Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions

Abstract This paper deals with the stability analysis of a class of uncertain switched systems on non-uniform time domains. The considered class consists of dynamical systems which commute between an uncertain continuous-time subsystem and an uncertain discrete-time subsystem during a certain period of time. The theory of dynamic equations on time scale is used to study the stability of these systems on non-uniform time domains formed by a union of disjoint intervals with variable length and variable gap. Using the concept of common Lyapunov function, sufficient conditions are derived to guarantee the asymptotic stability of this class of systems on time scale with bounded graininess function. The proposed scheme is used to study the leader–follower consensus problem under intermittent information transmissions.

[1]  J. J. Dacunha.  Stability for time varying linear dynamic systems on time scales , 2005 .

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

[3]  J. Daafouz,et al.  Stabilization of linear impulsive systems through a nearly-periodic reset , 2013 .

[4]  Zbigniew Bartosiewicz,et al.  On stabilisability of nonlinear systems on time scales , 2013, Int. J. Control.

[5]  Do Duc Thuan,et al.  Stability radius of implicit dynamic equations with constant coefficients on time scales , 2011, Syst. Control. Lett..

[6]  John M. Davis,et al.  STABILITY OF SIMULTANEOUSLY TRIANGULARIZABLE SWITCHED SYSTEMS ON HYBRID DOMAINS , 2014 .

[7]  A. Michel,et al.  Analysis and design of switched normal systems , 2006 .

[8]  Michael Defoort,et al.  Stability analysis of a class of switched linear systems on non-uniform time domains , 2014, Syst. Control. Lett..

[9]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[10]  Robert J. Marks,et al.  Algebraic and dynamic Lyapunov equations on time scales , 2009, 2010 42nd Southeastern Symposium on System Theory (SSST).

[11]  Fabian R. Wirth,et al.  A spectral characterization of exponential stability for linear time-invariant systems on time scales , 2003 .

[12]  Mario di Bernardo,et al.  Minimal control synthesis adaptive control of continuous bimodal piecewise affine systems , 2012, CDC.

[13]  Fabian R. Wirth,et al.  Stability radii for positive linear time-invariant systems on time scales , 2010, Syst. Control. Lett..

[14]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[15]  Michael Defoort,et al.  Hybrid sliding mode observer for switched linear systems with unknown inputs , 2014, J. Frankl. Inst..

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

[17]  Youssef N. Raffoul,et al.  Exponential stability of dynamic equations on time scales , 2005 .

[18]  Michael Defoort,et al.  Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics , 2015 .

[19]  Josep M. Olm,et al.  Model reference adaptive control of discrete‐time piecewise linear systems , 2013 .

[20]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

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

[22]  J. Geromel,et al.  Stability and stabilization of discrete time switched systems , 2006 .

[23]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[24]  G. Zhai.  Quadratic stabilizability of discrete-time switched systems via state and output feedback , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[25]  Robert J. Marks,et al.  Stability of switched linear systems on non-uniform time domains , 2010, 2010 42nd Southeastern Symposium on System Theory (SSST).

[26]  Stefano Di Gennaro,et al.  Self-triggered control for multi-agent systems with unknown non-linear inherent dynamics , 2014 .

[27]  Mario di Bernardo,et al.  Hybrid Model Reference Adaptive Control of Piecewise Affine Systems , 2013, IEEE Transactions on Automatic Control.

[28]  Mandeep Singh,et al.  Existence of solutions for three-point BVPs arising in bridge design , 2014 .

[29]  Stefano Di Gennaro,et al.  Self-triggered control for multi-agent systems under a directed switching graph , 2013, 52nd IEEE Conference on Decision and Control.