This article focuses on the design of model predictive control (MPC) systems for nonlinear processes that utilize an ensemble of recurrent neural network (RNN) models to predict nonlinear dynamics. Specifically, RNN models are initially developed based on a data set generated from extensive open-loop simulations within a desired process operation region to capture process dynamics with a sufficiently small modeling error between the RNN model and the actual nonlinear process model. Subsequently, Lyapunov-based MPC (LMPC) that utilizes RNN models as the prediction model is developed to achieve closed-loop state boundedness and convergence to the origin. Additionally, machine learning ensemble regression modeling tools are employed in the formulation of LMPC to improve prediction accuracy of RNN models and overall closed-loop performance while parallel computing is utilized to reduce computation time. Computational implementation of the method and application to a chemical reactor example is discussed in the second article of this series.

Machine learning is receiving more attention in classical engineering fields, and in particular, recurrent neural networks (RNNs) coupled with ensemble regression tools have demonstrated the capability of modeling nonlinear dynamic processes. In Part I of this two-article series, the Lyapunov-based model predictive control (LMPC) method using a single RNN model and an ensemble of RNN models, respectively, was rigorously developed for a general class of nonlinear systems. In the present article, computational implementation issues of this new control method ranging from training of the RNN models, ensemble regression of the RNN models, and parallel computation for accelerating the real-time model calculations are addressed. Furthermore, a chemical reactor example is used to demonstrate the implementation and effectiveness of these machine-learning tools in LMPC as well as compare them with standard state-space model identification tools.

Recurrent neural networks (RNNs) have been widely used to model nonlinear dynamic systems using time-series data. While the training error of neural networks can be rendered sufficiently small in many cases, there is a lack of a general framework to guide construction and determine the generalization accuracy of RNN models to be used in model predictive control systems. In this work, we employ statistical machine learning theory to develop a methodological framework of generalization error bounds for RNNs. The RNN models are then utilized to predict state evolution in model predictive controllers (MPC), under which closed-loop stability is established in a probabilistic manner. A nonlinear chemical process example is used to investigate the impact of training sample size, RNN depth, width, and input time length on the generalization error, along with the analyses of probabilistic closed-loop stability through the closed-loop simulations under Lyapunov-based MPC.

We present a machine learning-based predictive control scheme that integrates an online update of the recurrent neural network (RNN) models to capture process nonlinear dynamics in the presence of model uncertainty. Specifically, an ensemble of the RNN models are initially obtained for the nominal system, for which Lyapunov-based model predictive control (LMPC) is utilized to drive the state to the steady-state, and economic Lyapunov-based MPC (LEMPC) is applied to achieve closed-loop stability and economic optimality simultaneously. Subsequently, an event-trigger mechanism based on the decreasing rate of Lyapunov function and an error-trigger mechanism that relies on prediction errors are developed for an online model update, in which the most recent process data are utilized to derive a new ensemble of RNN models with enhanced prediction accuracy. By incorporating the event and error-triggered online RNN update within real-time machine learning-based LMPC and LEMPC, process dynamic performance is improved in terms of guaranteed closed-loop stability, optimality, and smoothness of control actions. The proposed methodology is applied to a chemical process example with time-varying disturbances under LMPC and LEMPC, respectively, to demonstrate the effectiveness of an online update of machine learning models in real-time control problems.

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