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 to characterize energy and power functions 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. QDFs allow us to derive a clear characterization of the FFDIs using some inequality in terms of them as a main result. This characterization leads to a physical interpretation in terms of the dissipation inequality with the compensating rate which guarantees dissipativity of a behavior with some rate constraints. Such an 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.
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 the dissipation theory based on the behavioral approach. In this paper, we consider a time domain characterization of the finite frequency domain inequalities (FFDIs) using QDFs. We first derive an LMI condition which is equivalent to the FFDIs. Using the condition, we derive a time domain characterization of the FFDIs in terms of QDFs as a main result of this paper. This condition recovers the previous result in the sense that our result can characterize the FFDIs for the system of nonproper transfer functions.
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