LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability

The robust stability of uncertain systems in polytopic domains is investigated by means of homogeneous polynomially parameter-dependent Lyapunov (HPPDL) functions which are quadratic with respect to the state variables. A systematic procedure to construct linear matrix inequality (LMI) conditions whose solutions assure the existence of HPPDL functions of increasing degree is given. For each degree, a sequence of relaxations based on real algebraic methods provides sufficient LMI conditions of increasing precision for the existence of an HPPDL function which tend asymptotically to the necessity. As a result, families of LMI conditions parametrized on the degree of the HPPDL functions and on the relaxation level provide efficient numerical tests of different complexities to assess the robust stability of both continuous and discrete-time uncertain systems.

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

[2]  Etienne de Klerk,et al.  Approximation of the Stability Number of a Graph via Copositive Programming , 2002, SIAM J. Optim..

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

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

[5]  Robert E. Skelton,et al.  Stability tests for constrained linear systems , 2001 .

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

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

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

[9]  Pedro Luis Dias Peres,et al.  A less conservative LMI condition for the robust stability of discrete-time uncertain systems , 2001, Syst. Control. Lett..

[10]  B. Reznick,et al.  A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra , 2001 .

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

[12]  C. Scherer.  Higher-order relaxations for robust LMI problems with verifications for exactness , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  P. Gahinet,et al.  Affine parameter-dependent Lyapunov functions and real parametric uncertainty , 1996, IEEE Trans. Autom. Control..

[14]  Pedro Luis Dias Peres,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, IEEE Trans. Autom. Control..

[15]  J. Geromel,et al.  LMI characterization of structural and robust stability: the discrete-time case , 1999 .

[16]  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).

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

[18]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[19]  Graziano Chesi,et al.  Robust analysis of linear systems affected by time-invariant hypercubic parametric uncertainty , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[20]  J. Bernussou,et al.  A new robust D-stability condition for real convex polytopic uncertainty , 2000 .

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

[22]  J. Lasserre,et al.  On parameter-dependent Lyapunov functions for robust stability of linear systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).