Unified initial condition response analysis of Lur'e systems and linear time-invariant systems

Abstract

A unified framework to the initial condition response (ICR) analysis of non-linear time-varying systems of the Lur'e type and linear time-invariant (LTI) systems is considered. To quantify the transient behaviour resulting from initial conditions, an ICR measure is defined. An appropriate upper bound for the ICR measure can be calculated based upon the condition number of a positive definite matrix, associated with a quadratic Lyapunov function. Owing to the particular structure of the Lur'e systems, bounding the ICR measure is transformed into a minimization problem, constrained by either two simultaneous Lyapunov matrix inequalities or a single algebraic Riccati inequality. In the LTI case, the ICR measure is merely the supremum of the induced norm of the state transition matrix and its calculation involves solving a Lyapunov matrix equation. The ICR measure bound is significant from the engineering standpoint since it enables the calculation of the non-saturating domain of the state variables. In the LTI case, this bound also provides a quantification of impulse and unit step responses. The results are illustrated by several examples.