$\mathcal{H}_2$-optimal Model Reduction of Linear Quadratic Output Systems in Finite Frequency Range

TL;DR

This work extends frequency-limited model order reduction to linear-quadratic-output (LQO) systems by defining the frequency-limited $\mathcal{H}_{2,\omega}$ norm and deriving local optimality conditions for the reduced-order model. It shows that exact satisfaction of all conditions via Petrov-Galerkin projection is generally impossible in the frequency-limited setting, and proposes the Frequency-limited $\mathcal{H}_2$ Near-optimal Iterative Algorithm (FLHNOIA), a projection-based method that approximately fulfills the optimality requirements while avoiding costly Gramian computations and direct matrix logarithm evaluations. A practical, bandpass-filter based approach approximates the matrix logarithm $F_\omega$, enabling scalable computation through sparse-dense Sylvester equations. Numerical experiments on small, medium, and very large systems (including a 1,000,000-state model) demonstrate that FLHNOIA achieves superior accuracy within the frequency band $[\omega_1,\omega_2]$ compared to FLBT and HOMORA, while maintaining computational efficiency. These results offer a scalable, high-fidelity MOR tool for applications requiring accurate dynamics in a finite frequency range, such as notch filtering, stability analysis, and large-scale structural models.

Abstract

In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Using the optimality conditions and the Petrov-Galerkin projection, a stationary point iteration algorithm is proposed, which approximately satisfies these optimality conditions upon convergence. The main computational effort in the proposed algorithm involves solving sparse-dense Sylvester equations. These equations are frequently encountered in $\mathcal{H}_2$ model order reduction algorithms and can be solved efficiently. Moreover, the algorithm bypasses the requirement of matrix logarithm computation, which is typically necessary for most frequency-limited reduction methods and can be computationally demanding for high-order systems. An illustrative example is provided to numerically validate the developed theory. The proposed algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency range is demonstrated through the reduction of an advection-diffusion equation-based model, commonly used in model reduction literature for testing algorithms. Additionally, the algorithm's computational efficiency is highlighted by successfully reducing a flexible space structure model of order one million.

Figures (6)

• Figure 1: Relative Error Comparison within $[5,6]$ rad/sec
• Figure 2: Decay in Singular values of $\hat{P}_v$ and Relative Error $\frac{|\hat{F}_\omega-F_\omega|_2}{|F_\omega|_2}$
• Figure 3: Relative Error $\frac{||G-G_k||_{\mathcal{H}_{2,\omega}}}{||G||_{\mathcal{H}_{2,\omega}}}$
• Figure 4: Relative Error $\frac{|G_1(j\nu)-G_{k,1}(j\nu)|}{|G_1(j\nu)|}$ Comparison within $[10,12]$ rad/sec
• Figure 5: Relative Error $\frac{|G_2(j\nu)-G_{k,2}(j\nu)|}{|G_2(j\nu)|}$ Comparison within $[10,12]$ rad/sec

Theorems & Definitions (10)

• Definition 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Remark 1