Feedback stabilization of linear autonomous time lag systems

Abstract

A stabilization theory which employs well-established finite-dimensional control system tools is developed for the stabilization of linear autonomous time lag systems. The main ideas include 1) a set whose elements are matrices each of which is a left zero of the system characteristic quasi-polynomial matrix, and 2) a linear transformation which reduces the delay system to a delay-free system whose state matrix is a direct sum of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> elements of the matrix set where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is some positive integer. From the definition of this matrix set, it is shown that each of its elements inherits its spectrum from that of the delay system so that by design, the system unstable poles may be embedded in the spectrum of the delay-free system. Under the assumption of spectral stabilizability, it is then shown how to obtain a stabilizing feedback control law on the basis of the delay-free system. Numerical examples are presented to confirm the theory.