Solutions to Linear Bimatrix Equations with Applications to Pole Assignment of Complex-Valued Linear Systems

TL;DR

This paper provides explicit solutions to various linear bimatrix equations relevant to pole assignment and stability in complex-valued linear systems, with applications demonstrated on second-order systems like spacecraft rendezvous.

Contribution

It introduces complete solutions to generalized Sylvester, Stein, and Lyapunov bimatrix equations and applies these to pole assignment and stability problems in complex-valued systems.

Findings

Explicit solutions to bimatrix equations are derived.

Solutions enable pole assignment in complex-valued systems.

Applications include spacecraft rendezvous control.

Abstract

We study in this paper solutions to several kinds of linear bimatrix equations arising from pole assignment and stability analysis of complex-valued linear systems, which have several potential applications in control theory, particularly, can be used to model second-order linear systems in a very dense manner. These linear bimatrix equations include generalized Sylvester bimatrix equations, Sylvester bimatrix equations, Stein bimatrix equations, and Lyapunov bimatrix equations. Complete and explicit solutions are provided in terms of the bimatrices that are coefficients of the equations/systems. The obtained solutions are then used to solve the full state feedback pole assignment problem for complex-valued linear system. For a special case of complex-valued linear systems, the so-called antilinear system, the solutions are also used to solve the so-called anti-preserving (the closed-loop system is still an antilinear system) and normalization (the closed-loop system is a normal linear system) problems. Second-order linear systems, particularly, the spacecraft rendezvous control system, are used to demonstrate the obtained theoretical results.