The Stability Analysis of Linear Dynamical Systems with Time-Delays

Time-delay systems, which are also sometimes known as hereditary systems or systems with memory, aftereffects or time-lag, represent a class of infinite-dimensional systems, and are used to describe, among other types of systems, propagation and transport phenomena, population dynamics, economic systems, communication net- works and neural network models. A key method for the stability analysis of time- delay dynamical systems is the Lyapunov’s second method, applied to functional differential equations. Specifically, stability of a given linear time-delay dynamical system is typically shown using a Lyapunov-Krasovskii functional, which involves a quadratic part and an integral part. The quadratic part is usually associated with the stability of the forward delay-independent part of the retarded dynamical system, but the integral part of the functional is less understood. We present a concrete method of arriving at the Lyapunov-Krasovskii functional using dissipativity theory. The stability analysis of time-delay systems has been mainly classified into two categories: delay-dependent and delay-independent analysis. Delay-independent stability criteria provide sufficient conditions for stability of time-delay dynamical systems in- dependent of the amount of delay, whereas delay-dependent stability criteria provide sufficient conditions that are dependent on an upper bound of the delay. In systems where the time delay is known to be bounded, delay-dependent criteria usually give far less conservative stability predictions as compared to delay-independent results. Hence, for such systems it is of paramount importance to derive the sharpest possible delay-dependent stability margins. We show how the stability criteria may also be interpreted in the frequency domain in terms of a feedback interconnection of a matrix transfer function and a phase uncertainty block. We develop and present the general framework for a robust stability analysis method to account for phase uncertainties in linear systems. We present new robust stability results for time-delay systems based on pure phase information, and then, using this approach, we derive new and improved time-domain delay-dependent stability criteria for stability analysis of both retarded and neutral type time-delay systems, which we then compare with existing results in the literature.