Optimal pole assignment for discrete-time systems via Stein equations

This study is concerned with designing a feedback gain to minimise a quadratic performance index with guaranteed pole locations for closed-loop discrete-time linear systems. Firstly, a method that shifts the open-loop poles to desired locations by using a parametric linear Stein equation is presented. Then a recursive approach is proposed to shift every eigenvalue of a discrete-time linear system separately without mode decomposition in each step. By using such method, it is required to solve a linear Stein matrix equation of low order in each step. The presented method yields a solution which is optimal with respect to a quadratic performance index that can be obtained explicitly. The attractive feature of this method comparing with existing results is that it enables solutions to complex problems to be easily found without solving any non-linear algebraic Riccati equations. Moreover, analytical solutions can be obtained which may have advantages in some design problems. Numerical examples illustrate the proposed approach.