Dynamic optimization for switched time-delay systems with state-dependent switching conditions. (English) Zbl 1400.49042

Liu, Chongyang; Loxton, Ryan; Lin, Qun; Teo, Kok Lay

SIAM J. Control Optim. 56, No. 5, 3499-3523 (2018).

Summary: This paper considers a dynamic optimization problem for a class of switched systems characterized by two key attributes: (i) the switching mechanism is invoked automatically when the state variables satisfy certain switching conditions; and (ii) the subsystem dynamics involve time-delays in the state variables. The decision variables in the problem, which must be selected optimally to minimize system cost, consist of a set of time-invariant system parameters in the initial state functions. To solve the dynamic optimization problem, we first show that the partial derivatives of the system state with respect to the system parameters can be expressed in terms of the solution of a set of variational switched systems. Then, on the basis of this result, we develop a gradient-based optimization algorithm to determine the optimal parameter values. Finally, we validate the proposed algorithm by solving an example problem arising in the production of 1,3-propanediol.

MSC:

49M37	Numerical methods based on nonlinear programming
34K34	Hybrid systems of functional-differential equations
65K10	Numerical optimization and variational techniques

Keywords:

switched system; time-delay system; dynamic optimization; state-dependent switching; nonlinear optimization

Software:

NLPQLP; ParamEDE; Colsol-DDE

References:

[1]	N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, Essex, 1988. · Zbl 0658.93002
[2]	N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006. · Zbl 1127.93001
[3]	M. U. Akhmet, On the smoothness of solutions of impulsive autonomous systems, Nonlinear Anal., 60 (2005), pp. 311–324. · Zbl 1067.34006
[4]	D. K. Arrowsmith and C. M. Place, Ordinary Differential Equations, Chapman and Hall, London, 1982. · Zbl 0481.34005
[5]	R. G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley, New York, 1966. · Zbl 0146.28201
[6]	S. C. Bengea and R. A. Decarlo, Optimal control of switching systems, Automatica J. IFAC, 41 (2005), pp. 11–27. · Zbl 1088.49018
[7]	E. Blanchard, R. Loxton, and V. Rehbock, Dynamic optimization of dual-mode hybrid systems with state-dependent switching conditions, Optim. Methods Softw., 33 (2018), pp. 297–310. · Zbl 1390.49039
[8]	M. Boccadoro, Y. Wardi, M. Egerstedt, and E. Verriest, Optimal control of switching surfaces in hybrid dynamical systems, Discrete Event Dyn. Syst., 15 (2005), pp. 433–448. · Zbl 1101.93054
[9]	C. G. Cassandras, The event-driven paradigm for control, communication and optimization, J. Control Decis., 1 (2014), pp. 3–17.
[10]	K. G. Dishlieva, Differentiability of solutions of impulsive differential equations with respect to the impulsive perturbations, Nonlinear Anal. Real World Appl., 12 (2011), pp. 3541–3551. · Zbl 1231.34018
[11]	S. Galan, W. F. Feehery, and P. I. Barton, Parametric sensitivity functions for hybrid discrete/continuous systems, Appl. Numer. Math., 31 (1999), pp. 17–47. · Zbl 0937.65137
[12]	J. Hespanha, D. Liberzon, and A. S. Morse, Overcoming the limitations of adaptive control by means of logic-based switching, Systems Control Lett., 49 (2003), pp. 49–56. · Zbl 1157.93440
[13]	A. C. Hindmarsh, Large ordinary differential equation systems and software, IEEE Control Syst. Mag., 2 (1982), pp. 24–30.
[14]	V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[15]	S. M. Lenz, Impulsive Hybrid Discrete-Continuous Delay Differential Equations, Ph.D. dissertation, Heidelberg University, 2014. · Zbl 1294.65073
[16]	S. M. Lenz, J. P. Schloder, and H. G. Bock, Numerical computation of derivatives in systems of delay differential equations, Math. Comput. Simulation, 96 (2014), pp. 124–156.
[17]	D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, 2003.
[18]	Q. Lin, R. Loxton, and K. L. Teo, Optimal control of nonlinear switched systems: Computational methods and applications, J. Oper. Res. Soc. China, 1 (2013), pp. 275–311. · Zbl 1277.49044
[19]	Q. Lin, R. Loxton, and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), pp. 275–309. · Zbl 1276.49025
[20]	Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pac. J. Optim., 7 (2011), pp. 63–81. · Zbl 1211.49041
[21]	Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 18 (2011), pp. 59–76. · Zbl 1210.49035
[22]	Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica J. IFAC, 48 (2012), pp. 2116–2129. · Zbl 1258.49051
[23]	C. Y. Liu, Z. Gong, E. Feng, and H. Yin, Optimal switching control of a fed-batch fermentation process, J. Global Optim., 52 (2012), pp. 265–280. · Zbl 1241.49022
[24]	C. Y. Liu, R. Loxton, and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, J. Optim. Theory Appl., 163 (2014), pp. 957–988. · Zbl 1304.49063
[25]	R. Loxton, Q. Lin, and K. L Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pac. J. Optim., 10 (2014), pp. 537–560. · Zbl 1305.49042
[26]	R. Loxton, K. L. Teo, V. Rehbock, and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica J. IFAC, 45 (2009), pp. 973–980. · Zbl 1162.49044
[27]	T. R. Mehta and M. Egerstedt, Multi-modal control using adaptive motion description languages, Automatica J. IFAC, 44 (2008), pp. 1912–1917. · Zbl 1149.93315
[28]	Y. Mu, D. J. Zhang, H. Teng, W. Wang, and Z. L. Xiu, Microbial production of 1,3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation, Biotechnol. Lett., 28 (2006), pp. 1755–1759.
[29]	J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer-Verlag, New York, 2006. · Zbl 1104.65059
[30]	A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
[31]	K. Schittkowski, NLPQLP: A Fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search—User’s guide, University of Bayreuth, Bayreuth, 2007.
[32]	C. Seatzu, D. Corona, A. Giua, and A. Bemporad, Optimal control of continuous-time switched affine systems, IEEE Trans. Automat. Control, 51 (2006), pp. 726–741. · Zbl 1366.49038
[33]	H. J. Sussmann, A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, 1999, pp. 3972–3977.
[34]	K. L. Teo, C. J. Goh, and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991.
[35]	R. Von Schwerin, M. Winckler, and V. Schulz, Parameter estimation in discontinuous descriptor models, in Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems, D. Bestle and W. Schiehlen, eds., 1996, pp. 269–276. · Zbl 0875.70019
[36]	X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. Control, 49 (2004), pp. 2–15. · Zbl 1365.93308
[37]	C. Yu, Q. Lin, R. Loxton, K. L. Teo, and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, J. Optim. Theory Appl., 169 (2016), pp. 876–901. · Zbl 1342.49003