Bilinear Transformation for Discrete-Time Positive Real and Negative Imaginary Systems

This paper studies the connection between discrete-time and continuous-time negative imaginary systems. First, we analyze differences between two statements that are claimed to provide equivalent conditions for systems to be discrete-time positive real. Our conclusion is that one is equivalent to the definition of discrete-time positive real transfer matrices, the other is not. Second, by means of the bilinear transformation, a connection between discrete-time and continuous-time negative imaginary transfer matrices is established. Third, the concept of discrete-time lossless negative imaginary systems is introduced, and a discrete-time lossless negative imaginary lemma is developed to characterize the lossless negative imaginary properties in terms of minimal state-space realization. Some properties of discrete-time lossless negative imaginary transfer matrices are also studied. Several numerical examples illustrate the developed theory.

[1]  Ian R. Petersen,et al.  Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response , 2008, IEEE Transactions on Automatic Control.

[2]  Sourav Patra,et al.  Absolute stability analysis for negative-imaginary systems , 2016, Autom..

[3]  Lorenzo Ntogramatzidis,et al.  Some new results in the theory of negative imaginary systems with symmetric transfer matrix function , 2013, Autom..

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

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

[6]  Ian R. Petersen,et al.  Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures With Free Body Motion , 2013, IEEE Transactions on Automatic Control.

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

[8]  Sourav Patra,et al.  On LTI output strictly negative-imaginary systems , 2017, Syst. Control. Lett..

[9]  Brian D. O. Anderson,et al.  Recursive algorithm for spectral factorization , 1974 .

[10]  Junlin Xiong,et al.  Connections between discrete- and continuous-time results for positive real and negative imaginary systems , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[11]  Robert W. Newcomb,et al.  Linear multiport synthesis , 1966 .

[12]  Sourav Patra,et al.  On input-output negative-imaginary systems and an output strict negative-imaginary lemma , 2016, 2016 Indian Control Conference (ICC).

[13]  D. Hill,et al.  Generalizations and new proof of the discrete-time positive real lemma and bounded real lemma , 1999 .

[14]  Lorenzo Ntogramatzidis,et al.  Foundations of Not Necessarily Rational Negative Imaginary Systems Theory: Relations Between Classes of Negative Imaginary and Positive Real Systems , 2016, IEEE Transactions on Automatic Control.

[15]  Panos J. Antsaklis,et al.  On relationships among passivity, positive realness, and dissipativity in linear systems , 2014, Autom..

[16]  Robert Grino,et al.  A New Passive Repetitive Controller For Discrete-Time Finite-Frequency Positive-Real Systems , 2006, CDC.

[17]  Kamal Premaratne,et al.  Some corrections on "Discrete-time positive-real lemma revisited: the discrete-time counterpart of the Kalman-Yakubovitch lemma" , 1996 .

[18]  Lorenzo Ntogramatzidis,et al.  Discrete-time negative imaginary systems , 2017, Autom..

[19]  Ian R. Petersen,et al.  A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems , 2010, IEEE Transactions on Automatic Control.

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

[21]  Junlin Xiong,et al.  On non-proper negative imaginary systems , 2016, Syst. Control. Lett..

[22]  Kui Yao,et al.  Design and Implementation of Feedback Resonance Compensator in Hard Disk Drive Servo System: A Mixed Passivity, Negative-Imaginary and Small-Gain Approach in Discrete Time , 2015 .

[23]  Junlin Xiong,et al.  Properties and stability analysis of discrete-time negative imaginary systems , 2017, Autom..

[24]  Ian R. Petersen,et al.  On lossless negative imaginary systems , 2009, 2009 7th Asian Control Conference.

[25]  W. Haddad,et al.  Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , 2008 .