On State Feedback H∞ Control for Discrete-Time Singular Systems

This technical note deals with the state feedback H∞ control problem for linear time-invariant discrete-time singular systems. Relied on the use of auxiliary matrices and a positive scalar, a novel necessary and sufficient condition for the bounded real lemma is derived for discrete-time singular systems. The characterization is reduced to a strict linear matrix inequality (LMI) when the scalar is fixed, and the resulting LMI is non-conservative as long as the scalar is chosen sufficiently large. Moreover, the result is further expanded to H∞ controller design, and a numerically efficient and reliable design procedure is given. Since no particular restriction is imposed on the auxiliary matrices, the proposed result outperforms the existing methods in the literature. Numerical examples are included to illustrate the effectiveness of the present result.

[1]  Philippe Chevrel,et al.  Dilated LMI characterisations for linear time-invariant singular systems , 2010, Int. J. Control.

[2]  Tomomichi Hagiwara,et al.  New dilated LMI characterizations for continuous-time multiobjective controller synthesis , 2004, Autom..

[3]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[4]  Daniel W. C. Ho,et al.  Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation , 2010, Autom..

[5]  Yuanqing Xia,et al.  New bounded real lemma for discrete-time singular systems , 2008, Autom..

[6]  A. Rantzer.  On the Kalman-Yakubovich-Popov lemma , 1996 .

[7]  Olivier Bachelier,et al.  Robust state feedback admissibilisation of discrete linear polytopic descriptor systems: a strict linear matrix inequality approach , 2012 .

[8]  W. Marsden.  I and J , 2012 .

[9]  C. Yung.  H∞ Control for Linear Discrete-Time Descriptor Systems: State Feedback and Full Information Cases , 2008 .

[10]  Kan-Lin Hsiung,et al.  Lyapunov inequality and bounded real lemma for discrete-time descriptor systems , 1999 .

[11]  Shengyuan Xu,et al.  Robust Control and Filtering of Singular Systems , 2006 .

[12]  B. Jamison.  Reciprocal processes , 1974 .

[13]  Alan S. Willsky,et al.  Algorithms for the incorporation of predictive information in surveillance theory , 1985 .

[14]  A. Krener,et al.  Modeling and estimation of discrete-time Gaussian reciprocal processes , 1990 .

[15]  Chengwu Yang,et al.  H∞ state feedback control for discrete singular systems , 2000, IEEE Trans. Autom. Control..

[16]  Shengyuan Xu,et al.  Reduced-order H∞ filtering for singular systems , 2007, Syst. Control. Lett..

[17]  Philippe Chevrel,et al.  Extended ${\rm H}_{2}$ Controller Synthesis for Continuous Descriptor Systems , 2012, IEEE Transactions on Automatic Control.

[18]  Gabriela Iuliana Bara.  Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case , 2011, Int. J. Control.

[19]  Shengyuan Xu,et al.  H∞ filtering for singular systems , 2003, IEEE Trans. Autom. Control..

[20]  O. Bachelier,et al.  Robust S-regularity of matrix pencils applied to the analysis of descriptor models , 2011 .

[21]  Mohamed Yagoubi,et al.  On multiobjective synthesis for parameter-dependent descriptor systems , 2010 .

[22]  Mohamed Darouach,et al.  Novel bounded real lemma for discrete-time descriptor systems: Application to H∞ control design , 2012, Autom..

[23]  Izumi Masubuchi.  Output feedback controller synthesis for descriptor systems satisfying closed-loop dissipativity , 2007, Autom..

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

[25]  Alan J. Laub,et al.  The linear-quadratic optimal regulator for descriptor systems , 1985, 1985 24th IEEE Conference on Decision and Control.

[26]  F. Allgöwer,et al.  CAUSAL H ∞ CONTROL OF DESCRETE-TIME DESCRIPTOR SYSTEMS : AN LMI APPROACH IN TWO STEPS , 2004 .

[27]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[28]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in System and Control Theory , 1994, Studies in Applied Mathematics.

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

[30]  Langford B. White,et al.  Optimal Smoothing for Finite State Hidden Reciprocal Processes , 2011, IEEE Transactions on Automatic Control.

[31]  Xiaoming Chen.  An Equivalent LMI Representation of Bounded Real Lemma for Continuous-Time Systems , 2008 .

[32]  X. Ji,et al.  Robust state feedback H/spl infin/ control for uncertain linear discrete singular systems , 2007 .

[33]  Huijun Gao,et al.  State Estimation and Sliding-Mode Control of Markovian Jump Singular Systems , 2010, IEEE Transactions on Automatic Control.

[34]  Philippe Chevrel,et al.  Hinfinity control with unstable and nonproper weights for descriptor systems , 2012, Autom..