Projection-based model reduction for time-varying descriptor systems: New results

Abstract

We have presented a Krylov-based projection method for model reduction of linear time-varying descriptor systems in [13] which was based on earlier ideas in the work of J. Philips [17] and others. This contribution continues that work by presenting more details of linear time-varying descriptor systems and new results coming from real fields of application. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential-algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macro-models, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate both the performance and accuracy of the reduced-order model.

Stabilization of linear time‐varying reduced‐order models: A feedback controller approach

“…In this example, the reduced system matrices of k = 4 are constructed at each time step using the two‐sided Lanczos moment matching algorithm 63 . More compact representation of the projection can be built considering that the projection matrices span over the union of all the projection subspaces 32 ; however, its application is beyond the scope of this article.…”

“…Many of the reduction methods can be extended to the reduction of time‐varying systems, for a review see Reference 32 and the references therein. Balanced truncation methods, with a priori stability guarantee, are generalized to linear time‐varying systems 33‐36 .…”

Comparative analysis of model reduction strategies for circuit based periodic control problems

“…To ease the computational complexity, we can convert the linear time‐varying system () into its equivalent LTI form. This reformulation also allows us to extend theory dedicated to time‐invariant systems to the LTV setting [4–6]. It is important to note that this type of LTI conversion is possible for a system which has near linear response but is nonlinear in nature (sometimse called weakly nonlinear).…”

“…It is important to note that this type of LTI conversion is possible for a system which has near linear response but is nonlinear in nature (sometimse called weakly nonlinear). That means systems have small periodic perturbation [5–7].…”

“…Model reduction of () has been the topic of intensive research in recent years. Attempts have been made to reduce the dimensions of large‐scale systems using balanced truncation [8–12], Krylov‐based projection methods [1, 5, 13–16], the interpolatory method, optimal Hankel norm method, and polynomial method, as seen in [17]. Model reduction algorithms linking both balanced truncation and Krylov‐based methods have also been proposed in [2, 18–21].…”