Dwell time-based stabilisation of switched delay systems using free-weighting matrices

ABSTRACT In this paper, we present a quasi-convex optimisation method to minimise an upper bound of the dwell time for stability of switched delay systems. Piecewise Lyapunov–Krasovskii functionals are introduced and the upper bound for the derivative of Lyapunov functionals is estimated by free-weighting matrices method to investigate non-switching stability of each candidate subsystems. Then, a sufficient condition for the dwell time is derived to guarantee the asymptotic stability of the switched delay system. Once these conditions are represented by a set of linear matrix inequalities , dwell time optimisation problem can be formulated as a standard quasi-convex optimisation problem. Numerical examples are given to illustrate the improvements over previously obtained dwell time bounds. Using the results obtained in the stability case, we present a nonlinear minimisation algorithm to synthesise the dwell time minimiser controllers. The algorithm solves the problem with successive linearisation of nonlinear conditions.

[1]  Chengzhi Yuan,et al.  Hybrid Control for Switched Linear Systems With Average Dwell Time , 2015, IEEE Transactions on Automatic Control.

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

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

[4]  Robert Shorten,et al.  A result on common quadratic Lyapunov functions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[5]  Huijun Gao,et al.  H∞ control of switched delayed systems with average dwell time , 2013, Int. J. Control.

[6]  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).

[7]  Sina Y. Caliskan,et al.  Dwell-time computation for stability of switched systems with time delays , 2013 .

[8]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[9]  Michael Margaliot,et al.  On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws , 2009, IEEE Transactions on Automatic Control.

[10]  Zhengzhi Han,et al.  Stability and stabilization of positive switched systems with mode-dependent average dwell time , 2013 .

[11]  Wei Xing Zheng,et al.  Delay-Independent Minimum Dwell Time for Exponential Stability of Uncertain Switched Delay Systems , 2010, IEEE Transactions on Automatic Control.

[12]  Daniel Liberzon,et al.  Quantized control via locational optimization , 2002, IEEE Transactions on Automatic Control.

[13]  Jun Zhao,et al.  Stability and L2-gain analysis for switched delay systems: A delay-dependent method , 2006, Autom..

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Peng Yan,et al.  Robust Stabilization of Parameter Varying Time Delay Systems by Switched Controllers , 2014 .

[16]  Diana Baader,et al.  Stability Analysis And Robust Control Of Time Delay Systems , 2016 .

[17]  Karolin Papst,et al.  Stability Theory Of Switched Dynamical Systems , 2016 .

[18]  Uri Shaked,et al.  Robust stability and stabilization of linear switched systems with dwell time , 2010, 2010 Conference on Control and Fault-Tolerant Systems (SysTol).

[19]  Patrizio Colaneri,et al.  A Nonconservative LMI Condition for Stability of Switched Systems With Guaranteed Dwell Time , 2012, IEEE Transactions on Automatic Control.

[20]  A. Morse.  Supervisory control of families of linear set-point controllers Part I. Exact matching , 1996, IEEE Trans. Autom. Control..

[21]  Peng Yan,et al.  Stability analysis of switched time-delay systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[22]  L. Ghaoui,et al.  A cone complementarity linearization algorithm for static output-feedback and related problems , 1997, IEEE Trans. Autom. Control..

[23]  Daniel Liberzon,et al.  Stability of hybrid automata with average dwell time: an invariant approach , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[24]  Peng Yan,et al.  Dwell time optimization in switching control of parameter varying time delay systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[25]  Lihua Xie,et al.  Further Improvement of Free-Weighting Matrices Technique for Systems With Time-Varying Delay , 2007, IEEE Transactions on Automatic Control.

[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]  Bruce A. Francis,et al.  Stabilizing a linear system by switching control with dwell time , 2002, IEEE Trans. Autom. Control..

[28]  Peng Shi,et al.  Stability of switched positive linear systems with average dwell time switching , 2012, Autom..

[29]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[30]  Ahmet Taha Koru,et al.  On dwell time minimization for switched delay systems: Free-weighting matrices method , 2014, 53rd IEEE Conference on Decision and Control.

[31]  Guang-Ren Duan,et al.  Global stabilization of switched stochastic nonlinear systems in strict-feedback form under arbitrary switchings , 2013, Autom..

[32]  Magdi S. Mahmoud,et al.  Switched Time-Delay Systems , 2010 .

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