System Identification and Control Using Lyapunov-Based Deep Neural Networks without Persistent Excitation: A Concurrent Learning Approach

Deep Neural Networks (DNNs) are increasingly used in control applications due to their powerful function approximation capabilities. However, many existing formulations focus primarily on tracking error convergence, often neglecting the challenge of identifying the system dynamics using the DNN. This paper presents the first result on simultaneous trajectory tracking and online system identification using a DNN-based controller, without requiring persistent excitation. Two new concurrent learning adaptation laws are constructed for the weights of all the layers of the DNN, achieving convergence of the DNN's parameter estimates to a neighborhood of their ideal values, provided the DNN's Jacobian satisfies a finite-time excitation condition. A Lyapunov-based stability analysis is conducted to ensure convergence of the tracking error, weight estimation errors, and observer errors to a neighborhood of the origin. Simulations performed on a range of systems and trajectories, with the same initial and operating conditions, demonstrated 40.5% to 73.6% improvement in function approximation performance compared to the baseline, while maintaining a similar tracking error and control effort. Simulations evaluating function approximation capabilities on data points outside of the trajectory resulted in 58.88% and 74.75% improvement in function approximation compared to the baseline.

Lyapunov-Based Dropout Deep Neural Network (Lb-DDNN) Controller

Deep neural network (DNN)-based adaptive controllers can be used to compensate for unstructured uncertainties in nonlinear dynamic systems. However, DNNs are also very susceptible to overfitting and co-adaptation. Dropout regularization is an approach where nodes are randomly dropped during training to alleviate issues such as overfitting and co-adaptation. In this paper, a dropout DNN-based adaptive controller is developed. The developed dropout technique allows the deactivation of weights that are stochastically selected for each individual layer within the DNN. Simultaneously, a Lyapunov-based real-time weight adaptation law is introduced to update the weights of all layers of the DNN for online unsupervised learning. A non-smooth Lyapunov-based stability analysis is performed to ensure asymptotic convergence of the tracking error. Simulation results of the developed dropout DNN-based adaptive controller indicate a 38.32% improvement in the tracking error, a 53.67% improvement in the function approximation error, and 50.44% lower control effort when compared to a baseline adaptive DNN-based controller without dropout regularization.