Frequency domain conditions via Linear Matrix Inequalities

This paper revisits a pair of linear matrix inequalities (LMIs) that are related to checking a frequency domain inequality (FDI) over a finite interval. The first contribution is to show that the proposed pair of LMIs contain the original formulation of the Kalman-Yakubovich-Popov Lemma when the coefficient matrix is constant. The coefficient matrix can be made affine on the frequency variable at no extra computational cost. The second contribution is to show how to transform the frequency variable in order to extend the proposed results to infinite frequency intervals. In applications such as robustness analysis, allowing for frequency dependent coefficient matrices can be significant in reducing conservatism, a feature which is illustrated with a simple numerical example.