Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems

Abstract

From the necessary and sufficient conditions for complete reachability and observability of periodic time-varying descriptor systems, the symmetric positive semi-definite reachability/observability Gramians are defined. These Gramians can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and an illustrative numerical example is given. As an application of our results, the balanced realization of periodic descriptor systems is discussed.

Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations

“…k=0 corresponding to the eigenvalue at infinity (Benner et al (2011b);Chu et al (2007);Stykel (2008)). This type of equations arises in the context of periodic state feedback problems and in model reduction of periodic descriptor systems when the solutions of the noncausal matrix equations associated with the systems (Kuo et al (2004);Benner et al (2011a);Chu et al (2007)) are sought.…”

“…This type of equations arises in the context of periodic state feedback problems and in model reduction of periodic descriptor systems when the solutions of the noncausal matrix equations associated with the systems (Kuo et al (2004);Benner et al (2011a);Chu et al (2007)) are sought. Note that in that case Q l (k) = I n −P l (k) and Q r (k) = I n − P r (k), where P l (k), P r (k) are the spectral projectors onto the k-th left and right deflating subspaces of the periodic matrix pairs {(E k , A k )} K−1 k=0 corresponding to the finite eigenvalues.…”

“…In Section 2, we briefly review the direct methods that have been considered in (Chu et al (2007); Benner et al (2011b)) to solve the PPDALEs (2). The computational complexities and the drawbacks of these methods for large-scale sparse problems are also discussed.…”

“…The numerical solution of (2) has been considered in (Chu et al (2007)) for time-varying matrix coefficients. The method proposed there extends the periodic Schur method (Bojanczyk et al (1992);Varga (2004)) and the generalized Schur-Hammarling method (Stykel (2002)) developed for periodic standard and projected generalized Lyapunov equations, respectively.…”

“…Computing the Kronecker-like canonical forms of the periodic matrix pairs and solving the resulting periodic Sylvester equations are the most computationally expensive tasks in this algorithm (Algorithm 5.1 of Chu et al (2007)). …”