Dynamic optimization of dual-mode hybrid systems with state-dependent switching conditions. (English) Zbl 1390.49039

Summary: This paper presents a computational approach for optimizing a class of hybrid systems in which the state dynamics switch between two distinct modes. The times at which the mode transitions occur cannot be specified directly, but are instead governed by a state-dependent switching condition. The control variables, which should be chosen optimally by the system designer, consist of a set of continuous-time input signals. By introducing an auxiliary binary-valued control function to represent the system’s current mode, we show that any dual-mode hybrid system with state-dependent switching conditions can be transformed into a standard dynamic system subject to path constraints. We then develop a computational algorithm, based on control parameterization, the time-scaling transformation, and an exact penalty method, for determining the optimal piecewise constant input signals for the original hybrid system. A numerical example on cancer chemotherapy is included to demonstrate the effectiveness of the proposed algorithm.


49M37 Numerical methods based on nonlinear programming
65K10 Numerical optimization and variational techniques
90C30 Nonlinear programming
92C50 Medical applications (general)
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[1] Boccadoro, M.; Wardi, Y.; Egerstedt, M.; Verriest, E., Optimal control of switching surfaces in hybrid dynamical systems, Discret. Event Dyn. Syst., 15, 433-448 (2005) · Zbl 1101.93054
[2] Cassandras, C. G.; Pepyne, D. L.; Wardi, Y., Optimal control of a class of hybrid systems, IEEE Trans. Automat. Control, 46, 398-415 (2001) · Zbl 0992.93052
[3] Chai, Q.; Loxton, R.; Teo, K. L.; Yang, C., A max-min control problem arising in gradient elution chromatography, Ind. Eng. Chem. Res., 51, 6137-6144 (2012)
[4] Hedlund, S. and Rantzer, A., Optimal control of hybrid systems, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, USA, 7-10 December 1999, pp. 3972-3977.
[5] Jennings, L.S., Fisher, M.E., Teo, K.L., and Goh, C.J., MISER 3.3 Optimal Control Software - Theory and User Manual, University of Western Australia, Perth, Australia, 2004.
[6] Lee, H. W.J.; Teo, K. L.; Rehbock, V.; Jennings, L. S., Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica J. IFAC, 35, 1401-1407 (1999) · Zbl 0942.93025
[7] Lin, Q.; Loxton, R.; Teo, K. L., Optimal control of nonlinear switched systems: Computational methods and applications, J. Oper. Res. Soc. China, 1, 275-311 (2013) · Zbl 1277.49044
[8] Lin, Q.; Loxton, R.; Teo, K. L., The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10, 275-309 (2014) · Zbl 1276.49025
[9] Lin, Q.; Loxton, R.; Teo, K. L.; Wu, Y. H.; Yu, C., A new exact penalty method for semi-infinite programming problems, J. Comput. Appl. Math., 261, 271-286 (2014) · Zbl 1278.90410
[10] Loxton, R.; Lin, Q.; Teo, K. L., Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pac. J. Optim., 10, 537-560 (2014) · Zbl 1305.49042
[11] Loxton, R.; Teo, K. L.; Rehbock, V., Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44, 2923-2929 (2008) · Zbl 1160.49033
[12] Martin, R. B., Optimal control drug scheduling of cancer chemotherapy, Automatica J. IFAC, 28, 1113-1123 (1992)
[13] Martin, R. B.; Teo, K. L., Optimal Control of Drug Administration in Cancer Chemotherapy (1994), World Scientific: World Scientific, Singapore · Zbl 0870.92006
[14] Passenberg, B., Theory and algorithms for indirect methods in optimal control of hybrid systems, Ph.D. thesis, Technical University of Munich, Munich, Germany, 2012.
[15] Ruby, T.; Rehbock, V.; Lawrance, W. B., Optimal control of hybrid power systems, Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algor., 10, 429-439 (2003) · Zbl 1025.49024
[16] Shin, D. K.; Gürdal, Z.; Griffin, O. H., A penalty approach for nonlinear optimization with discrete design variables, Eng. Optim., 16, 29-42 (1990)
[17] Stursberg, O. and Panek, S., Control of switched hybrid systems based on disjunctive formulations, in Hybrid Systems: Computation and Control, C.J. Tomlin and M.R. Greenstreet, eds., Springer, Berlin, 2002, pp. 421-435. · Zbl 1046.49029
[18] Wang, L.; Lin, Q.; Loxton, R.; Teo, K. L.; Cheng, G., Optimal 1,3-propanediol production: Exploring the trade-off between process yield and feeding rate variation, J. Process Control, 32, 1-9 (2015)
[19] Woon, S. F.; Rehbock, V.; Loxton, R., Towards global solutions of optimal discrete-valued control problems, Optimal Control Appl. Methods, 33, 576-594 (2012) · Zbl 1275.49057
[20] Yu, C.; Li, B.; Loxton, R.; Teo, K. L., Optimal discrete-valued control computation, J. Global Optim., 56, 503-518 (2013) · Zbl 1272.49067
[21] Yu, C.; Teo, K. L.; Bai, Y., An exact penalty function method for nonlinear mixed discrete programming problems, Optim. Lett., 7, 23-38 (2013) · Zbl 1261.90031
[22] Yu, C.; Teo, K. L.; Zhang, L.; Bai, Y., On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, J. Ind. Manag. Optim., 8, 485-491 (2012) · Zbl 1364.90015
[23] Zhu, F.; Antsaklis, P. J., Optimal control of hybrid switched systems: A brief survey, Discret. Event Dyn. Syst., 25, 345-364 (2015) · Zbl 1328.93137
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