# Learning Sparse Dynamical Systems from a Single Sample Trajectory

## TL;DR

This paper proposes a Lasso-based method for identifying sparse linear systems from a single trajectory, providing finite-time guarantees and improved sample complexity bounds, with applications demonstrated in power systems.

## Contribution

Introduces a novel sparse system identification approach with sharp finite-time guarantees and polynomial sample complexity bounds, extending existing methods.

## Key findings

- High-probability correct sparsity pattern recovery
- Sample complexity scales polynomially with nonzero elements
- Effective bounds on estimation error influenced by system properties

## Abstract

This paper addresses the problem of identifying sparse linear time-invariant (LTI) systems from a single sample trajectory generated by the system dynamics. We introduce a Lasso-like estimator for the parameters of the system, taking into account their sparse nature. Assuming that the system is stable, or that it is equipped with an initial stabilizing controller, we provide sharp finite-time guarantees on the accurate recovery of both the sparsity structure and the parameter values of the system. In particular, we show that the proposed estimator can correctly identify the sparsity pattern of the system matrices with high probability, provided that the length of the sample trajectory exceeds a threshold. Furthermore, we show that this threshold scales polynomially in the number of nonzero elements in the system matrices, but logarithmically in the system dimensions --- this improves on existing sample complexity bounds for the sparse system identification problem. We further extend these results to obtain sharp bounds on the $\ell_{\infty}$-norm of the estimation error and show how different properties of the system---such as its stability level and \textit{mutual incoherency}---affect this bound. Finally, an extensive case study on power systems is presented to illustrate the performance of the proposed estimation method.