Spectral conditions for symmetric positive real and negative imaginary systems

Non-Hamiltonian spectral conditions for the class of symmetric multivariable strictly positive real and strictly negative imaginary systems are derived. They represent generalizations of known ones for strict positive realness to the cases with singular feedthrough matrix, and are novel in the context of strict negative imaginariness. Moreover, we propose a concept of strong negative imaginariness and establish its links to strict positive realness of symmetric systems. The proposed spectral conditions are useful in the corresponding assessment and enforcement procedures, as well as in quadratic stability analysis of uncertain and switched systems.

[1]  Jan C. Willems,et al.  REALIZATION OF SYSTEMS WITH INTERNAL PASSIVITY AND SYMMETRY CONSTRAINTS , 1976 .

[2]  Peter Benner,et al.  Spectral characterization and enforcement of negative imaginariness for descriptor systems , 2013 .

[3]  Robert Shorten,et al.  Quadratic Stability and Singular SISO Switching Systems , 2009, IEEE Transactions on Automatic Control.

[4]  P. Fuhrmann.  On symmetric rational transfer functions , 1983 .

[5]  V. Mehrmann,et al.  An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems , 2012 .

[6]  A. Semlyen,et al.  A Half-Size Singularity Test Matrix for Fast and Reliable Passivity Assessment of Rational Models , 2010, IEEE Transactions on Power Delivery.

[7]  Daniel Hershkowitz,et al.  On the inertia of intervals of matrices , 1990 .

[8]  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.

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

[10]  F. M. Callier,et al.  Dissipative Systems Analysis and Control: Theory and Applications (2nd Edition)-[Book review; B. Brogliato, R. Lozano, B. Maschke] , 2007, IEEE Transactions on Automatic Control.

[11]  Von der Fakultat,et al.  Dissipativity of linear quadratic systems , 2011 .

[12]  V. Mehrmann,et al.  A new method for computing the stable invariant subspace of a real Hamiltonian matrix , 1997 .

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

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

[15]  Peter Benner,et al.  Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils , 2002, SIAM J. Matrix Anal. Appl..

[16]  Fuzhen Zhang.  The Schur complement and its applications , 2005 .

[17]  T. Dhaene,et al.  Generalized eigenvalue passivity assessment of descriptor systems with applications to symmetric and singular systems , 2013 .

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