Stabilization of linear systems of delay differential equations by the delayed feedback method

Abstract

This paper consists of two folds. At first, we deal with the stability analysis of a linear system of delay differential equations. It is shown that the direct and cluster treatment methods are not applicable if there are some purely imaginary roots of the characteristic equation with multiplicity greater than one. To overcome the above difficulty, the system is decomposed into several subsystems. For the decomposition of a system, an invertible transformation is required to convert the matrices of the system into a block triangular (diagonal) form simultaneously. To achieve this goal, a necessary and sufficient condition is established. The second part concerns the stabilization of a linear system of delay differential equations using the delayed feedback method and design a controller for generating the desired response. More precisely, the unstable poles of the linear system of delay differential equations are moved to the left-half of the complex plane by the delayed feedback method. It is shown that the performance of the linear system of delay differential equations can be improved by applying the delayed feedback method.