Robust Stability Analysis against Perturbations of Smoothly Time-Varying Parameters

Robust stability analysis problem is considered for linear time-invariant systems subject to perturbations of parameters that vary smoothly in time. The problem is formulated by describing the uncertainty in the form of a region-of-variation that covers all possible trajectories obtained by plotting the variation of the parameter at a certain time versus the parameter itself at that same instant of time. In this fashion, the possible and very probable correlation between the parameter and its variation is taken into account. The problem is elaborated on via an integral quadratic constraints approach and, based on a main result relying on the swapping lemma, robust stability analysis tests are derived for polytopic regions of variation as well as for regions with generic descriptions expressed in terms of matrix inequalities. The tests for polytopic regions-of-variation are based on the convex hull and Pólya relaxation approaches, whereas the test for general regions relies on the sum-of-squares relaxation approach. The new tests offer significant reduction in conservatism at the cost of extra computational complexity.