Monotonic relaxations for robust control: new characterizations

Parameterized linear matrix inequalities (PLMIs), that is LMIs depending on a parameter confined to a compact set frequently arise in both analysis and synthesis problems of robust control. As a major difficulty, PLMIs are equivalent to an infinite family of LMI constraints and consequently are very hard to solve numerically. Known approaches to find solutions exploit relaxations inferred from convexity arguments. These relaxations involve a finite family of LMIs the number of which grows exponentially with the number of scalar parameters. In this note, we propose a novel approach based on monotonicity concept which allows us to solve PLMIs via a finite and of polynomial order family of LMIs. The effectiveness and viability of our approach are demonstrated by numerical examples such as robust stability analysis and linear parameter varying (LPV) synthesis for which we clearly show that no additional conservatism is entailed as compared to earlier techniques.

[1]  P. Gahinet,et al.  Affine parameter-dependent Lyapunov functions for real parametric uncertainty , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[2]  P. Apkarian,et al.  Relaxations of parameterized LMIs with control applications , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[3]  Pierre Apkarian,et al.  Parameterized LMIs in Control Theory , 2000, SIAM J. Control. Optim..

[4]  B. R. Barmish,et al.  Probabilistic enhancement of classical robustness margins: the unirectangularity concept , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[5]  R. T. Reichert.  Robust autopilot design using μ-synthesis , 1990 .

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

[7]  Minyue Fu,et al.  Improved upper bounds of the structured singular value , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[8]  R.T. Reichert,et al.  Robust autopilot design using μ-synthesis , 1990, 1990 American Control Conference.

[9]  Toshiharu Sugie,et al.  New upper bound of the real /spl mu/ based on the parameter dependent multiplier , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[10]  Alan J. Laub,et al.  The LMI control toolbox , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[11]  D. Siljak.  Parameter Space Methods for Robust Control Design: A Guided Tour , 1988, 1988 American Control Conference.

[12]  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.

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