The Boundary of the Discrete LQ Optimal Closed-loop Poles

In this paper we determine the region in which the closed-loop eigenvalues of an LQ-optimal discrete time system are located. It has been shown that this region is bounded by four lines parallel to the real and imaginary axes, creating a square region. This region is symmetric with respect to the real axis of z-plane, but not with respect to the imaginary axis. Inequality constraints, which bound the closed-loop eigenvalues in a discrete-time LQ-optimal system, are also presented. Two numerical examples are presented which demonstrate the workings of the proposed analysis.