(196h) Lyapunov-Stable Neural Network for Model-Based Control of Nonlinear Systems

Machine learning (ML) methods such as neural networks provide an efficient way to build nonlinear dynamic models from data that can be used in the model predictive control (MPC) system. The incorporation of neural networks into MPC is generally known as neural network-based MPC (NN-MPC), and has been studied extensively in the MPC literature [1, 2]. In NN-MPC, it is often assumed that there exists a stabilizing controller for the neural network model which also renders the original nonlinear system closed-loop stable. This enables the MPC formulation and the optimal control algorithm to guarantee that the state trajectory eventually converges to a small set around the origin while simultaneously minimizing the MPC loss function [3]. Such a stabilizing controller is usually not guaranteed for NN-MPC and is required as a prerequisite assumption. There are several research works revolving around NN-MPC where a neural network is developed while simultaneously ensuring that it has some form of guaranteed Lyapunov stability [4, 5]. However, to our best knowledge, these controllers still do not provably guarantee closed-loop stability, and this could pose a serious issue in performance critical applications.

In this work, we develop -- for the first time -- a pioneering methodology for the neural network modeling of nonlinear processes with Lyapunov stability guarantees, and the theory of its generalization performance and closed-loop stability. By designing a novel loss function that accounts for Lyapunov stability conditions in the training phase, the resulting Lyapunov-stable neural network (LSNN) is able to capture the nonlinear dynamics and guarantee closed-loop stability when incorporated in a model-based controller simultaneously. A key feature of the LSNNs is that they retain the stability region of the original nonlinear system as long as the training error is sufficiently small, which implies that the LSNN can be utilized for the entire operating domain. Provable closed-loop stability properties are derived based on the analysis of its generalization performance using statistical learning theory. Finally, a chemical reactor example is used to demonstrate the efficacy of the proposed ML modeling method.

References:

[1] Draeger, A., Engell, S., & Ranke, H. (1995). Model predictive control using neural networks. IEEE Control Systems Magazine, 15(5), 61-66.

[2] Wu, Z., Tran, A., Rincon, D., & Christofides, P.D. (2019). Machine learning‐based predictive control of nonlinear processes. Part I: theory. AIChE Journal, 65(11), 16729.

[3] Wu, Z., Tran, A., Rincon, D., & Christofides, P.D. (2019). Machine‐learning‐based predictive control of nonlinear processes. Part II: Computational implementation. AIChE Journal, 65(11), 16734.

[4] Åkesson, B.M., Toivonen, H.T., Waller, J.B., & Nyström, R.H. (2005). Neural network approximation of a nonlinear model predictive controller applied to a pH neutralization process. Computers & chemical engineering, 29(2), 323-335.

[5] Dai, H., Landry, B., Yang, L., Pavone, M., & Tedrake, R. (2021). Lyapunov-stable neural-network control. arXiv preprint arXiv:2109.14152.