(176f) Statistical Machine Learning in Model Predictive Control of Nonlinear Processes

Machine learning techniques such as recurrent neural networks (RNN) have been recently applied to model chemical processes using industrial process operation or numerical simulations data, and have been successfully incorporated in model predictive control (MPC) schemes when first-principles models are unavailable [1, 2]. Although neural networks are able to fit any data set well using a sufficient number of layers and neurons according to universal approximation theorem, one of the major challenges involving neural networks is their generalization performance when implemented to real chemical processes. Statistical machine learning such as probably approximately correct (PAC) learning provides a useful tool to address the question that when a low training error leads to a low generalization error. For example, in [3, 4, 5], generalization error bounds, which depend on a number of factors such as neural network complexity and number of data samples, were developed for feed-forward neural networks. In [6], a margin-based data dependent generalization error bound was developed for recurrent neural networks that were designed for multi-class classification problems. Additionally, in [7], generalization performance for a single-output, one-hidden-layer RNN model was studied. However, at this point, the generalization performance for modeling a general class of nonlinear dynamical systems using RNN models and the stability properties of RNN-based MPC have not been investigated.

In this work, we take advantage of statistical machine learning theory and develop a generalization error bound for RNN model that is developed for multiple-input multiple-output nonlinear systems. The generalization error bound provides an insight on how well this RNN model will behave in real processes, and also provides a guide showing how to improve its generalization performance by optimizing neural network size and data collection process. Subsequently, we incorporate the statistical machine learning model within model predictive control (MPC), for which probabilistic stability analysis is carried out to demonstrate that the nonlinear system can be stabilized at the steady-state under MPC with a certain probability provided that the RNN model satisfies the generalization error bound. Finally, a chemical process example was used to demonstrate the relationship between RNN generalization error and training error along with the dependence on network complexity and data set size. Closed-loop simulation was also carried out to demonstrate probabilistic closed-loop stability of nonlinear systems under RNN-based MPC.

References:

[1] 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), e16729.

[2] 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), e16734.

[3] Bartlett, P., Foster, D. J., & Telgarsky, M. (2017). Spectrally-normalized margin bounds for neural networks. arXiv preprint arXiv:1706.08498.

[4] Zou, D., & Gu, Q. (2019). An improved analysis of training over-parameterized deep neural networks. arXiv preprint arXiv:1906.04688.

[5] Golowich, N., Rakhlin, A., & Shamir, O. (2018, July). Size-independent sample complexity of neural networks. In Conference On Learning Theory (pp. 297-299). PMLR.

[6] Chen, M., Li, X., & Zhao, T. (2019). On generalization bounds of a family of recurrent neural networks. arXiv preprint arXiv:1910.12947.

[7] Hanson, J., Raginsky, M., & Sontag, E. (2020). Learning Recurrent Neural Net Models of Nonlinear Systems. arXiv preprint arXiv:2011.09573.