Time-scaling transformation for optimal control problem with time-varying delay

Di Wu; Yanqin Bai; Fusheng Xie

doi:10.3934/dcdss.2020098

Article Contents

2020, Volume 13, Issue 6: 1683-1695. Doi: 10.3934/dcdss.2020098

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Time-scaling transformation for optimal control problem with time-varying delay

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China

^* Corresponding author: Yanqin Bai
^* Corresponding author: Yanqin Bai

Received: January 31, 2018

Revised: April 30, 2018

Published: April 2020

This research was supported by grants from the National Natural Science Foundation of China (No.11771275, No.11871039)

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Abstract

This paper focuses on the solution to nonlinear time-delay optimal control problems with time-varying delay subject to canonical equality and inequality constraints. Traditional control parameterization in conjunction with time-scaling transformation could optimize control parameters and switching times at the same time when the time-delays in the dynamic system are taken as constants. The purpose of this paper is to extend this method to solve the dynamic systems with time-varying delay. We introduce a hybrid time-scaling transformation that converts the given time-delay system into an equivalent system defined on a new time horizon with fixed switching times. Meanwhile, we obtain the value of time-delay state utilizing the relationship between the new time scale and the original one. After computing the gradients of the cost and constraints with respect to the control heights and its durations, we could solve the equivalent optimal control problem using gradient based optimization method.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. Optimal control for Problem 1 for the case of $ q = 7 $

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Figure 2. Optimal state trajectory for Problem 1 using hybrid time-scaling transformation for the case of $ q = 7 $

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Figure 3. Optimal state trajectory for Problem 1 using traditional control parameterization with fixed switching times ($ q = 7 $)

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Figure 4. Optimal control for Problem 2 for the case of $ q = 8 $

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Figure 5. Optimal state trajectory for Problem 2 using hybrid time-scaling transformation for the case of $ q = 8 $

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Figure 6. Optimal state trajectory for Problem 2 using traditional control parameterization with fixed switching times ($ q = 8 $)

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Table 1. Optimal cost of Problem 1

(a) using hybrid time-scaling transformation		(b) using traditional control parameterization
partition number	$ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $	partition number	$ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=7 $	2.1275	$ q=7 $	2.1347
$ q=5 $	2.1333	$ q=5 $	2.1446
$ q=3 $	2.1340	$ q=3 $	2.1812

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Table 2. Parameters in Problem 2

a	b	c	$ t_{f} $	Q	R	S
0.2	0.5	0.2	1.5	$ I_{2\times2} $	$ I_{2\times2} $	$ 10^{4}I_{2\times2} $

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Table 3. Optimal cost of Problem 2

(a) Using hybrid time-scaling transformation		(b) Using traditional control parameterization
partition number	$ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $	partition number	$ g_{0}(\mathit{\boldsymbol{u}}^{q,*}) $
$ q=8 $	3.8750	$ q=8 $	5.3252
$ q=6 $	4.2319	$ q=6 $	6.1942
$ q=4 $	6.9578	$ q=4 $	7.9119

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