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

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Abstract: This paper focuses on exploring efficient ways to find $mathcal{H} 2$optimal Structure-Preserving Model Order Reduction (SPMOR) of the second-ordersystems via interpolatory projection-based method Iterative Rational KrylovAlgorithm (IRKA). To get the reduced models of the second-order systems, theclassical IRKA deals with the equivalent first-order converted forms andestimates the first-order reduced models. The drawbacks of that of thetechnique are failure of structure preservation and abolishing the propertiesof the original models, which are the key factors for some of the physicalapplications. To surpass those issues, we introduce IRKA based techniques thatenable us to approximate the second-order systems through the reduced modelsimplicitly without forming the first-order forms. On the other hand, there arevery challenging tasks to the Model Order Reduction (MOR) of the large-scalesecond-order systems with the optimal $mathcal{H} 2$ error norm and attain therapid rate of convergence. For the convenient computations, we discusscompetent techniques to determine the optimal $mathcal{H} 2$ error normsefficiently for the second-order systems. The applicability and efficiency ofthe proposed techniques are validated by applying them to some large-scalesystems extracted form engineering applications. The computations are donenumerically using MATLAB simulation and the achieved results are discussed inboth tabular and graphical approaches.