# American Institute of Mathematical Sciences

April  2019, 15(2): 817-832. doi: 10.3934/jimo.2018072

## On the global optimal solution for linear quadratic problems of switched system

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

* Corresponding author: Zhi Guo Feng

Received  December 2016 Revised  March 2018 Published  June 2018

The global optimal solution for the optimal switching problem is considered in discrete time, where these subsystems are linear and the cost functional is quadratic. The optimal switching problem is a discrete optimization problem. Complete enumeration search is always required to find the global optimal solution, which is very expensive. Relaxation method is an effective method to transform the discrete optimization problem into the continuous optimization problem, while the optimal solution is always not the feasible solution of the discrete optimization problem. In this paper, we propose a special class of relaxation method to transform the optimal switching problem into a relaxed optimization problem. We prove that the optimal solution of this modified relaxed optimization problem is exactly that of the optimal switching problem. Then, the global optimal solution can be obtained by solving the continuous optimization problem easily. Numerical examples are demonstrated to show that the modified relaxation method is efficient and effective to obtain the global optimal solution.

Citation: Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang. On the global optimal solution for linear quadratic problems of switched system. Journal of Industrial & Management Optimization, 2019, 15 (2) : 817-832. doi: 10.3934/jimo.2018072
##### References:
 [1] H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest, Gradient descent approach to optiomal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186. doi: 10.1007/s10957-007-9305-y. [2] S. C. Bengea and R. A. DeCarlo, Optimal control of switching systems, Automatica, 41 (2005), 11-27. doi: 10.1016/j.automatica.2004.08.003. [3] M. Egerstedt, Y. Wardi and H. Axelsson, Transition-time optimization for switched-mode dynamical systems, IEEE Transactions on Automatic Control, 51 (2006), 110-115. doi: 10.1109/TAC.2005.861711. [4] Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024. [5] Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. doi: 10.3934/jimo.2005.1.499. [6] Z. G. Feng, Z.G. Feng, K. L. Teo and V. Rehbock, A discrete filled function method for the optimal control of switched systems in discrete time, Optimal Control, Applications and Methods, 30 (2009), 583-593. doi: 10.1002/oca.885. [7] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parametrization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. [8] R. Li, Z. G. Feng, K. L. Teo and G. R. Duan, Optimal Piecewise State Feedback Control for Impulsive Switched Systems, Mathematical and Computer Modelling, 48 (2008), 468-479. doi: 10.1016/j.mcm.2007.06.028. [9] 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. doi: 10.1007/s40305-013-0021-z. [10] C. Liu, R. Loxton and K. L. Teo, Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306. doi: 10.1007/s10589-013-9632-x. [11] C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process, Nonlinear Analysis Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. [12] C. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7. [13] C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43793-3. [14] Q. Long and C. Wu, A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization, Journal of Industrial and Management Optimization, 10 (2014), 1279-1296. doi: 10.3934/jimo.2014.10.1279. [15] R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031. [16] B. Piccoli, Hybrid systems and optimal control, in Proceedings of the 37th IEEE Conference on Decision and Control, 1998, 13–18. doi: 10.1109/CDC.1998.760582. [17] E. Rentsen, J. Zhou and K. L. Teo, A global optimization approach to fractional optimal control, Journal of Industrial and Management Optimization, 12 (2016), 73-82. doi: 10.3934/jimo.2016.12.73. [18] M. Soler, A. Olivares and E. Staffetti, Hybrid optimal control approach to commercial aircraft trajectory planning, Journal of Guidance, Control, and Dynamics, 33 (2010), 985-991. [19] H. J. Sussmann, A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 425–430. doi: 10.1109/CDC.1999.832814. [20] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parametrization enhancing transform for constrained optimal control problems, Journal of Australian Mathematical Society, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. [21] C. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. [22] C. Wu, K. L. Teo and S. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica, 49 (2013), 1809-1815. doi: 10.1016/j.automatica.2013.02.052. [23] C. Wu, K. L. Teo, R. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. doi: 10.1016/j.aml.2005.11.018. [24] X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16. doi: 10.1109/TAC.2003.821417. [25] W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193. doi: 10.1016/j.automatica.2016.12.002. [26] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual miser: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. [27] J. Zhai, T. Niu, J. Ye and E. Feng, Optimal control of nonlinear switched system with mixed constraints and its parallel optimization algorithm, Nonlinear Analysis Hybrid Systems, 25 (2017), 21-40. doi: 10.1016/j.nahs.2017.02.001. [28] C. Zhao, C. Wu, J. Chai, X. Wang, X. Yang, J. M. Lee and M. J. Kim, Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty, Applied Soft Computing, 55 (2017), 549-564. doi: 10.1016/j.asoc.2017.02.009. [29] X. L. Zhu, Z. G. Feng and J. W. Peng, Robust design of sensor fusion problem in discrete time, Journal of Industrial and Management Optimization, 13 (2017), 825-834. doi: 10.3934/jimo.2016048.

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##### References:
 [1] H. Axelsson, Y. Wardi, M. Egerstedt and E. I. Verriest, Gradient descent approach to optiomal mode scheduling in hybrid dynamical systems, Journal of Optimization Theory and Applications, 136 (2008), 167-186. doi: 10.1007/s10957-007-9305-y. [2] S. C. Bengea and R. A. DeCarlo, Optimal control of switching systems, Automatica, 41 (2005), 11-27. doi: 10.1016/j.automatica.2004.08.003. [3] M. Egerstedt, Y. Wardi and H. Axelsson, Transition-time optimization for switched-mode dynamical systems, IEEE Transactions on Automatic Control, 51 (2006), 110-115. doi: 10.1109/TAC.2005.861711. [4] Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024. [5] Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. doi: 10.3934/jimo.2005.1.499. [6] Z. G. Feng, Z.G. Feng, K. L. Teo and V. Rehbock, A discrete filled function method for the optimal control of switched systems in discrete time, Optimal Control, Applications and Methods, 30 (2009), 583-593. doi: 10.1002/oca.885. [7] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parametrization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. [8] R. Li, Z. G. Feng, K. L. Teo and G. R. Duan, Optimal Piecewise State Feedback Control for Impulsive Switched Systems, Mathematical and Computer Modelling, 48 (2008), 468-479. doi: 10.1016/j.mcm.2007.06.028. [9] 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. doi: 10.1007/s40305-013-0021-z. [10] C. Liu, R. Loxton and K. L. Teo, Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306. doi: 10.1007/s10589-013-9632-x. [11] C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process, Nonlinear Analysis Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. [12] C. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7. [13] C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43793-3. [14] Q. Long and C. Wu, A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization, Journal of Industrial and Management Optimization, 10 (2014), 1279-1296. doi: 10.3934/jimo.2014.10.1279. [15] R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031. [16] B. Piccoli, Hybrid systems and optimal control, in Proceedings of the 37th IEEE Conference on Decision and Control, 1998, 13–18. doi: 10.1109/CDC.1998.760582. [17] E. Rentsen, J. Zhou and K. L. Teo, A global optimization approach to fractional optimal control, Journal of Industrial and Management Optimization, 12 (2016), 73-82. doi: 10.3934/jimo.2016.12.73. [18] M. Soler, A. Olivares and E. Staffetti, Hybrid optimal control approach to commercial aircraft trajectory planning, Journal of Guidance, Control, and Dynamics, 33 (2010), 985-991. [19] H. J. Sussmann, A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 425–430. doi: 10.1109/CDC.1999.832814. [20] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parametrization enhancing transform for constrained optimal control problems, Journal of Australian Mathematical Society, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. [21] C. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. [22] C. Wu, K. L. Teo and S. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica, 49 (2013), 1809-1815. doi: 10.1016/j.automatica.2013.02.052. [23] C. Wu, K. L. Teo, R. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. doi: 10.1016/j.aml.2005.11.018. [24] X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16. doi: 10.1109/TAC.2003.821417. [25] W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193. doi: 10.1016/j.automatica.2016.12.002. [26] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual miser: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. [27] J. Zhai, T. Niu, J. Ye and E. Feng, Optimal control of nonlinear switched system with mixed constraints and its parallel optimization algorithm, Nonlinear Analysis Hybrid Systems, 25 (2017), 21-40. doi: 10.1016/j.nahs.2017.02.001. [28] C. Zhao, C. Wu, J. Chai, X. Wang, X. Yang, J. M. Lee and M. J. Kim, Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty, Applied Soft Computing, 55 (2017), 549-564. doi: 10.1016/j.asoc.2017.02.009. [29] X. L. Zhu, Z. G. Feng and J. W. Peng, Robust design of sensor fusion problem in discrete time, Journal of Industrial and Management Optimization, 13 (2017), 825-834. doi: 10.3934/jimo.2016048.
Optimal weight of the first subsystem obtained by relaxation method and modified relaxation method in Example 1
Optimal weight of the second subsystem obtained by relaxation method and modified relaxation method in Example 1
Optimal weight of the third subsystem obtained by relaxation method and modified relaxation method in Example 1
Optimal solution and truncated solution in Example 1
Optimal weight of the first subsystem obtained by relaxation method and modified relaxation method in Example 2
Optimal weight of the second subsystem obtained by relaxation method and modified relaxation method in Example 2
Optimal weight of the third subsystem obtained by relaxation method and modified relaxation method in Example 2
Optimal solution and truncated solution in Example 2
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