The use of iterative dynamic programming employing systematic region contraction and accessible grid points is investigated for the optimal control of time-delay systems. At the time of generating the grid points for the state variables, the corresponding delayed variables at each time stage are also generated and stored in memory. Then, when applying dynamic programming, a linear approximation is used to obtain the initial profile for the delayed variables during integration. This procedure was tested with four problems of different complexity. In each case the optimal control policy is easily obtained and the results compare very favourably with those reported in the literature using other computational procedures.

Optimal control of time-delay systems by dynamic programming

In the first part of this paper, we summarize and complete earlier results of Delfour-Mitter [3] on the optimal control problem for linear hereditary differential systems (HDS) with a linear-quadratic cost function The properties of the operator Π(t) which characterizes the feedback gains and the reference functionr(t) are fully detailed In a second part, we construct an approximation to the linear HDS in state form and to the linear adjoint state equation and prove convergence In a third part we construct and solve the approximate optimal control problem following the method of J C Nedelec In the last part we construct the approximation to Π(t) andr(t) and prove convergence Finally we give a number of typical examples to illustrate the main features of the kernel of Π(t)

The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution

A hierarchical scheme for the optimal control of large-scale nonlinear time-delay systems is proposed. A linear approximation of the original system is defined about a nominal trajectory. The system is decomposed into N subsystems in such a way that the delay terms as well as interaction terms can be treated as extra inputs to linear nondelay subsystems. A numerical example illustrates this approach.

A computational algorithm for large-scale nonlinear time-delay systems

The paper presents a parameter-embedding approach to the optimal control of linear time-delay systems, and develops a simple computational algorithm via Walsh functions. The algorithm employs the concept of Walsh operational matrices for delay and advance. These operational matrices provide us with a means of numerically integrating differential equations with delayed and advanced arguments; a major task in the process of computing optimal controls for time-delay systems. An attractive feature of the present method is its ultimate simplicity and the resulting piecewise constant controls are convenient in practical implementation.

Optimal control of linear systems with delays in state and control via Walsh functions

This work presents a numerical method to solve the optimal control problem with time-delayed arguments and a fixed terminal time. A series of auxiliary states obtained from the linearly truncated Taylor series expansion are used to represent the status of a time-delayed state at different time intervals. The backward iterative dynamic programming (IDP) technique can thus be directly employed to solve the delay-free optimal control problem with augmented states. Five numerical examples are provided, demonstrating the proposed method's effectiveness in solving the time-delayed optimal control problems. Copyright © 2000 John Wiley & Sons, Ltd.

H. C. Chan

Papers

Optimization of time delay systems using parameter imbedding

Optimization of Time Delay Systems Using Parameter Imbedding