Robust Stability Analysis against Perturbations of Smoothly Time-Varying Parameters

Abstract

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 regionof-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.

An IQC approach to robust estimation against perturbations of smoothly time-varying parameters

“…For a precise definition see [13]. The preceding concept of a timevarying operator and its rate-of-variation motivates to introduce the concept of the so called region-of-variation (ROV) R := {(∆(t), V ∆ (t)) , t ∈ [0, ∞)} where we assume R to be compact and star-shaped with star-center zero, i.e.…”

“…Since we are concerned with LTV perturbations we employ the IQC-test from [13]. Let us therefore consider the following variant of the swapping lemma:…”

“…One could think, e.g., of uncertain time-delays, multiple static nonlinearities or uncertain LTI dynamics. Our special attention is dedicated to IQC tests for smoothly time-varying parametric uncertainties with bounded rates-of-variation, as was shown in [11], [12], and as was recently extended in [13] and [14]. Until very recently, the dynamic IQC-framework was mostly suitable for analysis purposes only.…”

“…The first goal of this paper is to merge the new results of [15] with the special IQC test for uncertain smoothly time-varying parameters from [13] and to show that we can systematically reduce conservatism with respect to the PDLF based approaches for a broader class of systems. The second goal is to show that the estimator synthesis problem from [15] can be further generalized to the use of a weighting filter at the output.…”

“…After introducing some preliminaries in Section II, we formally state the problem in Section III. Then in Section IV, we give a short recap of IQC theory and describe the special IQC test for smoothly time-varying parameter uncertainties as it was presented in [13]. This is then merged with an IQC test for H 2 -performance.…”