Adaptive Tracking Control of Uncertain Nonlinear Systems via Self-Organizing Deep Neural Networks With Learning Rate Adaptation

This paper investigates the adaptive tracking control problem for uncertain nonlinear systems using self-organizing deep neural networks (SODNNs). Conventional DNN-based control approaches typically employ fixed depth architectures with multiple hidden-layers to model uncertain nonlinear dynamics. Such fixed structures often face an inherent trade-off between approximation accuracy and computational efficiency, resulting in either excessive computational burden or insufficient representational capability. To address these issues, an event-triggered self-organizing mechanism is proposed, which enables the DNN to dynamically adjust its network depth by adding or removing hidden-layers when prescribed triggering conditions are violated. In this way, the network structure is dynamically adapted to the evolving system dynamics, achieving an effective balance between approximation precision and computational burden. Furthermore, an adaptive learning rate strategy is developed, in which the learning rates of individual layers are adjusted online based on the state tracking error and the norm of the corresponding weight gradients. This mechanism enhances approximation efficiency, control performance and robustness against uncertainties. Rigorous Lyapunov-based analysis is provided to guarantee that all closed-loop signals are ultimately bounded under the proposed control framework. The effectiveness and practical feasibility of the proposed method are validated through numerical simulations and real-time experiments conducted on a Franka Emika Panda robotic manipulator.