Sparse solution of the Lyapunov equation for large-scale interconnected systems

TL;DR

This paper presents efficient methods for approximating solutions to large-scale Lyapunov equations in interconnected systems by exploiting the decay properties of the solution, enabling scalable computation.

Contribution

It introduces two new methods that leverage the decay of Lyapunov solutions for large, sparse, and well-conditioned matrices, improving computational efficiency.

Findings

Methods scale linearly with system size for well-conditioned matrices.

Numerical experiments confirm the efficiency and accuracy of the methods.

Decay properties enable sparse approximations of large Lyapunov solutions.

Abstract

We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation $\underline{A}\underline{X}+\underline{X}\underline{A}^{T}=\underline{P}$, where the coefficient matrices $\underline{A}$ and $\underline{P}$ are large, symmetric banded matrices. The (sparsity) pattern of $\underline{A}$ describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution $\underline{X}$ are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of $\underline{X}$ is faster if the condition number of $\underline{A}$ is smaller. By exploiting the decay of entries of $\underline{X}$, we develop two computationally efficient methods for approximating $\underline{X}$ by a banded matrix. For a well-conditioned and sparse banded $\underline{A}$, the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of $\underline{A}$ and $\underline{P}$ that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.