# American Institute of Mathematical Sciences

• Previous Article
Neutral and indifference pricing with stochastic correlation and volatility
• JIMO Home
• This Issue
• Next Article
Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints
January  2018, 14(1): 183-198. doi: 10.3934/jimo.2017042

## Optimal control of switched systems with multiple time-delays and a cost on changing control

 1 School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai 264005, China 2 Department of Mathematics and Statistics, Curtin University, Perth 6845, Australia

* Corresponding author: Chongyang Liu

Received  July 2016 Revised  October 2016 Published  January 2018 Early access  April 2017

In this paper, we consider a class of optimal switching control problems with multiple time-delays and a cost on changing control and subject to terminal state constraints. A computational method involving three stages is developed to solve this class of optimal control problems. First, by parameterizing the control function with piecewise-constant functions, the optimal switching control problem is approximated by a sequence of finite-dimensional optimization problems, where the original switching times, the control heights and the control switching times are decision variables. Second, by introducing new variables, the total variation of the control variables is transformed into an equivalently smooth function. Third, we convert the constrained optimization problem into one only with box constraints by an exact penalty function method. The gradients of the cost functional are then derived, which can be combined with any gradient-based optimization method to determine the optimal solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

Citation: Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial and Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042
##### References:
 [1] N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, Essex, 1988. [2] S. C. Bengea and A. D. Raymond, Optimal control of switching systems, Automatica, 41 (2005), 11-27.  doi: 10.1016/j.automatica.2004.08.003. [3] J. M. Blatt, Optimal control with a cost of switching control, Journal of the Australian Mathematical Society-Series B, 19 (1976), 316-332.  doi: 10.1017/S0334270000001181. [4] F. Delmotte, E. I. Verriest and M. Egerstedt, Optimal impulsive control of delay systems, ESAIM Control Optimisation and Calculus of Variations, 14 (2008), 767-779.  doi: 10.1051/cocv:2008009. [5] P. Howlett, Optimal strategies for the control of a train, Automatica, 32 (1996), 519-532.  doi: 10.1016/0005-1098(95)00184-0. [6] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012. [7] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [8] Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems-Series B, 18 (2011), 59-76. [9] C. Liu, Z. Gong, B. Shen and E. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706.  doi: 10.1016/j.apm.2012.02.044. [10] C. Liu, Z. Gong, K. L. Teo and E. Feng, Multi-objective optimization of nonlinear switched time-delay systems in fed-batch process, Applied Mathematical Modelling, 40 (2016), 10533-10548.  doi: 10.1016/j.apm.2016.07.010. [11] 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. [12] 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. [13] 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. [14] R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.  doi: 10.1016/j.automatica.2013.05.027. [15] J. Matula, On an extremum problem, Journal of the Australian Mathematical Society-Series B, 28 (1987), 376-392.  doi: 10.1017/S0334270000005464. [16] J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999. doi: 10.1007/b98874. [17] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5. [18] T. I. Seidman, Optimal control for switching systems, Proceedings of the 21st Annual Conference on Information Science and Systems, 1987. [19] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. [20] K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control, Journal of Optimization Theory and Applications, 68 (1991), 336-357.  doi: 10.1007/BF00941572. [21] E. I. Verriest, Optimal control for switched point delay systems with refractory period, The 16th IFAC World Congress, 38 (2005), 413-418.  doi: 10.3182/20050703-6-CZ-1902.00930. [22] E. I. Verriest, F. Delmotte and M. Egerstedt, Optimal impulsive control of point delay systems with refractory period, Proceedings of the 5th IFAC Workshop on Time Delay Systems, 2004. [23] L. Wang, Q. Lin, R. Loxton, K. L. Teo and G. Cheng, Optimal 1, 3-propanediol production: Exploring the trade-off between process yield and feeding rate variation, Journal of Process Control, 32 (2015), 1-9.  doi: 10.1016/j.jprocont.2015.04.011. [24] S. F. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.  doi: 10.1002/oca.1015. [25] 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. [26] 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. [27] C. Yu, B. Li, R. Loxton and K. L. Teo, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.

show all references

##### References:
 [1] N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, Essex, 1988. [2] S. C. Bengea and A. D. Raymond, Optimal control of switching systems, Automatica, 41 (2005), 11-27.  doi: 10.1016/j.automatica.2004.08.003. [3] J. M. Blatt, Optimal control with a cost of switching control, Journal of the Australian Mathematical Society-Series B, 19 (1976), 316-332.  doi: 10.1017/S0334270000001181. [4] F. Delmotte, E. I. Verriest and M. Egerstedt, Optimal impulsive control of delay systems, ESAIM Control Optimisation and Calculus of Variations, 14 (2008), 767-779.  doi: 10.1051/cocv:2008009. [5] P. Howlett, Optimal strategies for the control of a train, Automatica, 32 (1996), 519-532.  doi: 10.1016/0005-1098(95)00184-0. [6] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012. [7] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [8] Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems-Series B, 18 (2011), 59-76. [9] C. Liu, Z. Gong, B. Shen and E. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706.  doi: 10.1016/j.apm.2012.02.044. [10] C. Liu, Z. Gong, K. L. Teo and E. Feng, Multi-objective optimization of nonlinear switched time-delay systems in fed-batch process, Applied Mathematical Modelling, 40 (2016), 10533-10548.  doi: 10.1016/j.apm.2016.07.010. [11] 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. [12] 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. [13] 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. [14] R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.  doi: 10.1016/j.automatica.2013.05.027. [15] J. Matula, On an extremum problem, Journal of the Australian Mathematical Society-Series B, 28 (1987), 376-392.  doi: 10.1017/S0334270000005464. [16] J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999. doi: 10.1007/b98874. [17] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5. [18] T. I. Seidman, Optimal control for switching systems, Proceedings of the 21st Annual Conference on Information Science and Systems, 1987. [19] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. [20] K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control, Journal of Optimization Theory and Applications, 68 (1991), 336-357.  doi: 10.1007/BF00941572. [21] E. I. Verriest, Optimal control for switched point delay systems with refractory period, The 16th IFAC World Congress, 38 (2005), 413-418.  doi: 10.3182/20050703-6-CZ-1902.00930. [22] E. I. Verriest, F. Delmotte and M. Egerstedt, Optimal impulsive control of point delay systems with refractory period, Proceedings of the 5th IFAC Workshop on Time Delay Systems, 2004. [23] L. Wang, Q. Lin, R. Loxton, K. L. Teo and G. Cheng, Optimal 1, 3-propanediol production: Exploring the trade-off between process yield and feeding rate variation, Journal of Process Control, 32 (2015), 1-9.  doi: 10.1016/j.jprocont.2015.04.011. [24] S. F. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594.  doi: 10.1002/oca.1015. [25] 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. [26] 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. [27] C. Yu, B. Li, R. Loxton and K. L. Teo, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.
Optimal control.
Optimal state trajectories.
Cost, terminal constraint and total variation for different weighting coefficients
 Weight$\gamma$ Cost$x_1(1.5)-2$ Terminal constraint$x_2(1.5)-1$ Total variation$\bigvee\limits_0^{1.5}u$ 0 $4.3054\times10^{-5}$ $3.7923\times10^{-8}$ 185.4279 0.01 0.0024 $4.9779\times10^{-8}$ 95.1071 0.05 0.0173 $2.9718\times10^{-7}$ 53.5099 0.1 0.0716 $1.3383\times10^{-7}$ 31.3635 0.5 0.0234 $3.8322\times10^{-5}$ 5.4157
 Weight$\gamma$ Cost$x_1(1.5)-2$ Terminal constraint$x_2(1.5)-1$ Total variation$\bigvee\limits_0^{1.5}u$ 0 $4.3054\times10^{-5}$ $3.7923\times10^{-8}$ 185.4279 0.01 0.0024 $4.9779\times10^{-8}$ 95.1071 0.05 0.0173 $2.9718\times10^{-7}$ 53.5099 0.1 0.0716 $1.3383\times10^{-7}$ 31.3635 0.5 0.0234 $3.8322\times10^{-5}$ 5.4157
 [1] Di Wu, Yin Chen, Changjun Yu, Yanqin Bai, Kok Lay Teo. Control parameterization approach to time-delay optimal control problems: A survey. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022108 [2] Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1825-1843. doi: 10.3934/dcdss.2020108 [3] Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2469-2482. doi: 10.3934/jimo.2021076 [4] Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251 [5] Chongyang Liu, Meijia Han. Time-delay optimal control of a fed-batch production involving multiple feeds. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1697-1709. doi: 10.3934/dcdss.2020099 [6] Canghua Jiang, Cheng Jin, Ming Yu, Zongqi Xu. Direct optimal control for time-delay systems via a lifted multiple shooting algorithm. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021135 [7] Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021 [8] Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure and Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567 [9] Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297 [10] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [11] Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial and Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471 [12] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations and Control Theory, 2022, 11 (1) : 177-197. doi: 10.3934/eect.2020107 [13] 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 and Management Optimization, 2019, 15 (2) : 817-832. doi: 10.3934/jimo.2018072 [14] Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023 [15] Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159 [16] Lei Liu, Shaoying Lu, Cunwu Han, Chao Li, Zejin Feng. Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 367-379. doi: 10.3934/naco.2020008 [17] K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050 [18] Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1569-1589. doi: 10.3934/dcdss.2020357 [19] Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106 [20] Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435

2021 Impact Factor: 1.411