Robust stability of polytopic systems via affine parameter-dependent Lyapunov functions

This paper studies robust stability of linear systems with polytopic uncertainty. New necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz or Schur stability of a polytopic system, which are composed of a family of linear matrix inequality conditions of increasing precision. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function, and necessity is asymptotically attained. Compared with the existing results in the literature, it is shown that the new stability conditions provide less conservative tests at each step. A numerical example is given to illustrate the effectiveness of the new results.

[1]  Zongli Lin,et al.  A descriptor system approach to robust stability analysis and controller synthesis , 2004, IEEE Trans. Autom. Control..

[2]  Valter J. S. Leite,et al.  D-stability of polytopes of polynomial matrices: characterization through LMIs , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[3]  Guang-Hong Yang,et al.  Robust Stability of Polytopic Systems via Affine Parameter-Dependent Lyapunov Functions , 2008, SIAM J. Control. Optim..

[4]  Uri Shaked,et al.  Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty , 2001, IEEE Trans. Autom. Control..

[5]  Long Wang.  Robust stability of polynomial families under nonlinearly correlated perturbations: Boundary results with applications , 1997, 1997 European Control Conference (ECC).

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

[7]  P. Peres,et al.  Stability of polytopes of matrices via affine parameter-dependent lyapunov functions : Asymptotically exact LMI conditions , 2005 .

[8]  Ricardo C. L. F. Oliveira,et al.  LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions , 2006, Syst. Control. Lett..

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

[10]  Shaosheng Zhou,et al.  New characterization of positive realness and control of a class of uncertain polytopic discrete-time systems , 2005, Syst. Control. Lett..

[11]  Tong Heng Lee,et al.  A less conservative robust stability test for linear uncertain time-delay systems , 2006, IEEE Trans. Autom. Control..

[12]  Vladimir L. Kharitonov,et al.  Robust stability of nested polynomial families , 1996, Autom..

[13]  Graziano Chesi,et al.  Robust stability of polytopic systems via polynomially parameter-dependent Lyapunov functions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[14]  Munther A. Dahleh,et al.  Polytopic control Lyapunov functions for robust stabilization of a class of nonlinear systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[15]  Valter J. S. Leite,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

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

[17]  Pedro Luis Dias Peres,et al.  An LMI condition for the robust stability of uncertain continuous-time linear systems , 2002, IEEE Trans. Autom. Control..

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

[19]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[20]  Valter J. S. Leite,et al.  Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems , 2004 .

[21]  R. Jackson.  Inequalities , 2007, Algebra for Parents.