Abstract
In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems adopting Filippov’s solution concept and using non-smooth ISS Lyapunov functions. The main motivation for adopting non-smooth ISS Lyapunov functions is that “multiple Lyapunov functions” are commonly used in the stability theory for hybrid systems. We will show that the existence of a non-smooth (but Lipschitz continuous) ISS Lyapunov function for a discontinuous system implies ISS. Next, we will prove an ISS interconnection theorem for two discontinuous dynamical systems that both admit an ISS Lyapunov function. The interconnection will be shown to be globally asymptotically stable under a small gain condition. The developed ISS theory will be applied to observer-based controller design for a class of piecewise linear systems using an observer structure proposed by the authors. The LMI-based design of the state feedback and the observer can be performed separately.
This work is partially supported by European project grants SICONOS (IST2001-37172) and HYCON Network of Excellence, contract number FP6-IST-511368.
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References
Sontag, E.D.: The ISS philosophy as a unifying framework for stability-like behavior. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds.) Nonlinear Control in the Year 2000. Lecture Notes in Control and Information Sciences, pp. 443–468. Springer, Heidelberg (2000)
Sontag, E., Wang, Y.: On characterisations of the input-to-state stability property. System and Control Letters (1995)
Jiang, Z.P., Mareels, I., Wang, Y.: A Lyapunov formulation of hte nonlinear small-gain theorem for interconnected ISS systems. Automatica 32, 1211–1215 (1996)
Lin, Y., Sontag, E., Wang, Y.: A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim. 34(1), 124–160 (1996)
Jiang, Z., Teel, A., Praly, L.: Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals & Systems 7, 95–120 (1994)
Arcak, M., Kokotović, P.: Observer based control of systems with slope-restricted nonlinearities. IEEE Transactions on Automatic Control 46(7), 1146–1150 (2001)
Arcak, M.: Certainty equivalence output feedback design with circle criterion observers. IEEE Transactions on Automatic Control 50 (2005)
Branicky, M.S.: Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control 43, 475–482 (1998)
DeCarlo, R., et al.: Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 1069–1082 (2000)
Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control 43(4), 555–559 (1998)
Pettersson, S., Lennartson, B.: LMI for stability and robustness of hybrid systems. In: Proc. of the American Control Conference, Albuquerque, New Mexico, June 1997, pp. 1714–1718 (1997)
Filippov, A.: Differential Equations with Discontinuous Righthand Sides. In: Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (1988)
Shevitz, D., Paden, B.: Lyapunov theory for nonsmooth systems. IEEE Transactions on automatic control 39, 1910–1914 (1994)
Cai, C., Teel, A.R.: Results on input-to-state stability for hybrid systems. In: Proc. Conference Decision and Control, pp. 5403–5408 (2005)
Vu, L., Chatterjee, D., Liberzon, D.: ISS of switched systems and applications to switching adaptive control. In: 44th IEEE Conference on Decision and Control, Seville, Spain, pp. 120–125. IEEE Computer Society Press, Los Alamitos (2005)
Liberzon, D., Nesic, D.: Stability analysis of hybrid systems via small-gain theorems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 421–435. Springer, Heidelberg (2006)
Juloski, A., Heemels, W., Weiland, S.: Observer design for a class of piecewise affine systems. In: Proc. of Conference on Decision and Control 2002, Las Vegas, USA, pp. 2606–2611 (2002)
Pettersson, S.: Switched state jump observers for switched systems. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005)
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Heemels, W.P.M.H., Weiland, S., Juloski, A.L. (2007). Input-to-State Stability of Discontinuous Dynamical Systems with an Observer-Based Control Application. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds) Hybrid Systems: Computation and Control. HSCC 2007. Lecture Notes in Computer Science, vol 4416. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71493-4_22
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DOI: https://doi.org/10.1007/978-3-540-71493-4_22
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