Adaptive modeling of systems with uncertain dynamics via continuous long-short term memories

This study presents an innovative modeling approach for systems with uncertain dynamics, utilizing a novel long -short-term memory (LSTM) architecture featuring continuous temporal evolution. The proposed approximate model is resolved by introducing a non -parametric identifier, the trajectories converge to the states of systems exhibiting uncertain dynamics in an ultimately bounded manner. Given the multi -layered nature of continuous dynamics, we introduce two identifiers, considering the possibility of accessing or not accessing the LSTM's internal state. Consequently, two identifiers have been devised, considering the influence of the internal state of the LSTM. Applying a control Lyapunov function to each identifier enables the derivation of weight evolution in the LSTM's two layers: the output and the hidden layers. These weights are linked to input states' short- and long-term effects on the LSTM's dynamics. To validate the proposed identifiers' effectiveness compared to traditional Hopfield-like differential neural networks, we provide a numerical example and conduct complementary experimental validations. These results confirm the theoretical findings regarding the impact of short and long-term temporal information on the LSTM's current state and demonstrate superior identification quality compared to traditional neural networks exhibiting continuous dynamics adhering to the Hopfield form. Collectively, these findings substantiate the advancements presented in this study.