Discretized Lyapunov-Krasovskii Functional for Systems With Multiple Delay Channels

Many practical systems have a large number of state variables but only a few components have time delays. These delay components are often scalar or low dimensional, and involve single time delay in each component. A coupled differential-difference equation is well suited to formulate such systems. It is known that such a formulation is very general. Systems with multiple related or independent delays can be transformed into this standard form. Similar to regular time-delay systems, the existence of a quadratic Lyapunov-Krasovkii functional is necessary and sufficient for stability. This article discusses the discretization of such a quadratic Lyapunov-Krasovskii functional. Even for time-delay systems of retarded type, the formulation has significant advantage over the traditional formulation, as the size of the resulting linear matrix inequalities are drastically reduced for such systems. Indeed, the computational effort needed for checking stability of such a large system with a few low dimensional delays is quite reasonable.