On the representation of switched systems with inputs by perturbed control systems

This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem. 2004 Elsevier Ltd. All rights reserved.

[1]  Karl Henrik Johansson,et al.  Dynamical properties of hybrid automata , 2003, IEEE Trans. Autom. Control..

[2]  J. L. Mancilla-Aguilar,et al.  A converse Lyapunov theorem for nonlinear switched systems , 2000 .

[3]  Eduardo D. Sontag,et al.  Uniform stability properties of switched systems with switchings governed by digraphs , 2005 .

[4]  Michael Malisoff.  Further Results on Lyapunov Functions and Domains of Attraction for Perturbed Asymptotically Stable Systems , 2003 .

[5]  António Ornelas.  Parametrization of Carathéodory multifunctions , 1990 .

[6]  Eduardo D. Sontag,et al.  Input-Output-to-State Stability , 2001, SIAM J. Control. Optim..

[7]  Eduardo D. Sontag,et al.  An infinite-time relaxation theorem for differential inclusions , 2001 .

[8]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[9]  J. Carifio,et al.  Nonlinear Analysis , 1995 .

[10]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[11]  Eduardo Sontag,et al.  New characterizations of input-to-state stability , 1996, IEEE Trans. Autom. Control..

[12]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[13]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[14]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

[15]  J. L. Mancilla-Aguilar,et al.  On converse Lyapunov theorems for ISS and iISS switched nonlinear systems , 2001 .

[16]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[17]  K. Deimling.  Multivalued Differential Equations , 1992 .

[18]  J. L. Mancilla-Aguilar,et al.  State-norm estimation of switched nonlinear systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[19]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[20]  Andrzej Fryszowski,et al.  Continuous version of Filippov-Wažewski relaxation theorem , 1991 .

[21]  Eduardo D. Sontag,et al.  Lyapunov Characterizations of Input to Output Stability , 2000, SIAM J. Control. Optim..

[22]  Eduardo Sontag,et al.  Notions of input to output stability , 1999, Systems & Control Letters.

[23]  James R. Munkres,et al.  Topology; a first course , 1974 .

[24]  J. Aubin.  Set-valued analysis , 1990 .

[25]  S. Sastry,et al.  Zeno hybrid systems , 2001 .

[26]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[27]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .