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.

Abstract Measurement of certain variables within the industrial sector remains a challenge due to the prohibitive costs of sensors, the intricate installation processes, or the continuous nature of production demands. Moreover, if a backup sensor is required in case the main sensor fails, the installation and maintenance difficulties are further increased. A possibility to address this issue is the indirect estimation of the desired variable by leveraging other correlated measures within the operational process. Data-driven techniques are well-suited for this aim, given their capacity to model potentially complex industrial processes. This paper proposes the implementation of a virtual flow sensor for its integration in the control loop of an industrial process. More specifically, four different data-driven methods have been tested to obtain the virtual sensor: multiple linear regression (MLR), multilayer perceptron (MLP), long-short term memory (LSTM) and deep long-short term memory (DeepLSTM). MAE, RMSE and $$R^2$$ R 2 have been chosen as evaluation metrics for model selection and testing. Furthermore, the robustness of the virtual flow sensor is not only evaluated under ideal operating conditions, but it is also tested under adverse conditions with various noise levels added to the measured signals. Additionally, the performance of the flow control loop using the real and virtual sensors is also evaluated in both ideal and adverse conditions. IAE, ITAE, and IAVU indices are used to assess the control performance. The results prove the robustness of the LSTM-based virtual flow sensor and the effectiveness of the control loop using it, avoiding the modification of the controller and interrupting the process when the real flow sensor fails.