Robustness analysis of uncertain time‐varying systems with unknown initial conditions

Abstract

SummaryThis paper focuses on the robustness analysis of discrete‐time, linear time‐varying (LTV) systems subject to various uncertainties, such as static and dynamic, time‐invariant and time‐varying, linear perturbations, and unknown initial conditions. The proposed approach is based on integral quadratic constraint theory and allows for a potentially more accurate characterization of the set in which the initial state resides by imposing separate constraints on the initial values of the state variables as opposed to simply requiring the initial state to lie in some ellipsoid. The adopted problem formulation facilitates the analysis of uncertain LTV systems subject to disturbance inputs that are bounded pointwise in time, and the developed results enable determining useful pointwise bounds on the performance outputs given such inputs. The main analysis result is given for eventually periodic nominal systems, which include linear time‐invariant, finite horizon, and periodic systems as special cases. The analysis conditions are expressed as linear matrix inequalities. Two additional results stemming from the main analysis theorem are provided that can be used to determine overapproximated ellipsoidal reachable sets. Finally, the utility of the proposed approach is demonstrated in an illustrative example.