Lyapunov-based Economic Model Predictive Control Research Articles

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.

Economic optimization and predictive control for nonlinear systems using Lyapunov based nonlinear cost function design

Optimal control of physical processes depends on the formulation of objective functions and contemplation of appropriate predictive control. In this article, we propose a new framework, to blend the concepts of the Lyapunov-based economic model predictive control (Ly-EMPC) scheme with the novel cost-efficient profit function design. Furthermore, Lyapunov’s stability criteria is employed to verify the existence of duality and optimality for the designed economic stage costs. This method differs from the existing EMPC techniques by implementing Lagrange multipliers in the rotated cost function as a measure of stability. The closed-loop performance using the regularized Ly-EMPC is validated against a standard tracking objective-based nonlinear MPC for three benchmark examples through an econometrics gauge factor. Interestingly, in contrast to tracking NMPC, the proposed Ly-EMPC scheme excels in economic performance, while the formation of nonlinear economic stage cost is conceptually simpler, novel, and practically profitable.

Economic Model Predictive Control of Nonlinear Systems Using Online Learning of Neural Networks

This work focuses on the development of a Lyapunov-based economic model predictive control (LEMPC) scheme that utilizes recurrent neural networks (RNNs) with an online update to optimize the economic benefits of switched non-linear systems subject to a prescribed switching schedule. We first develop an initial offline-learning RNN using historical operational data, and then update RNNs with real-time data to improve model prediction accuracy. The generalized error bounds for RNNs updated online with independent and identically distributed (i.i.d.) and non-i.i.d. data samples are derived, respectively. Subsequently, by incorporating online updating RNNs within LEMPC, probabilistic closed-loop stability, and economic optimality are achieved simultaneously for switched non-linear systems accounting for the RNN generalized error bound. A chemical process example with scheduled mode transitions is used to demonstrate that the closed-loop economic performance under LEMPC can be improved using an online update of RNNs.

Online-Learning-Based Economic MPC of Switched Nonlinear Systems*

This work develops a Lyapunov-based economic model predictive control (LEMPC) scheme that utilizes recurrent neural networks (RNNs) with online update to optimize the economic benefits of switched nonlinear systems subject to a prescribed switching schedule. We first develop an initial offline-learning RNN using historical operational data, and then update RNN models using real-time data to improve model prediction accuracy. The generalized error bound for RNNs updated online with non-independent and identically distributed (non-i.i.d.) data samples is first derived. Subsequently, by incorporating the online update of RNNs within LEMPC, probabilistic closed-loop stability and economic optimality are achieved simultaneously for switched nonlinear systems accounting for the RNN generalized error bound. A chemical process example with scheduled mode transitions is used to demonstrate that the closed-loop economic performance under LEMPC can be improved using online learning of RNNs.

Pareto Multiobjective Lyapunov-Based Economic Model-Predictive Control for Nonlinear Time-Delayed System

This article proposes a dual-mode multiobjective Lyapunov-based economic model-predictive control (MOLEMPC) for constrained nonlinear time-delay systems with external disturbances. First, considering the conflicting objectives, the Pareto compromise approach is used to calculate the compromised economic stability point and the multiobjective optimization function. To compensate for the impact of input delay, a model-based estimator is designed by using the actual state, historical inputs, and nominal models of the system. By the proposed dual-mode Lyapunov-based economic model-predictive control (LEMPC) scheme, the feasible domain of the initial point of a closed-loop system is increased, in the meantime, the stability of the closed-loop system with disturbances and delay is guaranteed. The recursive feasibility and stability analysis of the MOLEMPC controller is given in detail. We illustrate that the closed-loop system state converges in a robust invariant set after a finite time. Finally, simulations demonstrate the reliability and advantages of the proposed strategy.

Development of directed randomization for discussing a minimal security architecture

Strategies for mitigating the impacts of cyberattacks on control systems using a control-oriented perspective have become of greater interest in recent years. Our group has contributed to this trend by developing several methods for detecting cyberattacks on process sensors, actuators, or both sensors and actuators simultaneously using an advanced optimization-based control strategy known as Lyapunov-based economic model predictive control (LEMPC). However, each technique comes with benefits and limitations, both with respect to one another and with respect to traditional information technology and computer science-type approaches to cybersecurity. An important question to ask, therefore, is what the goal should be of the development of new control-based techniques for handling cyberattacks on control systems, and how we will be able to benchmark these as “successful” compared to other techniques to drive development or signal when the research in this direction has reached maturity. In this paper, we propose that the goal of research in control system cybersecurity for next-generation manufacturing should be the development of a security architecture that provides flexibility and safety with lowest cost, and seek to clarify this concept by re-analyzing some of the security techniques from our prior work in such a context. We also show how new methods can be developed and analyzed within this “minimum security architecture” context by proposing a technique which we term “directed randomization” that may require less sensors to be secured in a system than some of our prior methods, potentially adding flexibility to the system while still maintaining security. Directed randomization seeks to utilize the existence of two possible stabilizing inputs at every sampling time to attempt to create a challenge for an attacker for setting up an arbitrary sensor attack policy without being detected within a finite number of sampling periods. We discuss benefits and limitations of this technique with respect to our prior cybersecurity strategies and also with respect to extended versions of these prior concepts, such as image-based control and distributed control, to provide further insights into the minimum security concept.

Lyapunov-Based Economic Model Predictive Control for Detecting and Handling Actuator and Simultaneous Sensor/Actuator Cyberattacks on Process Control Systems

The controllers for a cyber-physical system may be impacted by sensor measurement cyberattacks, actuator signal cyberattacks, or both types of attacks. Prior work in our group has developed a theory for handling cyberattacks on process sensors. However, sensor and actuator cyberattacks have a different character from one another. Specifically, sensor measurement attacks prevent proper inputs from being applied to the process by manipulating the measurements that the controller receives, so that the control law plays a role in the impact of a given sensor measurement cyberattack on a process. In contrast, actuator signal attacks prevent proper inputs from being applied to a process by bypassing the control law to cause the actuators to apply undesirable control actions. Despite these differences, this manuscript shows that we can extend and combine strategies for handling sensor cyberattacks from our prior work to handle attacks on actuators and to handle cases where sensor and actuator attacks occur at the same time. These strategies for cyberattack-handling and detection are based on the Lyapunov-based economic model predictive control (LEMPC) and nonlinear systems theory. We first review our prior work on sensor measurement cyberattacks, providing several new insights regarding the methods. We then discuss how those methods can be extended to handle attacks on actuator signals and then how the strategies for handling sensor and actuator attacks individually can be combined to produce a strategy that is able to guarantee safety when attacks are not detected, even if both types of attacks are occurring at once. We also demonstrate that the other combinations of the sensor and actuator attack-handling strategies cannot achieve this same effect. Subsequently, we provide a mathematical characterization of the “discoverability” of cyberattacks that enables us to consider the various strategies for cyberattack detection presented in a more general context. We conclude by presenting a reactor example that showcases the aspects of designing LEMPC.

Lyapunov-based economic model predictive control for online model discrimination

Economic model predictive control (EMPC) is a flexible control design strategy that can be modified to achieve many operating goals while also ensuring safe operation (e.g., by adding Lyapunov-based stability constraints to form Lyapunov-based EMPC, or LEMPC). Prior works have investigated LEMPC capabilities for achieving goals online beyond optimizing process economics, including aiding in model structure selection to benefit model-based control system design since the accuracy and quality of the process model are important for achieving an expected performance from such systems. This work further probes the capabilities of LEMPC to accomplish multiple objectives during process operation, including aiding in the discrimination between mechanistic models online. In particular, several rival mechanistic models may explain the existing data. To discard models from this set that do not fully represent the actual process, a new set of “online experiments” can be conducted to collect more information. However, additional experimentation may be costly and unsafe to be performed. LEMPC can aid in performing online data collection when discrimination between mechanistic models is needed, with the flexibility to ensure safety as the data is gathered and trade off the data-gathering goal for cost considerations. Motivated by this, we discuss how LEMPC can be designed to automatically and dynamically collect data that is useful for the selection of mechanistic models from among a set of possibilities. A chemical process example is used to clarify benefits and limitations of LEMPC for promoting online model discrimination.

Actuator Cyberattack Handling Using Lyapunov-based Economic Model Predictive Control

Cybersecurity has gained increasing interest as a consequence of the potential impacts of cyberattacks on profits and safety. While attacks can affect various components of a plant, prior work from our group has focused on the impact of cyberattacks on control components such as process sensors and actuators and the development of detection strategies for cybersecurity derived from control theory. In this work, we provide greater focus on actuator attacks; specifically, we extend a detection and control strategy previously applied for sensor attacks and based on an optimization-based control technique called Lyapunov-based economic model predictive control (LEMPC) to detect attacks impacting the control action applied by the actuators when the state measurements provided to the controller are accurate. Closed-loop stability guarantees are rigorously derived. A continuous stirred tank reactor is simulated to elucidate aspects of the detection strategy proposed.

Integrated cyberattack detection and handling for nonlinear systems with evolving process dynamics under Lyapunov-based economic model predictive control

Safety-critical processes are becoming increasingly automated and connected. While automation can increase efficiency, it brings new challenges associated with guaranteeing safety in the presence of uncertainty especially in the presence of control system cyberattacks. One of the challenges for developing control strategies with guaranteed safety and cybersecurity properties under sufficient conditions is the development of appropriate detection strategies that work with control laws to prevent undetected attacks that have immediate closed-loop stability consequences. Achieving this, in the presence of uncertainty brought about by plant/model mismatch and process dynamics that can change with time, requires a fundamental understanding of the characteristics of attacks that can be detected with reasonable detection mechanisms and characterizing and verifying system safety properties when cyberattacks and changing system behavior cannot be distinguished. Motivated by this, this paper discusses three cyberattack detection strategies for nonlinear processes whose dynamics change with time when these processes are operated under an optimization-based control strategy known as Lyapunov-based economic model predictive control (LEMPC) until the closed-loop state either leaves a characterizable region of state-space or an attack detection threshold related to state estimates or state predictions is exceeded. Following this, the closed-loop state is maintained within a larger region of operation under an updated cyberattack detection strategy for a characterizable time period. A Taylor series-based model is used for making state predictions to allow theoretical guarantees to be explicitly tied to the numerical approximation of the model used within the LEMPC. A process example illustrates the Taylor series-based model concept.

Integrated cyberattack detection and resilient control strategies using Lyapunov‐based economic model predictive control

AbstractThe use of an integrated system framework, characterized by numerous cyber/physical components (sensor measurements, signals to actuators) connected through wired/wireless networks, has not only increased the ability to control industrial systems but also the vulnerabilities to cyberattacks. State measurement cyberattacks could pose threats to process control systems since feedback control may be lost if the attack policy is not thwarted. Motivated by this, we propose three detection concepts based on Lyapunov‐based economic model predictive control (LEMPC) for nonlinear systems. The first approach utilizes randomized modifications to an LEMPC formulation online to potentially detect cyberattacks. The second method detects attacks when a threshold on the difference between state measurements and state predictions is exceeded. Finally, the third strategy utilizes redundant state estimators to flag deviations from “normal” process behavior as cyberattacks.

Lyapunov-based economic model predictive control for nonlinear descriptor systems

A large number of physical applications such as electrical circuits, chemical processes and multibody systems can be accurately described by a set of differential algebraic equations (DAEs). For instance, sustainable water desalination techniques such as membrane distillation (MD) is an example of a chemical process that is described by DAEs. Optimal control strategies such as economic model predictive control (EMPC) has many operational advantages (e.g., maximizing process economics and enhancing operational safety) for chemical applications. However, constructing EMPC schemes for a general class of nonlinear DAEs systems while ensuring closed-loop stability and recursive feasibility is an open research challenge that has not been considered. Motivated by the above considerations, this note introduces a Lyapunov-based economic model predictive control (LEMPC) design that can economically operate nonlinear descriptor systems while satisfying input and states constraints. To guarantee closed-loop stability and recursive feasibility of the proposed control design, a Lyapunov-based control law is introduced to characterize the stability region at which the LEMPC can maximize the process economics. Finally, a chemical batch process modeled by nonlinear differential algebraic equations (DAEs) is utilized to demonstrate the applicability of the proposed framework.

Anomaly-Handling in Lyapunov-Based Economic Model Predictive Control via Empirical Models

A question that faces data-driven autonomous systems is verification that they will perform in a safe manner despite changes in the environment on which they act over time or incomplete knowledge of the system model. This work analyzes closed-loop stability of nonlinear systems under Lyapunov-based economic model predictive control (LEMPC) with data-driven models in the case where it is desirable to have the ability to detect when the data-driven model is or becomes insufficiently accurate for maintaining the closed-loop state in an expected region of state-space. Implications of the results for false sensor measurement cyberattacks seeking to impact the fidelity of models derived from model identification are discussed and illustrated through a chemical process example.

Stability of nonlinear system under distributed Lyapunov-based economic model predictive control with time-delay

In industrial processes, time-delay often occurs in many operations and increases difficulty to control these dynamical systems. In this paper, we propose a distributed Lyapunov economic model predictive control (DLEMPC) method for continuous-time nonlinear systems with input time-delay. Next, we establish sufficient conditions to make the system feasible and stable under our proposed DLEMPC strategy, and bound system state in a region which contains the provided original value. Finally, a chemical process simulation is given to illustrate our theoretical results.

Distributed Economic Model Predictive Control for a Wind–Photovoltaic–Battery Microgrid Power System

This paper proposed a dual-mode distributed economic model predictive control (EMPC) for a nonlinear wind–photovoltaic–battery microgrid power system (WPB-MPS). The communications between each subsystem are taken into consideration and the economic objectives that are established based on the aspects that: satisfy the total demand; make full use of the power generated by the WPB-MPS; optimize the battery's state of charge; and reduce the fluctuation of power exchange with the grid. The subsystem-based EMPCs work iteratively and cooperatively to solve the economic objective functions, which reflect economies of the system and not necessary to drive the system to a certain steady state as hierarchical control. A dual-mode Lyapunov-based EMPC has been formulated to guarantee the stability of the system with disturbance, and under certain conditions, the closed-loop system state can be driven and hold in a safe region. Simulations demonstrate the advantages and efficiency of the proposed method.

Economic Machine-Learning-Based Predictive Control of Nonlinear Systems

In this work, a Lyapunov-based economic model predictive control (LEMPC) method is developed to address economic optimality and closed-loop stability of nonlinear systems using machine learning-based models to make predictions. Specifically, an ensemble of recurrent neural network (RNN) models via a k-fold cross validation is first developed to capture process dynamics in an operating region. Then, the LEMPC using an RNN ensemble is designed to maintain the closed-loop state in a stability region and optimize process economic benefits simultaneously. Parallel computing is employed to improve computational efficiency of real-time implementation of LEMPC with an RNN ensemble. The proposed machine-learning-based LEMPC method is demonstrated using a nonlinear chemical process example.

Distributed economic model predictive control for operational safety of nonlinear processes

Achieving operational safety of chemical processes while operating them in an economically‐optimal manner is a matter of great importance. Our recent work integrated process safety with process control by incorporating safety‐based constraints within model predictive control (MPC) design; however, the safety‐based MPC was developed with a centralized architecture, with the result that computation time limitations within a sampling period may reduce the effectiveness of such a controller design for promoting process safety. To address this potential practical limitation of the safety‐based control design, in this work, we propose the integration of a distributed model predictive control architecture with Lyapunov‐based economic model predictive control (LEMPC) formulated with safety‐based constraints. We consider both iterative and sequential distributed control architectures, and the partitioning of inputs between the various optimization problems in the distributed structure based on their impact on process operational safety. Moreover, sufficient conditions that ensure feasibility and closed‐loop stability of the iterative and sequential safety distributed LEMPC designs are given. A comparison between the proposed safety distributed EMPC controllers and the safety centralized EMPC is demonstrated via a chemical process example. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3404–3418, 2017

Error‐triggered on‐line model identification for model‐based feedback control

In industry, it may be difficult in many applications to obtain a first‐principles model of the process, in which case a linear empirical model constructed using process data may be used in the design of a feedback controller. However, linear empirical models may not capture the nonlinear dynamics over a wide region of state‐space and may also perform poorly when significant plant variations and disturbances occur. In the present work, an error‐triggered on‐line model identification approach is introduced for closed‐loop systems under model‐based feedback control strategies. The linear models are re‐identified on‐line when significant prediction errors occur. A moving horizon error detector is used to quantify the model accuracy and to trigger the model re‐identification on‐line when necessary. The proposed approach is demonstrated through two chemical process examples using a model‐based feedback control strategy termed Lyapunov‐based economic model predictive control (LEMPC). The chemical process examples illustrate that the proposed error‐triggered on‐line model identification strategy can be used to obtain more accurate state predictions to improve process economics while maintaining closed‐loop stability of the process under LEMPC. © 2016 American Institute of Chemical Engineers AIChE J, 63: 949–966, 2017

Economic model predictive control designs for input rate-of-change constraint handling and guaranteed economic performance

Abstract Economic model predictive control (EMPC) has been a popular topic in the recent chemical process control literature due to its potential to improve process profit by operating a system in a time-varying manner. However, time-varying operation may cause excessive wear of the process components such as valves and pumps. To address this issue, input magnitude constraints and input rate-of-change constraints can be added to the EMPC optimization problem to prevent possible frequent and extreme changes in the requested inputs. Specifically, we develop input rate-of-change constraints that can be incorporated in Lyapunov-based EMPC (LEMPC) that ensure controller feasibility and closed-loop stability. Furthermore, we develop a terminal equality constraint for LEMPC that can ensure that the performance of LEMPC is at least as good as that of a Lyapunov-based controller in finite-time and in infinite-time. Chemical process examples demonstrate the incorporation of input rate-of-change constraints and terminal state constraints in EMPC.

Economic model predictive control of nonlinear time‐delay systems: Closed‐loop stability and delay compensation

Closed‐loop stability of nonlinear time‐delay systems under Lyapunov‐based economic model predictive control (LEMPC) is considered. LEMPC is initially formulated with an ordinary differential equation model and is designed on the basis of an explicit stabilizing control law. To address closed‐loop stability under LEMPC, first, we consider the stability properties of the sampled‐data system resulting from the nonlinear continuous‐time delay system with state and input delay under a sample‐and‐hold implementation of the explicit controller. The steady‐state of this sampled‐data closed‐loop system is shown to be practically stable. Second, conditions such that closed‐loop stability, in the sense of boundedness of the closed‐loop state, under LEMPC are derived. A chemical process example is used to demonstrate that indeed closed‐loop stability is maintained under LEMPC for sufficiently small time‐delays. To cope with performance degradation owing to the effect of input delay, a predictor feedback LEMPC methodology is also proposed. The predictor feedback LEMPC design employs a predictor to compute a prediction of the state after the input delay period and an LEMPC scheme that is formulated with a differential difference equation (DDE) model, which describes the time‐delay system, initialized with the predicted state. The predictor feedback LEMPC is also applied to the chemical process example and yields improved closed‐loop stability and economic performance properties. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4152–4165, 2015