(432d) Statistical Machine-Learning-Based Predictive Control of Nonlinear Time-Delay Processes

Machine-learning–based model predictive controllers (MPCs) have demonstrated their superior closed-loop performance when compared with the MPCs using (usually linear) data-driven models in traditional industrial process control systems. Most existing works on learning-based MPC derived closed-loop stability properties based on the assumption that the generalization error of machine learning (ML) models is bounded [1,2]. However, this assumption may not hold in practice. While the training error of ML models could be rendered sufficiently low with good-quality datasets and a careful tuning of model hyper-parameters, a fundamental challenge for the implementation of ML models in chemical process control is the generalization performance on unseen data. Among the several types of neural networks, long-short term memory networks (LSTMs) have shown to be effective when considering problems that require remembering past data and long-term memory [3]. Additionally, time-delays are a common phenomenon that occur in many chemical processes. Usually, these time delays are an expression of the material transit throughout the chemical process [4]. Therefore, it is important to study ML-based MPCs for systems with time delays taking into account the advantages of LSTM neural networks.

Motivated by the above considerations, we develop a long-short term memory neural network (LSTM)-based MPC schemes for time-delay nonlinear systems in this work. While MPC of stochastic nonlinear systems has been extensively studied in literature, for example, Ref. [5,6], very few research works study ML–based MPC for stochastic nonlinear systems. In this study, we perform a probabilistic closed-loop stability analysis for the time-delayed nonlinear systems under LSTM-MPC. Specifically, a generalization error bound is first derived for LSTM RNN through Rademacher complexity approach [7]. Then, closed-loop stability results are developed for the time-delay nonlinear system. The theoretical study provides a guidance showing how to improve machine learning models in a systematic way in order to achieve desired accuracy in both open-loop and closed-loop simulations. Finally, a chemical reactor is used as an example to illustrate the relation between training sample size and the RNN generalization error as well as the probability of closed-loop system stability.

[1] Limon, D., Calliess, J., & Maciejowski, J. M. (2017). Learning-based nonlinear model predictive control. IFAC-PapersOnLine, 50(1), 7769-7776.

[2] 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.

[3] Graves, A. (2012). Long short-term memory. Supervised sequence labelling with recurrent neural networks, 37-45.

[4] Ellis, M., & Christofides, P. D. (2015). Economic model predictive control of nonlinear time‐delay systems: Closed‐loop stability and delay compensation. AIChE Journal, 61(12), 4152-4165.

[5] Deng, H., Krstic, M., & Williams, R. J. (2001). Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Transactions on automatic control, 46(8), 1237-1253.

[6] Mahmood, M., & Mhaskar, P. (2012). Lyapunov-based model predictive control of stochastic nonlinear systems. Automatica, 48(9), 2271-2276.

[7] Wu, Z., Rincon, D., Gu, Q., & Christofides, P. D. (2021). Statistical Machine Learning in Model Predictive Control of Nonlinear Processes. Mathematics, 9(16), 1912.