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Stabilization of finite automata with application to hybrid systems control. (English) Zbl 1235.93207
Summary: This paper discusses the state feedback stabilization problem of a Deterministic Finite Automaton (DFA), and its application to stabilizing Model Predictive Control (MPC) of hybrid systems. In the modeling of a DFA, a linear state equation representation recently proposed by the authors is used. First, this representation is briefly explained. Next, after the notion of equilibrium points and stabilizability of the DFA are defined, a necessary and sufficient condition for the DFA to be stabilizable is derived. Then a characterization of all stabilizing state feedback controllers is presented. Third, a simple example is given to show how to follow the proposed procedure. Finally, control Lyapunov functions for hybrid systems are introduced based on the above results, and the MPC law is proposed. The effectiveness of this method is shown by a numerical example.

MSC:
93D15 Stabilization of systems by feedback
68Q80 Cellular automata (computational aspects)
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Software:
MATISSE; UMDES
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[1] Alur R, Henzinger TA, Lafferriere G, Pappas GJ (2000) Discrete abstraction of hybrid systems. Proc IEEE 88(7):971–984
[2] Batt G, Ropers D, de Jong H, Geiselmann J, Mateescul R, Page M, Schneider D (2005) Validation of qualitative models of genetic regulatory networks by model checking: analysis of the nutritional stress response in Escherichia coli. Bioinformatics 21(1):19–28 · Zbl 1078.93555
[3] Bemporad A, Morari M (1999) Control of systems integrating logic, dynamics, and constraints. Automatica 35:407–427 · Zbl 1049.93514
[4] Brave Y, Heymann M (1989) On stabilization of discrete-event processes. In: Proc. 28th IEEE conf. on decision and control, pp 2737–2742 · Zbl 0704.93054
[5] Cassandras CG, Lafortune S (2008) Introduction to discrete event systems, 2nd edn. Springer, New York · Zbl 1165.93001
[6] Chaves M, Eissing T, Allgöwer F (2009) Regulation of apoptosis via the NF{\(\kappa\)}B pathway: modeling and analysis. In: Ganguly N, Deutsch A, Mukherjee A (eds) Dynamics on and of complex networks: applications to biology, computer science and the social sciences. Birkhauser, Boston, pp 19–34
[7] Di Cairano S, Lazar M, Bemporad A, Heemels WPMH (2008) A control Lyapunov approach to predictive control of hybrid systems. In: Proc. 11th int’l conf. on hybrid systems: computation and control, LNCS 4981. Springer, New York · Zbl 1144.93323
[8] Girard A, Julius AA, Pappas GJ (2008) Approximate simulation relations for hybrid systems. Discrete Event Dyn Syst 18(2):163–179 · Zbl 1395.93113
[9] Kobayashi K, Imura J (2006) Modeling of discrete dynamics for computational time reduction of model predictive control. In: Proc. 17th int’l symp. on mathematical theory of networks and systems, pp 628–633
[10] Kobayashi K, Imura J (2007) Minimality of finite automata representation in hybrid systems control. In: Proc. 10th int’l conf. on hybrid systems: computation and control, LNCS 4416. Springer, New York, pp 343–356 · Zbl 1221.93116
[11] Kumar R, Garg VK, Marcus SI (1993) Language stability and stabilizability of discrete event dynamical systems. SIAM J Control Optim 31(5):1294–1320 · Zbl 0785.93006
[12] LazarWillsky AS, Antsaklis PJ M (2009) Flexible control Lyapunov functions. In: 2009 American control conference, pp 102–107
[13] Megretski A (2002) Robustness of finite state automata. In: Multidisciplinary research in control: the Mohammed Dahleh symp., pp 147–160
[14] Ozveren CM, Willsky AS, Antsaklis PJ (1991) Stability and stabilizability of discrete event dynamic systems. J Assoc Comput Mach 38(3):730–752 · Zbl 0812.93002
[15] Sontag ED (1983) A Lyapunov-like characterization of asymptotic controllability. SIAM J Control Optim 21(3):462–471 · Zbl 0513.93047
[16] Tarraf DC, Dahleh MA, Megretski A (2005) Stability of deterministic finite state machines. In: Proc. American control conf., pp 3932–3936
[17] Tarraf DC, Megretski A, Dahleh MA (2008) A framework for robust stability of systems over finite alphabets. IEEE Trans Automat Contr 54(5):1133–1146 · Zbl 1367.93482
[18] Tazaki Y, Imura J (2008) Bisimilar finite abstractions of interconnected systems. In: Proc. 11th int’l conf. on hybrid systems: computation and control, LNCS 4981. Springer, New York · Zbl 1144.93301
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