Abstract
This paper mainly deals with switched linear systems defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function. Its aim is to give sufficient conditions for such a system to be GUAS.
We show that this property of being GUAS is equivalent to the uniform observability on \([0,+\infty)\) of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system.
Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.
The results are partially extended to nonlinear analytic systems.
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References
Agrachev, A.A., Liberzon, D.: Lie-algebraic stability criteria for switched systems. SIAM J. Control Optim. 40, 253–269 (2001)
Bacciotti, A., Mazzi, L.: An invariance principle for nonlinear switched systems. Syst. Control Lett. 54, 1109–1119 (2005)
Balde, M., Boscain, U., Mason, P.: A note on stability conditions for planar switched systems. Int. J. Control 82(10), 1882–1888 (2009)
Balde, M., Jouan, P.: Geometry of the limit sets of linear switched systems. SIAM J. Control Optim. 49(3), 1048–1063 (2011)
Dayawansa, W.P., Martin, C.F.: A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans. Autom. Control 44, 751–760 (1999)
Hespanha, J.P.: Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle. IEEE Trans. Autom. Control 49(4), 470–482 (2004)
Hespanha, J.P., Liberzon, D., Angeli, D., Sontag, E.D.: Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Autom. Control 50(2), 154–168 (2005)
Jouan, P., Naciri, S.: Asymptotic stability of uniformly bounded nonlinear switched systems. Math. Control Relat. Fields 3(3), 323–345 (2013)
Liberzon, D.: Switching in Systems and Control. Systems & Control: Foundations & Applications. Birkhäuser, Boston (2003)
Mancilla-Aguillar, J.L., Garcia, R.A.: A converse Lyapunov theorem for nonlinear switched systems. Syst. Control Lett. 41, 67–71 (2000)
Margaliot, M., Liberzon, D.: Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions. Syst. Control Lett. 55(1), 8–16 (2006)
Mason, P., Boscain, U., Chitour, Y.: Common polynomial Lyapunov functions for linear switched systems. SIAM J. Control Optim. 45(1), 226–245 (2006)
Molchanov, A.P., Piatnitskii, E.S.: Lyapunov functions that specify necessary and sufficient conditions of absolute continuity of nonlinear nonstationary control systems, Part I. Autom. Remote Control 47, 344–354 (1986); Part II: 443–451; Part III: 620–630
Riedinger, P., Sigalotti, M., Daafouz, J.: On the algebraic characterization of invariant sets of switched linear systems. Automatica 46, 1047–1052 (2010)
Serres, U., Vivalda, J.C., Riedinger, P.: On the convergence of linear switched systems. IEEE Trans. Autom. Control 56(2), 320–332 (2011)
Sontag, E.D.: Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edn. Springer, New York (1998)
Trentelman, H.L., Stoorvogel, A.A., Hautus, M.: Control Theory for Linear Systems. Springer, Berlin (2001)
Whitney, H.: Local properties of analytic varieties. In: Differential and Combinatorial Topology, pp. 205–244. Princeton University Press, Princeton (1965)
Acknowledgements
The authors wish to express their thanks to Paolo Mason for the example of Sect. 3.6.5.
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Balde, M., Jouan, P. & Naciri, S. Stability of Uniformly Bounded Switched Systems and Observability. Acta Appl Math 144, 55–75 (2016). https://doi.org/10.1007/s10440-015-0039-9
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DOI: https://doi.org/10.1007/s10440-015-0039-9