Switching time and parameter optimization in nonlinear switched systems with multiple time-delays. (English) Zbl 1304.49063

Summary: In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.


49M37 Numerical methods based on nonlinear programming
65K10 Numerical optimization and variational techniques
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
92E20 Classical flows, reactions, etc. in chemistry


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[1] van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Springer, Berlin (2000) · Zbl 0940.93004
[2] Ramadge, P., Wonham, W.: The control of discrete event systems. Proc. IEEE 77, 81-98 (1989)
[3] Astrom, K.J., Puruta, K.: Swing up a pendulum by energy control. Autom. J. IFAC 36, 287-295 (2000) · Zbl 0941.93543
[4] Hespanha, J., Liberzon, D., Morse, A.S.: Overcoming the limitations of adaptive control by means of logic-based switching. Syst. Control Lett. 49, 49-56 (2003) · Zbl 1157.93440
[5] Howlett, P.: Optimal strategies for the control of a train. Autom. J. IFAC 32, 519-532 (1996) · Zbl 0848.93041
[6] Liu, C.Y., Feng, E.M., Yin, H.C.: Optimal switching control for microbial fed-batch culture. Nonlinear Anal. Hybrid Syst. 2, 1168-1174 (2008) · Zbl 1163.93347
[7] Loxton, R., Teo, K.L., Rehbock, V., Ling, W.K.: Optimal switching instants for a switched-capacitor DC/DC power converter. Autom. J. IFAC 45, 973-980 (2009) · Zbl 1162.49044
[8] Woon, S.F., Rehbock, V., Loxton, R.: Towards global solutions of optimal discrete-valued control problems. Optim. Control Appl. Methods 33, 576-594 (2012) · Zbl 1275.49057
[9] Seidman, T.I.: Optimal control for switching systems. In: Proceedings of the 21st Annual Conference on Information Science and Systems, Baltimore, MD, pp. 485-489 (1987).
[10] Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control 43, 31-45 (1998) · Zbl 0951.93002
[11] Sussmann, H.: A maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, pp. 3972-3977 (1999).
[12] Luus, R., Chen, Y.Q.: Optimal switching control via direct search optimization. Asian J. Control 6, 302-306 (2004)
[13] Xu, X.P., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Autom. Control 49, 2-15 (2004) · Zbl 1365.93308
[14] Bengea, S.C., Raymond, A.D.: Optimal control of switching systems. Autom. J. IFAC 41, 11-27 (2005) · Zbl 1088.49018
[15] Giua, A., Seatzu, C., Van Der Mee, C.: Optimal control of autonomous linear systems switched with a preassigned finite sequence. In: Proceedings of the 2001 IEEE International Symposium on Intelligent Control, México City, México, pp. 144-146 (2001).
[16] Xu, X.P., Antsaklis, P.J.: Optimal control of switched autonomous systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, pp. 4401-4406 (2002).
[17] Egerstedt, M., Wardi, Y., Delmotte, F.: Optimal control of switching times in switched dynamical systems. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, pp. 2138-2143 (2003). · Zbl 1145.93302
[18] Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H.: A new computational method for optimizing nonlinear impulsive systems. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 18, 59-76 (2011) · Zbl 1210.49035
[19] Loxton, R., Teo, K.L., Rehbock, V.: Computational method for a class of switched system optimal control problems. IEEE Trans. Autom. Control 54, 2455-2460 (2009) · Zbl 1367.93287
[20] Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Autom. J. IFAC 29, 1667-1694 (2003) · Zbl 1145.93302
[21] Meyer, C., Schroder, S., De Doncker, R.W.: Solid-state circuit breakers and current limiters for medium-voltage systems having distributed power systems. IEEE Trans. Power Electr. 19, 1333-1340 (2004)
[22] Kim, D.K., Park, P.G., Ko, J.W.: Output-feedback \[H_\infty \]∞ control of systems over communication networks using a deterministic switching system approach. Autom. J. IFAC 40, 1205-1212 (2004) · Zbl 1056.93527
[23] 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
[24] Verriest, E.I., Delmotte, F., Egerstedt, M.: Optimal impulsive control of point delay systems with refractory period. In: Proceedings of the 5th IFAC Workshop on Time Delay Systems, Leuven, Belgium (2004). · Zbl 1148.49017
[25] Verriest, E.I.: Optimal control for switched point delay systems with refractory period. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005). · Zbl 1139.49030
[26] Delmotte, F., Verriest, E.I., Egerstedt, M.: Optimal impulsive control of delay systems. ESAIM Control Optim. Calc. Var. 14, 767-779 (2008) · Zbl 1148.49017
[27] Wu, C.Z., Teo, K.L., Li, R., Zhao, Y.: Optimal control of switched systems with time delay. Appl. Math. Lett. 19, 1062-1067 (2006) · Zbl 1123.49030
[28] Ahmed, N.U.: Elements of Finite-dimensional Systems and Control Theory. Longman Scientific and Technical, Essex (1988) · Zbl 0658.93002
[29] Li, R., Teo, K.L., Wong, K.H., Duan, G.R.: Control parameterization enhancing transform for optimal control of switched systems. Math. Comput. Model. 43, 1393-1403 (2006) · Zbl 1139.49030
[30] Farhadinia, B., Teo, K.L., Loxton, R.: A computational method for a class of non-standard time optimal control problems involving multiple time horizons. Math. Comput. Model. 49, 1682-1691 (2009) · Zbl 1165.49309
[31] Liu, C.Y., Loxton, R., Teo, K.L.: Optimal parameter selection for nonlinear multistage systems with time-delays. Comput. Optim. Appl. doi:10.1007/s10589-013-9632-x · Zbl 1326.90099
[32] Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, Essex (1991) · Zbl 0747.49005
[33] Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) · Zbl 0930.65067
[34] Schittkowski, K.: A Fortran Implementation of a Sequential Quadratic Programming Algorithm with Distributed and Non-monotone Line Search—User’s Guide. University of Bayreuth, Bayreuth (2007)
[35] Hindmarsh, A.C.: Large ordinary differential equation systems and software. IEEE Control Syst. Mag. 2, 24-30 (1982)
[36] Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980) · Zbl 0423.65002
[37] Xiu, Z.L., Song, B.H., Sun, L.H., Zeng, A.P.: Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process. Biochem. Eng. J. 11, 101-109 (2002)
[38] Mu, Y., Zhang, D.J., Teng, H., Wang, W., Xiu, Z.L.: Microbial production of 1,3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation. Biotechnol. Lett. 28, 1755-1759 (2006)
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