A Modified GMRES Method for Solving a Symmetric Solution to Lyapunov Equation for Multi-Agent Systems

Abstract

We consider solving a large-scale Lyapunov equation for a multi-agent system. As is well known, the Lyapunov equation can be solved by equivalently rewriting it as a system of linear equations. The difficulties in solving this system are memory requirement and computational complexity due to the large-scale coefficient matrix involving a number of Kronecker products. This paper presents a modified GMRES method for solving the aforementioned system of linear equations taking account of its tensor structure and the symmetry of the unknown matrix in the Lyapunov equation. Through numerical experiments, the improvement in memory requirement and computational time by the present algorithms is verified in comparison with the previous GMRES-based methods.

Keywords

• lyapunov equation
• symmetric solution
• multi-agent system
• tensor
• n-mode product