Interpolatory Projection Techniques for $\mathcal{H}_2$ Optimal Structure-Preserving Model Order Reduction of Second-Order Systems

TL;DR

This paper introduces interpolatory projection techniques based on IRKA that efficiently achieve $ ext{H}_2$ optimal structure-preserving model reduction for second-order systems without converting to first-order forms, maintaining physical properties.

Contribution

The paper proposes IRKA-based methods that implicitly approximate second-order systems directly, preserving structure and properties, and introduces efficient techniques for computing optimal $ ext{H}_2$ error norms.

Findings

Successfully applied to large-scale engineering systems

Achieved rapid convergence and high accuracy in $ ext{H}_2$ norm

Validated effectiveness through numerical MATLAB simulations

Abstract

This paper focuses on exploring efficient ways to find $\mathcal{H}_2$ optimal Structure-Preserving Model Order Reduction (SPMOR) of the second-order systems via interpolatory projection-based method Iterative Rational Krylov Algorithm (IRKA). To get the reduced models of the second-order systems, the classical IRKA deals with the equivalent first-order converted forms and estimates the first-order reduced models. The drawbacks of that of the technique are failure of structure preservation and abolishing the properties of the original models, which are the key factors for some of the physical applications. To surpass those issues, we introduce IRKA based techniques that enable us to approximate the second-order systems through the reduced models implicitly without forming the first-order forms. On the other hand, there are very challenging tasks to the Model Order Reduction (MOR) of the large-scale second-order systems with the optimal $\mathcal{H}_2$ error norm and attain the rapid rate of convergence. For the convenient computations, we discuss competent techniques to determine the optimal $\mathcal{H}_2$ error norms efficiently for the second-order systems. The applicability and efficiency of the proposed techniques are validated by applying them to some large-scale systems extracted form engineering applications. The computations are done numerically using MATLAB simulation and the achieved results are discussed in both tabular and graphical approaches.