Predictive Control Of Nonlinear Processes Research Articles

Machine learning-based input-augmented Koopman modeling and predictive control of nonlinear processes

Koopman-based modeling and model predictive control have been a promising alternative for optimal control of nonlinear processes. Good Koopman modeling performance significantly depends on an appropriate nonlinear mapping from the original state-space to a lifted state space. In this work, we propose an input-augmented Koopman modeling and model predictive control approach. Both the states and the known inputs are lifted using two deep neural networks (DNNs), and a Koopman model with nonlinearity in inputs is trained within the higher-dimensional state space. A Koopman-based model predictive control problem is formulated. To bypass non-convex optimization induced by the nonlinearity in the Koopman model, we further present an iterative implementation algorithm, which approximates the optimal control input via solving a convex optimization problem iteratively. The proposed method is applied to a chemical process and a biological water treatment process via simulations. The efficacy and advantages of the proposed modeling and control approach are demonstrated.

Explicit machine learning-based model predictive control of nonlinear processes via multi-parametric programming

Machine learning-based model predictive control (ML-MPC) has been developed to control nonlinear processes with unknown first-principles models. While ML models can capture nonlinear dynamics of complex systems, the complexity of ML models leads to increased computation time for real-time implementation of ML-MPC. To address this issue, in this work, we propose an explicit ML-MPC framework for nonlinear processes using multi-parametric programming. Specifically, a self-adaptive approximation algorithm is first developed to obtain a piecewise linear affine function that approximates the behaviors of ML models. Then, multi-parametric quadratic programming (mpQP) problems are formulated to generate the solution map for states in discretized state–space. Furthermore, to accelerate the implementation of explicit ML-MPC, a neighbor-first search algorithm is developed. Finally, an example of a chemical reactor is used to demonstrate the effectiveness of the explicit ML-MPC.

Model predictive control of nonlinear processes using transfer learning-based recurrent neural networks

Artificial neural networks (ANNs), one of the deep learning techniques that has sparked a lot of attention recently for its exceptional modeling capabilities of nonlinear systems, are an essential candidate for model-based control systems. In particular, recurrent neural networks (RNN) have shown remarkable capacity to forecast the dynamic behavior of nonlinear processes using time-series data. In order to regulate the rate at which chemical species are produced in intricate processes, RNNs have successfully served as the predictive model in model-based controllers. However, the necessity for massive amounts of data to capture complex processes is a drawback of typical RNN models. With the goal of efficient utilization of the available data, a different, transfer learning-based training approach for RNNs is presented in this work. The transfer learning strategy is taken into consideration to overcome the challenges stemming from lack of data for the construction of RNN models. In particular, weight-sharing RNNs are developed using a priori physical knowledge. Next, a comprehensive analysis of a complex chemical process on a large scale is conducted to showcase the benefits of the suggested approach across various data sizes and its effectiveness when combined with MPC in contrast to conventional RNNs.

Encrypted decentralized model predictive control of nonlinear processes with delays

This work focuses on enhancing the operational safety, cybersecurity, computational efficiency, and closed-loop performance of large-scale nonlinear time-delay systems. This is achieved by employing a decentralized model predictive controller (MPC) with encrypted networked communication. Within this decentralized setup, the nonlinear process is partitioned into multiple subsystems, each controlled by a distinct Lyapunov-based MPC. These controllers take into account the interactions between subsystems by utilizing full state feedback, while computing the control inputs only corresponding to their respective subsystem. To address the performance degradation associated with input delays, we integrate a predictor with each LMPC to compute the states after the input delay period. The LMPC model is initialized with these predicted states. To cope with state delays, the LMPC model is formulated using differential difference equations (DDEs) that describe the state-delays in the system. Further, to enhance cybersecurity, all signals transmitted to and received from each subsystem are encrypted. A stability analysis is carried out for the encrypted decentralized MPC when it is utilized in a time-delay system. Bounds are set up for the errors arising from encryption, state-delays, and sample-and-hold implementation of the controller. Guidelines are established to implement this proposed control structure in any nonlinear time-delay system. The simulation results, conducted on a nonlinear chemical process network, illustrate the effective closed-loop performance of the decentralized MPCs alongside the predictor with encrypted communication when dealing with input and state delays in a large-scale process.

Reduced-order Koopman modeling and predictive control of nonlinear processes

In this paper, we propose an efficient data-driven predictive control approach for general nonlinear processes based on a reduced-order Koopman operator. A Kalman-based sparse identification of nonlinear dynamics method is employed to select lifting functions for Koopman identification. The selected lifting functions are used to project the original nonlinear state–space into a higher-dimensional linear function space, in which Koopman-based linear models can be constructed for the underlying nonlinear process. To curb the significant increase in the dimensionality of the resulting full-order Koopman models caused by the use of lifting functions, we propose a reduced-order Koopman modeling approach based on proper orthogonal decomposition. A computationally efficient linear robust predictive control scheme is established based on the reduced-order Koopman model. A case study on a benchmark chemical process is conducted to illustrate the effectiveness of the proposed method. Comprehensive comparisons are conducted to demonstrate the advantage of the proposed method.

Model predictive control of nonlinear processes using neural ordinary differential equation models

Neural Ordinary Differential Equation (NODE) is a recently proposed family of deep learning models that can perform a continuous approximation of a linear/nonlinear dynamic system using time-series data by integrating the neural network model with classical ordinary differential equation solvers. Modeling nonlinear dynamic processes using data has historically been a critical challenge in the field of chemical engineering research. With the development of computer science and neural network technology, recurrent neural networks (RNN) have become a popular black-box approach to accomplish this task and have been utilized to design model predictive control (MPC) systems. However, as a discrete-time approximation model, RNN requires strictly uniform step sequence data to operate, which makes it less robust to an irregular sampling scenario, such as missing data points during operation due to sensor failure or other types of random errors. This paper aims to develop a NODE model and construct an MPC based on this novel continuous-time neural network model. In this context, closed-loop stability is established and robustness to noise is addressed using a variety of data analysis techniques. An example of a chemical process is utilized to evaluate the performance of NODE-based MPC. Furthermore, the performance of the NODE-based MPC under Gaussian and non-Gaussian noise is investigated, and the subsampling method is found to be effective in building models for MPC that are suitable for handling the presence of non-Gaussian noise in the data.

Real-time adaptive sparse-identification-based predictive control of nonlinear processes

This study introduces a sparse identification-based model predictive control (MPC) framework that incorporates on-line updates of the sparse-identified model to account for nonlinear dynamics and model uncertainty in process systems. The methodology involves obtaining a nonlinear first-order ordinary differential equation model using sparse identification for nonlinear dynamics (SINDy), which is integrated into two control schemes: Lyapunov-based MPC (LMPC) for achieving steady-state operation and Lyapunov-based economic MPC (LEMPC) for achieving both closed-loop stability and optimal economic performance. To improve prediction accuracy, an on-line model update scheme is proposed for the SINDy models. Specifically, an error-trigger mechanism that utilizes prediction errors and then uses the most recent process data to update the parameters of the SINDy model in real-time is designed. By incorporating the error-triggered on-line model updates in the SINDy-based LMPC and LEMPC, the dynamic performance of the process is enhanced, ensuring closed-loop stability, optimality, and smooth control actions. Following theoretical results on the boundedness of the closed-loop states and detailed discussions on the selection criteria for parameters of the error-triggered SINDy update scheme, the effectiveness of the proposed methodology is demonstrated through a chemical process example with time-varying disturbances under the LEMPC framework.

Physics-informed recurrent neural network modeling for predictive control of nonlinear processes

In this work, we present a physics-informed recurrent neural network (PIRNN) modeling approach, and a PIRNN-based predictive control scheme for a general class of nonlinear dynamic systems. Specifically, we first develop a hybrid data-driven and physics-guided modeling framework that integrates measurement data and mechanistic mathematical models to construct high-fidelity RNN models. Then, we derive a generalization error bound of the PIRNN model based on a nominal system model via the Rademacher complexity technique from statistical machine learning theory. Subsequently, the PIRNN model is utilized in Lyapunov-based model predictive controllers and applied to a chemical reactor example with Gaussian measurement noise to demonstrate its improved noise rejection and generalization performance in comparison to the purely data-driven and the purely physics-guided RNN-based predictive control schemes.

Modeling and predictive control of nonlinear processes using transfer learning method

AbstractThis work develops a transfer learning (TL) framework for modeling and predictive control of nonlinear systems using recurrent neural networks (RNNs) with the knowledge obtained in modeling one process transferred to another. Specifically, transfer learning uses a pretrained model developed based on a source domain as the starting point, and adapts the model to a target process with similar configurations. The generalization error for TL‐based RNN (TL‐RNN) is first derived to demonstrate the generalization capability on the target process. The theoretical error bound that depends on model capacity and the discrepancy between source and target domains is then utilized to guide the development of pretrained models for improved model transferability. Subsequently, the TL‐RNN model is utilized as the prediction model in model predictive controller (MPC) for the target process. Finally, the simulation study of chemical reactors via Aspen Plus Dynamics is used to demonstrate the benefits of transfer learning.

Physics-Informed Online Machine Learning and Predictive Control of Nonlinear Processes with Parameter Uncertainty

Physics-Informed Online Machine Learning and Predictive Control of Nonlinear Processes with Parameter Uncertainty

A contraction-constrained economic Model Predictive Control for nonlinear processes using multi-timescale models

Many chemical processes exhibit diverse timescale dynamics with a strong coupling between timescale sensitive variables. Model predictive control with a non-uniformly spaced optimisation horizon is an effective approach to multi-timescale control and offers opportunities for reduced computational complexity. In such an approach, dynamics with different timescales can be included in an optimisation problem by implementing faster timescale models (with shorter time intervals) earlier in the prediction horizon and increasingly slower timescale models (with longer time intervals) towards the end. In this paper, a reference-flexible condition is developed based on contraction theory to provide a stability guarantee for cost minimising control of multiple timescale nonlinear systems using multiple predictive models.

On generalization error of neural network models and its application to predictive control of nonlinear processes

In order to approximate nonlinear dynamic systems utilizing time-series data, recurrent neural networks (RNNs) and long short-term memory (LSTM) networks have frequently been used. The training error of neural networks may often be made suitably modest; however, the accuracy can be further improved by incorporating prior knowledge in the construction of machine learning-based models. Specifically, physics-based RNN modeling has yielded more reliable RNN models than traditional RNNs. Yet, a framework for constructing and assessing the generalization ability of such RNN models as well as LSTM models to be utilized in model predictive control (MPC) systems is lacking. In this work, we develop a methodological framework to quantify the generalization error bounds for partially-connected RNNs and LSTM models. The partially-connected RNN model is then utilized to predict the state evolution in a MPC scheme. We illustrate through open-loop and closed-loop simulations of a nonlinear chemical process of two reactors-in-series that the proposed approach provides a flexible framework for leveraging both prior knowledge and data, thereby improving the performance significantly when compared to a fully-connected modeling approach under Lyapunov-based MPC.

Feature selection-based machine learning modeling for distributed model predictive control of nonlinear processes

In this work, we develop reduced-order machine learning models using feature selection methods for distributed model predictive control (DMPC) of nonlinear processes. Specifically, filter, wrapper, and embedded methods for feature selection are first utilized to select a subset of input features that significantly impacts the prediction of system output. The feature selection methods are then integrated with the development of reduced-order recurrent neural network (RNN) models that capture the system dynamics using only the selected input features. Subsequently, the reduced-order RNN models are incorporated into sequential and iterative DMPCs to stabilize the nonlinear system at the steady-state. Finally, a reactor–reactor–separator process example is used to demonstrate that the DMPC using reduced-order RNN models achieves the desired closed-loop performance with improved computational efficiency.

Machine learning-based reduced-order modeling and predictive control of nonlinear processes

In this work, we develop a model predictive control scheme for nonlinear systems using autoencoder-based reduced-order machine learning models. First, an autoencoder is developed for model order reduction by projecting the process states onto a low-dimensional space using data generated from open-loop simulations of the nonlinear system in the original high-dimensional space. Subsequently, reduced-order recurrent neural networks (RNN) are developed to capture the dominant dynamics of the nonlinear system using the low-dimensional data. Lyapunov-based model predictive control (MPC) scheme using RNN models in low-dimensional space is developed to stabilize the nonlinear system. Finally, a diffusion-reaction process example is used to demonstrate the effectiveness of the proposed reduced-order RNN modeling approach and RNN-based predictive control method.

Statistical Machine Learning in Model Predictive Control of Nonlinear Processes

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.

Machine learning modeling and predictive control of nonlinear processes using noisy data

AbstractThis work focuses on machine learning modeling and predictive control of nonlinear processes using noisy data. We use long short‐term memory (LSTM) networks with training data from sensor measurements corrupted by two types of noise: Gaussian and non‐Gaussian noise, to train the process model that will be used in a model predictive controller (MPC). We first discuss the LSTM training with noisy data following a Gaussian distribution, and demonstrate that the standard LSTM network is capable of capturing the underlying process dynamic behavior by reducing the impact of noise. Subsequently, given that the standard LSTM performs poorly on a noisy data set from industrial operation (i.e., non‐Gaussian noisy data), we propose an LSTM network using Monte Carlo dropout method to reduce the overfitting to noisy data. Furthermore, an LSTM network using co‐teaching training method is proposed to further improve its approximation performance when noise‐free data from a process model capturing the nominal process state evolution is available. A chemical process example is used throughout the manuscript to illustrate the application of the proposed modeling approaches and demonstrate their open‐ and closed‐loop performance under a Lyapunov‐based model predictive controller with state measurements corrupted by industrial noise.

Machine-learning-based state estimation and predictive control of nonlinear processes

Machine learning techniques have demonstrated their capability in capturing dynamic behavior of complex, nonlinear chemical processes from operational data. As full state measurements may be unavailable in chemical plants, this work proposes two machine-learning-based state estimation approaches. The first approach integrates recurrent neural networks (RNN) within the extended Luenberger observer framework to develop data-based state estimators. The second approach utilizes a hybrid model that integrates feed-forward neural networks with first-principles models to capture process dynamics in the state estimator. Then, an output feedback model predictive controller is designed based on the state estimates provided by the machine-learning-based estimators to stabilize the closed-loop system at the steady-state. A chemical process example is utilized to illustrate the effectiveness of the proposed machine-learning-based state estimation and control approaches.

Fractional-order multi-model predictive control for nonlinear processes

Fractional-order multi-model predictive control for nonlinear processes

Co-Teaching Approach to Machine Learning-based Predictive Control of Nonlinear Processes

Abstract Machine learning modeling of chemical processes using noisy data is practically a challenging task due to the occurrence of overfitting during learning. In this work, we propose a co-teaching learning algorithm that develops Long short-term memory (LSTM) networks to capture the ground truth (i.e., underlying process dynamics) from noisy data. We consider an industrial chemical reactor example and use Aspen Plus Dynamics to generate process operational data that is corrupted by sensor noise generated by industrial noisy measurements. An LSTM model is developed using the co-teaching method with additional noise-free data generated from simulations of the reactor first-principles model. Through open-loop and closed-loop simulations, we demonstrate that compared to the LSTM model developed from the standard training process, the co-teaching LSTM model is more accurate in predicting process dynamics, and therefore, achieves better closed-loop performance under model predictive control.

Fractional-order multi-model predictive control for nonlinear processes

This paper proposes a novel fractional-order multi-model predictive control (FO-MMPC) design to deal with a class of nonlinear fractional order systems. Based on the assumption that the plant is governed by a fractional order nonlinear dynamical model, we are able for each operating region, to determine the linear portion of the nonlinear process as a single fractional order pole transfer function. The predictive model is then approximated by rational transfer functions using the singularity function approach in the frequency domain. The main contribution in this control approach is the use of a switching algorithm between fractional order prediction models that are approximating the nonlinear system dynamics for different operating points and input range intervals. By using the FO-MMPC for controlling the level of a conical tank system with nonlinear dynamics using a multiple fractional-order predictive models, it is shown by means of numerical simulations the effectiveness of the proposed control strategy.