Distributed economic MPC: Application to a nonlinear chemical process network

In the present work, we focus on the development and application of Lyapunov-based economic model predictive control (LEMPC) designs to a catalytic alkylation of benzene process network, which consists of four continuously stirred tank reactors and a flash separator. We initially propose a new economic measure for the entire process network which accounts for a broad set of economic considerations on the process operation including reaction conversion, separation quality and energy efficiency. Subsequently, steady-state process optimization is first carried out to locate an economically optimal (with respect to the proposed economic measure) operating steady-state. Then, a sequential distributed economic model predictive control design method, suitable for large-scale process networks, is proposed and its closed-loop stability properties are established. Using the proposed method, economic, distributed as well as centralized, model predictive control systems are designed and are implemented on the process to drive the closed-loop system state close to the economically optimal steady-state. Extensive simulations are carried out to demonstrate the application of the proposed economic MPC (EMPC) designs and compare them with a centralized Lyapunov-based model predictive control design, which uses a conventional, quadratic cost function that includes penalty on the deviation of the states and inputs from their economically optimal steady-state values, from computational time and closed-loop performance points of view.     

Elucidation of the role of constraints in economic model predictive control

Economic model predictive control (EMPC) is a predictive feedback control methodology that unifies economic optimization and control. EMPC uses a stage cost that reflects the process/system economics. In general, the stage cost used is not a quadratic stage cost like that typically used in standard tracking model predictive control. In this paper, a brief overview of EMPC methods is provided. In particular, the role of constraints imposed in the optimization problem of EMPC for feasibility, closed-loop stability, and closed-loop performance is explained. Three main types of constraints are considered including terminal equality constraints, terminal region constraints, and constraints designed via Lyapunov-based techniques. The paper closes with a well-known chemical engineering example (a non-isothermal CSTR with a second-order reaction) to illustrate the effectiveness of time-varying operation to improve closed-loop economic performance compared to steady-state operation and to demonstrate the impact of economically motivated constraints on optimal operation.     

On finite-time and infinite-time cost improvement of economic model predictive control for nonlinear systems

A novel two-layer economic model predictive control (EMPC) structure that addresses provable finite-time and infinite-time closed-loop economic performance of nonlinear systems in closed-loop with EMPC is presented. In the upper layer, a Lyapunov-based EMPC (LEMPC) scheme is formulated with performance constraints by taking advantage of an auxiliary Lyapunov-based model predictive control (LMPC) problem solution formulated with a quadratic cost function. The lower layer LEMPC uses a shorter prediction horizon and smaller sampling period than the upper layer LEMPC and involves explicit performance-based constraints computed by the upper layer LEMPC. Thus, the two-layer architecture allows for dividing dynamic optimization and control tasks into two layers for a computationally manageable control scheme at the feedback control (lower) layer. A chemical process example is used to demonstrate the performance and stability properties of the two-layer LEMPC structure.     

Economic model predictive control of parabolic PDE systems: Addressing state estimation and computational efficiency

In a previous work [20], an economic model predictive control (EMPC) system for parabolic partial differential equation (PDE) systems was proposed. Through operating the PDE system in a time-varying fashion, the EMPC system demonstrated improved economic performance over steady-state operation. The EMPC system assumed the knowledge of the complete state spatial profile at each sampling period. From a practical point of view, measurements of the state variables are typically only available at a finite number of spatial positions. Additionally, the basis functions used to construct a reduced-order model (ROM) for the EMPC system were derived using analytical sinusoidal/cosinusoidal eigenfunctions. However, constructing a ROM on the basis of historical data-based empirical eigenfunctions by applying Karhunen-Loève expansion may be more computationally efficient. To address these issues, several EMPC systems are formulated for both output feedback implementation and with ROMs based on analytical sinusoidal/cosinusoidal eigenfunctions and empirical eigenfunctions. The EMPC systems are evaluated using a non-isothermal tubular reactor example, described by two nonlinear parabolic PDEs, where a second-order reaction takes place. The model accuracy, computational time, input and state constraint satisfaction, and closed-loop economic performance of the closed-loop tubular reactor under the different EMPC systems are compared.     

约束非线性系统输入到状态稳定经济型模型预测控制

考虑约束非线性系统经济型最优控制问题,提出一种关于经济性能输入到状态稳定的经济型模型预测控制(EMPC)策略.通过离线计算系统的最优经济稳态点,构建关于该稳态点跟踪的稳定最优控制问题.在此基础上,利用稳定最优控制问题的最优值函数和关于经济性能函数的松弛量构造EMPC优化问题的收缩约束,再结合不变集原理和输入到状态稳定性...     

Economic optimization using model predictive control with a terminal cost

In the standard model predictive control implementation, first a steady-state optimization yields the equilibrium point with minimal economic cost. Then, the deviation from the computed best steady state is chosen as the stage cost for the dynamic regulation problem. The computed best equilibrium point may not be the global minimum of the economic cost, and hence, choosing the economic cost as the stage cost for the dynamic regulation problem, rather than the deviation from the best steady state, offers potential for improving the economic performance of the system. It has been previously shown that the existing framework for MPC stability analysis, which addresses to the standard class of problems with a regulation objective, does not extend to economic MPC. Previous work on economic MPC developed new tools for stability analysis and identified sufficient conditions for asymptotic stability. These tools were developed for the terminal constraint MPC formulation, in which the system is stabilized by forcing the state to the best equilibrium point at the end of the horizon. In this work, we relax this constraint by imposing a region constraint on the terminal state instead of a point constraint, and adding a penalty on the terminal state to the regulator cost. We extend the stability analysis tools, developed for terminal constraint economic MPC, to the proposed formulation and establish that strict dissipativity is sufficient for guaranteeing asymptotic stability of the closed-loop system. We also show that the average closed-loop performance outperforms the best steady-state performance. For implementing the proposed formulation, a rigorous analysis for computing the appropriate terminal penalty and the terminal region is presented. A further extension, in which the terminal constraint is completely removed by modifying the regulator cost function, is also presented along with its stability analysis. Finally, an illustrative example is presented to demonstrate the differences between the terminal constraint and the proposed terminal penalty formulation.     

双层结构预测控制研究进展

现代工业大系统的优化控制采用递阶结构,其中以预测控制为代表的先进过程控制已经成为重要的一级.目前,主流的工业预测控制技术均采用双层结构,即包含稳态优化层和动态控制层.双层结构预测控制技术可以有效解决复杂工业过程常见的多目标优化、多变量控制的难点问题.本文简要总结了双层结构预测控制的算法,并从控制输入与被控输出稳态关系入手分析了多变量预测控制稳态解的相容性和唯一性,说明了稳态优化的重要性.针对双层结构预测控制与区间预测控制的性能比较、稳态模型的奇异性以及闭环系统动态特性等提出了一些见解,并指出了需要重点研究的主题.     

Economic model predictive control with triggered evaluations: State and output feedback

In this work, we focus on the computation load reduction in the optimization of economic model predictive control (EMPC) for nonlinear systems. Specifically, event-based triggering approach is adopted to significantly reduce the number of evaluations of the EMPC. First, we consider the case that state feedback is available and design a triggering condition based on the difference between the actual system state and its predicted value. At a sampling time, if the triggering condition is satisfied, the EMPC is re-evaluated. Subsequently, we consider the case that only output feedback is available. In this case, a robust moving horizon estimator is used to reconstruct the state information from output measurements and the corresponding triggering condition is based on the difference between the measured and predicted output as well as its time derivatives. For both cases, the EMPC is redesigned to account for potential open-loop operations. Sufficient conditions that ensure the closed-loop stability are provided for both cases. A chemical process is used to illustrate the effectiveness of the proposed designs.     

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.     

Production scheduling in dynamic real-time optimization with closed-loop prediction

Process plants are operating in an increasingly dynamic environment, fueled largely by globalization and deregulation of energy markets, resulting in fluctuating market conditions and large variations in electricity prices. Such conditions pose challenges for traditional hierarchical plant decision-making systems, leading to efforts toward integration across the decision-making layers. This paper proposes a formulation for integration of production scheduling decisions within a dynamic real-time optimization (DRTO) framework. The DRTO formulation utilizes a closed-loop prediction of the plant response under the action of constrained model predictive control (MPC). The integrated scheduling and DRTO system communicates decisions to the underlying MPC system through time-varying set-point trajectories, thereby permitting the standard MPC implementation to be retained. The efficacy of the prosed system is illustrated through application to both single-input single-output (SISO) and multi-input multi-output (MIMO) case studies.     

Robust MPC for systems with output feedback and input saturation

In this work, it is proposed an MPC control algorithm with proved robust stability for systems with model uncertainty and output feedback. It is assumed that the operating strategy is such that system inputs may become saturated at transient or steady state. The developed strategy aims at the case in which the controller performs in the output-tracking scheme following an optimal set point that is provided by an upper optimization layer of the plant control structure. In this case, the optimal operating point usually lies at the boundary of the region where the input is defined. Assuming that the system remains stabilizable in the presence of input saturation, the design of the robust controller is performed off-line and an on-line implementation strategy is proposed. At each sampling step, a sub optimal control law is obtained by combining control configurations that correspond to particular subsets of available manipulated inputs. Stability of the closed-loop system is forced by considering in the off-line step of the controller design, a state contracting restriction for the closed-loop system. To produce an offset free controller and to attend the case of unknown steady state, the method is developed for a state-space model in the incremental form. The method is illustrated with simulation examples extracted from the process industry.     

Multiobjective model predictive control

This paper proposes a novel model predictive control (MPC) scheme based on multiobjective optimization. At each sampling time, the MPC control action is chosen among the set of Pareto optimal solutions based on a time-varying, state-dependent decision criterion. Compared to standard single-objective MPC formulations, such a criterion allows one to take into account several, often irreconcilable, control specifications, such as high bandwidth (closed-loop promptness) when the state vector is far away from the equilibrium and low bandwidth (good noise rejection properties) near the equilibrium. After recasting the optimization problem associated with the multiobjective MPC controller as a multiparametric multiobjective linear or quadratic program, we show that it is possible to compute each Pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular Pareto optimal solution. Furthermore, we provide conditions for selecting Pareto optimal solutions so that the MPC control loop is asymptotically stable, and show the effectiveness of the approach in simulation examples.     

Convergence in economic model predictive control with average constraints

In this paper, we thoroughly investigate various aspects of economic model predictive control with average constraints, i.e., constraints on average values of state and input variables. In particular, we first show that a certain time-varying output constraint has to be included into the MPC problem formulation in order to ensure fulfillment of these average constraints. Optimizing a general (possibly economic) performance criterion may result in a non-converging behavior of the corresponding closed-loop system. While such a behavior might be acceptable in some cases, it may be undesirable for other types of applications. Hence as a second contribution, we provide a Lyapunov-like analysis to conclude that indeed asymptotic convergence to the optimal steady-state follows if the system satisfies a certain dissipativity condition. Finally, for the case that this dissipativity property is not satisfied but still a convergent behavior of the closed-loop is required, we examine two different methods how convergence can be enforced within an economic MPC setup by imposing additional average constraints on the system. In the first method, an additional average constraint is defined which results in the system being dissipative, while the second consists of imposing an additional even zero-moment average constraint. We illustrate our results with various examples.     

约束非线性系统稳定经济模型预测控制

考虑约束非线性系统,提出一种新的具有稳定性保证的经济模型预测控制（Economic model predictive control,EMPC）策略.由于经济性能指标的非凸性和非正定性,引入关于经济最优平衡点的正定辅助函数.利用辅助函数的最优值函数定义原始EMPC优化问题的稳定性约束.应用终端约束集、终端代价函数和局部控制器三要素,建立闭环系统关于经济最优平衡点的渐近稳定性和渐近平均性能.进一步,结合多目标理想点概念,将提出的控制策略用于多个经济性能指标的优化控制,得到稳定多目标EMPC策略.最后,以连续搅拌反应器为例,比较仿真结果验证本文策略的有效性.     

Fuel efficiency‐oriented platooning control of connected nonlinear vehicles: A distributed economic MPC approach

This paper considers the fuel efficiency‐oriented platooning control problem of connected vehicles. We present a novel distributed economic model predictive control (EMPC) approach to solve the problem of the vehicle platoon subject to nonlinear dynamics and safety constraints. In order to improve fuel economy of the whole vehicle platoon, the fuel consumption criterion is used to design the distributed EMPC strategy for the platoon. Meanwhile, the car‐tracking performance is exploited to guarantee stability and string stability of the platoon. Then the fuel efficiency control problem of the platoon is formulated as a distributed dual‐layer economic optimal control problem, which is solved in a fashion of receding horizon. It is proved that the proposed strategy guarantees asymptotic stability and predecessor‐follower string stability as well as fuel economy of the whole platoon by minimizing the fuel consumption cost. Finally, the effectiveness of the proposed strategy is highlighted by comparing its performance with that of the traditional distributed MPC strategy in numerical simulations.     

Steady-state target optimization designs for integrating real-time optimization and model predictive control

In industrial practice, the optimal steady-state operation of continuous-time processes is typically addressed by a control hierarchy involving various layers. Therein, the real-time optimization (RTO) layer computes the optimal operating point based on a nonlinear steady-state model of the plant. The optimal point is implemented by means of the model predictive control (MPC) layer, which typically uses a linear dynamical model of the plant. The MPC layer usually includes two stages: a steady-state target optimization (SSTO) followed by the MPC dynamic regulator. In this work, we consider the integration of RTO with MPC in the presence of plant-model mismatch and constraints, by focusing on the design of the SSTO problem. Three different quadratic program (QP) designs are considered: (i) the standard design that finds steady-state targets that are as close as possible to the RTO setpoints; (ii) a novel optimizing control design that tracks the active constraints and the optimal inputs for the remaining degrees of freedom; and (iii) an improved QP approximation design were the SSTO problem approximates the RTO problem. The main advantage of the strategies (ii) and (iii) is in the improved optimality of the stationary operating points reached by the SSTO-MPC control system. The performance of the different SSTO designs is illustrated in simulation for several case studies.     

Robust MPC for polytopic uncertain systems with time-varying delays

This paper addresses robust model predictive control (MPC) for time-delay systems with polytopic uncertainty. Uncertain time-varying input delay and state delays are considered, and the infinite horizon control moves are parametrised into an augmented state feedback law at each time instant. A receding horizon implementation of this state feedback law renders satisfaction of input/state constraints and closed-loop stability. For time-invariant delays and known delays, simplified results are obtained. A numerical example and a benchmark problem on continuous stirred tank reactor (CSTR) are given to illustrate the effectiveness of the proposed techniques.     

A general framework to design stabilizing nonlinear model predictive controllers

We propose a new model predictive control (MPC) framework to generate feedback controls for time-varying nonlinear systems with input constraints. We provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered. The supplied sufficient conditions for stability can also be used to analyse the stability of most previous MPC schemes. The class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system. Some important classes of nonlinear systems, including some nonholonomic systems, can now be stabilized by MPC. In addition, we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem, and thereby reduce the computational effort in the optimization algorithms used to implement MPC.     

Interactions between control and process design under economic model predictive control

Economic model predictive control (EMPC) is a model-based control scheme that integrates process control and economic optimization, which can potentially allow for time-varying operating policies to maximize economic performance. The manner in which an EMPC operates a process to optimize economics depends on the process dynamics, which are fixed by the process design. This raises the question of how process and EMPC designs interact. Works which have addressed process and control design interactions for steady-state operation have sought to simultaneously develop process designs and control law parameters to find the most profitable way to operate a process that is able to prevent process constraints from being violated and to optimize capital costs in the presence of disturbances. Because EMPC has the potential to operate a process in a transient fashion, this work first focuses on how EMPC and process design interact in the absence of disturbances. Using small-scale process examples, we seek to understand the fundamental nature of the interactions between EMPC and process design, including how these interactions can impact computational complexity of the controller and the design procedure. We subsequently utilize the insights gained to suggest controller design variables which might be considered as decision variables for a simultaneous process and control design problem when disturbances are considered.     

Economic model predictive control of nonlinear singularly perturbed systems

We focus on the development of a Lyapunov-based economic model predictive control (LEMPC) method for nonlinear singularly perturbed systems in standard form arising naturally in the modeling of two-time-scale chemical processes. A composite control structure is proposed in which, a “fast” Lyapunov-based model predictive controller (LMPC) using a quadratic cost function which penalizes the deviation of the fast states from their equilibrium slow manifold and the corresponding manipulated inputs, is used to stabilize the fast dynamics while a two-mode “slow” LEMPC design is used on the slow subsystem that addresses economic considerations as well as desired closed-loop stability properties by utilizing an economic (typically non-quadratic) cost function in its formulation and possibly dictating a time-varying process operation. Through a multirate measurement sampling scheme, fast sampling of the fast state variables is used in the fast LMPC while slow-sampling of the slow state variables is used in the slow LEMPC. Appropriate stabilizability assumptions are made and suitable constraints are imposed on the proposed control scheme to guarantee the closed-loop stability and singular perturbation theory is used to analyze the closed-loop system. The proposed control method is demonstrated through a nonlinear chemical process example.