(235a) Barrier Function-Based Predictive Controllers Compared with Lyapunov-Based Economic Model Predictive Control

Barrier functions have been of interest for safety-critical control applications (e.g., [1,2]). These are Lyapunov-like functions used for certifying safety in the sense of set invariance [3,4], rather than tracking of the origin. These functions are Lyapunov-like in the sense that they have a requirement on both the form of the barrier function and on its time derivative. The motivation for considering a barrier function in safety-critical control, instead of a Lyapunov function, is that Lyapunov functions are required to be strictly decreasing along the process state trajectory under a stabilizing input, whereas barrier functions can have greater flexibility (i.e., they may not be required to change monotonically along the closed-loop state trajectory). In chemical process control, barrier functions have been explored for their ability to block out certain unsafe regions of state-space and enable controllers to drive the closed-loop state to regions of state-space around, but not inside, such regions [5,6]. Machine learning-based techniques for generating barrier functions for chemical processes have been developed in [7]. In this talk, we will focus on a type of barrier function termed a zeroing barrier function in [4], which is required to be positive in a safe operating region, zero at the boundary of the safe operating region, and negative outside this region. The time derivative of the barrier function should be greater than the negative of a strictly increasing function of the state. This gives some flexibility to the barrier function to either increase (indicating that the closed-loop state is moving toward the origin) or decrease (indicating that the closed-loop state is approaching the boundary of the safe operating region).

This talk will focus on examining relationships between this type of barrier function and Lyapunov-based economic model predictive control (LEMPC) [8]. The motivation for this work is that LEMPC is a control law designed with two operating modes such that in the first mode of operation, the closed-loop state is able to either move toward the origin or toward the boundary of a subset of a safe operating region, but then in the second mode of operation, which is triggered when the closed-loop state leaves the subset, it is driven back into the subset. The somewhat similar behavior of the closed-loop state under a zeroing barrier function and in the first mode of operation of LEMPC inspires looking in more depth at when LEMPC and an LEMPC-inspired control design that replaces Lyapunov function-based constraints with zeroing barrier function-based constraints are equivalent, and how they differ otherwise. Specifically, we will present two formulations of economic model predictive control (EMPC) with safety-based constraints derived from barrier functions [9]. One will be derived to replace the two constraints used in LEMPC with a single constraint requiring that the change in the barrier function along the closed-loop state trajectory under the input computed by the barrier function-based EMPC must be no less than the change under an auxiliary control law (where this auxiliary control law can cause a change in the sign of the rate of change of the barrier function in different regions of state-space). We demonstrate closed-loop stability under this control law and clarify similarities and differences compared with LEMPC. Subsequently, we present a second barrier function-based EMPC with two constraints on the barrier function and discuss the conditions under which the constraints would be equivalent to those in LEMPC. We compare LEMPC and the first barrier function-based control law using a continuous stirred tank reactor (CSTR) that indicates various factors that affect which of the control laws is more profitable.

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[2] Choi, J. J., Lee, D., Sreenath, K., Tomlin, C. J., & Herbert, S. L. (2021). Robust control barrier-value functions for safety-critical control. arXiv preprint arXiv:2104.02808.

[3] Ames, A. D., Grizzle, J. W., & Tabuada, P. (2014, December). Control barrier function based quadratic programs with application to adaptive cruise control. In 53rd IEEE Conference on Decision and Control (pp. 6271-6278). IEEE.

[4] Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2016). Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 62(8), 3861-3876.

[5] Wu, Z., Durand, H., & Christofides, P. D. (2018). Safe economic model predictive control of nonlinear systems. Systems & Control Letters, 118, 69-76.

[6] Wu, Z., Albalawi, F., Zhang, Z., Zhang, J., Durand, H., & Christofides, P. D. (2019). Control Lyapunov-Barrier function-based model predictive control of nonlinear systems. Automatica, 109, 108508.

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

[8] Heidarinejad, M., Liu, J., & Christofides, P. D. (2012). Economic model predictive control of nonlinear process systems using Lyapunov techniques. AIChE Journal, 58(3), 855-870.

[9] Durand, H. and A. D. Ames, “A Control Barrier Function Perspective on Lyapunov-Based Economic Model Predictive Control,” Proceedings of the American Control Conference, Atlanta, Georgia, in press.