Real-time estimation of the switching signal for perturbed switched linear systems

We extend previous works of Fliess et al. [2008] on the estimation of the switching signal and of the state for switching linear systems to the perturbed case when the perturbation is structured that is when the perturbation is unknown but known to satisfy a certain differential equation (for example if the perturbation is constant then its time-derivative is zero). We characterize also singular inputs and/or perturbations for which the switched systems become undistinguishable. Several convincing numerical experiments are illustrating our techniques which are easily implementable.

[1]  Romain Bourdais,et al.  Stabilisation de systèmes non linéaires à commutation à l’aide de fonctions de Lyapunov contrôlées , 2006 .

[2]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[3]  Hebertt Sira-Ramírez,et al.  Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques , 2007 .

[4]  M. Fliess,et al.  An algebraic framework for linear identification , 2003 .

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

[6]  Shuzhi Sam Ge,et al.  Switched Linear Systems , 2005 .

[7]  Daniel Liberzon,et al.  On Invertibility of Switched Linear Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[8]  Ugo V. Boscain,et al.  Stability of Planar Switched Systems: The Linear Single Input Case , 2002, SIAM J. Control. Optim..

[9]  Alberto L. Sangiovanni-Vincentelli,et al.  Design of Observers for Hybrid Systems , 2002, HSCC.

[10]  M. Branicky.  Stability of switched and hybrid systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[11]  Long Wang,et al.  Linear matrix inequality approach to quadratic stabilisation of switched systems , 2004 .

[12]  Robin J. Evans,et al.  Stability results for switched controller systems , 1999, Autom..

[13]  Long Wang,et al.  On Controllability of Switched Linear Systems , 2002, IEEE Transactions on Automatic Control.

[14]  Eduardo D. Sontag,et al.  On the representation of switched systems with inputs by perturbed control systems , 2005 .

[15]  Romain Bourdais,et al.  Stabilization of nonlinear switched systems using control Lyapunov functions , 2007 .

[16]  Cédric Join,et al.  Numerical differentiation with annihilators in noisy environment , 2009, Numerical Algorithms.

[17]  M. Branicky.  Topology of hybrid systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[18]  Romain Bourdais,et al.  TOWARDS A MODEL-FREE OUTPUT TRACKING OF SWITCHED NONLINEAR SYSTEMS , 2007 .

[19]  Stefan Pettersson,et al.  Analysis and Design of Hybrid Systems , 1999 .

[20]  Raymond A. DeCarlo,et al.  Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems , 1998, Eur. J. Control.

[21]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[22]  S. Pettersson,et al.  Stability and robustness for hybrid systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[23]  R. Decarlo,et al.  Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[24]  Daniel Liberzon,et al.  Lie-Algebraic Stability Criteria for Switched Systems , 2001, SIAM J. Control. Optim..

[25]  M. Fliess,et al.  Compression différentielle de transitoires bruités , 2004 .

[26]  G. Zhai,et al.  Quadratic stabilizability of switched linear systems with polytopic uncertainties , 2003 .

[27]  K. Ito,et al.  On State Estimation in Switching Environments , 1970 .

[28]  M. Fliess,et al.  A revised look at numerical differentiation with an application to nonlinear feedback control , 2007, 2007 Mediterranean Conference on Control & Automation.

[29]  A. Stephen Morse,et al.  Supervisory control with state-dependent dwell-time logic and constraints , 2004, Autom..

[30]  Noureddine Manamanni,et al.  Exact differentiation and sliding mode observers for switched Lagrangian systems , 2006 .

[31]  Daniel Liberzon,et al.  Common Lyapunov functions for families of commuting nonlinear systems , 2005, Syst. Control. Lett..

[32]  Mille Millnert.  An approach to recursive identification of abruptly changing systems , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[33]  Michel Fliess,et al.  Analyse non standard du bruit , 2006, ArXiv.

[34]  L. Schwartz.  Théorie des distributions , 1966 .

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

[36]  Harry L. Trentelman,et al.  Essays on control : perspectives in the theory and its applications , 1993 .

[37]  E. De Santis,et al.  On observability and detectability of continuous-time linear switching systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[38]  Jean Buisson,et al.  On the Stabilisation of Switching Electrical Power Converters , 2005, HSCC.

[39]  S. Shankar Sastry,et al.  Observability of Linear Hybrid Systems , 2003, HSCC.

[40]  Roger W. Brockett,et al.  Hybrid Models for Motion Control Systems , 1993 .

[41]  R. Decarlo,et al.  Construction of piecewise Lyapunov functions for stabilizing switched systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[42]  Didier Maquin,et al.  Switching Time Estimation of Piecewise Linear Systems. Application to Diagnosis , 2003 .

[43]  George J. Pappas,et al.  Observability of Switched Linear Systems in Continuous Time , 2005, HSCC.

[44]  A. Stephen Morse,et al.  Switched nonlinear systems with state-dependent dwell-time , 2003, Syst. Control. Lett..

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

[46]  Mireille E. Broucke,et al.  Stability and controllability of planar, conewise linear systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[47]  Wilfrid Perruquetti,et al.  Real-time estimation for switched linear systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[48]  Romain Bourdais,et al.  Stabilité et stabilisation d'une classe de systèmes dynamiques hybrides , 2007 .

[49]  Shuzhi Sam Ge,et al.  Controllability and reachability criteria for switched linear systems , 2002, Autom..

[50]  Franco Blanchini,et al.  Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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