A comprehensive study of incremental input-to-state stability in echo state networks: Hyperparameter influence and stability promotion in reduced models

Echo State Networks (ESNs) are a powerful tool for modeling complex nonlinear dynamics in data-driven control applications. However, their high dimensionality poses significant challenges for reliable deployment and further analysis. This work presents a comprehensive analysis of incremental Input-to-State Stability ( ) for leaky integrator ESNs with feedback connections, focusing on deriving practical, computationally affordable stability conditions. We demonstrate that a joint condition on standard spectral metrics provides a way to promote , serving as a viable alternative to finding Lyapunov functions through computationally expensive techniques during the hyperparameter optimization. Through extensive numerical studies, we characterize how hyperparameters influence both these spectral metrics and . Furthermore, we have extended the available formal guarantees to models reduced through Proper Orthogonal Decomposition (POD), in order to formally ensure stability in the finally deployed models. The practical utility of this approach is validated on a quadruple-tank benchmark system, where we successfully identify stable, reduced-order ESNs that maintain high prediction accuracy. Our results provide ESN practitioners a simple yet effective framework for both promoting and ensuring stability in their models, facilitating their application in control scenarios. • Incremental Input-to-State Stability ( ) in Echo State Networks (ESNs) is evaluated. • A method to promote based on spectral norm and spectral radius is proposed. • The influence of hyperparameters on is evaluated through grid-search experiments. • Reduction methods are used to make ESNs tractable through Lyapunov arguments. • A quadruple-tank benchmark is used to validate the proposed methods.