Discrete-time negative imaginary systems

Abstract In this paper we introduce the notion of a discrete-time negative imaginary system and we investigate its relations with discrete-time positive real system theory. In the framework presented here, discrete-time negative imaginary systems are defined in terms of a sign condition that must be satisfied in a domain of analyticity of the transfer function, in analogy with the case of discrete-time positive real functions, as well as analogously to the continuous-time case. This means in particular that we do not need to restrict our notions and definitions to systems with rational transfer functions. We also provide a discrete-time counterpart of the different notions that have appeared so far in the literature within the framework of strictly positive real and in the more recent theory of strictly negative imaginary systems, and to show how these notions are characterized and linked to each other. Stability analysis results for the feedback interconnection of discrete-time negative imaginary systems are also derived.

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

[2]  Sourav Patra,et al.  Stability Analysis of Interconnected Systems With “Mixed” Negative-Imaginary and Small-Gain Properties , 2011, IEEE Transactions on Automatic Control.

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

[4]  P. Olver.  Nonlinear Systems , 2013 .

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

[6]  Bharath Bhikkaji,et al.  A New Scanning Method for Fast Atomic Force Microscopy , 2011, IEEE Transactions on Nanotechnology.

[7]  Alexander Lanzon,et al.  Feedback Control of Negative-Imaginary Systems , 2010, IEEE Control Systems.

[8]  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.

[9]  Arjan van der Schaft,et al.  Positive feedback interconnection of Hamiltonian systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[10]  Ian R. Petersen,et al.  Robust cooperative control of multiple heterogeneous Negative-Imaginary systems , 2015, Autom..

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

[12]  Ian R. Petersen,et al.  A strongly strict negative-imaginary lemma for non-minimal linear systems , 2011, Commun. Inf. Syst..

[13]  Robert Shorten,et al.  On the Characterization of Strict Positive Realness for General Matrix Transfer Functions , 2010, IEEE Transactions on Automatic Control.

[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]  Mattia Zorzi,et al.  AR Identification of Latent-Variable Graphical Models , 2014, IEEE Transactions on Automatic Control.

[16]  Ian R. Petersen,et al.  Negative Imaginary Lemmas for Descriptor Systems , 2016, IEEE Transactions on Automatic Control.

[17]  Ian R. Petersen,et al.  Spectral Conditions for Negative Imaginary Systems With Applications to Nanopositioning , 2014, IEEE/ASME Transactions on Mechatronics.

[18]  S. O. Reza Moheimani,et al.  A Negative Imaginary Approach to Modeling and Control of a Collocated Structure , 2012, IEEE/ASME Transactions on Mechatronics.

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

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

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

[22]  Mark R. Opmeer,et al.  Infinite-Dimensional Negative Imaginary Systems , 2011, IEEE Transactions on Automatic Control.

[23]  Mattia Zorzi,et al.  Rational approximations of spectral densities based on the Alpha divergence , 2013, Math. Control. Signals Syst..

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

[25]  W. Rudin.  Real and complex analysis , 1968 .

[26]  Chaohong Cai,et al.  Stability Analysis for a String of Coupled Stable Subsystems With Negative Imaginary Frequency Response , 2010, IEEE Transactions on Automatic Control.

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

[28]  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.

[29]  B. Bhikkaji,et al.  Fast scanning using piezoelectric tube nanopositioners: A negative imaginary approach , 2009, 2009 IEEE/ASME International Conference on Advanced Intelligent Mechatronics.

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

[31]  Ian R. Petersen,et al.  Robust Output Feedback Consensus for Networked Negative-Imaginary Systems , 2015, IEEE Transactions on Automatic Control.

[32]  Ian R. Petersen,et al.  A negative-imaginary lemma without minimality assumptions and robust state-feedback synthesis for uncertain negative-imaginary systems , 2012, Syst. Control. Lett..

[33]  Alessandro Chiuso,et al.  A Bayesian approach to sparse plus low rank network identification , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).