Online machine learning modeling and predictive control of nonlinear systems with scheduled mode transitions

This work develops a model predictive control (MPC) scheme using online learning of recurrent neural network (RNN) models for nonlinear systems switched between multiple operating regions following a prescribed switching schedule. Specifically, an RNN model is initially developed offline to model process dynamics using the historical operational data collected in a small region around a certain steady-state. After the system is switched to another operating region under a Lyapunov-based MPC with suitable constraints to ensure satisfaction of the prescribed switching schedule policy, RNN models are updated using real-time process data to improve closed-loop performance. A generalization error bound is derived for the updated RNN models using the notion of regret, and closed-loop stability results are established for the switched nonlinear system under RNN-based MPC. Finally, a chemical process example with the operation schedule that requires switching between two steady-states is used to demonstrate the effectiveness of the proposed RNN-MPC scheme.     

Machine learning-based model predictive control of hybrid dynamical systems

This article presents a machine learning-based model predictive control (MPC) scheme for stabilization of hybrid dynamical systems, for which the evolution of states exhibits both continuous and discrete dynamics described by differential and difference equations, respectively. We first present the development of two recurrent neural networks (RNNs) for approximating continuous- and discrete-time dynamics of hybrid dynamical systems, respectively, and then construct a unified hybrid RNN by integrating the two RNN models to capture both continuous and discrete dynamics. The hybrid RNN is used as the prediction model in Lyapunov-based MPC (RNN-LMPC), under which closed-loop stability of hybrid dynamical systems is established. Finally, two case studies including a bouncing ball example and a chemical process are utilized to illustrate the open- and closed-loop performance of the proposed RNN-LMPC scheme.     

Modeling and predictive control of nonlinear processes using transfer learning method

This 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.     

Machine-learning-based construction of barrier functions and models for safe model predictive control

In this paper, we propose a control Lyapunov-barrier function-based model predictive control method utilizing a feed-forward neural network specified control barrier function (CBF) and a recurrent neural network (RNN) predictive model to stabilize nonlinear processes with input constraints, and to guarantee that safety requirements are met for all times. The nonlinear system is first modeled using RNN techniques, and a CBF is characterized by constructing a feed-forward neural network (FNN) model with unique structures and properties. The FNN model for the CBF is trained based on data samples collected from safe and unsafe operating regions, and the resulting FNN model is verified to demonstrate that the safety properties of the CBF are satisfied. Given sufficiently small bounded modeling errors for both the FNN and the RNN models, the proposed control system is able to guarantee closed-loop stability while preventing the closed-loop states from entering unsafe regions in state-space under sample-and-hold control action implementation. We provide the theoretical analysis for bounded unsafe sets in state-space, and demonstrate the effectiveness of the proposed control strategy using a nonlinear chemical process example with a bounded unsafe region.     

Machine learning-based distributed model predictive control of nonlinear processes

This work explores the design of distributed model predictive control (DMPC) systems for nonlinear processes using machine learning models to predict nonlinear dynamic behavior. Specifically, sequential and iterative DMPC systems are designed and analyzed with respect to closed-loop stability and performance properties. Extensive open-loop data within a desired operating region are used to develop long short-term memory (LSTM) recurrent neural network models with a sufficiently small modeling error from the actual nonlinear process model. Subsequently, these LSTM models are utilized in Lyapunov-based DMPC to achieve efficient real-time computation time while ensuring closed-loop state boundedness and convergence to the origin. Using a nonlinear chemical process network example, the simulation results demonstrate the improved computational efficiency when the process is operated under sequential and iterative DMPCs while the closed-loop performance is very close to the one of a centralized MPC system.     

Statistical machine-learning–based predictive control of uncertain nonlinear processes

In this study, we present machine-learning–based predictive control schemes for nonlinear processes subject to disturbances, and establish closed-loop system stability properties using statistical machine learning theory. Specifically, we derive a generalization error bound via Rademacher complexity method for the recurrent neural networks (RNN) that are developed to capture the dynamics of the nominal system. Then, the RNN models are incorporated in Lyapunov-based model predictive controllers, under which we study closed-loop stability properties for the nonlinear systems subject to two types of disturbances: bounded disturbances and stochastic disturbances with unbounded variation. A chemical reactor example is used to demonstrate the implementation and evaluate the performance of the proposed approach.     

Koopman Lyapunov-based model predictive control of nonlinear chemical process systems

In this work, we propose the integration of Koopman operator methodology with Lyapunov-based model predictive control (LMPC) for stabilization of nonlinear systems. The Koopman operator enables global linear representations of nonlinear dynamical systems. The basic idea is to transform the nonlinear dynamics into a higher dimensional space using a set of observable functions whose evolution is governed by the linear but infinite dimensional Koopman operator. In practice, it is numerically approximated and therefore the tightness of these linear representations cannot be guaranteed which may lead to unstable closed-loop designs. To address this issue, we integrate the Koopman linear predictors in an LMPC framework which guarantees controller feasibility and closed-loop stability. Moreover, the proposed design results in a standard convex optimization problem which is computationally attractive compared to a nonconvex problem encountered when the original nonlinear model is used. We illustrate the application of this methodology on a chemical process example.     

Lyapunov‐based MPC with robust moving horizon estimation and its triggered implementation

In this work, we consider moving horizon state estimation (MHE)‐based model predictive control (MPC) of nonlinear systems. Specifically, we consider the Lyapunov‐based MPC (LMPC) developed in (Mhaskar et al., IEEE Trans Autom Control. 2005;50:1670–1680; Syst Control Lett. 2006;55:650–659) and the robust MHE (RMHE) developed in (Liu J, Chem Eng Sci. 2013;93:376–386). First, we focus on the case that the RMHE and the LMPC are evaluated every sampling time. An estimate of the stability region of the output control system is first established; and then sufficient conditions under which the closed‐loop system is guaranteed to be stable are derived. Subsequently, we propose a triggered implementation strategy for the RMHE‐based LMPC to reduce its computational load. The triggering condition is designed based on measurements of the output and its time derivatives. To ensure the closed‐loop stability, the formulations of the RMHE and the LMPC are also modified accordingly to account for the potential open‐loop operation. A chemical process is used to illustrate the proposed approaches. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4273–4286, 2013     

Data-driven prediction model of indoor air quality in an underground space

Several data-driven prediction methods based on multiple linear regression (MLR), neural network (NN), and recurrent neural network (RNN) for the indoor air quality in a subway station are developed and compared. The RNN model can predict the air pollutant concentrations at a platform of a subway station by adding the previous temporal information of the pollutants on yesterday to the model. To optimize the prediction model, the variable importance in the projection (VIP) of the partial least squares (PLS) is used to select key input variables as a preprocessing step. The prediction models are applied to a real indoor air quality dataset from telemonitoring systems data (TMS), which exhibits some nonlinear dynamic behaviors show that the selected key variables have strong influence on the prediction performances of the models. It demonstrates that the RNN model has the ability to model the nonlinear and dynamic system, and the predicted result of the RNN model gives better modeling performance and higher interpretability than other data-driven prediction models.     

Nonlinear model predictive control: current status and future directions

Linear model predictive control (LMPC) is well established as the industry standard for controlling constrained multivariable processes. A major limitation of LMPC is that plant behavior is described by linear dynamic models. As a result, LMPC is inadequate for highly nonlinear processes and moderately nonlinear processes which have large operating regimes. This shortcoming coupled with increasingly stringent demands on throughput and product quality has spurred the development of nonlinear model predictive control (NMPC). NMPC is conceptually similar to its linear counterpart except that nonlinear dynamic models are used for process prediction and optimization. The purpose of this paper is to provide an overview of current NMPC technology and applications, as well as to propose topics for future research and development. The review demonstrates that NMPC is well suited for controlling multivariable nonlinear processes with constraints, but several theoretical and practical issues must be resolved before widespread industrial acceptance is achieved.     

Data‐driven model predictive quality control of batch processes

The problem of driving a batch process to a specified product quality using data‐driven model predictive control (MPC) is described. To address the problem of unavailability of online quality measurements, an inferential quality model, which relates the process conditions over the entire batch duration to the final quality, is required. The accuracy of this type of quality model, however, is sensitive to the prediction of the future batch behavior until batch termination. In this work, we handle this “missing data” problem by integrating a previously developed data‐driven modeling methodology, which combines multiple local linear models with an appropriate weighting function to describe nonlinearities, with the inferential model in a MPC framework. The key feature of this approach is that the causality and nonlinear relationships between the future inputs and outputs are accounted for in predicting the final quality and computing the manipulated input trajectory. The efficacy of the proposed predictive control design is illustrated via closed‐loop simulations of a nylon‐6,6 batch polymerization process with limited measurements. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2852–2861, 2013     

Application of state estimation based NMPC to an unstable nonlinear process

Model predictive control (MPC) has become very popular both in process industry and academia due to its effectiveness in dealing with nonlinear, multivariable and/or hard-constrained plants.Although linear MPC can be applied for controlling nonlinear processes by obtaining a linearized model of the plant, this is only valid in a limited region. Therefore, a substantial improvement can be achieved by using the whole knowledge of the process dynamics, specially in the presence of marked nonlinearities. This effect can be strong if the process to control is open-loop unstable.The purpose of this paper is to introduce a nonlinear model predictive controller (NMPC) based on nonlinear state estimation, in order to exploit the knowledge of the nonlinear dynamics and to avoid modeling simplifications or linearization.A state-space formulation is proposed to achieve the control objective. To update the optimization involved in NMPC strategy, state estimation based on the measured outputs is proposed.As a particular application, we consider an open-loop unstable jacketed exothermic chemical reactor. This CSTR is widely recognized as a difficult problem for the purpose of control. In order to achieve the control goal, a NMPController coupled with a state observer are designed. The observer is also used to estimate some unmeasured disturbances. Finally, computer simulations are developed for showing the performance of both the nonlinear observer and the control strategy.     

Closed-loop dynamic real-time optimization with stabilizing model predictive control

Dynamic real-time optimization (DRTO) is a supervisory strategy at the upper level of the industrial process automation architecture that computes economically optimal set-point trajectories that are in turn passed on to the lower-level model predictive control (MPC) for tracking. The economically optimal solution, in several process industries, could lead to operating the plant at or around an unstable steady state. The present article accounts for this by developing a closed-loop DRTO (CL-DRTO) formulation that enables handling unstable operating points via an underlying MPC with stability constraints. To this end, a stabilizing MPC that handles trajectory tracking for unstable systems is embedded within the upper-level DRTO. The resulting CL-DRTO problem is reformulated by applying a simultaneous solution approach. The economic benefits realized by the proposed strategy are illustrated through applications to both linearized and nonlinear dynamic models for single-input single-output and multi-input multi-output continuous stirred tank reactor case studies.     

Encrypted model predictive control design for security to cyberattacks

In recent years, cyber-security of networked control systems has become crucial, as these systems are vulnerable to targeted cyberattacks that compromise the stability, integrity, and safety of these systems. In this work, secure and private communication links are established between sensor–controller and controller–actuator elements using semi-homomorphic encryption to ensure cyber-security in model predictive control (MPC) of nonlinear systems. Specifically, Paillier cryptosystem is implemented for encryption-decryption operations in the communication links. Cryptosystems, in general, work on a subset of integers. As a direct consequence of this nature of encryption algorithms, quantization errors arise in the closed-loop MPC of nonlinear systems. Thus, the closed-loop encrypted MPC is designed with a certain degree of robustness to the quantization errors. Furthermore, the trade-off between the accuracy of the encrypted MPC and the computational cost is discussed. Finally, two chemical process examples are employed to demonstrate the implementation of the proposed encrypted MPC design.     

Actuator stiction compensation via model predictive control for nonlinear processes

The problem of valve stiction is addressed, which is a nonlinear friction phenomenon that causes poor performance of control loops in the process industries. A model predictive control (MPC) stiction compensation formulation is developed including detailed dynamics for a sticky valve and additional constraints on the input rate of change and actuation magnitude to reduce control loop performance degradation and to prevent the MPC from requesting physically unrealistic control actions due to stiction. Although developed with a focus on stiction, the MPC‐based compensation method presented is general and has potential to compensate for other nonlinear valve dynamics which have some similarities to those caused by stiction. Feasibility and closed‐loop stability of the proposed MPC formulation are proven for a sufficiently small sampling period when Lyapunov‐based constraints are incorporated. Using a chemical process example with an economic model predictive controller (EMPC), the selection of appropriate constraints for the proposed method is demonstrated. The example verified the incorporation of the stiction dynamics and actuation magnitude constraints in the EMPC causes it to select set‐points that the valve output can reach and causes the operating constraints to be met. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2004–2023, 2016     

Robust PID controller design for nonlinear processes using JITL technique

A robust PID controller design methodology for nonlinear processes is proposed based on the just-in-time learning (JITL) technique. To do so, a composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter to approximate the inevitable modeling error caused by the process nonlinearity, is employed to model the process dynamics in the operating space of interest. The state space realizations of this composite model and PID controller are then reformulated as an uncertain closed-loop system, by which the corresponding robust stability condition is developed. Literature examples are employed to illustrate the proposed methodology and a comparison with the previous result is made.     

Bounded robust control of constrained multivariable nonlinear processes

This work focuses on control of multi-input multi-output (MIMO) nonlinear processes with uncertain dynamics and actuator constraints. A Lyapunov-based nonlinear controller design approach that accounts explicitly and simultaneously for process nonlinearities, plant-model mismatch, and input constraints, is proposed. Under the assumption that all process states are accessible for measurement, the approach leads to the explicit synthesis of bounded robust multivariable nonlinear state feedback controllers with well-characterized stability and performance properties. The controllers enforce stability and robust asymptotic reference-input tracking in the constrained uncertain closed-loop system and provide, at the same time, an explicit characterization of the region of guaranteed closed-loop stability. When full state measurements are not available, a combination of the state feedback controllers with high-gain state observes and appropriate saturation filters, is employed to synthesize bounded robust multivariable output feedback controllers that require only measurements of the outputs for practical implementation. The resulting output feedback design is shown to inherit the same closed-loop stability and performance properties of the state feedback controllers and, in addition, recover the closed-loop stability region obtained under state feedback, provided that the observer gain is sufficiently large. The developed state and output feedback controllers are applied successfully to non-isothermal chemical reactor examples with uncertainty, input constraints, and incomplete state measurements. Finally, we conclude the paper with a discussion that attempts to put in perspective the proposed Lyapunov-based control approach with respect to the nonlinear model predictive control (MPC) approach and discuss the implications of our results for the practical implementation of MPC, in control of uncertain nonlinear processes with input constraints.     

A cyber-secure control-detector architecture for nonlinear processes

This work presents a detector-integrated two-tier control architecture capable of identifying the presence of various types of cyber-attacks, and ensuring closed-loop system stability upon detection of the cyber-attacks. Working with a general class of nonlinear systems, an upper-tier Lyapunov-based Model Predictive Controller (LMPC), using networked sensor measurements to improve closed-loop performance, is coupled with lower-tier cyber-secure explicit feedback controllers to drive a nonlinear multivariable process to its steady state. Although the networked sensor measurements may be vulnerable to cyber-attacks, the two-tier control architecture ensures that the process will stay immune to destabilizing malicious cyber-attacks. Data-based attack detectors are developed using sensor measurements via machine-learning methods, namely artificial neural networks (ANN), under nominal and noisy operating conditions, and applied online to a simulated reactor-reactor-separator process. Simulation results demonstrate the effectiveness of these detection algorithms in detecting and distinguishing between multiple classes of intelligent cyber-attacks. Upon successful detection of cyber-attacks, the two-tier control architecture allows convenient reconfiguration of the control system to stabilize the process to its operating steady state.     

Use of bifurcation analysis for development of nonlinear models for control applications

Nonlinear system identification poses challenging questions because a closed general theory is not available for this field. Particularly, nonlinear models based on neural networks (NN) may present incompatible general dynamic process behavior, leading to improper closed-loop responses, even when they allow for satisfactory one step ahead prediction of process dynamics, as required by traditional validation methods. It is shown here that performing detailed bifurcation and stability analysis may be very helpful for the adequate development and implementation of nonlinear models and model based controllers. The study of many parameters that are defined a priori during the training of the NN shows that the spurious dynamic behavior is related mostly to the use of incomplete data sets during the learning process. This is an indication that, for each kind of process, the number, range and distribution of the data points in the operation region of interest are of paramount importance for proper training of the nonlinear model. Strategies to improve the quality of the training procedure are provided and analyzed both theoretically and experimentally, using the solution polymerization of styrene in a tubular reactor as a case study.