Data-driven adaptive gain tuning for the augmented homogeneous differentiator
Derivative estimation from noisy sensor measurements is a fundamental requirement in feedback control, and any such estimator is subject to an inherent trade-off between tracking speed and noise rejection. In the augmented homogeneous differentiator (AHD), this trade-off is governed by a single scalar gain. A large gain tracks fast transients but amplifies noise, whereas a small gain attenuates noise at the cost of a slower response, and no fixed value remains suitable when the signal statistics vary over time. This paper presents an adaptive-gain AHD in which a network combining convolutional, recurrent, and attention layers is trained offline and deployed online to predict, from a short window of the raw noisy signal, the smallest gain that meets a prescribed velocity-accuracy target. Using the raw noisy signal rather than the differentiator output as the network input avoids a direct feedback path from that output to the predictor. On a held-out window-level test set, the predictor attains R2=0.9834. In a frequency-sweep test, the adaptive scheme reduces the first-order velocity-estimation error by 99.3 % relative to a fixed gain tuned for the low-frequency regime. In a step test, it reduces the steady-state third-order derivative noise by 87.2 %. In closed-loop control of a variable-length pendulum, it reduces the control-input total variation by 32 % while preserving tracking accuracy.