# American Institute of Mathematical Sciences

September  2020, 16(5): 2531-2549. doi: 10.3934/jimo.2019068

## 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

Received  October 2018 Revised  January 2019 Published  July 2019

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.

Citation: Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2531-2549. doi: 10.3934/jimo.2019068
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##### References:
Global optimal switching sequences under different penalty parameters in Example 4.1
Global optimal switching sequences under different penalty parameters in Example 4.2
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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
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
 $\alpha$ 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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