MATHEMATICAL ENGINEERING TECHNICAL REPORTS Characterizations of Finite Frequency Properties Using Quadratic Differential Forms

Many of practical design specifications are provided by finite frequency properties described by inequalities over restricted finite frequency intervals. A quadratic differential form (QDF) is a useful algebraic tool when we consider dissipation theory based on the behavioral approach. In this paper, we investigate time domain characterizations of the finite frequency domain inequalities (FFDIs) using QDFs. Based on QDFs, we derive a characterization of the FFDIs using quadratic differential forms as a main result. This condition leads to a physical interpretation in terms of the compensating rate, which guarantees dissipativity of some behavior with some rate constraints. Such interpretation has not been clarified by the previous studies of finite frequency properties. The aforementioned characterization yields an LMI condition whose solvability is equivalent to the FFDIs. This can be regarded as the finite frequency KYP lemma in the behavioral framework.