Frequency-Limited Balanced Truncation with Low-Rank Approximations

Abstract

In this article we investigate model order reduction of large-scale systems using frequency-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerical realization of this model reduction approach. We discuss numerical methods to take care of the involved matrix-valued functions. The occurring large-scale Lyapunov equations are solved for low-rank approximations for which we also establish results regarding the eigenvalues of their solutions. These results, and also numerical experiments, will show that the eigenvalues of the Lyapunov solutions in frequency-limited balanced truncation often decay faster than those in standard balanced truncation. Moreover, we show in further numerical examples that frequency-limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region. Algorithm 1.1. Square-root balanced truncation (low-rank version).Input : System matrices A, B, C defining an asymptotically stable dynamical system (1.1). Output: MatricesÃ,B,C of the reduced system. 1 Compute low-rank solution factors Z P , Z Q of the solutions of (1.2), such that P ≈ Z P Z T P , Q ≈ Z Q Z T Q . 2 Compute and partition a (thin) singular value decompositionwhere Σ 1 = diag(σ 1 , . . . , σ r ) contains the largest r (approximate) Hankel singular values.1 . 4 Generate reduced order model 1.2. Goals and overview of this article. By looking at the error bound (1.4), the above BT framework generates reduced order models that are accurate for all values ω ∈ R, which, from an application oriented view, are typically considered as Downloaded 03/03/16 to 193.175.53.21. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.FLBT WITH LOW-RANK APPROXIMATIONS A473 frequencies. In several applications, however, the underlying physical or technical system operates only in a small frequency interval [ω 1 , ω 2 ] of interest. Restricting the BT procedure to this frequency interval has lead to frequency-limited balanced truncation (FLBT), which was proposed in [19]. One motivation for FLBT is that, compared to ordinary BT, by restricting to a small interval [ω 1 , ω 2 ], we hope to obtain higher accuracies with reduced order models of the same dimension, or to achieve a comparable accuracy with smaller reduced order models inside the interval, while allowing for larger errors outside.The main purpose of this article is to provide a numerically efficient framework for carrying out FLBT for high-dimensional systems. We start by reviewing the concept of frequency-limited Gramians and show how to formulate a procedure similar to the square root approach in Algorithm 1.1. In section 3, we investigate the eigenvalue decay of the frequency-limited Gramians. For the occurring CALEs, we will, as in the unlimited case, employ low-rank approximations of the solu...

Computation of controllability and observability Gramians in modeling of discrete‐time noncommensurate fractional‐order systems

“…For this reason, research works which aim to obtain a balanced system in a numerically efficient way have still been carried out (see e.g. [44][45][46][47][48][49][50][51][52]). The preliminary results of approximation of a fractional-order system by use of the expanded state space model and the BT method have been presented in [33].…”

“…One way to lower the computational cost is to employ algorithms that compute approximate solutions of controllability and observability Gramians [45][46][47]…”

Frequency-limited pseudo-optimal rational Krylov algorithm for power system reduction

“…FLBT [25] becomes computationally expensive in a largescale setting because it requires the solution of two large-scale Lyapunov equations. In [34], FLBT is generalized using a low-rank approximation of Lyapunov equations to extend its applicability to large-scale systems. FLPORK and O-FLPORK do not involve large-scale Lyapunov equations like FLBT [25].…”

“…FLPORK and O-FLPORK do not involve large-scale Lyapunov equations like FLBT [25]. However, as argued in [34], F (A) may become computationally expensive in a large-scale setting. In [34], various computationally efficient approaches to compute F (A) are discussed which also includes quadrature rules for numerical integration.…”

“…However, as argued in [34], F (A) may become computationally expensive in a large-scale setting. In [34], various computationally efficient approaches to compute F (A) are discussed which also includes quadrature rules for numerical integration. The computational cost in these methods directly depends on the number of quadrature nodes.…”