Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

Abstract

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Iterative solvers for periodic matrix equations and model reduction for periodic control systems

“…In control theory, (2) and (3) are well known as causal and noncausal reachability Lyapunov equations, respectively [1], [5]. Note that Q l (k) = I − P l (k) and Q r (k) = I − P r (k) in (2). Similar equations arise for the causal and noncausal observability Lyapunov equations [5].…”

“…[5], [16], [2]). In control theory, (2) and (3) are well known as causal and noncausal reachability Lyapunov equations, respectively [1], [5]. Note that Q l (k) = I − P l (k) and Q r (k) = I − P r (k) in (2).…”

“…LIFTED REPRESENTATION OF PPDALES Stability analysis and numerical solution of PPDALEs (2) and (3) for dimension-varying system matrices are often described by the corresponding lifted representations of system (1) (see, e.g. [18], [1], [4]).…”

“…If the pencil λE − A has no zero eigenvalues, then A is nonsingular and (6) can be transformed to a standard discretetime Lyapunov equation by the multiplication from the left and right with A −1 and (A −1 ) T . In this case, however, both the ADI and Smith iterations fail to converge for the resulting Lyapunov equation [2]. This problem can be circumvented by considering a generalized Cayley transformation given by…”

Gradient Based Iterative Algorithm to Solve General Coupled Discrete-Time Periodic Matrix Equations Over Generalized Reflexive Matrices

“…The applications of linear matrix equations have motivated both mathematicians and engineers to construct methods catering to solve linear matrix equations [1,4,6,7,8,9,19,23,25]. Based on Smith iterations [24], iterative methods were developed for periodic standard Lyapunov matrix equations and projected generalized Lyapunov matrix equations [27,28].…”