Robust stability and stabilization of linear switched systems with dwell time

Sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, which is non-increasing at the switching instants is assigned to each subsystem. During the dwell time, this function varies piecewise linearly in time after switching occurs. It becomes time invariant afterwards. This function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties. The method proposed is then applied to stabilization via state-feedback both for the nominal and the uncertain cases.

[1]  J.P. Hespanha,et al.  Estimation under uncontrolled and controlled communications in Networked Control Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[2]  Liu Hsu,et al.  LMI characterization of structural and robust stability , 1998, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[3]  P. Bougerol.  Kalman filtering with random coefficients and contractions , 1993 .

[4]  Long Wang,et al.  Stabilization of switched linear systems with time-delay in detection of switching signal , 2005 .

[5]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[6]  P. Gahinet,et al.  A convex characterization of gain-scheduled H∞ controllers , 1995, IEEE Trans. Autom. Control..

[7]  Patrizio Colaneri.  Dwell time analysis of deterministic and stochastic switched systems , 2009, 2009 European Control Conference (ECC).

[8]  Andrea Censi,et al.  On the performance of Kalman filtering with intermittent observations: A geometric approach with fractals , 2009, 2009 American Control Conference.

[9]  Bruno Sinopoli,et al.  A characterization of the critical value for Kalman filtering with intermittent observations , 2008, 2008 47th IEEE Conference on Decision and Control.

[10]  Bruno Sinopoli,et al.  Kalman filtering with intermittent observations , 2004, IEEE Transactions on Automatic Control.

[11]  Ling Shi,et al.  Probabilistic performance of state estimation across a lossy network , 2008, Autom..

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

[13]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[14]  Orjan Stenflo.  Iterated Function Systems Controlled by a Semi-Markov Chain , 1996 .

[15]  A. Farina,et al.  A Gradient Descent Algorithm for evaluating the Karcher-Fréchet mean in the manifold of the symmetric positive definite matrices and its application in target detection , 2010 .

[16]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[17]  Shuzhi Sam Ge,et al.  Analysis and synthesis of switched linear control systems , 2005, Autom..

[18]  Wilfrid S. Kendall,et al.  Riemannian barycentres and geodesic convexity , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  M. Barnsley,et al.  A new class of markov processes for image encoding , 1988, Advances in Applied Probability.

[20]  Patrizio Colaneri,et al.  Stability and Stabilization of Continuous-Time Switched Linear Systems , 2006, SIAM J. Control. Optim..

[21]  Toshihisa Tanaka,et al.  An Algorithm to Compute Averages on Matrix Lie Groups , 2009, IEEE Transactions on Signal Processing.

[22]  Francesco Bullo,et al.  On Kalman Filtering for Detectable Systems With Intermittent Observations , 2009, IEEE Transactions on Automatic Control.

[23]  Robin J. Evans,et al.  Hybrid Dynamical Systems: Controller and Sensor Switching Problems , 2012 .

[24]  Michael Margaliot,et al.  Necessary and sufficient conditions for absolute stability: the case of second-order systems , 2003 .

[25]  L. A. Mcgee,et al.  Discovery of the Kalman filter as a practical tool for aerospace and industry , 1985 .

[26]  João Pedro Hespanha,et al.  Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle , 2004, IEEE Transactions on Automatic Control.

[27]  Uri Shaked,et al.  Time-convexity and time-gain-scheduling in finite-horizon robust H∞-control , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[28]  Subhrakanti Dey,et al.  Stability of Kalman filtering with Markovian packet losses , 2007, Autom..

[29]  Liu Hsu,et al.  LMI characterization of structural and robust stability , 1998 .

[30]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.