# American Institute of Mathematical Sciences

• Previous Article
Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints
• JIMO Home
• This Issue
• Next Article
Neutral and indifference pricing with stochastic correlation and volatility
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  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 & 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] 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 & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567 [2] Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297 [3] Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471 [4] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [5] 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 [6] Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159 [7] 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 & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435 [8] B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641 [9] Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737 [10] Alex Bombrun, Jean-Baptiste Pomet. Asymptotic behavior of time optimal orbital transfer for low thrust 2-body control system. Conference Publications, 2007, 2007 (Special) : 122-129. doi: 10.3934/proc.2007.2007.122 [11] Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010 [12] Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285 [13] Chongyang Liu, Zhaohua Gong, Enmin Feng, Hongchao Yin. Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture. Journal of Industrial & Management Optimization, 2009, 5 (4) : 835-850. doi: 10.3934/jimo.2009.5.835 [14] Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369 [15] Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016 [16] Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783 [17] Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723 [18] Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619 [19] Jingtao Shi, Juanjuan Xu, Huanshui Zhang. Stochastic recursive optimal control problem with time delay and applications. Mathematical Control & Related Fields, 2015, 5 (4) : 859-888. doi: 10.3934/mcrf.2015.5.859 [20] Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591

2017 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables