Full delayed state feedback pole assignment of discrete-time time-delay systems

This paper studies the problem of pole assignment of discrete-time time delay system with delayed state feedback. The problem is solved in this paper by requiring that the maximal delay in the feedback equals the maximal delay of the open-loop system. A necessary and sufficient condition guaranteeing the existence of a solution is presented. By using the augmentation technique, the pole assignment problem is then transformed to the problem of solving a linear matrix equation such that certain conditions are satisfied. To solve the linear equation problem, when the desired closed-loop eigenvalues are not prescribed, a parametric approach using real arithmetic is presented by using polynomial matrices associated with the system matrices. When the desired closed-loop eigenvalues are prescribed, singular value decomposition can be adopted to solve the linear matrix equation. Both approaches can provide full degree of freedom, which can be further utilized to accomplish some other design objects. The robust pole assignment problem is considered to demonstrate the advantages of the method. Numerical examples are employed to illustrate the effectiveness of the proposed approaches.