Real-time machine learning for operational safety of nonlinear processes via barrier-function based predictive control

doi:10.1016/j.cherd.2020.01.007

Chemical Engineering Research and Design

Volume 155, March 2020, Pages 88-97

https://doi.org/10.1016/j.cherd.2020.01.007 Get rights and content

Highlights

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Operational safety of nonlinear processes via barrier function-based model predictive control.
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Data-based recurrent neural network modeling for model predictive control.
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Real-time update of the recurrent neural network model using process measurements.
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Evaluation of approach using a chemical reactor example.

Abstract

This work proposes a real-time model predictive control (MPC) system using control Lyapunov–barrier functions (CLBF) and recurrent neural network (RNN) models to ensure simultaneous closed-loop stability and operational safety for a general class of nonlinear systems subject to time-varying disturbances. An RNN model is first developed for the nominal system (i.e., without disturbances) and incorporated in the designs of CLBF-based MPC and of CLBF-based economic MPC (EMPC) to provide state predictions for the optimization problems of MPCs. Subsequently, to improve the closed-loop performance in terms of operational safety and stability in the presence of disturbances, online learning of RNN models is incorporated within the real-time implementation of CLBF-MPC and of CLBF-EMPC to update the RNN models using the most recent process measurement data. The proposed adaptive machine-learning-based CLBF-MPC and CLBF-EMPC schemes are evaluated using a nonlinear chemical process example.

Introduction

Recurrent neural networks (RNN) have demonstrated their capability in approximating complex, nonlinear dynamic systems in the discipline of data-driven modeling, (e.g., You and Nikolaou, 1993, Kosmatopoulos et al., 1995, Trischler and D’Eleuterio, 2016). As RNNs are able to capture temporal dynamic behavior via the feedback loops that bring past neuron information into the current network, they have been utilized to construct dynamic process models for model-based optimization of chemical processes when a first-principles model is unavailable. For example, RNN modeling methods have been recently incorporated in the design of model predictive control (MPC) to operate a nonlinear process at the steady-state while optimizing process dynamic performance. In Wu et al. (2019c), it is demonstrated that the desired closed-loop performance in terms of guaranteed stability, optimal response, and smooth control actions can be achieved under the MPC using an ensemble of RNN models that well capture process nonlinear dynamics. In Wong et al. (2018), RNN models were utilized to provide a dynamic process model for MPC to operate a continuous reactor for pharmaceutical manufacturing. In addition to process stability, another issue that is of significant importance in chemical process operation and has attracted a lot of attention in the engineering community is process operational safety. To ensure that a process is being operated in safe operating conditions for all times, safety protection systems including process control systems, alarms systems and emergency shutdown systems have been developed and widely-used in industry. Specifically, advanced process control systems at the lowest level of safety protection system need to be designed to account for safety considerations such as operating the system safely and avoiding triggering alarms systems too frequently.

To that end, control Lyapunov–barrier functions (CLBF) (Romdlony and Jayawardhana, 2016) that are designed based on control Lyapunov functions and control barrier functions (Ames et al., 2016; Jankovic, 2017) have been utilized to design the safety constraints for MPC to stabilize the closed-loop state at its steady-state and avoid unsafe operating regions in state-space. It has been demonstrated in Wu et al. (2019a) that under CLBF-MPC, closed-loop stability and process operational safety can be achieved for nonlinear processes with guaranteed recursive feasibility of the optimization problem of CLBF-MPC. Additionally, in Wu and Christofides (2019b), CLBF-based constraints are incorporated in the design of machine learning-based EMPC to achieve process operational safety and economic optimality simultaneously by dynamically operating the system in a stability region.

Despite the successful applications of machine-learning-based controllers for closed-loop stability and process operational safety, there is an increasing need to address online learning of machine learning models as all real-world processes are changing over time due to external disturbances and internal variations (e.g., catalyst deactivation) (Valappil and Georgakis, 2000, Marquardt, 2002). For example, an event-triggered feedback system was design in Wang and Lemmon (2008) to stabilize a nonlinear system by updating control actions when the stability condition is violated. Additionally, the event-triggered mechanism has also been incorporated in the design of neural network-based control schemes in Sahoo et al. (2015), Liu and Yang (2018), Wu et al. (2019b). For example, adaptive machine-learning-based MPC has been developed in Wu et al. (2019b) to improve closed-loop performances by updating RNN models using real-time process data for nonlinear processes subject to time-varying disturbances.

Motivated by the above, in this work, we develop a real-time adaptive CLBF-based MPC scheme that updates RNN models following the event-triggered and error-triggered mechanisms that have been proposed in Wu et al. (2019b). Specifically, an RNN-based CLBF-MPC and an RNN-based CLBF-EMPC are first developed following Wu et al. (2019a) and Wu and Christofides (2019b) to derive closed-loop stability and safety for the nominal system. Subsequently, online learning of RNN models is incorporated in the real-time implementation of CLBF-MPC and of CLBF-EMPC to adapt the RNN models to time-varying disturbances such that the system can be operated safely in the sense that the closed-loop state can be driven to its steady-state under CLBF-MPC, and can be maintained in the stability region under CLBF-EMPC, while avoiding the unsafe region in state-space in the presence of disturbances. Finally, the real-time machine learning-based CLBF-MPC and CLBF-EMPC are applied to a chemical process example to demonstrate their improved closed-loop performances compared to those under the CLBF-MPC and CLBF-EMPC without online learning of RNN models.

Section snippets

Notation

The notation $|\cdot|$ is used to denote the Euclidean norm of a vector. $x^{T}$ denotes the transpose of $x$ . The notation $L_{f} V (x)$ denotes the standard Lie derivative $L_{f} V (x) ≔ \frac{\partial V (x)}{\partial x} f (x)$ . Set subtraction is denoted by “ $∖$ ”, i.e., $A ∖ B ≔ {x \in R^{n} | x \in A, x \notin B}$ . $\emptyset$ signifies the empty set. The function $f (\cdot)$ is of class $C^{1}$ if it is continuously differentiable in its domain. A continuous function $α : [0, a) \to [0, \infty)$ is said to belong to class $K$ if it is strictly increasing and is zero only when evaluated at zero.

Class of systems

The class of

Real-time CLBF-based predictive controllers using RNNs

In this section, we first present the formulation of CLBF-based predictive controllers using an RNN model and demonstrate that closed-loop stability and safety can be achieved simultaneously for the nominal system of Eq. (1) (i.e., $w (t) \equiv 0$ ). Subsequently, online learning of RNN models is employed within CLBF-based predictive controllers via the event-triggered and the error-triggered mechanisms to improve RNN prediction accuracy for the uncertain system with time-varying disturbances $w (t)$ .

Application to a chemical process example

A chemical process example is provided to illustrate the application of machine-learning-based CLBF-MPC and CLBF-EMPC to operate the system in the safe stability region. Specifically, a well-mixed, non-isothermal continuous stirred tank reactor (CSTR) where an irreversible second-order exothermic reaction takes place is considered. The reaction transforms a reactant $A$ to a product $B$ ( $A \to B$ ). The inlet concentration of $A$ , the inlet temperature and feed volumetric flow rate of the reactor are $C_{A 0}$ , $T$

Conclusion

In this work, we proposed real-time machine learning-based CLBF-MPC and CLBF-EMPC schemes to optimize process operational safety and closed-loop stability for nonlinear systems subject to time-varying disturbances. The RNN models were first developed for a general class of nonlinear systems and incorporated in the designs of CLBF-MPC and of CLBF-EMPC to provide future state predictions. The event-triggered and the error-triggered mechanisms were then integrated within the real-time

Conflict of interest

None declared.

Acknowledgments

Financial support from the National Science Foundation and the Department of Energyis gratefully acknowledged.

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Real-time machine learning for operational safety of nonlinear processes via barrier-function based predictive control

Highlights

Abstract

Introduction

Section snippets

Notation

Class of systems

Real-time CLBF-based predictive controllers using RNNs

Application to a chemical process example

Conclusion

Conflict of interest

Acknowledgments

Syst. Control Lett.

Neurocomputing

Automatica

Syst. Control Lett.

Neural Netw.

Automatica

Chem. Eng. Res. Des.

Chem. Eng. Res. Des.

Control barrier function based quadratic programs for safety critical systems

IEEE Trans. Autom. Control

Combining control Lyapunov and barrier functions for constrained stabilization of nonlinear systems

High-order neural network structures for identification of dynamical systems

IEEE Trans. Neural Netw.