Eigenvalue Assignment for Control of Time-Delay System

Abstract

Delay is the property by which the response to an input is delayed in its effect. It is an inherent phenomenon in many control engineering practices. Mathematical description of a system generally assumes that the behavior of considered process depends only on present state. This assumption is not satisfied in the cases where considerable delay is present which has to be accounted. Thus, information about former states is included in the mathematical description of the system. Such systems are called time-delay systems. Presence of delay complicates system analysis. They degrade system performance and may make the system unstable. So, controlling time-delay systems has gained much importance, and many control methodologies have been developed over the years. A newly designed method is presented for control of proportional-integral (PI) controllers of first-order plants in presence of time delay. Here, the solutions are based on delay differential equations, which are derived in terms of the Lambert W function. PI controllers for first-order plants with time delays are designed by obtaining the rightmost eigenvalues in the infinite eigen spectrum of time-delay systems and assigning them to desired positions in the complex plane. The process is possible due to a novel property of the Lambert W function. The controllers designed using the presented method can improve the system performance and successfully stabilize an unstable plant.