Positive Operators and Stable Truncation

Abstract We introduce a notion of balanced truncation for generalized Lyapunov operators and show that it preserves asymptotic stability. The proof relies on the theory of positive mappings and a result by Hans Schneider. Applications of our result can be found in model order reduction of stochastic linear systems.

[1]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .

[2]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[3]  P. Benner,et al.  Balanced Truncation for Stochastic Linear Systems with Guaranteed Error Bound , 2014 .

[4]  H. Schneider.  Positive operators and an inertia theorem , 1965 .

[5]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[6]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[7]  Peter Benner,et al.  Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems , 2011, SIAM J. Control. Optim..

[8]  Peter Benner,et al.  Model reduction for stochastic systems , 2015 .

[9]  M. A. Krasnoselʹskii,et al.  Positive Linear Systems, the Method of Positive Operators , 1989 .

[10]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[11]  Hans Schneider,et al.  Matrices leaving a cone invariant , 2007 .

[12]  Lyapunov Revisited: Variations on a Matrix Theme , 1997 .

[13]  L. Elsner,et al.  Monotonie und randspektrum bei vollstetigen operatoren , 1970 .

[14]  T. Damm.  Rational Matrix Equations in Stochastic Control , 2004 .

[15]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[16]  Wolfgang Arendt,et al.  Resolvent Positive Operators , 1987 .

[17]  O. Perron.  Zur Theorie der Matrices , 1907 .

[18]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[19]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[20]  T. Damm.  Positive groups on Hn are completely positive , 2004 .

[21]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[22]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .