Analysis of Uncertain Discrete-Time Linear Periodic Systems based on System Lifting and LMIs

In this article, we propose novel linear matrix inequality (LMI) conditions for the stability and l 2 gain performance analysis of discrete-time linear periodically time-varying (LPTV) systems. These LMIs are convex with respect to all of the coefficient matrices of the LPTV systems and this property is promising when we deal with several control system analysis and synthesis problems. For example, we can apply those LMIs straightforwardly to robust performance analysis problems of LPTV systems that are affected by polytopic-type uncertainties. Even though our approach for robust performance analysis is conservative in general, we can reduce the conservatism gradually by artificially regarding the original N-periodic system as pN-periodic and increasing p. In addition, thanks to the simple structure of the LMI conditions, we can readily derive a viable test to verify the exactness of the computation results.

[1]  S. Bittanti,et al.  Analysis of discrete-time linear periodic systems , 1996 .

[2]  Yoshio Ebihara,et al.  Robustness analysis of uncertain discrete-time linear systems based on system lifting and LMIs , 2009, 2009 ICCAS-SICE.

[3]  Dimitri Peaucelle,et al.  Periodically time-varying dynamical controller synthesis for polytopic-type uncertain discrete-time linear systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[4]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[5]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

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

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

[8]  Dimitri Peaucelle,et al.  LMI-based Periodically Time-Varying Dynamical Controller Synthesis for Discrete-Time Uncertain Linear Systems , 2008 .

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

[10]  Masayuki Sato,et al.  LMI Tests for Positive Definite Polynomials: Slack Variable Approach , 2009, IEEE Transactions on Automatic Control.

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

[12]  Tomomichi Hagiwara,et al.  Robust Performance Analysis of Uncertain LTI Systems: Dual LMI Approach and Verifications for Exactness , 2007, IEEE Transactions on Automatic Control.

[13]  J. Geromel,et al.  Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems , 2002 .

[14]  Dimitri Peaucelle,et al.  Robust Hinfinity performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs , 2007, Syst. Control. Lett..

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

[16]  Venkataramanan Balakrishnan,et al.  Semidefinite programming duality and linear time-invariant systems , 2003, IEEE Trans. Autom. Control..

[17]  Patrizio Colaneri,et al.  The extended periodic lyapunov lemma , 1985, Autom..

[18]  Geir E. Dullerud,et al.  A new approach for analysis and synthesis of time-varying systems , 1999, IEEE Trans. Autom. Control..

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

[20]  D. Peaucelle,et al.  Robust Performance Analysis of Linear Time-Invariant Uncertain Systems by Taking Higher-Order Time-Derivatives of the State , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[21]  Karolos M. Grigoriadis,et al.  A unified algebraic approach to linear control design , 1998 .