Stability of Non-linear Neural Feedback Loops using Sum of Squares

Neural network controllers have the potential to improve the performance of feedback systems compared to traditional controllers, due to their ability to act as general function approximators. However, quantifying their safety and robustness properties has proven challenging due to the non-linearities of the activation functions inside the neural network. A key robustness indicator is certifying the stability properties of the feedback system and providing a region of attraction, which has been addressed in previous literature. However, these works only address linear systems or require one to abstract the plant non-linearities and bound them using slope and sector constraints. In this paper we use a Sum of Squares programming framework to compute the stability of non-linear systems with neural network controllers directly. Within this framework, we can propose higher order candidate Lyapunov functions with richer structures that are able to better capture the dynamics of the non-linear system and the nonlinearities in the neural network. We are also able to analyse these systems in continuous time, whereas other methods rely on discretising the system. These higher order Lyapunov functions are used in conjunction with higher order multipliers on the inequality and equality constraints that bound the neural network input-output properties. The volume of the region of attraction computed is increased compared to other methods, allowing for better safety guarantees on the stability of the system. We are also able to easily analyse non-linear polynomial systems, which is not possible to do with other methods. We are also able to conduct robustness analysis on the parameter uncertainty. We show the benefits of our method using numerical examples.