Model Order Reduction: Techniques and Tools

Model order reduction (MOR) is here understood as a computational technique to reduce the order of a dynamical system described by a set of ordinary or differential-algebraic equations (ODEs or DAEs) to facilitate or enable its simulation, the design of a controller, or optimization and design of the physical system modeled. It focuses on representing the map from inputs into the system to its outputs, while its dynamics are treated as a blackbox so that the large-scale set of describing ODEs/DAEs can be replaced by a much smaller set of ODEs/DAEs without sacrificing the accuracy of the input-to-output behavior. 1 Problem description This survey is concerned with linear timeinvariant (LTI) systems in state-space form Eẋ(t) = Ax(t)+Bu(t), y(t) =Cx(t)+Du(t), (1) where E,A∈Rn×n are the system matrices, B∈Rn×m is the input matrix, C ∈Rp×n is the output matrix, and D ∈ Rp×m is the feedthrough (or input-output) matrix. The size n of the matrix A is often referred to as the order of the LTI system. It mainly determines the amount of time needed to simulate the LTI system. Such LTI systems often arise from a finite element modeling using commerical software such as ANSYS or NASTRAN which results in a second-order differential equation of the form Mẍ(t)+Dẋ(t)+Kx(t)=Fu(t),y(t)=Cpx(t)+Cvẋ(t), where the mass matrix M, the stiffness matrix K and the damping matrix D are square matrices in Rs×s, F ∈Rs×m, Cp,Cv ∈Rq×s, x(t)∈Rs, u(t)∈Rm, y(t)∈ Rq. Such second-order differential equations are typically transformed to a mathematically equivalent firstorder differential equation [ I 0 0 M ] } {{ } E [ ẋ(t) ẍ(t) ] } {{ } ż(t) = [ 0 I −K −D ] } {{ } A [ x(t) ẋ(t) ] } {{ }