On the Stability Analysis of Deep Neural Network Representations of an Optimal State Feedback

• Published in: IEEE transactions on aerospace and electronic systems Vol. 57; no. 1; pp. 145 - 154
• Main Authors: Izzo, Dario, Tailor, Dharmesh, Vasileiou, Thomas
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Artificial neural networks; asymptotic stability; Computer architecture; Control stability; deep learning; guidance and control network; Initial conditions; Motion stability; Neural networks; neurocontrollers; Nonlinear control; nonlinear control systems; Nonlinear systems; Optimal control; optimal state feedback; Perturbation; Representations; Stability analysis; State feedback; time-delay systems; Training; Trajectory
• ISSN: 0018-9251; 1557-9603
• DOI: 10.1109/TAES.2020.3010670

Recent works have shown that the optimal state feedback for deterministic, nonlinear autonomous systems can be approximated by deep neural networks. In this article, we consider the stability of nonlinear systems controlled by such a network representation of the optimal feedback. First, we show that principal methods from stability theory readily applies. We then propose a novel method based on differential algebra techniques to study the robustness of a nominal trajectory with respect to perturbations of the initial conditions. It is, to the best of our knowledge, the first time that differential algebraic techniques are shown to allow for the high-order analysis of motion stability for a nonlinear system in general and for a neurocontrolled system in particular. We exemplify the proposed method in the 2-D case of the optimal control of a quadcopter and demonstrate it for different neural network architectures.