Abstract
The optimal control problem for a switched system whose state vector includes both continuous and discrete components is considered. The continuous part of the system is governed by differential equations, while its discrete part, which models the operation of a switching device with memory, is governed by recurrence inclusions. The discrete part switches the operation modes of the continuous part of the system, and is itself affected by the continuous part. The switching times and their number are not specified in advance. They are found as a result of optimizing a performance index. This problem is a generalization of a classical optimal control problem for continuous–discrete systems. It is shown that the cost function is constructed from certain auxiliary functions—the so-called conditional cost functions. Equations for the conditional cost functions are derived and sufficient optimality conditions are proved. The application of these conditions is demonstrated by the example of designing a time optimal switched system that is similar to A.A. Fel’dbaum’s classical example of the time optimal continuous system.
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References
S. N. Vasil’ev and A. I. Malikov, “On some results on stability of switched and hybride systems,” in Actual Problems of Continuous Media Mechanics, Collection of Articles to 20 Years of Inst. Math. Mech. Kazan. Sci. Center of RAS (Foliant, Kazan, 2011), Vol. 1, pp. 23–81 [in Russian].
K. Yu. Kotov and O. Ya. Shpilevaya, “Switched systems: stability and design (review),” Avtometriya 44 (5), 71–87 (2008).
D. Liberzon, Switching in Systems and Control (Birkhäuser, Boston, 2003).
Z. Ji, L. Wang, and X. Guo, “Design of switching sequences for controllability realization of switched linear systems,” Automatica 43, 662–668 (2007).
H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: A survey of recent results,” IEEE Trans. Autom. Control. 54, 308–322 (2009).
Z. Sun and S. Ge, “Analysis and synthesis of switched linear control systems,” Automatica 41, 181–195 (2005).
X. Xu and P. J. Antsaklis, “Optimal control of switched systems based on parameterization of the switching instants,” IEEE Trans. Autom. Control 49, 2–16 (2004).
R. W. Brockett, “Hybrid models for motion control systems,” in Perspectives in the Theory and Its Applications (Birkhäuser, Boston, 1993), pp. 29–53.
M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: Model and optimal control theory,” IEEE Trans. Autom. Control 43, 31–45 (1998).
S. Hedlund and A. Rantzer, “Optimal control of hybrid systems,” in Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, 1999, pp. 3972–3977.
C. G. Cassandras, D. L. Pepyne, and Y. Wardi, “Optimal control of a class of hybrid systems,” IEEE Trans. Autom. Control 46, 398–415 (2001).
V. I. Gurman, “Models and optimality conditions for hybrid controlled systems,” J. Comput. Syst. Sci. Int. 43, 560 (2004).
A. S. Bortakovskii, “Synthesis of logical–dynamical systems on the basis of sufficient optimality conditions,” J. Comput. Syst. Sci. Int. 49, 207 (2010).
A. S. Bortakovskii, “Synthesis of optimal switched systems,” J. Comput. Syst. Sci. Int. 54, 715 (2015).
A. S. Bortakovskii, “Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems,” J. Comput. Syst. Sci. Int. 55, 1 (2016).
A. S. Bortakovskii and A. V. Panteleev, “Sufficient conditions for optimal control of batch systems,” Avtom. Telemekh., No. 7, 57–66 (1987).
K. D. Zhuk and A. A. Timchenko, Automated Design of Logical-Dynamical Systems (Nauk. Dumka, Kiev, 1981) [in Russian].
V. V. Semenov, “Dynamic programming in the synthesis of logical-dynamical systems,” Priborostroenie, No. 9, 71–77 (1984).
A. I. Malikov, “On the stability of logical-dynamic control systems with structural alterations,” J. Comput. Syst. Sci. Int. 35, 169 (1996).
A. S. Bortakovskii, “Sufficient conditions of control optimality of deterministic logical-dynamical systems,” Inform., Ser. Avtomatiz. Proektir., Nos. 2–3, 72–79 (1992).
S. N. Vasil’ev, A. K. Zherlov, E. A. Fedosov, et al., Intelligent Control of Dynamical Systems (Fizmatlit, Moscow, 2000) [in Russian].
S. V. Emel’yanov, Variable Structure Automatic Control Systems (Nauka, Moscow, 1967) [in Russian].
R. Bellman, Dynamic Programming (Princeton Univ. Press, London, Oxford, 1957).
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974) [in Russian].
I. P. Natanson, Theory of Functions of a Real Variable, Dover Books on Mathematics (Gostekhteorizdat, Moscow, 1957; Dover, New York, 2016).
S. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (Nauka, Moscow, 1973) [in Russian].
A. A. Fel’dbaum, “Optimal processes in automatic control systems,” Avtom. Telemekh. 14, 712–728 (1953).
V. G. Boltyanskii, “Sufficient conditions for optimality and the justification of the dynamic programming method,” Izv. Akad. Nauk SSSR, Ser. Mat. 28, 481–514 (1964).
M. M. Khrustalev, “Necessary and sufficient conditions for optimality in Bellman equation form,” Dokl. Akad. Nauk SSSR, Ser. Mat. 242, 1023–1026 (1978).
V. V. Aleksandrov, V. G. Boltyanskii, S. S. Lemak, et al., Optimal Motion Control (Fizmatlit, Moscow, 2005) [in Russian].
V. F. Dem’yanov and A. M. Rubinov, Principles of Nonsmooth Analysis and Quasidifferential Calculus (Nauka, Moscow, 1990) [in Russian].
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, et al., Mathematical Theory of Optimal Processes, Vol. 4 of Classics of Soviet Mathematics (Fizmatgiz, Moscow, 1961; CRC, Boca Raton, FL, 1987).
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Original Russian Text © A.S. Bortakovskii, 2017, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2017, No. 4, pp. 86–103.
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Bortakovskii, A.S. Sufficient optimality conditions for controlling switched systems. J. Comput. Syst. Sci. Int. 56, 636–651 (2017). https://doi.org/10.1134/S1064230717040049
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DOI: https://doi.org/10.1134/S1064230717040049