Sparse approximate inverses of Gramians and impulse response matrices of large-scale interconnected systems

TL;DR

This paper demonstrates that inverses of well-conditioned finite-time Gramians and impulse response matrices of large-scale interconnected systems can be approximated by sparse matrices, enabling scalable distributed estimation and control.

Contribution

The paper introduces a novel method for approximating inverses of large-scale system matrices with sparse matrices, facilitating distributed control and estimation.

Findings

Sparse inverse approximations are computationally feasible for large systems.

The sparsity patterns depend mainly on the system matrices' sparsity.

Complexity of algorithms scales linearly with the number of subsystems.

Abstract

In this paper we show that inverses of well-conditioned, finite-time Gramians and impulse response matrices of large-scale interconnected systems described by sparse state-space models, can be approximated by sparse matrices. The approximation methodology established in this paper opens the door to the development of novel methods for distributed estimation, identification and control of large-scale interconnected systems. The novel estimators (controllers) compute local estimates (control actions) simply as linear combinations of inputs and outputs (states) of local subsystems. The size of these local data sets essentially depends on the condition number of the finite-time observability (controllability) Gramian. Furthermore, the developed theory shows that the sparsity patterns of the system matrices of the distributed estimators (controllers) are primarily determined by the sparsity patterns of state-space matrices of large-scale systems. The computational and memory complexity of the approximation algorithms are $O(N)$, where $N$ is the number of local subsystems of the interconnected system. Consequently, the proposed approximation methodology is computationally feasible for interconnected systems with an extremely large number of local subsystems.