×

Equivalence of switching linear systems by bisimulation. (English) Zbl 1122.93319

Summary: A general notion of hybrid bisimulation is proposed for the class of switching linear systems. Connections between the notions of bisimulation-based equivalence, state-space equivalence, algebraic and input-output equivalence are investigated. An algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived. Hybrid state space reduction is performed by hybrid bisimulation between the hybrid system and itself. By specializing the results obtained on bisimulation, also characterizations of simulation and abstraction are derived. Connections between observability, bisimulation-based reduction and simulation-based abstraction are studied.

MSC:

93B17 Transformations
93A30 Mathematical modelling of systems (MSC2010)
93C05 Linear systems in control theory
93B20 Minimal systems representations
93B25 Algebraic methods
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] DOI: 10.1109/5.871304 · doi:10.1109/5.871304
[2] Broucke, M. 1998. ”Regularity of solutions and homotopic equivalence for hybrid systems”. Proceeding of the 37th IEEE Conference on Decision and Control. December1998, Tampa, Florida, USA. pp.4283–4288.
[3] Callier FM, Linear System Theory (1991) · doi:10.1007/978-1-4612-0957-7
[4] Clarke EM, Model Checking (2002)
[5] DOI: 10.1109/9.754812 · Zbl 0960.93046 · doi:10.1109/9.754812
[6] De Santis, E, Di Benedetto, MD and Pola, G. 2003. ”On observability and detectability of continuous-time switching linear systems”. Proceedings of the 42nd IEEE Conference on Decision and Control. 2003, Hawaii, USA. pp.5777–5782. Maui.
[7] DOI: 10.1109/TAC.2003.822860 · Zbl 1365.93287 · doi:10.1109/TAC.2003.822860
[8] De Santis, E, Benedetto, MD and Pola, G. 2004b. ”Structural discrete state space decompositions for a class of hybrid systems”. Mediterranean Conference on Control and Automation (MED04). 6–9 June2004b, Kysadasi, Aydin, Turkey.
[9] De Santis, E, Di Benedetto, MD and Girasole, G. 2005. ”Digital idle speed control of automotive engines using hybrid models”. IFAC World Congress. 2005, Prague.
[10] Henzinger TA, ICIALP 95: Automata, Languages, and programming pp 324– (1995)
[11] Hermanns H, Interactive Markov Chain, Lecture Notes in Computer Science 2428 (2002) · Zbl 1012.68142 · doi:10.1007/3-540-45804-2
[12] Hopcroft JE, Introduction to Automata Theory, Languages, and Computation (1979)
[13] Kelley JL, Linear Topological Spaces (1963)
[14] Lafferriere G, Hybrid Systems V pp 186– (1998)
[15] DOI: 10.1007/PL00009858 · Zbl 1059.68073 · doi:10.1007/PL00009858
[16] Lygeros J, Automatica 35 pp 349– (1999) · Zbl 0943.93043 · doi:10.1016/S0005-1098(98)00193-9
[17] Milner R, Comunication and Concurrency (1989)
[18] DOI: 10.1109/9.539424 · Zbl 0872.93009 · doi:10.1109/9.539424
[19] Tabuada, P and Pappas, GJ. 2003. ”Finite bisimulations of controllable linear systems”. Proceedings of the 42st IEEE Conference on Decision and Control. December2003, Hawaii, Maui, USA. pp.634–639.
[20] DOI: 10.1016/j.automatica.2003.07.003 · Zbl 1045.93033 · doi:10.1016/j.automatica.2003.07.003
[21] Pappas GJ, Systems & Control Letters 52 pp 49– (2004) · Zbl 1157.93300 · doi:10.1016/j.sysconle.2003.09.013
[22] Park, D. ”Concurrency and automata on infinite sequences”. Fifth GI Conference on Theoretical Computer Science, Vol. 104 of Lecture Notes in Computer Science. Edited by: Deussen, P. pp.167–183. Karlsruhe, Germany: Springer. Ed.
[23] Pola, G, van der Schaft, AJ and Di Benedetto, MD. 2004. ”Bisimulation theory for switching linear systems”. Proc. of the 43rd IEEE Conference on Decision and Control (CDC 04). 14–17 December2004, Paradise Island, Bahamas, USA. pp.1406–1411.
[24] Tabuada P, Hybrid Systems: Computation and Control pp 436– (2002) · doi:10.1007/3-540-45873-5_34
[25] Tomlin C, Hybrid Systems: Computation and Control 1386 pp 360– (1998) · doi:10.1007/3-540-64358-3_51
[26] van der Schaft AJ, International Journal of Robust and Nonlinear Control 11 pp 417– (2001) · Zbl 0977.93007 · doi:10.1002/rnc.591
[27] van der Schaft AJ, Hybrid Systems: Computation and Control HSCC04 pp 555– (2004) · doi:10.1007/978-3-540-24743-2_37
[28] van der Schaft, AJ. 2004b. ”Equivalence of hybrid dynamical systems”. Proc. of Mathematical Theory of Networks and Systems. July 5–92004b, Leuven, Belgium. (MTNS 04) · Zbl 1365.93212
[29] DOI: 10.1109/TAC.2004.838497 · Zbl 1365.93212 · doi:10.1109/TAC.2004.838497
[30] Wicks MA, European J. Control 4 pp 140– (1998) · Zbl 0910.93062 · doi:10.1016/S0947-3580(98)70108-6
[31] Zefran, M and Burdick, JW. 1998. ”Design of switching controllers for systems with changing dynamics”. Proc. of the 37th IEEE Conference on Decision and Control. 1998. pp.2113–2118.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.