Numerical Methods for Optimal Control Problems with State ConstraintsWhile optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature. |
Contents
Introduction | 1 |
2 Optimal Control | 5 |
3 Numerical Methods for Optimal Control Problems | 7 |
Estimates on Solutions to Differential Equations and Their Approximations | 13 |
2 Lagrangian Hamiltonian and Reduced Gradients | 19 |
First Order Method | 27 |
2 Representation of Functional Directional Derivatives | 31 |
3 Relaxed Controls | 32 |
RungeKutta Based Procedure for Optimal Control of Differential Algebraic Equations | 129 |
2 The Method | 133 |
21 Implicit RungeKutta Methods | 134 |
22 Calculation of the Reduced Gradients | 137 |
3 Implementation of the Implicit RungeKutta Method | 144 |
32 Stopping Criterion for the Newton Method | 145 |
33 Stepsize Selection | 146 |
4 Numerical Experiments | 151 |
4 The Algorithm | 34 |
5 Convergence Properties of the Algorithm | 38 |
6 Proof of the Convergence Theorem etc | 41 |
7 Concluding Remarks | 52 |
Implementation | 55 |
11 Second Order Correction To the Line Search | 65 |
12 Resetting the Penalty Parameter | 66 |
3 Numerical Examples | 68 |
Second Order Method | 81 |
2 Function Space Algorithm | 84 |
3 SemiInfinite Programming Method | 86 |
4 Bounding the Number of Constraints | 92 |
41 Some Remarks on Direction Finding Subproblems | 94 |
42 The Nonlinear Programming Problem | 98 |
43 The Watchdog Technique for Redundant Constraints | 107 |
44 TwoStep Superlinear Convergence | 121 |
45 Numerical Experiments | 125 |
Concluding Remarks | 127 |
5 Some Remarks on Integration and Optimization Accuracies | 164 |
6 Concluding Remarks | 166 |
A Primal RangeSpace Method for PiecewiseLinear Quadratic Programming | 169 |
A2 A RangeSpace MethodIntroduction | 170 |
A3 The Basic Method | 171 |
A4 Efficient Implementation | 175 |
A41 Adding a Bound to the Working Set | 178 |
A42 Deleting a Bound from the Working Set | 182 |
A43 Adding a Vector a to the Working Set | 184 |
A44 Deleting a Vector a from the Working Set | 186 |
A5 Computation of the Lagrange Multipliers Corresponding to the Fixed Variables | 187 |
A6 Modifications and Extensions | 188 |
A7 Numerical Experiments | 191 |
References | 197 |
List of Symbols | 209 |
| 213 | |
Other editions - View all
Numerical Methods for Optimal Control Problems with State Constraints Radoslaw Pytlak Limited preview - 2006 |
Numerical Methods for Optimal Control Problems with State Constraints Radoslaw Pytlak No preview available - 2014 |
Common terms and phrases
active set adjoint equations algorithm applied approximation Assume bounded brachistochrone problem calculated calculus of variations Chapter computational control functions convergence analysis corresponding defined described differential differential-algebraic equations direction finding subproblem discretization equality constraints evaluated Example exists FD Algorithm finite number Fo(u follows formula fu(t gradient algorithms Hessian Hessian matrix implementation inequality constraints integration procedure Jacobians Lagrange multipliers Lemma linear linearly independent LSSOL matrix nonlinear programming number of constraints number of iterations objective function optimal control problems penalty parameter PLQP PNLP PNTSOL problem PN programming problem proof Proposition range-space rate of convergence Re,u reduced gradients relaxed controls Runge-Kutta methods satisfies sequence solution solved SQP algorithm stationary point stepsize stiff equations superlinearly convergent system equations Theorem tions trajectory Uk+1 updated variables vectors windshear xu(t นี้
Popular passages
Page 200 - Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems, J.
Page 202 - Pola* (1982). A superlinearly convergent algorithm for constrained optimization problems. Mathematical Programming Study 16, 45-61.
Page 199 - Pesch, HJ: Abort Landing in the Presence of a Windshear as a Minimax Optimal Control Problem, Part 2: Multiple Shooting and Homotopy. J.
Page 203 - Unified steerable phase I-phase II method of feasible directions for semi-infinite opti-mization.
Page 203 - On the Quadratic Programming Algorithm of Goldfarb and Idnani' Mathematical Programming Study.
Page 206 - Relaxed Controls and the Convergence of Optimal Control Algorithms, SIAM J.
Page 200 - El-Alem; A global convergence theory for the Celis-Dennis-Tapia trust region algorithm for constrained optimization, SIAM J.
Page 205 - A Feasible Directions Algorithm for Optimal Control Problems with State and Control Constraints: Convergence Analysis, Research Report C96-24, Centre for Process Systems Engineering, Imperial College, UK, also SIAM J.