Lyapunov-Stable Adaptive Neural PID Control for High-Precision Motor-Gear Systems Using a Symmetry-Oriented Structural Representation: Design and Experimental Validation

This article proposes an adaptive neural PID controller based on Lyapunov theory, called LAN-PID, for high-precision tracking of position and velocity reference trajectories in motor-gearbox systems subject to nonlinearities, friction, parametric uncertainties, and external load disturbances. A structural symmetry representation of the electromechanical system is introduced via a structural correspondence between the rotational variables of the motor shaft and those of the gearbox output shaft. This correspondence is established through a transformation determined by the gear ratio, which relates the angular position, angular velocity, and transmitted torque, providing an equivalent representation that preserves the structural relationships between the coupled rotational subsystems. In this context, structural symmetry is interpreted as a structural correspondence between the system’s dynamic variables, without assuming that the motor and gearbox have identical physical parameters or that the model is strictly invariant under a mathematical transformation. The proposed LAN-PID architecture combines adaptive PID control with a feedforward artificial neural network, which continuously adjusts the controller gains according to the system’s operating conditions. A Lyapunov-based adaptation law is derived for the neural network weights, and the resulting discrete-time Lyapunov analysis is used to evaluate the bounded behavior of the adaptive signals and the closed-loop system under the considered operating conditions. Initially, the controller is evaluated in MATLAB/Simulink using a nonlinear model of the motor-gearbox system that incorporates realistic parameters, friction effects, and load disturbances. Subsequently, the proposed strategy is implemented experimentally on a physical motor-gearbox platform, where signal acquisition and control are performed in real time using LabVIEW and a Raspberry Pi Pico W. The experimental evaluation considers different operating conditions and external loads corresponding to masses of up to 9kg (19.84lb). Compared to the conventional fixed-gain PID controller, the proposed LAN-PID reduced the experimental tracking RMSE by 66.7% under no-load conditions and by 81.1% under the evaluated load condition of 9kg, in addition to improving transient response and adaptation to changes in operating conditions. These results indicate that integrating a structural symmetry representation of the electromechanical system with a neural adaptation mechanism based on Lyapunov theory provides an effective framework for high-precision real-time control and trajectory tracking in motor-gearbox systems subject to nonlinearities, uncertainties, and external disturbances.