Robust Non-Linear Lyapunov Deep Learning Control Design For Chaotic Systems

Abstract

Despite their operational success, machine learning controllers lack theoretical guarantees in terms of system stability. In contrast, classic model-based controller design uses principled approaches such as Linear Quadratic Regulator (LQR) to synthesize stable controllers with verifiable proofs. In addition, deep learning controllers encounter feedback timing bottlenecks that increase exponentially with the system complexity. Deep learning is also dependent on the quality and diversity of the dataset to produce unbiased findings; therefore, the prediction of deep learning is not guaranteed. As a result, in this research, we develop and implement a guaranteed stability solution for safety critical and chaotic systems through the integration of Lyapunov Stability theory and deep machine learning. Three control methods are researched, leading to the development of the Deep Lyapunov-stable controller: the deep learning methodology, the Lyapunov control function, and controller parameters. In this research, we provide a generic method for synthesizing a Deep Lyapunov-stable control and a way to simultaneously confirm its stability. A unique Lyapunov control function is devised and shown to be effective in managing Duffing, Van der Pol, and Zohdy-Harb nonlinear systems, but with restrictions on the system's oscillation frequency, initial conditions and disturbances. Subsequently, Dynamic Lyapunov Deep Learning is introduced to alleviate the Lyapunov control’s shortcomings. Developing a deep learning architecture in combination with a customized Lyapunov control resolves the temporal delay and Lyapunov parameters calibration concern. Different datasets are also presented before establishing the one with the best accuracy. In addition to the dataset, the architecture of the deep learning model has a significant effect on the model's accuracy. A process for relearning is intended to accommodate the introduction of new system dynamics. Based on the correlation study, we also designed an optimization technique to improve the integration of the deep learning layer and controller layer. The proposed integration of Deep Learning and Lyapunov Control, referred to as Lyapunov Deep Learning (LDL) control, is applied in MATLAB / SIMULINK to the magnetic levitation chaotic nonlinear system to demonstrate its effectiveness in addressing sudden changes in system behavior, the environment, and demands in comparison to other methods of control.