Finite spectrum assignment problem for systems with delays

In this paper linear systems with delays in state and/or control variables are considered. The objective is to design a feedback law which yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned set of points in the complex plane. It is shown that in case of systems with delays in control only the problem is solvable if and only if some function space controllability criterion is met. The solution is then easily obtainable by standard spectrum assignment methods, while the resulting feedback law involves integrals over the past control. In case of delays in state variables it is shown that a technique based on the finite Laplace transform, related to a recent work on function space controllability, leads to a constructive design procedure. The resulting feedback consists of proportional and (finite interval) integral terms over present and past values of state variables. Some indications on how to combine these results in case of systems including both state and control delays are given. Sensitivity of the design to parameter variations is briefly analyzed.