Control parameterization approach to time-delay optimal control problems: A survey

Di Wu; Yin Chen; Changjun Yu; Yanqin Bai; Kok Lay Teo

doi:10.3934/jimo.2022108

Article Contents

2023, Volume 19, Issue 5: 3750-3783. Doi: 10.3934/jimo.2022108

This issue Previous Article A unified and tight linear convergence analysis of the relaxed proximal point algorithm Next Article Traffic flow modeling and optimization control in the approach airspace

Control parameterization approach to time-delay optimal control problems: A survey

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
School of Mathematical Sciences, Sunway University, Malaysia

^*Corresponding author: Changjun Yu (yuchangjun@126.com)
^*Corresponding author: Changjun Yu (yuchangjun@126.com)

Received: March 2022

Revised: April 2022

Early access: June 2022

Published: May 2023

This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039, No. 12171307 and Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900.

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Abstract

Control parameterization technique is an effective method to solve optimal control problems. It works by approximating the control function by piece-wise constant (or linear) function. In this way, the optimal control problems are approximated by optimal parameter selection problems, which can be regarded as finite-dimensional optimization problems, where the control heights and switching times of the piece-wise constant function are taken as the decision variables. They can be solved by using gradient-based optimization methods. For this, it requires the gradient formulas of the objective and constraint functions with respect to the decision variables. There are two methods for deriving the required gradient formulas. The aim of this paper is to survey various numerical methods developed based on control parameterization for solving optimal control problems governed by time-delay systems. Particular focus is on those numerical methods for optimizing switching time points. This is because the process of optimizing the switching time points is much more complicated for optimal control problems with time delays. In addition, we shall also review the computational methods developed based on control parameterization technique for solving two optimal control problems involving more complicated time delays.

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Mathematics Subject Classification: Primary: 49M37; Secondary: 65K10, 65P10, 90C30, 93C15.

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Figure 1. Piece-wise constant function approximation with equidistant switching times over the planning horizon $ (p=5) $

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Figure 2. The illustration of the time-scaling function $ (p=5) $

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Figure 3. The relationship between $ t $, $ t' $, $ s $ and $ s' $

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