(303e) Ensuring Convexity in Constrained Optimal Control Using Input Convex Neural Networks

Many common chemical and biochemical processes exhibit nonlinear behavior. In response to a dynamic market and increasing complexity of process operation, modeling the nonlinear dynamics of process systems has become crucial for process safety and profitability. Artificial neural networks (ANNs) are widely used to model nonlinear systems due to their universal approximation capability and simplicity in model development [1,2]. However, optimal control and decision-making on top of ANN models are difficult because the learned ANN models are usually non-convex from the input to output [3].

Input convex neural networks (ICNNs) are a family of deep learning models where the outputs are constructed to be convex functions of the inputs [4]. Similar to ordinary ANNs, ICNNs can efficiently approximate all continuous Lipschitz convex functions [5]. Additionally, ICNNs can be efficiently trained on GPUs using existing deep learning packages such as PyTorch and TensorFlow. By modeling systems using ICNNs, optimal control problems on top of the system models can be solved as convex optimization problems, leading to improved performance and robustness.

Current work discusses using ICNNs for constrained optimal control of nonlinear systems. This strategy uses Model Predictive Control (MPC) on top of ICNN-based system models to dynamically optimize system performance. The strategy consists of three basic steps: Firstly, the relationships that can be approximated by convex function are identified using field knowledge or simulation. Then, process data are used to train ICNN-based system and constraint models. Lastly, the system and constraint models are integrated into the MPC framework. Because both system dynamics and constraints are modeled using ICNNs, their Jacobians can be easily calculated using backpropagation, and the MPC problems can be solved efficiently using sequential quadratic programming (SQP).

The proposed approach will be illustrated through a case study based on the Van de Vusse reactor [6], which exhibits input multiplicity and nonminimum phase behavior. The simulation results demonstrate improved economic yield compared with normal ANNs. Additionally, the input convexity formulation is compared with simple regularization techniques, and unique benefits such as improved data efficiency and robustness of the proposed formulation are shown. By explicitly incorporating prior knowledge about convexity, this framework provides a good balance between the universal approximation power of deep learning and the computational feasibility required by control and optimization.

References

[1] M. Kuure-Kinsey, R. Cutright, B.W. Bequette, Computationally efficient neural predictive control based on a feedforward architecture, Ind. Eng. Chem. Res. 45 (2006) 8575–8582.

[2] N. Bhat, T.J. McAvoy, Use of neural nets for dynamic modeling and control of chemical process systems, Comput. Chem. Eng. 14 (1990) 573–582.

[3] K. Kawaguchi, Deep Learning without Poor Local Minima, in: D.D. Lee, M. Sugiyama, U. V Luxburg, I. Guyon, R. Garnett (Eds.), Adv. Neural Inf. Process. Syst. 29, Curran Associates, Inc., 2016: pp. 586–594.

[4] B. Amos, L. Xu, J. Zico Kolter, J.Z. Kolter, Input Convex Neural Networks, in: Proc. 34th Int. Conf. Mach. Learn. - Vol. 70, JMLR.org, 2017: pp. 146–155.

[5] Y. Chen, Y. Shi, B. Zhang, Optimal Control Via Neural Networks: A Convex Approach, in: Int. Conf. Learn. Represent., 2018.

[6] B.W. Bequette, Process control: modeling, design, and simulation, Prentice Hall Professional, Upper Saddle River, NJ, United States, 2003.