Dominant Subspaces of High-Fidelity Nonlinear Structured Parametric Dynamical Systems and Model Reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks

1 Model order reduction: basic concepts and notation

This is the first chapter of a three-volume series dedicated to the theory and ap plication of Model Order Reduction (MOR) .We motivate and introduce the basic con- cepts and notation, with reference to the two main cultural approaches to MOR: the system-theoretic approach employing state-space models and transfer functionc con- cepts (Volume1), and the numerical analysis approach as applied to partial differen- tial operators (Volume2) ,for which project ion and approximation in suitable function spaces provide a rich set of tools for MOR.The set two approaches are complementary but share.Despite the sometimes different opted language and notation,they also share the main ideas andkey concepts, which are briefly summarized in this chapter. The material is presented so that all chapters in this three-volume series are put into context, by high lighting the specific problems that they address. An overview of all MOR applications in volume 3 is also provided