${{H}_{\infty}}$ Model Reduction for Interval Frequency Negative Imaginary Systems

This paper studies an $H_{\infty }$ model reduction problem for interval frequency negative imaginary (IFNI) systems. For a given IFNI system, our goal is to find a reduced-order IFNI system satisfying a pre-specified $H_{\infty }$ approximation error bound over the finite-frequency interval. Necessary and sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an $H_{\infty }$ reduced-order IFNI system. An improved iterative algorithm is provided to solve the matrix inequalities and to minimize the $H_{\infty }$ approximation error. The proposed method is further clarified via the application to the electrical circuits, such as high-order Sallen–Key low-pass filter, piezoelectric tube scanner, and RLC circuit. The simulation results on these electrical circuits are compared with the finite-frequency interval Gramians-based model reduction method both in the frequency domain and time domain.