On Linear Quadratic Optimal Control of Discrete-Time Complex-Valued Linear Systems

TL;DR

This paper develops an iterative algorithm for solving the bimatrix Riccati equation in discrete-time complex-valued linear systems, enabling optimal control design with applications to systems with delays and special cases like anti-linear systems.

Contribution

It introduces a convergent iterative algorithm for the bimatrix Riccati equation and applies it to complex-valued, anti-linear, and delayed systems, expanding LQR control methods.

Findings

Algorithm converges to the unique positive definite solution.

Effective for complex-valued and anti-linear systems.

Demonstrated on a numerical example with time delay.

Abstract

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 shown to 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, 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 discrete-time time-delay system with one step state delay and a numerical example was used to illustrate the effectiveness of the proposed methods.