State-feedback and filtering problems using the generalized KYP lemma

The Kalman-Yakubovich-Popov (KYP) lemma is a useful tool in control theory, since it relates frequency-domain inequalities (FDIs) to linear matrix inequalities (LMIs). In the last decade, the standard KYP lemma was extended to cope with finite frequency intervals, namely low, middle and high frequencies. This extension is known as the generalized KYP lemma (gKYP). In this paper, a necessary and sufficient condition to assess middle-frequency specifications with an additional multiplier is proposed. This result is then used to address three problems in control theory: state-feedback control, observer-based estimation, and filter design. The drawback of the technique is that the extra multiplier must be complex, yielding complex matrices for the state feedback and observer-based estimation gains, as well as for the matrices of the filter realization. By imposing the multiplier to be real, sufficient design conditions are obtained, whose conservativeness is analyzed through numerical examples.