Optimal switching signal design with a cost on switching action

Wei Xu; Liying Yu; Gui-Hua Lin; Zhi Guo Feng

doi:10.3934/jimo.2019068

Article Contents

2020, Volume 16, Issue 5: 2531-2549. Doi: 10.3934/jimo.2019068

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Optimal switching signal design with a cost on switching action

1.
School of Management, Shanghai University, Shanghai, China
2.
Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, China
3.
College of Mathematics Science, Chongqing Normal University, Chongqing, China

^* Corresponding author: Gui-Hua Lin
^* Corresponding author: Gui-Hua Lin

Received: September 30, 2018

Revised: December 31, 2018

Early access: July 2019

Published: August 2020

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Abstract

In this paper, we consider a particular class of optimal switching problem for the linear-quadratic switched system in discrete time, where an optimal switching sequence is designed to minimize the quadratic performance index of the system with a switching cost. This is a challenging issue and studied only by few papers. First, we introduce a total variation function with respect to the switching sequence to measure the volatile switching action. In order to restrain the switching magnitude, it is added to the cost functional as a penalty. Then, the particular optimal switching problem is formulated. With the positive semi-definiteness of matrices, we construct a series of exact lower bounds of the cost functional at each time and the branch and bound method is applied to search all global optimal solutions. For the comparison between different global optimization methods, some numerical examples are given to show the efficiency of our proposed method.

Keywords:

Mathematics Subject Classification: 49M25.

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Figure 1. Global optimal switching sequences under different penalty parameters in Example 4.1

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Figure 2. Global optimal switching sequences under different penalty parameters in Example 4.2

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Table 1. Global optimal solutions under different penalty parameters in Example 4.1

$ \alpha $	Global optimal solution $ \sigma^{\ast} $	Switching times	Performance index	Switching cost	Optimal functional value $ J^{\ast} $
$ 0 $	(1 4 2 1 3 1 4 2 3 2)	9	40	0	40
	(1 4 2 1 3 1 4 2 1 1)	8	40	0	40
$ 0.5 $	(1 4 2 1 3 1 4 2 1 1)	8	40	8	48
	(1 1 3 4 4 4 2 1 1 1)	4	45	3	48
$ 1 $	(1 1 3 4 4 4 2 1 1 1)	4	45	6	51
$ 2 $	(2 3 2 2 2 2 1 1 1 1)	3	50	6	56
$ 5 $	(1 1 1 2 2 2 2 2 2 2)	1	59	5	64

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Table 2. Two kinds of B&B methods with $ \varepsilon = 10^{-5} $ in Example 4.1

$ \alpha $	Approximate B&B method				Exact B&B method
	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time
$ 0 $	(1 4 2 1 3 1 4 2 3 2)	40	116	0.0826s	(1 4 2 1 3 1 4 2 3 2)	40	128	0.1055s
	(1 4 2 1 3 1 4 2 1 1)				(1 4 2 1 3 1 4 2 1 1)
$ 0.5 $	(1 4 2 1 3 1 4 2 1 1)	48	144	0.1275s	(1 4 2 1 3 1 4 2 1 1)	48	152	0.1310s
	(1 1 3 4 4 4 2 1 1 1)				(1 1 3 4 4 4 2 1 1 1)
$ 1 $	(1 1 3 4 4 4 2 1 1 1)	51	376	0.2371s	(1 1 3 4 4 4 2 1 1 1)	51	392	0.2481s
$ 2 $	(2 3 2 2 2 2 1 1 1 1)	56	440	0.2799s	(2 3 2 2 2 2 1 1 1 1)	56	444	0.2806s
$ 5 $	(1 1 1 2 2 2 2 2 2 2)	64	360	0.2280s	(1 1 1 2 2 2 2 2 2 2)	64	372	0.2346s

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Table 3. Two kinds of B&B methods with $ \varepsilon = 1 $ in Example 4.1

$ \alpha $	Approximate B&B method				Exact B&B method
	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time
0	(1 4 2 1 1 1 1 1 3 1)	55	8	0.0115s	(1 4 2 1 3 1 4 2 3 2)	40	2796	10.4855s
					(1 4 2 1 3 1 4 2 1 1)
$ 0.5 $	(1 4 2 1 1 1 1 1 3 1)	60	8	0.0115s	(1 4 2 1 3 1 4 2 1 1)	48	2948	10.7044s
					(1 1 3 4 4 4 2 1 1 1)
$ 1 $	(1 1 3 4 4 4 2 1 1 1)	51	4	0.0091s	(1 1 3 4 4 4 2 1 1 1)	51	2768	10.4657s
$ 2 $	(1 1 3 4 4 4 2 1 1 1)	57	12	0.1379s	(2 3 2 2 2 2 1 1 1 1)	56	2504	10.3680s
	(1 1 3 4 4 4 2 2 2 2)
$ 5 $	(1 1 3 4 4 4 4 3 3 3)	70	4	0.0091s	(1 1 1 2 2 2 2 2 2 2)	64	2028	9.6311s

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Table 4. Global optimal solutions under different penalty parameters in Example 4.2

$ \alpha $	Global optimal solution $ \sigma^{\ast} $	Switching times	Performance index	Switching cost	Optimal functional value $ J^{\ast} $
0	(3 4 2 3 4 1 1 3 3 4)	7	5.1452	0	5.1452
0.1	(3 4 2 4 1 4 4 4 4 4)	5	5.2376	0.5	5.7376
0.5	(2 3 3 2 4 4 4 4 4 4)	3	5.6759	1.5	7.1759
2	(2 3 3 2 4 4 4 4 4 4)	3	5.6759	6	11.6759
5	(2 3 3 3 4 4 4 4 4 4)	2	7.7973	10	17.7973
10	(4 4 4 4 4 4 4 4 4 4)	0	18.5961	0	18.5961

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Table 5. Two kinds of B&B methods with $ \varepsilon = 10^{-20} $ in Example 4.2

$ \alpha $	Approximate B&B method				Exact B&B method
	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time
0	(3 4 2 3 4 1 1 3 3 4)	5.1452	164	0.5458s	(3 4 2 3 4 1 1 3 3 4)	5.1452	168	0.6742s
0.1	(3 4 2 4 1 4 4 4 4 4)	5.7376	108	0.6379s	(3 4 2 4 1 4 4 4 4 4)	5.7376	64	0.7968s
0.5	(2 3 3 2 4 4 4 4 4 4)	7.1759	96	0.7146s	(2 3 3 2 4 4 4 4 4 4)	7.1759	120	0.9573s
2	(2 3 3 2 4 4 4 4 4 4)	11.6759	132	1.5675s	(2 3 3 2 4 4 4 4 4 4)	11.6759	172	1.7968s
5	(2 3 3 3 4 4 4 4 4 4)	17.7973	36	1.2749s	(2 3 3 3 4 4 4 4 4 4)	17.7973	36	1.6238s
10	(4 4 4 4 4 4 4 4 4 4)	18.5961	24	0.8772s	(4 4 4 4 4 4 4 4 4 4)	18.5961	68	1.2210s

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Table 6. Two kinds of B&B methods with $ \varepsilon = 1 $ in Example 4.2

$ \alpha $	Approximate B&B method				Exact B&B method
	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time	$ \sigma^{\ast} $	$ J^{\ast} $	Searching times	Time
0	(2 3 3 2 4 1 4 3 4 4)	5.5347	12	0.0679s	(3 4 2 3 4 1 1 3 3 4)	5.1452	660	85.3819s
0.1	(2 3 3 2 4 1 4 4 4 4)	6.1436	4	0.0615s	(3 4 2 4 1 4 4 4 4 4)	5.7376	568	86.7247s
0.5	(2 3 3 2 4 4 4 4 4 4)	7.1759	4	0.0618s	(2 3 3 2 4 4 4 4 4 4)	7.1759	508	87.3288s
2	(2 3 3 2 4 4 4 4 4 4)	11.6759	24	0.1131s	(2 3 3 2 4 4 4 4 4 4)	11.6759	408	86.0455s
5	(2 3 3 3 4 4 4 4 4 4)	17.7973	4	0.0797s	(2 3 3 3 4 4 4 4 4 4)	17.7973	372	84.8442s
10	(2 3 3 3 4 4 4 4 4 4)	27.7973	4	0.2317s	(4 4 4 4 4 4 4 4 4 4)	18.5961	332	83.2846s

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