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Safety and Reachability of Piecewise Linear Hybrid Dynamical Systems Based on Discrete Abstractions

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Abstract

In this paper, a novel methodology for analysis of piecewise linear hybrid systems based on discrete abstractions of the continuous dynamics is presented. An important characteristic of the approach is that the available control inputs are taken into consideration in order to simplify the continuous dynamics. Control specifications such as safety and reachability specifications are formulated in terms of partitions of the state space of the system. The approach provides a convenient general framework not only for analysis, but also for controller synthesis of hybrid systems. The research contributions of this paper impact the areas of analysis, verification, and synthesis of piecewise linear hybrid systems.

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References

  • Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger. T., Ho, P-H., Nicollin. X., Oliveiro, A., Sifalus, J., and Yovine, S. 1995. The algorithmic analysis of hybrid systems. Theoretical and onzpiiter Science 138: 3-34.

    Google Scholar 

  • Alur. R., Henzinger. T., Lafferriere. G., and Pappas, G. 2000. Discrete abstractions of hybrid systems. Proceedings of IEEE 88(7): 971-984.

    Google Scholar 

  • Antsaklis, P. (ed.) 2000. Proceedings of the IEEE, Special Issue on Hybrid Systems: Theory and Applications, volume 88.

  • Antsaklis, P., Koutsoukos, X., and Zaytoon. J. 1998. On hybrid control of complex systems: A survey. European Journal of Autoniation 32(9-10): 1023-1045.

    Google Scholar 

  • Asarin, E., Boumez. 0., Dang, T., and Maler, 0. 2000a. Approximate reachability analysis of piecewise-linear dynamical systems. In N. Lynch and B. Krogh (eds). Hybrid Svsrenis-'oniputation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, pp. 20-31.

  • Asarin, E., Bournez, 0., Dang, T., Maler. 0., and Pnueli. A. 2000b. Effective synthesis of switching controllers for linear systems. Proceedings of IEEE 88(7): 1011-1025.

    Google Scholar 

  • Asarin, E., Maler, 0., and Pnueli, A. 1995. Symbolic controller synthesis for discrete and timed systems. In P. Antsaklis, W. Kohn. A. Nerode, and S. Sastry (eds). Hybrid Systems II, volume 999 of Lecture Notes in Computer Science. Springer, pp. 1-20.

  • Balluchi. A., Benvenuti, L., Villa, T., Wong-Toi, H., and Sangiovanni-Vincentelli, A. 1999. A case study of hybrid controller synthesis of a heating system. In Proceedings of the 5th European Control Conference, ECC99, Germany: Carlsruhe.

    Google Scholar 

  • Bemporad, A., and Morari. M. 1999a. Control of systems integrating logic, dynamics, and constraints. Autoniatica 35(3): 407.-427.

    Google Scholar 

  • Bemporacl, A., and Moran. M. 1999h. Verification of hybrid systems via mathematical programming. In F Vaandrager and J. van Schuppen (eds.). HSCC 99: Hybrid Systems-Computation and Control, volume 1569 of Lecture Notes in Computer Science. Springer-Verlag, pp. 31-45.

  • Bemporad, A., Torrisi, F, and Morari, M. 2000. Optimization-based verification and stability characterization of piecewise affine and hybrid systems. In N. Lynch and B. Krogh (eds), Hybrid Systems-Computation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, pp. 45-58.

  • Berardi, L., Santis, E. D., and Benedetto, M. D. 2000. Invariant sets and control synthesis for switching systems with safety specifications. In N. Lynch and B. Krogh (eds), HybridSystems-computation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, pp. 59-72.

  • Boumez, 0., Maler, 0., and Pnueli, A. 1999. Orthogonal polyhedra: Representation and comiiutation. In HSCC 99: Hybrid Systems-Computation and Control, volume 1569 of Lecture Notes in Coniputer Science. Springer-Verlag, pp. 46-60.

  • Broucke, M. 1999. A geometric approach to bisimulation and verification of hybrid systems. In F Vaandrager and J. van Schuppen (eds). HSCC 99: Hybrid Systems-Computation and Control, volume 1569 of Lecture Notes in Computer Science. Springer-Verlag, pp. 61-75.

  • Caines, P., and Wei, Y.-J. 1998. Hierarchical hybrid control systems: A lattice formulation. IEEE Transactions an Automatic Control 43(4): 501-508.

    Google Scholar 

  • Chang, C. 1990. Model Theory. Elsevier.

  • Chutinan, A., and Krogh, B. 1999a. Computing approximated aytomata for a class of linear hybrid systems. In P. Antsaklis, W. Kohn. M. Lemmon, A. Nerode, and S. Sastry (eds). Hybrid Systems V volume 1567 of Lecture Notes in Computer Science. Springer. pp. 16-37.

  • Chutinan, A., and Krogh, B. 1999b. Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In E Vaandrager and J. van Schuppen (eds), HSCC 99: Hybrid Systems-Computation and Control, volume 1569 of Lecture Notes in Computer Science. Spnnger-Verlag, pp. 76-90.

  • Cury, J., Krogh, B., and Niinomi. T. 1998. Synthesis of supervisory controllers for hybrid systems based on approximating automata. IEEE Transactions on Automatic Control 43(4): 564-568.

    Google Scholar 

  • Duffin, R. 1974. On Fourier's analysis of linear inequality systems. Mathematical Programming Study 171-95.

  • Henzinger, T. 1995. Hybrid automata with finite bisimulations. In Z. Fulop and G. Gécgeg (eds). IfJ'ALP'95: Autontata. Languages, and Programming. Springer-Verlag.

  • Johansson, M. 1999. Piecewise Linear Control Systems. PhD thesis, Sweden: Lund University.

  • Koutsoukos. X. 2000. Analysis and Design of Piecewise Linear Hybrid Dynamical Systems. PhD thesis, Department of Electrical Engineering, Noire Dame, IN: University of Notre Dame.

  • Koutsoukos, X., and Antsaklis, P. 1999a. Design of hybrid system regulators. In Proceedings oft/ic 38th iEEE Conference on Decision and Control. Phoenix. AZ, pp. 3990-3995.

  • Koutsoukos, X. and Antsaklis, P. 1999b. Hybrid control of a robotic manufacturing system. In Proceedings oft/ic7th IEEE Mediterranean Conference on Control and Automation. Israel: Haifa, pp. 144-159.

    Google Scholar 

  • Koutsoukos, X., and Antsaklis, P. 2000. A hybrid feedback regulator approach to control an automotive suspension system. In N. Lynch and B. Krogh (eds). Hybrid Systems-Computation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag. pp. 188-201.

  • Koutsoukos. X., Antsaklis, P., Stiver, J., and Lemmon, M. 2000. Supervisory control of hybrid systems. Proceedings of IEEE 88(7): 1026-1049.

    Google Scholar 

  • Lafferriere. 0., Pappas, 0., and Sastry, S. 1999. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn. M. Lemmon, A. Nerode, and S. Sastry (eds). Hybrid Systems V. volume 1567 of Lecture Notes in Computer Science. Springer, pp. 186-203.

  • Leenaerts. D., and van Bokhoven, W. 1998. Piecewise Linear Modeling and Analysis. Kluwer.

  • Lemmon, M. 2000. On the existence of solutions to controlled hybrid automata. In N. Lynch and B. Krogh (eds). Hybrid Systems-Computation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, pp. 229-242.

  • Lunze, J., Nixdorf, B., and Schroder, J. 1999. Deterministic discrete-event representations of linear contmuousvariable systems. Automatica 35(3): 396-406.

    Google Scholar 

  • Lygeros, J., Tomlin. C., and Sastry, S. 1999. Controllers for reachability specifications for hybrid systems. Autoniatica 35(3): 349-370.

    Google Scholar 

  • Lynch, N., Segala, R., Vaandrager, F., and Weinberg, H. 1996. Hybrid I/O automata. In R. Alur. T. A. Henzinger, and E. D. Sontag (eds), Hybrid Systems II!, Verification and Control, volume 1066 of Lecture Notes in Computer Science. Springer. pp. 496-510.

  • Maler, 0., Pnueli, A., and Sifakis, J. 1995. On the synthesis of discrete controllers for timed systems. In E. Mayr and C. Puech (eds). STACS 95: Theoretical Aspects of Computer Science, volume 900 of Lecture Notes in Computer Science. Springer-Verlag, pp. 229-242.

  • Morse, A. 1996. Supervisory control of families of linear set-point controllers-Part 1: Exact matching. IEEE Transactions on Automatic Control 41: 1413-1431.

    Google Scholar 

  • Morse, A. ed. 1997. Control using logic-based switching, volume 222 of Lecture Notes in Control and Information Sciences. Springer.

  • Motzkin, T. 1952. The Theory of Linear Inequalities. Santa Monica, CA: Rand Corp.

    Google Scholar 

  • Nash, S., and Sofer. A. 1996. Linear and Nonlinear Programming. McGraw-Hill.

  • Nerode, A., and Kohn, W. 1993. Models for hybrid systems: Automata, topologies, controllability, observability. Itt R. L. Grossman. A. Nerode, A. P. Ravn, and H. Rischel (eds), Hybrid Systems, volume 736 of Lecture Notes in Computer Science. Springer-Verlag, pp. 317-356.

  • Nerode, A., and Shore, R. 1993. Logic for Applications. Texts and Monographs in Computer Science, Springer-Verlag.

  • Raisch, J. and O'Young, S. 1998. Discrete approximation and supervisory control of continuous systems. IEEE Transactions on Automatic Control 43(4): 568-573.

    Google Scholar 

  • Ramadge, P. 1990. On the periodicity of symbolic observations of piecewise smooth discrete-time systems. IEEE Transactions on Automatic Control 35(7): 807-813.

    Google Scholar 

  • Sam, M. 1981. Introduction to Algebraic System Theory. Academic Press.

  • Sontag, E. 1981. Nonlinear regulation: The piecewise linear approach. IEEE Transactions on Automatic Control 26(2): 346-358.

    Google Scholar 

  • Sontag. E. 1982. Remarks on piecewise-linear algebra. Pacific Journal of Mathematics 92 (1): 183-210.

    Google Scholar 

  • Sontag, E. 1996. Interconnected automata and linear systems: A theoretical framework in discrete-time. In R. Alur, T. Henzinger, and E. Sontag (eds.), Hybrid Systems III, Verification and Control, volume 1066 of Lecture Notes in Computer Science. Springer, pp. 436-448.

  • Stiver, J. 1995. Analysis and design of hybrid control systems. PhD thesis, Department of Electrical Engineering. Notre Dame, IN: University of Notre Dame.

    Google Scholar 

  • Stiver, J., Antsaklis. P., and Lemmon. M. 1996a. An invariant based approach to the design of hybrid control systems. In IF,4C 13th Triennial World Congress, volume J. San Francisco, CA, pp. 467-472.

  • Stiver, J., Antsaklis, P., and Lemmon, M. 1996b. A logical DES approach to the design of hybrid control systems. Mat/il. Comput. Modelling 23(11/12): 55-76.

    Google Scholar 

  • Stiver. J., Koutsoukos. X., and Antsaklis, P. 2001. An invariant based approach to the design of hybrid control systems. International Journal of Robust and Nonlinear Control 11(5): 453-478.

    Google Scholar 

  • Tomlin. C., Lygeros, J., and Sastry. S. 1998. Synthesizing controllers for nonlinear hybnd systems. In T. Henzinger and S. Sastry (eds), HSCC 98: Hybrid Systents-Computation and Control, Lecture Notes in Computer Science 1386. Springer-Verlag, pp. 360-373.

  • Vidal, R., Schaffert. S., Lygeros, 3., and Sastry, S. 2000. Controlled invariance of discrete time systems. In N. Lynch and B. Krogh (eds), Hybrid Svstems.-Computation and Control, volume 1790 of Lecture Notes in Computer Science. Springer-Verlag, pp. 437-450.

  • Williams, H. 1986. Fourier's method of linear programming and its dual. American Mathematical Monthly 93: 681-695.

    Google Scholar 

  • Wong-Toi, H. 1997. The synthesis of controllers for linear hybrid automata. In Proceedings of the 36th IEEE Conference on Decision and Control. San Diego, CA, pp. 4607-4612.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, 37235, USA

    Xenofon D. Koutsoukos

  2. Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Panos J. Antsaklis

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  1. Xenofon D. Koutsoukos
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  2. Panos J. Antsaklis
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Koutsoukos, X.D., Antsaklis, P.J. Safety and Reachability of Piecewise Linear Hybrid Dynamical Systems Based on Discrete Abstractions. Discrete Event Dynamic Systems 13, 203–243 (2003). https://doi.org/10.1023/A:1024050108075

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