Positive realness of a set of matrix-valued time-varying uncertainties

The manuscript is concerned with characterizations of a set of matrix-valued, time-varying, uncertain multiplication operators. The set of uncertain multiplication operators arises from robust stability analysis of systems with aperiodic sampling-and-hold devices, where the sampling periods are assumed to be bounded within a given range. Conditions under which the uncertain operator is positive real are identified. Based on the positive real property, a number of quadratic constraints are derived for the uncertain operator. Such quadratic constraints are proven to be useful in analyzing robustness of aperiodic sampled-data systems against variation of sampling intervals.