Model order reduction via real Schur-form decomposition

Solving Stable Sylvester Equations via Rational Iterative Schemes

“…Therefore, we may apply the iterative schemes proposed for sign function computations in order to solve (2). This was first observed in [42] and re-discovered, e.g., in [7,36].…”

“…The cross-Gramian contains information about certain properties of the linear system [23] and can also be used for model reduction [2]. We apply Algorithm 4 to (28) for a system described in [1, Example 4.2] which comes from a model for heat control in a thin rod.…”

“…This is a very strong assumption as compared to the necessary ones (A−λC, D−λB both regular with disjoint spectra) that can be handled by the Bartels-Stewart and Hessenberg-Schur methods. Therefore, the methods developed here can only be applied to a limited set of equations of the forms (1) and (2). On the other hand, so far there is no Bartels-Stewart and Hessenberg-Schur based method to treat the special case of a factored right-hand side.…”

“…The focus in this paper is on a special class of iterative schemes to solve stable Sylvester equations that appear in the computation of the matrix sign function. The basic Newton iteration classically used in this context was first proposed for (2) in [42]; see also [7,36]. It can basically be described by the recursive application of the rational function f (z) = z + 1 z to a suitably chosen matrix.…”

“…Sylvester equations have numerous applications in control theory, signal processing, filtering, model reduction, image restoration, decoupling techniques for ordinary and partial differential equations, implementation of implicit numerical methods for ordinary differential equations, and block-diagonalization of matrices; see, e.g., [14,16,21,22,18,27,44] to name only a few references. In some applications, in particular in model reduction using crossGramians [2,23], image restoration [14], and observer design [17], the right-hand side of (2) is given in factored form, C = F G, where F ∈ R n×p , G ∈ R p×m . If p ≪ n, m then it can often be observed that the solution also presents a low (numerical) rank [30].…”