Learning the dynamics of nonlinear systems with regional stability guarantees through linear matrix inequality constraints

TL;DR

This paper introduces a method to learn regionally stable neural network models for nonlinear systems using LMI constraints, ensuring stability within specific state regions based on data.

Contribution

It develops a novel approach combining barrier functions and LMI conditions to guarantee regional stability of learned models from regional data.

Findings

The method guarantees regional stability for the learned model.

Numerical example demonstrates improved stability guarantees over global methods.

Compared to unconstrained methods, the approach ensures stability within a specified state set.

Abstract

This paper presents a method that learns a regionally stable recurrent neural network model from a set of input-output data generated by an unknown dynamical system. Relying on generalized sector conditions on the deadzone activation function, we first derive sufficient conditions that guarantee forward invariance on a compact set of the state space for any inputs from a given set. Such regional properties lead to less conservative conditions compared to variants that offer a global form of stability, and are in line with the system data that is only observed regionally. Our learning method derives conditions for regional stability using a barrier function approach, leading to models equipped with a certificate of regional stability in a subset of the state space and for a given input set. We illustrate our theoretical result with a numerical example and compare it to methods that impose a global form of stability, which fail to identify the system, and with a method that imposes no stability constraints at all, which does not guarantee a stable behavior within any state or input set.