Before revision:  In general , the classical IRKA can be applied to the second-order system by converting it into an equivalent first-order form and get the reduced model in a first-order form. In this case, the reduced model can not preserve the structure of the second-order system which is however necessary for further manipulation. Here we develop IRKA based algorithms that enable us to generate approximate reduced second-order systems without explicitly converting the systems into a first-order form .
After revision:  In general , the classical IRKA can be applied to the second-order system by converting it into an equivalent first-order form and get the reduced model in a 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 generate approximate reduced second-order systems without explicitly converting the systems into a first-order form .