Consistent hierarchical economic NMPC for a class of hybrid systems using neighboring-extremal updates

A hierarchical two-layer control algorithm is developed for a class of hybrid (discrete-continuous dynamic) systems to support economically optimal operation of batch or continuous processes with a predefined production schedule. For this class of hybrid systems, the optimal control moves as well as the controlled switching times between two adjacent modes are determined online. In contrast to closely related schemes for integrated scheduling and control, the sequence of modes is not optimized. On the upper layer, the economic optimal control problem is solved rigorously by a slow hybrid economic model predictive controller at a low sampling rate. On the lower layer, a fast hybrid neighboring-extremal controller is based on the same economic optimal control problem as the slow controller to ensure consistency between both layers. The fast neighboring-extremal controller updates rather than tracks the optimal trajectories from the upper layer to account for disturbances. Consequently, the fast controller steers the process to its operational bounds under disturbances and the economic potential of the process is exploited anytime. The suggested two-layer control algorithm provides fully consistent control action on the fast and slow time-scale and thus avoids performance degradation and even infeasibilities which are commonly encountered if inconsistent optimal control problems are formulated and solved.     

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.     

Reduction of the QP-MPC cascade structure to a single layer MPC

Some commercial MPC packages are implemented in two layers, the QP static layer and the MPC dynamic layer. In the absence of an upper Real Time Optimization layer, the static layer solves a simplified economic optimization problem, which defines optimum feasible targets for the dynamic layer. Since the LP/QP static layer and the MPC dynamic layer are usually executed within the same sampling period, it is not trivial to guarantee that the interaction between the two layers will not disrupt the stability of the whole structure. In this paper, it is proposed an approach to reduce the two-layer structure of some commercial MPCs to a single dynamic layer where the control cost function is extended to include the economic objective. In the proposed approach the convergence and stability of the closed-loop system can be obtained if the economic term of the cost function is properly weighted. A simulation example of a simple industrial system shows the efficiency of the proposed strategy.     

Model-based control of a simulated moving bed chromatographic process for the separation of fructose and glucose

Chromatographic separations are an expanding technology for the separation of high value products, particularly in the area of pharmaceutics, food, and fine chemicals. The simulated moving bed (SMB) process as a continuous chromatographic separation process is an interesting alternative to conventional batch chromatography, and gained more and more impact recently. The SMB process is realized by connecting several single chromatographic columns in series. A countercurrent movement of the bed is approximated by a cyclic switching of the inlet and outlet ports in the direction of the fluid stream. Because of its complex dynamics, the optimal operation and automatic control of SMB processes is a challenging task. This paper presents the design of a model-based optimization and control scheme for SMB chromatographic separation processes and its application to the separation of fructose and glucose. We propose a two-layer control architecture where the optimal operating trajectory is calculated off-line by dynamic optimization based on a rigorous process model. The parameters of the model are adapted based on online measurements. The low-level control task is to keep the process on the optimal trajectory despite disturbances and plant/model mismatch. Here identification models based on simulation data of the rigorous process model along the optimal trajectory are combined with a suitable local controller. The efficiency of the trajectory control algorithm is shown in a simulation study for the separation of fructose and glucose on an 8-column SMB plant.     

Nonlinear model predictive control of biodiesel production via transesterification of used vegetable oils

The economic performance of an industrial scale semi-batch reactor for biodiesel production via transesterification of used vegetable oils is investigated by simulation using nonlinear model predictive control (NMPC) technology. The objective is to produce biodiesel compliant to the biodiesel standards at the minimum costs. A first-principle model is formulated to describe the dynamics of the reactor mixture temperature and composition. The feed oil and mixture composition are characterized using a pseudo-component approach, and the thermodynamic properties are estimated from group contribution methods. The dynamic model is used by the NMPC framework to predict the optimal control profiles, where a multiple shooting based dynamic optimization problem is solved at every sampling time. Simulation results with the economic performance of an industrial scale semi-batch reactor are presented for control configurations manipulating the methanol feed flow rate and the heat duty.     

工业过程多速率分层运行优化控制

工业过程运行优化控制通常采用基础回路层和运行层两层结构,涉及不同时间尺度特性的被控对象,且由于检测装置采样周期不同难以统一控制与采样周期;此外,运行层动态往往机理复杂难以建模.因此针对这一多层次、多时间尺度且部分模型未知的复杂多速率控制问题,本文提出一种工业过程多速率分层运行优化控制方法.该方法在使用提升技术解决分层多速率问题的基础上,采用一种基于Q-!学习的数据驱动运行层设定值优化方法,更新基础回路层的设定值;并针对提升后的系统采用模型预测控制（Model predictive control,MPC）方法设计基础回路层控制器以跟踪设定值,从而实现运行指标的优化控制.对典型工业闭路磨矿过程进行了仿真实验,验证了本文所提方法的有效性.     

锅炉–汽轮机系统的模糊经济模型预测控制

现代电厂的优化控制通常采用双层控制结构,上层通过优化经济性能指标获得稳态设定值,传递到下层实现设定值跟踪.然而,传统的控制结构往往会忽略动态跟踪过程中的经济性能.本文针对锅炉–汽轮机系统设计了基于模糊模型的经济模型预测控制策略.通过离线设计稳定的线性反馈控制律和可行域,来保证优化问题的递推可行性和稳定性.通过在线求解双模态经济模型预测控制优化问题,实现锅炉汽轮机系统动态过程中经济性能的提高.大范围和小范围负荷变化情况下的仿真结果表明了本文提出的模糊经济模型预测控制的有效性.     

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.     

针铁矿法沉铁过程双层结构优化控制

针铁矿沉铁过程是锌冶炼过程中一个非常重要的环节,其中最重要的是控制氧气添加量,因此本文提出一种针铁矿沉铁过程双层结构优化控制方法.上层定义氧气利用率衡量理论消耗量与实际添加量的差别,以过程氧气利用率最高为目标优化设定级联反应器出口二价铁离子浓度下降梯度,下层以过程氧气消耗最少和出口离子浓度与上层设定值误差最小为优化目标,过程动态模型和工艺条件为约束,求解构造的非线性优化控制问题得到各反应器最优氧气添加速率.为减少不确定性干扰对系统的影响,采用一种模型参数自适应校正的方法对模型参数进行校正保证优化控制器的性能.最后根据过程离子浓度采样值计算过程实际氧气利用率作为上层优化参数重更新反应器出口二价铁离子浓度最优设定值.由于下层优化问题约束多且约束多呈非线性,采用Legendre伪谱法求解下层优化问题.仿真结果表明,所提出的双层结构优化控制方法能实现过程准确控制,减少过程氧气消耗.     

Application of economic MPC to the energy and demand minimization of a commercial building

This paper presents an application case study of an economic model predictive control (EMPC) method for optimizing the building demand and energy cost under the time-of-use price policy. The control strategy is comprised of an economic objective function that accounts for the combination of energy and demand costs with a time-of-use rate structure, a dynamic thermal process and power model of the building thermal mass dynamics, and a set of constraints to ensure the building is operated properly. The optimization is a min–max optimization problem and is converted to a linear program. The EMPC method is implemented in a commercial office building located in Milwaukee, Wisconsin, USA. An internet-based control architecture is developed to carry out tests with the EMPC controller at a remote location. The test results show that the EMPC strategy is capable of shifting the peak demand to off-peak hours and reducing energy costs compared to a baseline case for the building.     

An MPC approach to the design of two-layer hierarchical control systems

A methodology for the design of two-layer hierarchical control systems is presented. The high layer corresponds to a system with slow dynamics, whose control inputs must be provided by subsystems with faster dynamics placed at the low layer. Model Predictive Control laws are synthesized for both layers and overall convergence properties are established. The use of different control configurations is also considered by allowing the switching on/off of the subsystems at the low layer. A simulation example is reported to witness the potentialities of the proposed solution.     

Measurement-based optimization of batch and repetitive processes using an integrated two-layer architecture

This paper is concerned with optimal control of batch and repetitive processes in the presence of uncertainty. An integrated two-layer optimization strategy is proposed, whereby within-run corrections are performed using a neighboring-extremal update strategy and run-to-run corrections are based on a constraint-adaptation scheme. The latter is appealing since a feasible operating strategy is guaranteed upon convergence, and its combination with neighboring-extremal updates improves the reactivity and convergence speed. Moreover, these two layers are consistent in that they share the same objective function. The proposed optimization scheme is declined into two versions, namely an indirect version based on the Pontryagin maximum principle and a direct version that applies a control parameterization and nonlinear programming techniques. Although less rigorous, the latter approach can deal with singular extremals and path constraints as well as handle active-set changes more conveniently. Two case studies are considered. The indirect approach is demonstrated for a level-control problem in an experimental two-tank system, whereas the direct approach is illustrated in numerical simulation on a fed-batch reactor for acetoacetylation of pyrrole. The results confirm that faster adaptation is possible with the proposed integrated two-layer scheme compared to either constraint adaptation or neighboring-extremal update alone.     

New Formulation of Dynamic Output Feedback Robust Model Predictive Control with Guaranteed Quadratic Boundedness

The dynamic output feedback robust model predictive controller for a system with both polytopic uncertainty and bounded disturbance is addressed in this paper. This controller utilizes a main optimization problem to find the control law and a simple auxiliary optimization problem to refresh the bounds on the true state. The main optimization problem, which is not necessarily solved at each sampling instant, achieves the near‐optimal solution. The auxiliary optimization, which is solved at each sampling instant, is followed with a simple criterion which determines whether or not to solve the main optimization problem at the next sampling time. By applying the proposed method, the augmented state of the closed‐loop system is guaranteed to converge to the neighborhood of the equilibrium point.     

Neighboring-extremal updates for nonlinear model-predictive control and dynamic real-time optimization

Nonlinear model-predictive control (NMPC) and dynamic real-time optimization (DRTO) lead to a substantial improvement of the operation of complex nonlinear processes. Whereas the focus in NMPC is primarily on control performance by minimizing the deviation from a given set-point, the objective in DRTO is to achieve a profitable and flexible operation adapted to changing market conditions and process uncertainties by employing an economic optimization criterion. A method for solving dynamic optimization problems based on neighboring-extremal updates suitable for applications in NMPC and DRTO is presented in this paper. If process operation is affected by small perturbations, efficient techniques for updating the nominal trajectories based on parametric sensitivities can be applied [J. Kadam, W. Marquardt, Sensitivity-based solution updates in closed-loop dynamic optimization, in: Proceedings of the DYCOPS 7 Conference, Cambridge, USA, 2004]; these updates do not require the solution of the rigorous optimization problem but rely on first and second-order sensitivities computed by composite adjoints. However for larger perturbations and strong nonlinearities, the fast updates obtained by the neighboring-extremal solutions are not sufficiently accurate, and the solution of the nonlinear optimization problem requires further iterations with updated sensitivities to give a feasible and optimal solution. The application of the method to a simulated semi-batch reactor demonstrates its capabilities. The presented method is discussed in comparison to other methods in the literature.     

The infinite time near optimal decentralized regulator problem for singularly perturbed systems: a convex optimization approach

This paper presents LQ decentralized pole location for singularly perturbed systems. The poles are located in a sector included in the left-half complex plane. The singular perturbation method is used to define reduced and well-behaved problems. It is shown that the LQ control problem with pole location in a sector can be solved using the LMI tool. The associated parametrical optimization problem involves a linear cost objective under LMI constraints. The decentralized control problem is then solved in the reduced slow system by just introducing structure constraints on the matrix variables, constraints that do not destroy the linearity and then the convexity of the problem.     

Adaptive composite suboptimal control for linear singularly perturbed systems with unknown slow dynamics

In this article, using singular perturbation theory and adaptive dynamic programming (ADP) approach, an adaptive composite suboptimal control method is proposed for linear singularly perturbed systems (SPSs) with unknown slow dynamics. First, the system is decomposed into fast‐ and slow‐subsystems and the original optimal control problem is reduced to two subproblems in different time‐scales. Afterward, the fast subproblem is solved based on the known model of the fast‐subsystem and a fast optimal control law is designed by solving the algebraic Riccati equation corresponding to the fast‐subsystem. Then, the slow subproblem is reformulated by introducing a system transformation for the slow‐subsystem. An online learning algorithm is proposed to design a slow optimal control law by using the information of the original system state in the framework of ADP. As a result, the obtained fast and slow optimal control laws constitute the adaptive composite suboptimal control law for the original SPSs. Furthermore, convergence of the learning algorithm, suboptimality of the adaptive composite suboptimal control law and stability of the whole closed‐loop system are analyzed by singular perturbation theory. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed methods.     

基于数学规划的化工过程常规控制系统的结构选择

化工过程常规控制结构选择问题,由过程控制最底层的多回路PID控制器的结构选择和参数优化问题组成,会直接影响到整个化工过程控制系统的整体控制性能。本文作出常规控制结构选择问题的数学描述,将其转化为带有0-1变量约束的混合整数动态优化问题,并通过将0-1变量松弛化、引入附加等式约束求解此问题。以复杂的催化裂化装置反应-再生系统为例求解其常规控制结构选择问题,求解结果验证优化算法的有效性。     

Optimizing model predictive control of an industrial distillation column

The main scope of this work is the implementation of an MPC that integrates the control and the economic optimization of the system. The two problems are solved simultaneously through the modification of the control cost function that includes an additional term related to the economic objective. The optimizing MPC is based on a quadratic program (QP) as the conventional MPC and can be solved with the available QP solvers. The method was implemented in an industrial distillation system, and the results show that the approach is efficient and can be used, in several practical cases.     

Optimality and convergence of adaptive optimal control by reinforcement synthesis

Adaptive Optimal Control (AOC) by reinforcement synthesis is proposed to facilitate the application of optimal control theory in feedback controls. Reinforcement synthesis uses the critic–actor architecture of reinforcement learning to carry out sequential optimization. Optimality conditions for AOC are formulated using the discrete minimum principle. A proof of the convergence conditions for the reinforcement synthesis algorithm is presented. As the final time extends to infinity, the reinforcement synthesis algorithm is equivalent to the Dual Heuristic dynamic Programming (DHP) algorithm, a version of approximate dynamic programming. Thus, formulating DHP with the AOC approach has rigorous proofs of optimality and convergence. The efficacy of AOC by reinforcement synthesis is demonstrated by solving a linear quadratic regulator problem.     

5G超密集异构网络带内无线回传资源分配方案

为实现5G超密集异构网络中无线回传链路和接入链路之间的最优资源分配,研究多用户场景下双层异构网络的联合用户调度和功率分配问题,在队列稳定和无线回传资源有限的情况下,综合考虑用户调度、功率分配和干扰控制等因素,对带内无线回传的最优资源分配问题进行数学建模并求解,基于李雅普诺夫优化理论提出联合用户调度和功率分配的优化算法。将优化问题解耦为网络内各个用户的调度以及宏基站和小基站的功率分配过程,采用MOSEK求解器和二分类方法获得用户调度向量,利用拉格朗日乘子法求解功率分配问题,并通过队列的时刻更新过程实现最优资源分配。仿真结果表明,在多用户场景下,该方案能够有效提升网络总吞吐量以及网络效用,并且毫米波频段的通信性能优于传统蜂窝网络频段。