(12d) Nonlinear Model Predictive Control Using Statistical Machine-Learning-Based Control Lyapunov Barrier Functions

Safety-critical systems exist in many application domains, and safety-related constraints must be strictly met in order to prevent harm on process stability, economic gains, and/or operational safety. There has been extensive research on providing safety verification of a system as well as synthesizing control laws with provable safety properties. Control Barrier Functions (CBFs) are proposed as a tool to characterize the safety of dynamical systems by certifying whether a control law achieves forward invariance of a safe set, similar to the utility of Control Lyapunov Function (CLF) in certifying stability properties [1-2]. CBFs can be incorporated in the design of control laws for multi-objective control of safety-critical systems, e.g., controllers designed based on Control Lyapunov Barrier Functions (CLBF), where a CLF is used to characterize a stability region and ensure stability properties, and a CBF is used to characterize an unsafe region where state trajectory under the CLBF-based control law will not enter at all times. This approach has been further explored for nonlinear systems subject to constrained inputs, where CLBF-based control laws are used as contractive constraints in the design of a model predictive controller (MPC) to provide closed-loop stability and safety guarantees for nonlinear processes with embedded bounded and unbounded unsafe regions [3].

Designing a valid CBF to be used in safety-critical control systems is a challenge faced by many; as such, we proposed a machine-learning method to construct a CBF from data points. There has been some research into probabilistic safety certification of barrier functions, but not in the sense of analyzing the generalization error of the modeling method. In this work, we provide statistical analysis on the CBF construction method proposed in our previous work in [4] and model the CBF using a feed-forward neural network, which will be used to design a CLBF-based model predictive control system. Using statistical machine learning, an upper bound for the generalization error or expected error of a neural network model can be derived [5]. We first develop the generalization error bound on the FNN-CBF, and derive probablistic safety and stability guarantees for the control law designed using a CLBF with FNN-CBF under sufficient conditions. The sampling, modeling, and verification procedures of the FNN are discussed. Then, we extend the probablistic stability and safety properties to the FNN-CLBF MPC, and demonstrate that with high probability, the FNN-CLBF MPC is able to maintain the closed-loop state of a nonlinear process within a safe set and ultimately bounded within a terminal set around the origin.

[1] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada. Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 62:3861–3876, 2017

[2] X. Xu, P. Tabuada, J. W. Grizzle, and A. D. Ames. Robustness of control barrier functions for safety critical control. IFAC-PapersOnLine, 48(27):54–61, 2015.

[3] Z. Wu, F. Albalawi, Z. Zhang, H. Zhang, J.and Durand, and P. D. Christofides. Control lyapunov-barrier function-based model predictive control of nonlinear systems. Automatica,109:108508, 2019

[4] S. Chen, Z. Wu, and P. D. Christofides. Machine-learning-based construction of barrier functions and models for safe model predictive control. AIChE Journal e17456, 2021.

[5] N. Golowich, A. Rakhlin, and O. Shamir. Size-independent sample complexity of neural networks. In Proceedings of the Conference On Learning Theory, pages 297–299, Stockholm, Sweden, 2018