A Lyapunov-Gated Adaptive Law for Drift-Free Neural-Network Depth Control of a Remotely Operated Vehicle in Persistent Seas

This paper addresses the depth regulation of an inspection-class remotely operated vehicle whose hydrodynamic model is uncertain and whose environment applies a persistent, non-vanishing wave disturbance. A neural-network learning term added to a proportional-derivative feedback can compensate the uncertain dynamics online but, under a persistent disturbance, the classical gradient update never stops adjusting the network weights, so the weights drift without bound and eventually destabilise the loop. The problem solved here is to make the online learning provably bounded in exactly this regime. The proposed solution is an adaptive law with two mechanisms: a Lyapunov gate that admits a weight update only when it does not increase a control Lyapunov function and otherwise holds the update and a leakage term that bleeds the weights toward zero when the error carries no useful information. The weights are proved uniformly ultimately bounded without requiring the regressor to be persistently exciting or the disturbance to vanish, and the closed loop is proved input-to-state stable with an explicit ultimate bound. The controller is compared, on one common set of wave realisations, against six baselines spanning classical and modern adaptation, namely the naive gradient update, the sigma modification, the e modification, the projection operator, concurrent learning and composite adaptation, across four sea states and over 40 realisations per sea state, and its two mechanisms are separated by an ablation. In the most severe sea state the naive update and the concurrent-learning scheme diverge in every run with a weight norm near 100, whereas the proposed law keeps the weight norm at 20.1 and attains the smallest regulation error, 5.40 cm. The ablation shows that both mechanisms are needed: without the gate the divergence returns in 30 per cent of the runs and without the leakage the weight norm grows by a third.