Discrete-time Lossless Positive Real Lemma Based on Kalman Reachability Decomposition

This paper studies the discrete-time lossless positive real properties of transfer functions with minimal state-space realizations. Firstly, necessary and sufficiency conditions are established to characterize the discrete-time lossless positive real properties of a transfer function in terms of its behavior on the unit circle. Secondly, a DT-LPR lemma is given based on the QR decomposition and the Kalman reachability decomposition. This lemma provides necessary and sufficiency conditions for state-space systems to be DT-LPR. Finally, a numerical example is given to illustrate the developed theory.

[1]  Petros A. Ioannou,et al.  Necessary and sufficient conditions for strictly positive real matrices , 1990 .

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

[3]  Brian D. O. Anderson,et al.  Discrete positive-real fu nctions and their applications to system stability , 1969 .

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

[5]  D. S. Bernstein,et al.  Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[6]  Brian D. O. Anderson,et al.  Stability of adaptive systems: passivity and averaging analysis , 1986 .

[7]  Lihua Xie,et al.  Positive real analysis and synthesis of uncertain discrete time systems , 2000 .

[8]  B. Brogliato,et al.  Dissipative Systems Analysis and Control , 2000 .

[9]  A. Zinober,et al.  Adaptive Control: the Model Reference Approach , 1980 .

[10]  Guang-Hong Yang,et al.  New characterisations of positive realness and static output feedback control of discrete-time systems , 2009, Int. J. Control.

[11]  E. Jury,et al.  Discrete-time positive-real lemma revisited: the discrete-time counterpart of the Kalman-Yakubovitch lemma , 1994 .

[12]  Wei Lin,et al.  Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems , 1994, IEEE Trans. Autom. Control..

[13]  Brian D. O. Anderson,et al.  Network Analysis and Synthesis: A Modern Systems Theory Approach , 2006 .