Efficient Computation of Lyapunov Functions Using Deep Neural Networks for the Assessment of Stability in Controller Design

Abstract

Abstract This paper presents a deep neural network (DNN) based method to estimate approximate Lyapunov functions and their orbital derivatives, which are key to the stability of the system in control theory. Our approach addresses the challenge of the curse of dimensionality in control and optimization problems, demonstrating that the computational effort required grows only polynomically with the state dimension. This is a significant improvement over traditional methods. We emphasize that the calculated functions are approximations of Lyapunov functions and not exact representations. This distinction is important, as validating these approximations in high-dimensional environments is challenging and opens new avenues for future research. Our approach diverges from traditional grid-based approaches and moves away from relying on small-gain theorems and accurate subsystem knowledge. This flexibility is proven in deriving Lyapunov functions for the development of stabilizing feedback rules in nonlinear systems. A crucial aspect of our approach is the use of ReLU activation features in neural networks, which is steady with modern deep getting-to-know traits. We also explore the feasibility of using DNNs to estimate fairly constructive Lyapunov functions, despite the demanding situations posed through uncertainty. Our set of rules' outcomes aren't unique, highlighting the want to set up criteria for figuring out especially useful Lyapunov features. The paper culminates with the aid of emphasizing the capacity of DNNs to approximate compositional Lyapunov capabilities, especially underneath small-gain conditions, to mitigate the curse of dimensionality. Our contributions are manifold, including scalability in dealing with systems of various dimensionality, flexibility in accommodating both low and excessive-dimensional structures, and performance in computing Lyapunov features through deep learning strategies. However, challenges continue to be in the approximation accuracy and the verification of Lyapunov functions in better dimensions.