On the Positive Definite Solutions to the 2-D Continuous-timeLyapunov Equation

Authors:
Chengshan Xiao;P. Agathoklis;David J. Hill
Affiliations:
Department of Electrical Engineering,University of Sydney, NSW 2006, Australia. E-mail: xiao@ee.usyd.edu.au;Department of Electrical and Engineering, University of Victoria, BC, V8W 3P6, Canada. E-mail: pan@ece.uvic.ca;Department of Electrical Engineering,University of Sydney, NSW 2006, Australia. E-mail: davidh@ee.usyd.edu.au
Venue:
Multidimensional Systems and Signal Processing
Year:
1997

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Abstract

The very strict positive real lemma is further developedfor nonminimal 1-D continuous-time systems and is used to studythe 2-D continuous-time Lyapunov equation. Based on it, an extendedcondition for the bivariate characteristic polynomial of a matrixto be very strict Hurwitz is proposed for general 2-D analogsystems with characteristic polynomials involving 1-D factorpolynomials. It is also shown that in such a case the bivariatepolynomial can be decomposed into a 2-D bivariate polynomialwith the corresponding matrix satisfying certain controllabilityand observability conditions and into up to two 1-D polynomials.Further, two algorithms for computing the positive definite solutionsto the 2-D Lyapunov equation are presented.