Recurrent Neural Networks (RNNs) have shown remarkable performances in system identification, particularly in nonlinear dynamical systems such as thermal processes. However, stability remains a critical challenge in practical applications: although the underlying process may be intrinsically stable, there may be no guarantee that the resulting RNN model captures this behavior. This paper addresses the stability issue by deriving a sufficient condition for Input-to-State Stability based on the infinity-norm (ISS$_{\infty}$) for Long Short-Term Memory (LSTM) networks. The obtained condition depends on fewer network parameters compared to prior works. A ISS$_{\infty}$-promoted training strategy is developed, incorporating a penalty term in the loss function that encourages stability and an ad hoc early stopping approach. The quality of LSTM models trained via the proposed approach is validated on a thermal system case study, where the ISS$_{\infty}$-promoted LSTM outperforms both a physics-based model and an ISS$_{\infty}$-promoted Gated Recurrent Unit (GRU) network while also surpassing non-ISS$_{\infty}$-promoted LSTM and GRU RNNs.

翻译：循环神经网络（RNN）在系统辨识中展现出卓越性能，尤其在热工过程等非线性动力系统中。然而，实际应用中的稳定性仍是关键挑战：尽管底层过程可能本质稳定，但无法保证所构建的RNN模型能够捕捉这一特性。本文通过推导基于无穷范数的长短期记忆（LSTM）网络输入到状态稳定性（ISS$_{\infty}$）的充分条件来解决稳定性问题。与先前研究相比，所获条件依赖的网络参数更少。我们提出了一种ISS$_{\infty}$增强训练策略，在损失函数中引入促进稳定性的惩罚项，并采用专用早停方法。通过热系统案例研究验证了所提方法训练的LSTM模型质量，ISS$_{\infty}$增强型LSTM不仅优于物理模型和ISS$_{\infty}$增强型门控循环单元（GRU）网络，同时超越了非ISS$_{\infty}$增强型LSTM和GRU RNN。