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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. DelmotteE. 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.  Google Scholar

[5]

P. Howlett, Optimal strategies for the control of a train, Automatica, 32 (1996), 519-532.  doi: 10.1016/0005-1098(95)00184-0.  Google Scholar

[6]

R. LiK. L. TeoK. 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.  Google Scholar

[7]

Q. LinR. 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.  Google Scholar

[8]

Q. LinR. LoxtonK. 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.   Google Scholar

[9]

C. LiuZ. GongB. 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.  Google Scholar

[10]

C. LiuZ. GongK. 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.  Google Scholar

[11]

C. LiuR. 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.  Google Scholar

[12]

R. LoxtonK. 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.  Google Scholar

[13]

R. LoxtonK. L. TeoV. 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.  Google Scholar

[14]

R. LoxtonQ. 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.  Google Scholar

[15]

J. Matula, On an extremum problem, Journal of the Australian Mathematical Society-Series B, 28 (1987), 376-392.  doi: 10.1017/S0334270000005464.  Google Scholar

[16]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[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.  Google Scholar

[18]

T. I. Seidman, Optimal control for switching systems, Proceedings of the 21st Annual Conference on Information Science and Systems, 1987. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

L. WangQ. LinR. LoxtonK. 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.  Google Scholar

[24]

S. F. WoonV. 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.  Google Scholar

[25]

C. WuK. L. TeoR. 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.  Google Scholar

[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.  Google Scholar

[27]

C. YuB. LiR. 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.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, Essex, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. DelmotteE. 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.  Google Scholar

[5]

P. Howlett, Optimal strategies for the control of a train, Automatica, 32 (1996), 519-532.  doi: 10.1016/0005-1098(95)00184-0.  Google Scholar

[6]

R. LiK. L. TeoK. 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.  Google Scholar

[7]

Q. LinR. 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.  Google Scholar

[8]

Q. LinR. LoxtonK. 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.   Google Scholar

[9]

C. LiuZ. GongB. 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.  Google Scholar

[10]

C. LiuZ. GongK. 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.  Google Scholar

[11]

C. LiuR. 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.  Google Scholar

[12]

R. LoxtonK. 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.  Google Scholar

[13]

R. LoxtonK. L. TeoV. 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.  Google Scholar

[14]

R. LoxtonQ. 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.  Google Scholar

[15]

J. Matula, On an extremum problem, Journal of the Australian Mathematical Society-Series B, 28 (1987), 376-392.  doi: 10.1017/S0334270000005464.  Google Scholar

[16]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[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.  Google Scholar

[18]

T. I. Seidman, Optimal control for switching systems, Proceedings of the 21st Annual Conference on Information Science and Systems, 1987. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

L. WangQ. LinR. LoxtonK. 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.  Google Scholar

[24]

S. F. WoonV. 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.  Google Scholar

[25]

C. WuK. L. TeoR. 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.  Google Scholar

[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.  Google Scholar

[27]

C. YuB. LiR. 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.  Google Scholar

Figure 1.  Optimal control.
Figure 2.  Optimal state trajectories.
Table 1.  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
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