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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

Received  January 2022 Revised  March 2022 Early access April 2022

Fund Project: 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

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.

Citation: Xi Zhu, Changjun Yu, Kok Lay Teo. A new switching time optimization technique for multi-switching systems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022067
##### References:
 [1] F. Bencherki, S. Türkay and H. Akay, Deadbeat observer based switched linear system identification, preprint, 2021, arXiv: 2107.14571. [2] Z. Geng and Z. Sun, Sub-optimal aggregation of switched linear systems, Proceedings of the 31st Chinese Control Conference, Hefei, China, 2012, 2163–2166. [3] X. Guo and H. P. Ren, A switching control strategy based on switching system model of three-phase vsr under unbalanced grid conditions, IEEE Transactions on Industrial Electronics, 68 (2021), 5799-5809. [4] J. Jin, J. P. Ramirez, S. G. Wee, D. H. Lee, Y. G. Kim and N. Gans, A switched-system approach to formation control and heading consensus for multi-robot systems, Intelligent Service Robotics, 11 (2018), 207-224. [5] L. Li, C. J. Yu, N. Zhang, Y. Q. Bai and Z. Gao, A time-scaling technique for time-delay switched systems, Discrete & Continuous Dynamical Systems - S, 13 (2020), 1825-1843.  doi: 10.3934/dcdss.2020108. [6] Q. Lin, R. Loxton and K. L. Teo, Optimal control of nonlinear switched systems: computational methods and applications, Journal of the Operations Research Society of China, 1 (2013), 275-311. [7] C. Y. Liu and Z. H. Gong, Optimal control of switched systems arising in fermentation processes, Springer Optimization & Its Applications, 2014. doi: 10.1007/978-3-662-43793-3. [8] C. Y. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Nonlinear Analysis: Hybrid Systems, 163 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7. [9] R. Loxton, Q. Lin and K. L. Teo, Switching Time Optimization for Nonlinear Switched Systems: Direct Optimization and the Time-scaling Transformation, Pacific Journal of Optimization, 10 (2014), 537-560.  doi: 10.3934/jimo.2014.10.275. [10] R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [11] R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE transactions on automatic control, 54 (2009), 2455-2460.  doi: 10.1109/TAC.2009.2029310. [12] P. Mu, L. Wang and C. Y. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7. [13] T. Niu, J. G. Zhai, H. C. Yin, E. M. Feng and C. Y. Liu, The uncoupled microbial fed-batch fermentation optimization based on state-dependent switched system, International Journal of Biomathematics, 14 (2021), 2150025.  doi: 10.1142/S179352452150025X. [14] T. Niu, J. G. Zhai, H. C. Yin, E. M. Feng, C. Y. Liu and Z. L. Xiu, Multi-objective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation, International Journal of Systems Science, 51 (2020), 1798-1813.  doi: 10.1080/00207721.2020.1780338. [15] L. Wang, J. L. Yuan, C. Z. Wu and X. Y.Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z. [16] X. Xu and P. J. Antsaklis, An approach for solving general switched linear quadratic optimal control problems, Proceedings of the 40th IEEE Conference on Decision and Control, Florida, USA, 2001, 2478–2483. [17] X. Xu and P. J. Antsaklis, Optimal control of switched autonomous systems, Proceedings of the 41st IEEE Conference on Decision and Control, Florida, USA, 2002, 4401–4406. [18] X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, Automatic Control IEEE Transactions on, 49 (2004), 2-16.  doi: 10.1109/TAC.2003.821417. [19] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485. [20] D. T. Zeng, X. Yu, J. Huang and C. Tan, Numerical computation for a kind of time optimal control problem for the tubular reactor system, Mathematical Problems in Engineering, (2016), 1–9. doi: 10.1155/2018/9580470. [21] C. Zhang, M. Gan and C. Xue, Data-driven optimal switching and control of switched systems, Control Theory and Technology, 19 (2021), 299-314.  doi: 10.1007/s11768-021-00054-y. [22] N. Zhang, C. J. Yu and F. S. Xie, The time-scaling transformation technique for optimal control problems with time-varying time-delay switched systems, Journal of the Operations Research Society of China, 8 (2020), 581-600.  doi: 10.1007/s40305-020-00299-5. [23] Y. Zhang, Y. Q. Liu and Y. Liu, Stability analysis for a type of multiswitching system with parallel structure, Mathematical Problems in Engineering, 2018 (2018), 1-16.  doi: 10.1155/2018/3834601.

show all references

##### References:
 [1] F. Bencherki, S. Türkay and H. Akay, Deadbeat observer based switched linear system identification, preprint, 2021, arXiv: 2107.14571. [2] Z. Geng and Z. Sun, Sub-optimal aggregation of switched linear systems, Proceedings of the 31st Chinese Control Conference, Hefei, China, 2012, 2163–2166. [3] X. Guo and H. P. Ren, A switching control strategy based on switching system model of three-phase vsr under unbalanced grid conditions, IEEE Transactions on Industrial Electronics, 68 (2021), 5799-5809. [4] J. Jin, J. P. Ramirez, S. G. Wee, D. H. Lee, Y. G. Kim and N. Gans, A switched-system approach to formation control and heading consensus for multi-robot systems, Intelligent Service Robotics, 11 (2018), 207-224. [5] L. Li, C. J. Yu, N. Zhang, Y. Q. Bai and Z. Gao, A time-scaling technique for time-delay switched systems, Discrete & Continuous Dynamical Systems - S, 13 (2020), 1825-1843.  doi: 10.3934/dcdss.2020108. [6] Q. Lin, R. Loxton and K. L. Teo, Optimal control of nonlinear switched systems: computational methods and applications, Journal of the Operations Research Society of China, 1 (2013), 275-311. [7] C. Y. Liu and Z. H. Gong, Optimal control of switched systems arising in fermentation processes, Springer Optimization & Its Applications, 2014. doi: 10.1007/978-3-662-43793-3. [8] C. Y. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Nonlinear Analysis: Hybrid Systems, 163 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7. [9] R. Loxton, Q. Lin and K. L. Teo, Switching Time Optimization for Nonlinear Switched Systems: Direct Optimization and the Time-scaling Transformation, Pacific Journal of Optimization, 10 (2014), 537-560.  doi: 10.3934/jimo.2014.10.275. [10] R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [11] R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE transactions on automatic control, 54 (2009), 2455-2460.  doi: 10.1109/TAC.2009.2029310. [12] P. Mu, L. Wang and C. Y. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7. [13] T. Niu, J. G. Zhai, H. C. Yin, E. M. Feng and C. Y. Liu, The uncoupled microbial fed-batch fermentation optimization based on state-dependent switched system, International Journal of Biomathematics, 14 (2021), 2150025.  doi: 10.1142/S179352452150025X. [14] T. Niu, J. G. Zhai, H. C. Yin, E. M. Feng, C. Y. Liu and Z. L. Xiu, Multi-objective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation, International Journal of Systems Science, 51 (2020), 1798-1813.  doi: 10.1080/00207721.2020.1780338. [15] L. Wang, J. L. Yuan, C. Z. Wu and X. Y.Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z. [16] X. Xu and P. J. Antsaklis, An approach for solving general switched linear quadratic optimal control problems, Proceedings of the 40th IEEE Conference on Decision and Control, Florida, USA, 2001, 2478–2483. [17] X. Xu and P. J. Antsaklis, Optimal control of switched autonomous systems, Proceedings of the 41st IEEE Conference on Decision and Control, Florida, USA, 2002, 4401–4406. [18] X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, Automatic Control IEEE Transactions on, 49 (2004), 2-16.  doi: 10.1109/TAC.2003.821417. [19] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485. [20] D. T. Zeng, X. Yu, J. Huang and C. Tan, Numerical computation for a kind of time optimal control problem for the tubular reactor system, Mathematical Problems in Engineering, (2016), 1–9. doi: 10.1155/2018/9580470. [21] C. Zhang, M. Gan and C. Xue, Data-driven optimal switching and control of switched systems, Control Theory and Technology, 19 (2021), 299-314.  doi: 10.1007/s11768-021-00054-y. [22] N. Zhang, C. J. Yu and F. S. Xie, The time-scaling transformation technique for optimal control problems with time-varying time-delay switched systems, Journal of the Operations Research Society of China, 8 (2020), 581-600.  doi: 10.1007/s40305-020-00299-5. [23] Y. Zhang, Y. Q. Liu and Y. Liu, Stability analysis for a type of multiswitching system with parallel structure, Mathematical Problems in Engineering, 2018 (2018), 1-16.  doi: 10.1155/2018/3834601.
The structure of a switching system
The structure of a multi-switching system
The time-scaling function $\mu(s\mid\boldsymbol\theta)$
The variable switching times to be optimized
The first switching time transformation process
The second switching time transformation process
Optimal state trajectories for Example 1 obtained by using the two techniques
Optimal state trajectories for Case 1 of Example 2 obtained by using the two techniques
Optimal state trajectories for Case 2 and Case 3 of Example 2 obtained by using the traditional time scaling transformation and our proposed methods
Optimal state trajectories for Example 3
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] $$\boldsymbol\tau_2^\star=[0.20, 1.17]  Time-scaling method Proposed method Optimal cost g_0^\star 17.0752 14.2028 Optimal switching times \boldsymbol\tau_1^\star, \boldsymbol\tau_2^\star=[0.61, 0.86] \boldsymbol\tau_1^\star=[0.46, 0.94]$$ \boldsymbol\tau_2^\star=[0.20, 1.17]$
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$\times10^4$
 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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