Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-07T00:54:17.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Switching and stability properties of conewise linear systems

Published online by Cambridge University Press:  02 July 2009

Jinglai Shen ,
Lanshan Han  and
Jong-Shi Pang
Jinglai Shen
Affiliation:
Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA. shenj@umbc.edu
Lanshan Han
Affiliation:
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. hanlsh@illinois.edu; jspang@illinois.edu
Jong-Shi Pang
Affiliation:
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. hanlsh@illinois.edu; jspang@illinois.edu
Get access

Abstract

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.

Keywords

Variable structure systemsLyapunov and other classical stabilitiesasymptotic stability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 764 - 793
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arapostathis, A. and Broucke, M.E., Stability and controllability of planar conewise linear systems. Systems Control Lett. 56 (2007) 150–158. CrossRef
V.I. Arnold, Mathematical Methods of Classical Mechanics. Second Edition, Springer-Verlag, New York (1989).
S. Basu, R. Pollack and M.-F. Roy, Algorithms in Real Algebraic Geometry. Springer-Verlag (2003).
A. Berman, M. Neumann and R.J. Stern, Nonnegative Matrices in Dynamical Systems. John Wiley & Sons, New York (1989).
S.P. Bhat and D.S. Bernstein, Lyapunov analysis of semistability, in Proceedings of 1999 American Control Conference, San Diego (1999) 1608–1612.
J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry. Springer (1998).
N.K. Bose, Applied Multidimensional Systems Theory. Van Nostrand Reinhold (1982).
Brogliato, B., Some perspectives on analysis and control of complementarity systems. IEEE Trans. Automat. Contr. 48 (2003) 918–935. CrossRef
Çamlibel, M.K., Heemels, W.P.M.H. and Schumacher, J.M., On linear passive complementarity systems. European J. Control 8 (2002) 220–237. CrossRef
Çamlıbel, M.K., Pang, J.S. and Shen, J., Lyapunov stability of complementarity and extended systems. SIAM J. Optim. 17 (2006) 1056–1101. CrossRef
Çamlibel, M.K., Pang, J.S. and Shen, J., Conewise linear systems: non-Zenoness and observability. SIAM J. Control Optim. 45 (2006) 1769–1800. CrossRef
Çamlibel, M.K., Heemels, W.P.M.H. and Schumacher, J.M., Algebraic necessary and sufficient conditions for the controllability of conewise linear systems. IEEE Trans. Automat. Contr. 53 (2008) 762–774. CrossRef
C.T. Chen, Linear System Theory and Design. Oxford University Press, Oxford (1984).
R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press Inc., Cambridge (1992).
F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York (2003).
Goeleven, D. and Brogliato, B., Stability and instability matrices for linear evoluation variational inequalities. IEEE Trans. Automat. Contr. 49 (2004) 483–490. CrossRef
Han, L. and Pang, J.S., Non-Zenoness of a class of differential quasi-variational inequalities. Math. Program. Ser. A 121 (2009) 171–199. CrossRef
Heemels, W.P.M.H., Schumacher, J.M. and Weiland, S., Linear complementarity systems. SIAM J. Appl. Math. 60 (2000) 1234–1269. CrossRef
Hespanha, J.P., Uniform stability of switched linear systems: extension of LaSalle's invariance principle. IEEE Trans. Automat. Contr. 49 (2004) 470–482. CrossRef
Hespanha, J.P., Liberzon, D., Angeli, D. and Sontag, E.D., Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Automat. Contr. 50 (2005) 154–168. CrossRef
H. Khalil, Nonlinear Systems. Second Edition, Prentice Hall (1996).
Kurzweil, J., On the inversion of Lyapunov's second theorem on stability of motion. American Math. Soc. Translation 24 (1963) 19–77.
Liberzon, D., Hespanha, J.P. and Morse, A.S., Stability of switched systems: a Lie-algebraic condition. Systems Control Lett. 37 (1999) 117–122. CrossRef
Lygeros, J., Johansson, K.H., Simic, S.N., Zhang, J. and Sastry, S., Dynamic properties of hybrid automata. IEEE Trans. Automat. Contr. 48 (2003) 2–17. CrossRef
Mason, P., Boscain, U. and Chitour, Y., Common polynomial Lyapunov functions for linear switched systems. SIAM J. Control Optim. 45 (2006) 226–245. CrossRef
Molchanove, A.P. and Pyatnitskiy, Y.S., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Systems Control Lett. 13 (1989) 59–64. CrossRef
Pachter, M. and Jacobson, D.H., Observability with a conic observation set. IEEE Trans. Automat. Contr. 24 (1979) 632–633. CrossRef
Pang, J.S. and Shen, J., Strongly regular differential variational systems. IEEE Trans. Automat. Contr. 52 (2007) 242–255. CrossRef
Pang, J.S. and Stewart, D., Differential variational inequalities. Math. Program. Ser. A 113 (2008) 345–424. CrossRef
Parrilo, P.A., Semidefinite programming relaxations for semialgebraic problems. Math. Program. Ser. B 96 (2003) 293–320. CrossRef
S. Scholtes, Introduction to Piecewise Differentiable Equations. Habilitation thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Germany (1994).
Schumacher, J.M., Complementarity systems in optimization. Math. Program. Ser. B 101 (2004) 263–295. CrossRef
Shen, J. and Pang, J.S., Linear complementarity systems: Zeno states. SIAM J. Control Optim. 44 (2005) 1040–1066. CrossRef
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.
Shen, J. and Pang, J.S., Semicopositive linear complementarity systems. Internat. J. Robust Nonlinear Control 17 (2007) 1367–1386. CrossRef
G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41. American Mathematical Society, Providence (2002).