Positive real control for discrete-time singular systems with affine parameter dependence

Abstract This paper presents a less conservative and numerically tractable solution to the static state feedback positive real control problem for affinely parameter dependent discrete-time singular systems. Relied on the use of auxiliary matrices and a positive scalar decision variable, a novel necessary and sufficient condition of positive realness is first derived in terms of a strict matrix inequality for linear time-invariant discrete-time singular systems. This characterization leads to a numerically efficient and reliable way for the controller design synthesis. Then, the results are further expanded to parameter dependent singular systems whose coefficient matrices are affine functions of a time-invariant uncertain parameter vector. Both robust analysis and robust controller synthesis are addressed. Numerical examples are included to illustrate the effectiveness of the present results.

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

[2]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[3]  D. Bernstein,et al.  Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete-time theory , 1993 .

[4]  Fan-Ren Chang,et al.  The generalized state-space description of positive realness and bounded realness , 1996, Proceedings of the 39th Midwest Symposium on Circuits and Systems.

[5]  Delin Chu,et al.  Algebraic Characterizations for Positive Realness of Descriptor Systems , 2008, SIAM J. Matrix Anal. Appl..

[6]  Dan Zhang,et al.  Exponential H∞ filtering for discrete-time switched singular systems with time-varying delays , 2012, J. Frankl. Inst..

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

[8]  Masami Saeki,et al.  Synthesis of output feedback gain-scheduling controllers based on descriptor LPV system representation , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[9]  Chunyu Yang,et al.  Observer design for a class of nonlinear descriptor systems , 2013, J. Frankl. Inst..

[10]  A. Laub,et al.  The linear-quadratic optimal regulator for descriptor systems , 1987 .

[11]  L. Lee,et al.  Strictly positive real lemma and absolute stability for discrete-time descriptor systems , 2003 .

[12]  Mohamed Yagoubi,et al.  On State Feedback H∞ Control for Discrete-Time Singular Systems , 2013, IEEE Transactions on Automatic Control.

[13]  Gabriela Iuliana Bara,et al.  Robust analysis and control of parameter-dependent uncertain descriptor systems , 2011, Syst. Control. Lett..

[14]  Shaosheng Zhou,et al.  New characterization of positive realness and control of a class of uncertain polytopic discrete-time systems , 2005, Syst. Control. Lett..

[15]  Bin Zhou,et al.  Brief paper: Strict linear matrix inequality characterisation of positive realness for linear discrete-time descriptor systems , 2010 .

[16]  O. Brune.  Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency , 1931 .

[17]  Chengshan Xiao,et al.  Concepts of Strict Positive Realness and the Absolute Stability Problem of Continuous-Time Systems , 1998, Autom..

[18]  B. Anderson.  A SYSTEM THEORY CRITERION FOR POSITIVE REAL MATRICES , 1967 .

[19]  R. Kálmán.  LYAPUNOV FUNCTIONS FOR THE PROBLEM OF LUR'E IN AUTOMATIC CONTROL. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[20]  T. Başar.  Absolute Stability of Nonlinear Systems of Automatic Control , 2001 .

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

[22]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

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

[24]  Zhifeng Gao,et al.  Stabilization of discrete-time switched singular time-delay systems under asynchronous switching , 2012, J. Frankl. Inst..

[25]  L. Vandenberghe,et al.  Extended LMI characterizations for stability and performance of linear systems , 2009, Syst. Control. Lett..

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

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

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

[29]  Roland W. Freund,et al.  An extension of the positive real lemma to descriptor systems , 2004, Optim. Methods Softw..

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

[31]  Shengyuan Xu,et al.  On positive realness of descriptor systems , 2002 .

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