Passivity-preserving model reduction with finite frequency H∞H∞ approximation performance

This paper is concerned with model reduction for passive systems. For a given linear time-invariant system that is stable and positive real (PR), our goal is to find a PR reduced-order model to approximate it, and our attention is focused on reducing the error with respect to a finite frequency  H∞H∞ performance, which is the most remarkable difference between the proposed approach and the existing ones. First, by applying multiplier expansion, new conditions in terms of linear matrix inequalities are derived for characterizing the positive realness of the reduced-order model and the finite frequency H∞H∞ performance of the error system. A necessary and sufficient   condition is then established for parameterizing a PR reduced-order model with finite frequency H∞H∞ approximation performance, based on which, an iterative algorithm is constructed for numerically exploring such a reduced-order model. Particularly, a partial multiplier expansion treatment is introduced, which greatly reduces the decision variables but does not cause conservatism to the derived conditions. The proposed method is also extended to robust passivity-reserving model reduction with polytopic uncertainty. Finally, we provide two numerical examples about RLC circuits to show the effectiveness and advantages of the proposed model reduction method.