(382d) Learning-Based Nonlinear Model Predictive Control with Chance Constraints for Stochastic Systems

Design of model predictive controllers (MPC) faces an inherent trade-off between performance and robustness. In general, stability and robustness of MPC can be derived using approximate models, but MPC performance critically hinges on the model quality [1, 2]. In this work, we present a learning-based nonlinear MPC (NMPC) strategy with chance constraints for stochastic nonlinear systems. The learning-based NMPC strategy provides guarantees on stability and state chance constraint satisfaction, while the controller performance is enhanced by identifying more accurate models online without compromising the aforementioned guarantees. The key notion of the proposed strategy is that robustness guarantees can be decoupled from control performance by using two models of the system: the first is an approximate model derived offline with bounds on its uncertainty and is used for providing guarantees on stability and constraint satisfaction, and the second model is derived online using, for example, machine learning tools and is applied for computing the control cost function. Thus, the learning-based NMPC strategy minimizes a control cost subject to learned dynamics, while robustness is ensured by verifying that the designed control inputs preserve the stability and chance constraint satisfaction of the initial approximate model [3, 4].

To this end, we present a new constraint tightening approach by extending the concept of stochastic tubes [5] to a nonlinear setting. The constraint tightening approach is used for ensuring stability, recursive feasibility, and constraint satisfaction properties of the learning-based NMPC with individual chance constraints [6]. In this approach, state constraints are tightened recursively by constructing a sequence of sets that bound the one-step ahead disturbance propagation of the nominal model predictions. The sets are computed from an initial constraint set, which is obtained from the empirical cumulative distribution of the system uncertainty and is subsequently tightened with an appropriate backoff parameter to account for the individual chance constraints.

The proposed learning-based NMPC strategy has an online computational complexity comparable to that of nominal MPC. We demonstrate the performance of the learning-based NMPC on two nonlinear systems, including a benchmark DC-DC converter case study, where Gaussian process modeling [7] is used for online learning of the nonlinear system dynamics based on closed-loop data. Simulation results indicate that the learning-based NMPC leads to an enlarged domain of attraction and improved control performance due to, respectively, effective handling of chance constraints and computing the control cost based on learned dynamics.

References

[1]	D. Mayne, M. Seron and S. Rakovic, "Robust model predictive control of constrained linear systems with bounded disturbance.," Automatica, 2005.
[2]	A. Mesbah, "Stochastic model predictive control: An overview and perspectives for future research," IEEE Control Systems, 2016.
[3]	A. Aswani, H. Gonzalez, S. Shankar Sastry and C. Tomlin, "Provably safe and robust learning-based model predictive control," Automatica, 2013.
[4]	D. Limon, J. Calliess and J. Maciejowski, "Learning-based Nonlinear Model Predictive Control," in IFAC, 2017.
[5]	B. Kouvaritakis and M. Cannon, "Model predictive control: Classical, robust and stochastic.," Springer, 2015.
[6]	T. L. Santos, A. D. Bonzanini and A. Mesbah, "A Constraint-Tightening Approach to Nonlinear Model Predictive Control with Chance Constraints for Stochastic Systems," in Proceedings of IEEE Conference on Decision and Control, Submitted, 2018.
[7]	C. E. Rasmussen and K. I. Williams, Gaussian Processes for Machine Learning, Cambridge, Massachusetts; London, England: MIT Press, 2006.