A new switching time optimization technique for multi-switching systems

Xi Zhu; Changjun Yu; Kok Lay Teo

doi:10.3934/jimo.2022067

Article Contents

2023, Volume 19, Issue 4: 2838-2854. Doi: 10.3934/jimo.2022067

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A new switching time optimization technique for multi-switching systems

1.
Shanghai University, Shanghai, 200444, China
2.
Sunway University, Selangor, 473000, Malaysia

^*Corresponding author: Changjun Yu
^*Corresponding author: Changjun Yu

Received: January 2022

Revised: March 2022

Published: April 2023

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

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Abstract

For a multi-switching system, it consists of multiple parallel switching systems. The optimal control problem controlled by a multi-switching system is to determine the optimal time sequence for each of the parallel switching systems. The time-scaling transformation is a well-known switching time optimization approach which has been widely used in various problem settings. However, the time-scaling transformation requires that these parallel switching systems have the same number of subsystems and are designed to switch simultaneously. Thus, it is not applicable for solving optimal control problems of general multi-switching systems. This paper presents a new technique for optimizing the switching times of multi-switching systems.

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

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Figure 1. The structure of a switching system

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Figure 2. The structure of a multi-switching system

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Figure 3. The time-scaling function $ \mu(s\mid\boldsymbol\theta) $

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Figure 4. The variable switching times to be optimized

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Figure 5. The first switching time transformation process

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Figure 6. The second switching time transformation process

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Figure 7. Optimal state trajectories for Example 1 obtained by using the two techniques

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Figure 8. Optimal state trajectories for Case 1 of Example 2 obtained by using the two techniques

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Figure 9. Optimal state trajectories for Case 2 and Case 3 of Example 2 obtained by using the traditional time scaling transformation and our proposed methods

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Figure 10. Optimal state trajectories for Example 3

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Table 1. Optimal costs for Example 1 obtained by using the two techniques

	Time-scaling method	Proposed method
Optimal cost $ g_0^\star $	17.0752	14.2028
Optimal switching times	$ \boldsymbol\tau_1^\star, \boldsymbol\tau_2^\star=[0.61, 0.86] $	$ \boldsymbol\tau_1^\star=[0.46, 0.94] $ $ \boldsymbol\tau_2^\star=[0.20, 1.17] $

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Table 2. Optimal costs obtained by using the two techniques for Example 2

	Optimal cost $ g_0^* $
Using method	$ Case\; 1 $	$ Case \; 2 $	$ Case\; 3 $
Time-scaling method	1.6632$ \times 10^4 $	4.0932$ \times 10^4 $	-
Proposed method	1.6632$ \times10^4 $	4.0932$ \times 10^4 $	1.6650$ \times10^4 $

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