A Finite Spectrum Assignment Procedure for the Single-Input Linear Systems with Commensurate Time-Delays

Abstract

This paper presents a finite spectrum assignment (f.s.a.) procedure for the single-input linear systems with commensurate time-delays. For this purpose, an algorithm is given to construct the feedback for f.s.a. and it is shown that the algorithm is availlable under the spectral controllability of the system. Therefore, it turns out that the system is finite spectrally assignable if and only if it is spectrally controllable. The result is consistent with the existing one.
The algorithm consists of the following three steps.
(1) Construct a feedback formally by a state feedback with the future states.
(2) Construct a predictor for the future states.
(3) Realize the feedback for f.s.a. by the substitution of the predictor to that constructed in step (1).
In paticular, for step (2) a sufficient condition to predict the future states is given and an algorithm to construct the predictor is also presented. Finally, it is proven that the spectrum of the closed-loop system is in fact assigned to be a finite set by using the resulting feedback.