A Sparse Bayesian Approach to the Identification of Nonlinear State-Space Systems

TL;DR

This paper introduces a Bayesian sparse regression method for identifying nonlinear state-space systems from limited data, employing an iterative re-weighted $ ext{l}_1$-minimisation algorithm with convex constraints.

Contribution

It presents a novel convexification approach for Bayesian sparse system identification, enabling efficient estimation of nonlinear functions and parameters from limited data.

Findings

Effective identification on biological and physical systems

Handles nonlinearities with sparse Bayesian methods

Incorporates convex constraints for improved estimation

Abstract

This technical note considers the identification of nonlinear discrete-time systems with additive process noise but without measurement noise. In particular, we propose a method and its associated algorithm to identify the system nonlinear functional forms and their associated parameters from a limited number of time-series data points. For this, we cast this identification problem as a sparse linear regression problem and take a Bayesian viewpoint to solve it. As such, this approach typically leads to nonconvex optimisations. We propose a convexification procedure relying on an efficient iterative re-weighted $\ell_1$-minimisation algorithm that uses general sparsity inducing priors on the parameters of the system and marginal likelihood maximisation. Using this approach, we also show how convex constraints on the parameters can be easily added to our proposed iterative re-weighted $\ell_1$-minimisation algorithm. In the supplementary material \cite{appendix}, we illustrate the effectiveness of the proposed identification method on two classical systems in biology and physics, namely, a genetic repressilator network and a large scale network of interconnected Kuramoto oscillators.