Decentralized machine-learning-based predictive control of nonlinear processes

https://doi.org/10.1016/j.cherd.2020.07.019Get rights and content

Highlights

  • Machine-learning modeling of nonlinear chemical processes.

  • Decentralized predictive control using machine-learning models.

  • Characterization of closed-loop stability and performance.

  • Evaluation of the approach using a chemical process example.

Abstract

This work focuses on the design of decentralized model predictive control (MPC) systems for nonlinear processes, where the nonlinear process is broken down into multiple, yet coupled subsystems and the dynamic behavior of each subsystem is described by a machine learning model. One decentralized MPC is designed and used to control each subsystem while accounting for the interactions between subsystems through feedback of the entire process state. The closed-loop stability of the overall nonlinear process network and the performance properties of the decentralized model predictive control system using machine-learning prediction models are analyzed. More specifically, multiple recurrent neural network models suited for each different subsystem need to be trained with a sufficiently small modeling error from their respective actual nonlinear process models to ensure closed-loop stability. These recurrent neural network models are subsequently used as the prediction model in decentralized Lyapunov-based MPCs to achieve efficient real-time computation time while ensuring closed-loop state boundedness and convergence to the origin. The simulation results of a nonlinear chemical process network example demonstrate the effective closed-loop control performance when the process is operated under the decentralized MPCs using the independently-trained recurrent neural network models, as well as the improved computational efficiency compared to the closed-loop simulation of a centralized MPC system.

Introduction

Many industrial systems, such as power distribution grids, mechanical systems and chemical processes, supply chains, and urban traffic networks, are large-scale systems that cannot be handled by using a centralized controller because the system to be controlled is too large and the control problem to be solved is too complex (Bakule, 2008). These challenges cannot be simply solved by using faster computers with larger memory. Decentralized control systems have been proposed to address the concerns associated with control of large-scale systems, including challenges posed by high dimensionality, information structure constraints, uncertainties and delays in the system Bakule (2008). The analysis of the overall system is divided into independent sub-problems that are weakly related to each other to be regulated by separate controllers, which together constitute a decentralized control structure. As such, decentralized control structures provide a practical solution to decoupling large-scale processes and reducing the computational complexity of a centralized control problem. Many recent works have been done on the subject of decentralized control (Hudon and Bao, 2012, Liu et al., 2014, Hioe et al., 2014). For example, augmented estimators for subsystems were designed in Yin et al. (2018) to form a distributed state estimation network from decentralized estimators. A quasi-decentralized control framework was developed in Sun and El-Farra (2008) where the network utilization and communication costs were minimized. With respect to earlier works, in Magni and Scattolini (2006), a stabilizing decentralized model predictive control (MPC) algorithm was presented for nonlinear discrete-time systems subject to decaying disturbances. Moreover, a dynamically-coupled decentralized MPC system for large-scale linear processes subject to input constraints was presented in Alessio et al. (2011), where the global model of the process was approximated by multiple smaller subsystem models and the degree of decoupling was a tunable parameter in the design. Furthermore, a decentralized control strategy using the overlapping decomposition method and using H controllers was applied to a web transport system in Knittel et al. (2002). By only acquiring local output measurements and deciding local control inputs, the main advantages of a decentralized control scheme are the reduced communications and parallel computations.

In many practical problems, a mathematical model based on physical first-principles is not fully known or too complicated to be used as the internal model for model-based control. Nonlinear system identification and an efficient approach to such is a crucial step to designing nonlinear model predictive controllers. While there is no systematic way to parameterize general nonlinear dynamic systems, among existing techniques, the universal approximation properties of neural networks make them a powerful method for modeling nonlinear systems using data (Funahashi and Nakamura, 1993). There are many different structures of neural networks, such as feedforward neural networks (FNN), convolutional neural networks (CNN), and recurrent neural networks (RNN). RNN is itself a class of artificial neural networks (ANNs) that has proven its effectiveness in representing temporal dynamic behaviors of sequential data by taking advantage of feedback signals stored in its network units (Funahashi and Nakamura, 1993, Jin et al., 1995). Many classes and variants of RNNs have been proposed and utilized in various applications, such as model identification, speech recognition, signal processing, and intrusion detection, etc. (Debar and Dorizzi, 1992, Graves et al., 2013, Chung et al., 2014). A continuous-time recurrent neural network is developed in Al Seyab and Cao (2008) and used in the context of nonlinear model predictive control where automatic differentiation techniques are used in both training the neural network model as well as in solving the online optimization problem in the controller. A combination of FNN and RNN is used in Yan and Wang (2012) where unmodeled dynamics as well as unknown higher-order terms from the decomposition of the nonlinear system are modeled by an FNN via supervised learning, and the optimization problem of the nonlinear model predictive control is iteratively solved using a single-layer RNN called the simplified dual network. Three deep neural network structures are trained and evaluated in Ogunmolu et al. (2016) to demonstrate their effectiveness in modeling underlying characteristics of dynamical systems from input-output data. Moreover, in Wu and Christofides (2019), an ensemble of RNN models is developed and used in the design of a Lyapunov-based economic model predictive control to address economic optimality of nonlinear systems. The integration of machine-learning-based modeling methods and various advanced control architectures is a broad field with expanding research scope. Using state-of-the-art machine-learning methods to address the issues of model uncertainties in a decentralized control structure highlights the research interest of this study.

In this work, we introduce decentralized model-based control frameworks, where each decentralized controller employs a Long-Short-Term-Memory (LSTM) network – a particular class of RNN. One decentralized controller is designed and designated to one subsystem of the overall process, and each decentralized controller is designed via Lyapunov-based model predictive control (LMPC) theory (Mhaskar et al., 2006). We analyze the stability properties of the decentralized LMPC system that uses LSTM network models as the prediction model for each subsystem, and then compare the closed-loop performances of the decentralized LMPCs with those using first-principles models as the prediction model, and lastly compare with the closed-loop performance of a centralized LMPC system. The remainder of the paper is organized as follows. Section 2 presents preliminaries on notation, the general class of nonlinear systems considered, and the stabilizability assumptions. An introduction on RNN, specifically the structure and development of LSTM networks, as well as Lyapunov-based control using LSTM networks are presented in Section 3. The formulation and stability proofs of the decentralized LMPC systems using LSTM models are outlined in Section 4. In Section 5, closed-loop simulations of a two-CSTR-in-series process under the decentralized LMPC system are presented.

Section snippets

Notation

Throughout the manuscript, the notation xT is used to denote the transpose of x. · is used to denote the Euclidean norm of a vector. Set subtraction is denoted by “”, i.e., AB{xRn|xA,xB}. A continuous function α:[0,a)[0,) is said to belong to class K if it is strictly increasing and is zero only when evaluated at zero. The function f(·) is of class C1 if it is continuously differentiable in its domain. LfV(x) denotes the standard Lie derivative LfV(x)V(x)xf(x).

Class of systems

Consider a general

Long short-term memory neural network

It has been demonstrated in numerous examples in the literature that recurrent neural network (RNN) models are capable of modeling dynamic behaviors of time-series data and are an effective method to represent nonlinear processes (Su et al., 1992, Funahashi and Nakamura, 1993, Jin et al., 1995). With the use of feedback loops in the network, RNNs store outputs derived from past inputs, and use these previous output information together with current inputs to obtain a more accurate prediction of

Decentralized LMPC using LSTM models

When the optimization problem of a centralized MPC is too complex to solve within a reasonable time period (i.e., the sampling period), the control problem may be decoupled into smaller local optimization problems that are solved in separate processors/controllers to achieve improved computational efficiency. In a decentralized LMPC system, there is no communication between the different local controllers, therefore each controller does not have any knowledge on the control actions calculated

Application to a two-CSTR-in-series process

We use a chemical process example to demonstrate the closed-loop simulation of the nonlinear process under the decentralized Lyapunov-based model predictive control (LMPC) system using the trained LSTM models, and the simulation results will be compared to that of decentralized LMPC system using first-principles models, as well as a centralized model predictive control system using LSTM models. More specifically, two well-mixed, non-isothermal continuous stirred tank reactors (CSTRs) that

Conclusions

In this work, we presented the design of decentralized model predictive control systems for nonlinear processes using machine-learning models, which were developed separately to capture the nonlinear dynamic behavior of independent subsystems. The nonlinear process was divided into multiple subsystems, and the closed-loop stability and performance properties of the proposed decentralized framework with respect to the overall process were analyzed. Using Lypuanov-based control methods to

Declaration of Competing Interest

The authors report no declarations of interest.

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