA System | 133 |

10 STABILITY TUBES | 135 |

102 Constrained Finite Time Optimal Control of Piecewise Afﬁne Systems | 136 |

103 Stability Tubes for Nonlinear Systems | 139 |

104 Computation of Stability Tubes for Piecewise Affine Systems | 143 |

105 Comments on Stability Tubes | 144 |

106 Examples | 147 |

1062 Car on a PWA Hill | 148 |

EFFICIENT EVALUATION OF PIECEWISE CONTROL LAWS DEFINED OVER A LARGE NUMBER OF POLYHEDRA | 150 |

112 Software Implementation | 151 |

113 Point Location Problem | 152 |

115 Alternative Search Approaches | 154 |

116 The Proposed Search Algorithm | 156 |

1161 Bounding Box Search Tree | 157 |

1162 Local Search | 159 |

118 Examples | 160 |

1182 Constrained PWA System | 162 |

1183 Ball Plate System | 163 |

APPENDIX | 166 |

Alternative Proof of Lemma 77a | 167 |

BIBLIOGRAPHY | 171 |

Author Index | 182 |

187 | |

### Other editions - View all

Optimal Control of Constrained Piecewise Affine Systems Frank Christophersen No preview available - 2009 |

Optimal Control of Constrained Piecewise Affine Systems Frank Christophersen No preview available - 2007 |

### Common terms and phrases

¼nx affine function algorithm asymptotic stability based Lyapunov functions bounding boxes CFTOC problem CFTOC Problem 6.1 CFTOC solution Chapter CITOC closed-loop system complexity computation considered constrained finite control action convex convex set cost function defined Definition denotes dimension discrete-time dynamic programming eigenvalue equilibrium point example feasible state space feedback control law finite time optimal fPWA hybrid systems infeasibility infinite time solution invariant set Omax iteration step Lemma linear programs linear vector norm Lyapunov function Lyapunov stability Lyapunov stability region matrix maximal positively invariant mp-LP mp-MILP multi-parametric number of regions on-line optimal control optimal control problem optimization problem parameters piecewise affine systems polyhedral partition polyhedral regions polytope positively invariant set prediction horizon quadratic receding horizon control region of attraction search tree Section solved solver stability and feasibility stability tube Theorem tion trajectories value function vector norm x e A's µRH

### Popular passages

Page 175 - Linear systems with state and control constraints: the theory and application of maximal output admissible sets, IEEE Transactions on Automatic Control AC-36: 1008-1020.

Page 172 - FJ Christophersen, and M. Morari. A new Algorithm for Constrained Finite Time Optimal Control of Hybrid Systems with a Linear Performance Index.

Page 172 - B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer. Non-Linear Parametric Optimization.

Page 172 - Model Predictive Control Based on Linear Programming - The Explicit Solution," IEEE Transaction on Automatic Control, Vol.

Page 8 - Ax < b}, the hyperplane {x | cx = 6} is a supporting hyperplane of P. A subset F of P is called a face of P if either F = P, or F is the intersection of P with a supporting hyperplane of P.