Type II Singular Perturbation Approximation for Linear Systems with Lévy Noise

Abstract

To solve a stochastic linear evolution equation numerically, nite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations, we consider the singular perturbation approximation as a model order reduction technique in this paper. This approach is well-known from deterministic control theory and here we generalize it for controlled linear systems with Lévy noise. Additionally, we discuss properties of the reduced order model, provide an error bound, and give some examples to demonstrate the quality of this model order reduction technique.

Energy estimates and model order reduction for stochastic bilinear systems

“…There, a different reachability Gramian was considered which is defined as the solution to a matrix inequality. Energy estimates for linear stochastic systems for both the type I and the type II ansatz have recently been given in [32], such that MOR based on both approaches can be justified. As an alternative to BT, we want to refer to the singular perturbation approximation, where the work in [14,25] was extended to stochastic linear systems in [32,33].…”

“…An inequality is considered in (2.1), since the existence of a positive definite solution is not ensured when having an equality. The existence of a solution to (2.1) goes back to [12,32] and is given if…”

Singular perturbation approximation for linear systems with Lévy noise

“…In this section, we investigate the output error between two ROMs, in which the larger ROM has exactly one HSV than the smaller one. This concept of neighboring ROMs was first introduced in [31] but in the much simpler stochastic linear setting.…”

“…With this first extension, however, no L 2 -error bound can be achieved [6,12]. Therefore, an alternative approach based on a different reachability Gramian was studied for stochastic linear systems leading to an L 2 -error bound for BT [12] and for SPA [31].BT [1,5] and SPA [18] were also generalized to bilinear systems, which we refer to as the standard approach for these systems. Although bilinear terms are very weak nonlinearities, they can be seen as a bridge between linear and nonlinear systems.…”

“…where the existence of a positive definite solution to (1.3) goes back to [12,31]. We approximate the large scale system (1.1) by a system which has a much smaller state dimension r ≪ n. This reduced order model (ROM) is supposed be chosen, such that the corresponding output y r is close to the original one, i.e., y r ≈ y in some metric.…”