Switching and stability properties of conewise linear systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 764-793.

Being a unique phenomenon in hybrid systems, mode switch is of fundamental importance in dynamic and control analysis. In this paper, we focus on global long-time switching and stability properties of conewise linear systems (CLSs), which are a class of linear hybrid systems subject to state-triggered switchings recently introduced for modeling piecewise linear systems. By exploiting the conic subdivision structure, the “simple switching behavior” of the CLSs is proved. The infinite-time mode switching behavior of the CLSs is shown to be critically dependent on two attracting cones associated with each mode; fundamental properties of such cones are investigated. Verifiable necessary and sufficient conditions are derived for the CLSs with infinite mode switches. Switch-free CLSs are also characterized by exploring the polyhedral structure and the global dynamical properties. The equivalence of asymptotic and exponential stability of the CLSs is established via the uniform asymptotic stability of the CLSs that in turn is proved by the continuous solution dependence on initial conditions. Finally, necessary and sufficient stability conditions are obtained for switch-free CLSs.

DOI : 10.1051/cocv/2009021
Classification : 93B12, 93D05, 93D20
Mots-clés : variable structure systems, Lyapunov and other classical stabilities, asymptotic stability 
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Shen, Jinglai; Han, Lanshan; Pang, Jong-Shi. Switching and stability properties of conewise linear systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 764-793. doi : 10.1051/cocv/2009021. http://www.numdam.org/articles/10.1051/cocv/2009021/

[1] A. Arapostathis and M.E. Broucke, Stability and controllability of planar conewise linear systems. Systems Control Lett. 56 (2007) 150-158. | Zbl

[2] V.I. Arnold, Mathematical Methods of Classical Mechanics. Second Edition, Springer-Verlag, New York (1989). | Zbl

[3] S. Basu, R. Pollack and M.-F. Roy, Algorithms in Real Algebraic Geometry. Springer-Verlag (2003). | Zbl

[4] A. Berman, M. Neumann and R.J. Stern, Nonnegative Matrices in Dynamical Systems. John Wiley & Sons, New York (1989). | Zbl

[5] S.P. Bhat and D.S. Bernstein, Lyapunov analysis of semistability, in Proceedings of 1999 American Control Conference, San Diego (1999) 1608-1612.

[6] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry. Springer (1998). | Zbl

[7] N.K. Bose, Applied Multidimensional Systems Theory. Van Nostrand Reinhold (1982). | Zbl

[8] B. Brogliato, Some perspectives on analysis and control of complementarity systems. IEEE Trans. Automat. Contr. 48 (2003) 918-935.

[9] M.K. Çamlibel, W.P.M.H. Heemels and J.M. Schumacher, On linear passive complementarity systems. European J. Control 8 (2002) 220-237. | Zbl

[10] M.K. Çamlıbel, J.S. Pang and J. Shen, Lyapunov stability of complementarity and extended systems. SIAM J. Optim. 17 (2006) 1056-1101. | Zbl

[11] M.K. Çamlibel, J.S. Pang and J. Shen, Conewise linear systems: non-Zenoness and observability. SIAM J. Control Optim. 45 (2006) 1769-1800. | Zbl

[12] M.K. Çamlibel, W.P.M.H. Heemels and J.M. Schumacher, Algebraic necessary and sufficient conditions for the controllability of conewise linear systems. IEEE Trans. Automat. Contr. 53 (2008) 762-774.

[13] C.T. Chen, Linear System Theory and Design. Oxford University Press, Oxford (1984).

[14] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press Inc., Cambridge (1992). | Zbl

[15] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York (2003). | Zbl

[16] D. Goeleven and B. Brogliato, Stability and instability matrices for linear evoluation variational inequalities. IEEE Trans. Automat. Contr. 49 (2004) 483-490.

[17] L. Han and J.S. Pang, Non-Zenoness of a class of differential quasi-variational inequalities. Math. Program. Ser. A 121 (2009) 171-199. | Zbl

[18] W.P.M.H. Heemels, J.M. Schumacher and S. Weiland, Linear complementarity systems. SIAM J. Appl. Math. 60 (2000) 1234-1269. | Zbl

[19] J.P. Hespanha, Uniform stability of switched linear systems: extension of LaSalle's invariance principle. IEEE Trans. Automat. Contr. 49 (2004) 470-482.

[20] J.P. Hespanha, D. Liberzon, D. Angeli and E.D. Sontag, Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Automat. Contr. 50 (2005) 154-168. | Zbl

[21] H. Khalil, Nonlinear Systems. Second Edition, Prentice Hall (1996). | Zbl

[22] J. Kurzweil, On the inversion of Lyapunov's second theorem on stability of motion. American Math. Soc. Translation 24 (1963) 19-77. | Zbl

[23] D. Liberzon, J.P. Hespanha and A.S. Morse, Stability of switched systems: a Lie-algebraic condition. Systems Control Lett. 37 (1999) 117-122. | Zbl

[24] J. Lygeros, K.H. Johansson, S.N. Simic, J. Zhang and S. Sastry, Dynamic properties of hybrid automata. IEEE Trans. Automat. Contr. 48 (2003) 2-17.

[25] P. Mason, U. Boscain and Y. Chitour, Common polynomial Lyapunov functions for linear switched systems. SIAM J. Control Optim. 45 (2006) 226-245. | Zbl

[26] A.P. Molchanove and Y.S. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Systems Control Lett. 13 (1989) 59-64. | Zbl

[27] M. Pachter and D.H. Jacobson, Observability with a conic observation set. IEEE Trans. Automat. Contr. 24 (1979) 632-633. | Zbl

[28] J.S. Pang and J. Shen, Strongly regular differential variational systems. IEEE Trans. Automat. Contr. 52 (2007) 242-255.

[29] J.S. Pang and D. Stewart, Differential variational inequalities. Math. Program. Ser. A 113 (2008) 345-424. | Zbl

[30] P.A. Parrilo, Semidefinite programming relaxations for semialgebraic problems. Math. Program. Ser. B 96 (2003) 293-320. | Zbl

[31] S. Scholtes, Introduction to Piecewise Differentiable Equations. Habilitation thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Germany (1994).

[32] J.M. Schumacher, Complementarity systems in optimization. Math. Program. Ser. B 101 (2004) 263-295. | Zbl

[33] J. Shen and J.S. Pang, Linear complementarity systems: Zeno states. SIAM J. Control Optim. 44 (2005) 1040-1066. | Zbl

[34] J. Shen and J.S. Pang, Linear complementarity systems with singleton properties: non-Zenoness, in Proceedings of 2007 American Control Conference, New York (2007) 2769-2774. | Zbl

[35] J. Shen and J.S. Pang, Semicopositive linear complementarity systems. Internat. J. Robust Nonlinear Control 17 (2007) 1367-1386. | Zbl

[36] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41. American Mathematical Society, Providence (2002). | Zbl

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