Robust D stabilization of a polytope of matrices

The problem of stabilization of a polytope of matrices in a subregion D R of the complex plane is revisited. A new sufficient condition of robust D R stabilization is given. It implies the solution of an LMI involving matrix variables constrained by a non-linear algebraic relation. Some LMI relaxations are first proposed. Then, it is shown that a cone complementarity formulation of this condition allows us to associate an efficient iterative numerical procedure which leads to a low computational burden. This algorithm is tested on different numerical examples for which existing approaches in control literature fail.

[1]  José Claudio Geromel,et al.  H2-norm optimization with constrained dynamic output feedback controllers: decentralized and reliable control , 1999, IEEE Trans. Autom. Control..

[2]  Michael G. SafonovtS.  Robust Analysis, Sectors, and Quadratic Functionals , 1995 .

[3]  Stephen P. Boyd,et al.  Structured and Simultaneous Lyapunov Functions for System Stability Problems , 1989 .

[4]  P. Peres,et al.  On a convex parameter space method for linear control design of uncertain systems , 1991 .

[5]  C. DeMarco,et al.  The minimal dimension of stable faces required to guarantee stability of a matrix polytope , 1989 .

[6]  Pierre Apkarian,et al.  Robust pole placement in LMI regions , 1999, IEEE Trans. Autom. Control..

[7]  Hoang Duong Tuan,et al.  A sequential SDP/Gauss-Newton algorithm for rank-constrained LMI problems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[8]  H. Kokame,et al.  A polytopic quadratic Lyapunov functions approach to stability of a matrix polytope , 1990, 29th IEEE Conference on Decision and Control.

[9]  Y. Pyatnitskiy,et al.  An iterative method of Lyapunov function construction for differential inclusions , 1987 .

[10]  Denis Arzelier,et al.  Pole assignment of linear uncertain systems in a sector via a Lyapunov-type approach , 1993, IEEE Trans. Autom. Control..

[11]  Laurent El Ghaoui,et al.  Advances in linear matrix inequality methods in control: advances in design and control , 1999 .

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

[13]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

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

[15]  Qing-Guo Wang.  Necessary and sufficient conditions for stability of a matrix polytope with normal vertex matrices , 1991, Autom..

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

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

[18]  J. Geromel,et al.  An LMI optimization approach to multiobjective controller design for discrete-time systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[19]  Pierre Apkarian,et al.  Robust control via concave minimization local and global algorithms , 2000, IEEE Trans. Autom. Control..

[20]  Friedemann Leibfritz,et al.  An LMI-Based Algorithm for Designing Suboptimal Static H2/Hinfinity Output Feedback Controllers , 2000, SIAM J. Control. Optim..

[21]  Liao Xiao Xin,et al.  Necessary and sufficient conditions for stability of a class of interval matrices , 1987 .

[22]  H. Horisberger,et al.  Regulators for linear, time invariant plants with uncertain parameters , 1976 .

[23]  K. Goh,et al.  Robust synthesis via bilinear matrix inequalities , 1996 .

[24]  P. Gahinet,et al.  H∞ design with pole placement constraints: an LMI approach , 1996, IEEE Trans. Autom. Control..