Learning the dynamics of nonlinear systems with regional stability guarantees through linear matrix inequality constraints
For control engineers and machine learning practitioners modeling nonlinear systems from data, this provides a less conservative stability certificate that aligns with regionally observed data, enabling reliable model-based control.
This work learns a recurrent neural network model with regional stability guarantees from input-output data of an unknown dynamical system, using linear matrix inequality constraints derived from barrier functions and sector conditions. The method ensures forward invariance on a compact set, outperforming global stability approaches that fail to identify the system and unconstrained methods that lack stability guarantees.
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.