Stability criteria for uncertain linear time-varying systems

In this paper robust stability of continuous linear time-varying systems is addressed based on Lyapunov functions which are constructed by max-composition of continuously differentiable functions. The resulting Lyapunov functions are continuous but not necessarily differentiable and no individual component needs to be positive definite. When the components are quadratic functions it will be possible to prove robust stability of systems which fail the classic quadratic stability test. The resulting conditions are matrix inequalities which are linear after choosing a set of tuning parameters. The robust stability condition is also extended to provide upper-bounds on integral performance measures.

[1]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[2]  C. W. Scherer,et al.  Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness , 2005, SIAM J. Matrix Anal. Appl..

[3]  Jamal Daafouz,et al.  Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties , 2001, Syst. Control. Lett..

[4]  B. Barmish.  Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

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

[6]  Pierre-Alexandre Bliman,et al.  An existence result for polynomial solutions of parameter-dependent LMIs , 2004, Syst. Control. Lett..

[7]  Alexandre Trofino,et al.  Stabilizing switching rule design for affine switched systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[8]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

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

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

[11]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[12]  Geir E. Dullerud,et al.  Robust Stabilization and Disturbance Attenuation of Switched Linear Parameter-Varying Systems in Discrete Time , 2012 .

[13]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.