Higher-order relaxations for robust LMI problems with verifications for exactness

Robust semi-definite programming problems are know to have a wide range of applications, in particular in robust control. For rational uncertainty dependence, the full block S-procedure allows to systematically construct relaxations for the computation of guaranteed bounds. Typically these relaxations are conservative (causing a gap between actual and computed optimal values) since they involve the approximation of a set of so-called multipliers. The main purpose of this paper is to suggest a novel sequence of multiplier approximations which can be exploited in computations and which can be proved to be asymptotically exact. The second goal is to provide a numerical test for checking whether relaxations are exact (thus guaranteeing the absence of conservatism) which extends a recently formulated general principle to synthesis problems. We discuss the practical relevance of our results for LPV synthesis, and we illustrate them in terms of a numerical example.

[1]  Carsten W. Scherer,et al.  When are Multiplier Relaxations Exact , 2003 .

[2]  Etienne de Klerk,et al.  On the Convergence of the Central Path in Semidefinite Optimization , 2002, SIAM J. Optim..

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

[4]  Arkadi Nemirovski,et al.  On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty , 2002, SIAM J. Optim..

[5]  Carsten W. Scherer,et al.  LPV control and full block multipliers , 2001, Autom..

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

[7]  Graziano Chesi,et al.  An LMI-based approach for characterizing the solution set of polynomial systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[8]  Boris Polyak,et al.  Probabilistic robust design with linear quadratic regulators , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  C. Ferrier.  Computation of the distance to semi-algebraic sets , 2000 .

[10]  C.E. de Souza,et al.  Bi-quadratic stability of uncertain linear systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[11]  Tetsuya Iwasaki,et al.  LPV system analysis via quadratic separator for uncertain implicit systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[12]  C. Scherer,et al.  Robust stability analysis for parameter dependent systems using full block S-procedure , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[13]  S. Hara,et al.  Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations , 1998, IEEE Trans. Autom. Control..

[14]  Laurent El Ghaoui,et al.  Robust Solutions to Uncertain Semidefinite Programs , 1998, SIAM J. Optim..

[15]  Katya Scheinberg,et al.  Interior Point Trajectories in Semidefinite Programming , 1998, SIAM J. Optim..

[16]  C. Scherer.  A full block S-procedure with applications , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[17]  Tetsuya Iwasaki,et al.  Robust stability analysis with quadratic separator: parametric time-varying uncertainty case , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[18]  Carsten W. Scherer,et al.  Parametrically varying flight control system design with full block scalings , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[19]  M. Fu,et al.  A dual formulation of mixed μ and on the losslessness of (D, G) scaling , 1997, IEEE Trans. Autom. Control..

[20]  Jesús A. De Loera,et al.  An effective version of Pólya's theorem on positive definite forms , 1996 .

[21]  H. Ozbay,et al.  On the /spl Nscr//spl Pscr/-hardness of the purely complex /spl mu/ computation, analysis/synthesis, and some related problems in multidimensional systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[22]  Maarten Steinbuch,et al.  Parametric Uncertainty Modeling using LFTs , 1993, 1993 American Control Conference.

[23]  Andrew Packard,et al.  The complex structured singular value , 1993, Autom..

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