Economic model predictive control with time‐varying objective function for nonlinear process systems

Economic model predictive control (EMPC) is a control scheme that combines real‐time dynamic economic process optimization with the feedback properties of model predictive control (MPC) by replacing the quadratic cost function with a general economic cost function. Almost all the recent work on EMPC involves cost functions that are time invariant (do not explicitly account for time‐varying process economics). In the present work, we focus on the development of a Lyapunov‐based EMPC (LEMPC) scheme that is formulated with an explicitly time‐varying economic cost function. First, the formulation of the proposed two‐mode LEMPC is given. Second, closed‐loop stability is proven through a theoretical treatment. Last, we demonstrate through extensive closed‐loop simulations of a chemical process that the proposed LEMPC can achieve stability with time‐varying economic cost as well as improve economic performance of the process over a conventional MPC scheme. © 2013 American Institute of Chemical Engineers AIChE J 60: 507–519, 2014     

Real‐time economic model predictive control of nonlinear process systems

Closed‐loop stability of nonlinear systems under real‐time Lyapunov‐based economic model predictive control (LEMPC) with potentially unknown and time‐varying computational delay is considered. To address guaranteed closed‐loop stability (in the sense of boundedness of the closed‐loop state in a compact state‐space set), an implementation strategy is proposed which features a triggered evaluation of the LEMPC optimization problem to compute an input trajectory over a finite‐time prediction horizon in advance. At each sampling period, stability conditions must be satisfied for the precomputed LEMPC control action to be applied to the closed‐loop system. If the stability conditions are not satisfied, a backup explicit stabilizing controller is applied over the sampling period. Closed‐loop stability under the real‐time LEMPC strategy is analyzed and specific stability conditions are derived. The real‐time LEMPC scheme is applied to a chemical process network example to demonstrate closed‐loop stability and closed‐loop economic performance improvement over that achieved for operation at the economically optimal steady state. © 2014 American Institute of Chemical Engineers AIChE J, 61: 555–571, 2015     

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     

Composite fast‐slow MPC design for nonlinear singularly perturbed systems

The design of a composite control system for nonlinear singularly perturbed systems using model predictive control (MPC) is described. Specifically, a composite control system comprised of a “fast” MPC acting to regulate the fast dynamics and a “slow” MPC acting to regulate the slow dynamics is designed. The composite MPC system uses multirate sampling of the plant state measurements, i.e., fast sampling of the fast state variables is used in the fast MPC and slow‐sampling of the slow state variables is used in the slow MPC. Using singular perturbation theory, the stability and optimality of the closed‐loop nonlinear singularly perturbed system are analyzed. A chemical process example which exhibits two‐time‐scale behavior is used to demonstrate the structure and implementation of the proposed fast–slow MPC architecture in a practical setting. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1802–1811, 2012     

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     

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     

Distributed model predictive control of switched nonlinear systems with scheduled mode transitions

A method for the design of distributed model predictive control (DMPC) systems for a class of switched nonlinear systems for which the mode transitions take place according to a prescribed switching schedule is presented. Under appropriate stabilizability assumptions on the existence of a set of feedback controllers that can stabilize the closed‐loop switched, nonlinear system, a cooperative DMPC architecture using Lyapunov‐based model predictive control (MPC) in which the distributed controllers carry out their calculations in parallel and communicate in an iterative fashion to compute their control actions is designed. The proposed DMPC design is applied to a nonlinear chemical process network with scheduled mode transitions and its performance and computational efficiency properties in comparison to a centralized MPC architecture are evaluated through simulations. © 2013 American Institute of Chemical Engineers AIChE J, 59:860‐871, 2013     

Transition from Batch to Continuous Operation in Bio‐Reactors: A Model Predictive Control Approach and Application

This work considers the problem of determining the transition of ethanol‐producing bio‐reactors from batch to continuous operation and subsequent control subject to constraints and performance considerations. To this end, a Lyapunov‐based non‐linear model predictive controller is utilized that stabilizes the bio‐reactor under continuous mode of operation. The key idea in the predictive controller is the formulation of appropriate stability constraints that allow an explicit characterization of the set of initial conditions from where feasibility of the optimization problem and hence closed‐loop stability is guaranteed. Additional constraints are incorporated in the predictive control design to expand on the set of initial conditions that can be stabilized by control designs that only require the value of the Lyapunov function to decay. Then, the explicit characterization of the set of stabilizable initial conditions is used in determining the appropriate time for which the reactor must be run in batch mode. Specifically, the predictive control approach is utilized in determining the appropriate batch length that achieves stabilizable values of the state variables at the end of the batch. Application of the proposed method to the ethanol production process using Zymomonas mobilis as the ethanol producing micro‐organism demonstrates the effectiveness of the proposed model predictive control strategy in stabilizing the bio‐reactor.     

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     

Covariance‐based hardware selection‐part III: Distributed parameter systems

In this article, we propose a new method for the selection of control system hardware (sensors and actuators) for distributed parameter systems. The proposed design scheme seeks to minimize the capital cost of hardware while satisfying predefined performance constraints. Within this minimum capital cost framework, three design scenarios will be discussed; the closed‐loop full state information actuator selection problem, the open‐loop partial state information sensor selection problem, and the closed‐loop partial state information simultaneous sensor and actuator selection problem. The proposed method will be illustrated through application to a nonisothermal tubular reactor example. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

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.     

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     

A delay‐dependent robust fuzzy MPC approach for nonlinear CSTR

Based on Takagi–Sugeno (T–S) fuzzy models, a robust fuzzy model predictive control (MPC) algorithm is presented for a class of nonlinear time‐delay systems with input constraints. Delay‐dependent sufficient conditions for the robust stability of the closed‐loop system are derived, and the condition for the existence of the fuzzy model predictive controller is formulated in terms of nonlinear matrix inequality via the parallel distributed compensation (PDC) approach. By using a novel matrix transform technique, a receding optimization problem with linear matrix inequality (LMIs) constraints is constructed to design the desired controllers with an on‐line optimal receding horizon guaranteed cost. Finally, an example of continuous stirred tank reactors (CSTR) is given to demonstrate the effectiveness of the proposed results.     

Sequential and iterative architectures for distributed model predictive control of nonlinear process systems

In this work, we focus on distributed model predictive control of large scale nonlinear process systems in which several distinct sets of manipulated inputs are used to regulate the process. For each set of manipulated inputs, a different model predictive controller is used to compute the control actions, which is able to communicate with the rest of the controllers in making its decisions. Under the assumption that feedback of the state of the process is available to all the distributed controllers at each sampling time and a model of the plant is available, we propose two different distributed model predictive control architectures. In the first architecture, the distributed controllers use a one‐directional communication strategy, are evaluated in sequence and each controller is evaluated only once at each sampling time; in the second architecture, the distributed controllers utilize a bi‐directional communication strategy, are evaluated in parallel and iterate to improve closed‐loop performance. In the design of the distributed model predictive controllers, Lyapunov‐based model predictive control techniques are used. To ensure the stability of the closed‐loop system, each model predictive controller in both architectures incorporates a stability constraint which is based on a suitable Lyapunov‐based controller. We prove that the proposed distributed model predictive control architectures enforce practical stability in the closed‐loop system and optimal performance. The theoretical results are illustrated through a catalytic alkylation of benzene process example. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Economic model predictive control of nonlinear process systems using empirical models

Economic model predictive control (EMPC) is a feedback control technique that attempts to tightly integrate economic optimization and feedback control since it is a predictive control scheme that is formulated with an objective function representing the process economics. As its name implies, EMPC requires the availability of a dynamic model to compute its control actions and such a model may be obtained either through application of first principles or through system identification techniques. In industrial practice, it may be difficult in general to obtain an accurate first‐principles model of the process. Motivated by this, in the present work, Lyapunov‐based EMPC (LEMPC) is designed with a linear empirical model that allows for closed‐loop stability guarantees in the context of nonlinear chemical processes. Specifically, when the linear model provides a sufficient degree of accuracy in the region where time varying economically optimal operation is considered, conditions for closed‐loop stability under the LEMPC scheme based on the empirical model are derived. The LEMPC scheme is applied to a chemical process example to demonstrate its closed‐loop stability and performance properties as well as significant computational advantages. © 2014 American Institute of Chemical Engineers AIChE J, 61: 816–830, 2015     

On identification of well‐conditioned nonlinear systems: Application to economic model predictive control of nonlinear processes

The focus of this work is on economic model predictive control (EMPC) that utilizes well‐conditioned polynomial nonlinear state‐space (PNLSS) models for processes with nonlinear dynamics. Specifically, the article initially addresses the development of a nonlinear system identification technique for a broad class of nonlinear processes which leads to the construction of PNLSS dynamic models which are well‐conditioned over a broad region of process operation in the sense that they can be correctly integrated in real‐time using explicit numerical integration methods via time steps that are significantly larger than the ones required by nonlinear state‐space models identified via existing techniques. Working within the framework of PNLSS models, additional constraints are imposed in the identification procedure to ensure well‐conditioning of the identified nonlinear dynamic models. This development is key because it enables the design of Lyapunov‐based EMPC (LEMPC) systems for nonlinear processes using the well‐conditioned nonlinear models that can be readily implemented in real‐time as the computational burden required to compute the control actions within the process sampling period is reduced. A stability analysis for this LEMPC design is provided that guarantees closed‐loop stability of a process under certain conditions when an LEMPC based on a nonlinear empirical model is used. Finally, a classical chemical reactor example demonstrates both the system identification and LEMPC design techniques, and the significant advantages in terms of computation time reduction in LEMPC calculations when using the nonlinear empirical model. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3353–3373, 2015     

Economic model predictive control of stochastic nonlinear systems

This work focuses on the design of stochastic Lyapunov‐based economic model predictive control (SLEMPC) systems for a broad class of stochastic nonlinear systems with input constraints. Under the assumption of stabilizability of the origin of the stochastic nonlinear system via a stochastic Lyapunov‐based control law, an economic model predictive controller is proposed that utilizes suitable constraints based on the stochastic Lyapunov‐based controller to ensure economic optimality, feasibility and stability in probability in a well‐characterized region of the state‐space surrounding the origin. A chemical process example is used to illustrate the application of the approach and demonstrate its economic benefits with respect to an EMPC scheme that treats the disturbances in a deterministic, bounded manner. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3312–3322, 2018     

An economic model predictive control approach to integrated production management and process operation

Managing production schedules and tracking time‐varying demand of certain products while optimizing process economics are subjects of central importance in industrial applications. We investigate the use of economic model predictive control (EMPC) in tracking a production schedule. Specifically, given that only a small subset of the total process state vector is typically required to track certain scheduled values, we design a novel EMPC scheme, through proper construction of the objective function and constraints, that forces specific process states to meet the production schedule and varies the rest of the process states in a way that optimizes process economic performance. Conditions under which feasibility and closed‐loop stability of a nonlinear process under such an EMPC for schedule management can be guaranteed are developed. The proposed EMPC scheme is demonstrated through a chemical process example in which the product concentration is requested to follow a certain production schedule. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1892–1906, 2017