Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems

Abstract

Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of large-scale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bi-linearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reduced-order bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the Volterra-Wiener representation of the bilinear system. It is shown that the two-sided Krylov subspace technique matches significant more number of multimoments than the corresponding one-side technique.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…, N, up to a maximal order N forŝ ∈ C not a pole of H. This yields a reduced model whose transfer function H r (s) coincides in as many coefficients of its Taylor expansion (also called "moments") aboutŝ as possible for a given order of the reduced model. See, e.g., [22,101] for a review of this approach and its close connection to the (nonsymmetric) Lanczos process. The caseŝ = 0 is generally referred to as momentmatching, while forŝ = 0 we obtain shifted moments, andŝ = ∞ leads to matching of the Markov parameters of the full system.…”

Structure-Preserving Model Reduction

“…Suppose the matrices L, G, C, B in the transfer function (1.1) have some natural partitioning that is derived from, e.g., the physical layout of a VLSI circuit or a structural dynamical system: 2) where…”

“…Recent survey articles [1,2,7] provide in depth review of the subject and comprehensive references. Roughly speaking, these methods project the original system onto a smaller subspace to arrive at a (much) smaller system having properties, among others, that many leading terms (called moments) of the associated (matrix-valued) transfer functions expanded at given points for the original and reduced systems match.…”

“…Here G, C ∈ C N ×N , B ∈ C N ×m , L ∈ C N ×p . In today's applications of interests, such as VLSI circuit designs and structural dynamics, N can be up to millions [1,2,7]. Computations of H(s) usually have to be done through some kind of reduction on L, G, C and B.…”

Model reduction in large scale MIMO dynamical systems via the block Lanczos method

“…When dealing with high-order models, it is reasonable to look for an approximate stable model ẋ m (t) = A m x m (t) + B m u(t) y m (t) = C m x m (t), (1.2) in which A m ∈ R m×m , B m , C T m ∈ R m×s and x m (t), y m (t) ∈ R m , with m n. Hence, the reduction problem consists in approximating the triplet {A, B, C} by another one {Â,B,Ĉ} of small size. Several approaches in this area have been used as Padé approximation [15,33,34], balanced truncation [29,37], optimal Hankel norm [16,17] and Krylov subspace methods [3,6,11,12,21,22]. These approaches require the solution of coupled Lyapunov matrix equations [1,13,25,27] having the form A P + P A T + B B T = 0 A T Q + Q A + C T C = 0, (1.3) where P, Q are the controllability and the observability Grammians of the system (1.1).…”