On linear quadratic optimal control of discrete-time complex-valued linear systems

We study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete-time complex-valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete-time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex-valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti-Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete-time time-delay system with one-step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods.