A dynamic programming based approach for explicit model predictive control of hybrid systems

This work presents an algorithm for explicit model predictive control of hybrid systems based on recent developments in constrained dynamic programming and multi-parametric programming. By using the proposed approach, suitable for problems with linear cost function, the original model predictive control formulation is disassembled into a set of smaller problems, which can be efficiently solved using multi-parametric mixed-integer programming algorithms. It is also shown how the methodology is applied in the context of explicit robust model predictive control of hybrid systems, where model uncertainty is taken into account. The proposed developments are demonstrated through a numerical example where the methodology is applied to the optimal control of a piece-wise affine system with linear cost function.     

Mixed integer polynomial programming

The mixed integer polynomial programming problem is reformulated as a multi-parametric programming problem by relaxing integer variables as continuous variables and then treating them as parameters. The optimality conditions for the resulting parametric programming problem are given by a set of simultaneous parametric polynomial equations which are solved analytically to give the parametric optimal solution as a function of the relaxed integer variables. Evaluation of the parametric optimal solution for integer variables fixed at their integer values followed by screening of the evaluated solutions gives the optimal solutions.     

A multiparametric programming approach for mixed-integer quadratic engineering problems

In this work we propose algorithms for the solution of multiparametric quadratic programming (mp-QP) problems and multiparametric mixed-integer quadratic programming (mp-MIQP) problems with a convex and quadratic objective function and linear constraints. For mp-QP problems it is shown that the optimal solution, i.e. the vector of continuous variables and Lagrange multipliers, is an affine function of parameters. The basic idea of the algorithm is to use this affine expression for the optimal solution to systematically characterize the space of parameters by a set of regions of optimality. The solution of the mp-MIQP problems is approached by decomposing it into two subproblems, which converge based upon an iterative methodology. The first subproblem, which is an mp-QP, is obtained by fixing the integer variables and its solution represents a parametric upper bound. The second subproblem is formulated as a mixed-integer non-linear programming (MINLP) problem and its solution provides a new integer vector, which can be fixed to obtain a parametric solution, which is better than the current upper bound. The algorithm terminates with an envelope of parametric profiles corresponding to different optimal integer solutions. Examples are presented to illustrate the basic ideas of the algorithms and their application in model predictive and hybrid control problems.     

Optimal anticipatory control of ram velocity in injection molding

This paper discusses a computer control system for ram velocity of an injection molding machine using optimal state feedback based on the linear quadratic control theory. A new approach for the selection of appropriate weighting matrices is presented in this context. The simulation results reveal that the optimal controller has improved performance over the conventional PID controller presently used, having faster speed of response, significantly better tracking performance, and better noise filtering properties. The execution speed and the core storage requirements would allow implementation even on a small online computer.     

基于线性时变扰动模型的间歇过程最优迭代学习控制

A batch-to-batch optimal iterative learning control (ILC) strategy for the tracking control of product quality in batch processes is presented. The linear time-varying perturbation (LTVP) model is built for product quality around the nominal trajectories. To address problems of model-plant mismatches, model prediction errors in the previous batch run are added to the model predictions for the current batch run. Then tracking error transition models can be built, and the ILC law with direct error feedback is explicitly obtained. A rigorous theorem is proposed, to prove the convergence of tracking error under ILC. The proposed methodology is illustrated on a typical batch reactor and the results show that the performance of trajectory tracking is gradually improved by the ILC.     

From multi-parametric programming theory to MPC-on-a-chip multi-scale systems applications

An overview of multi-parametric programming and control is presented with emphasis on historical milestones, novel developments in the theory of multi-parametric programming and explicit MPC as well as their application to the design of advanced controller for complex multi-scale systems.     

Simultaneous design of explicit/multi-parametric constrained moving horizon estimation and robust model predictive control

In this work we present a rigorous methodology for the simultaneous design of moving horizon estimation (MHE) and robust model predictive control based on multi-parametric programming. First, an explicit/multi-parametric solution of the MHE is derived. Then, a novel method is presented that allows for the derivation of the estimation error dynamics, the bounding set of the estimation error, and the state estimate dynamic equations of constrained MHE. A framework is then presented for the design of robust explicit/multi-parametric model predictive control (MPC) controllers, based on tube-based MPC methods, which ensures that no constraints are violated due to the estimation error and the process noise in the system. This framework is first shown for the Kalman filter and unconstrained MHE and is then extended to the constrained MHE.     

基于Hammerstein-Wiener模型的连续搅拌反应釜神经网络预测控制

针对化工过程中广泛使用的连续搅拌反应釜(CSTR),提出一种基于神经网络的模型预测控制策略,采用分段最小二乘支持向量机辨识Hammerstein-Wiener模型系数的方法,在此基础上建立线性自回归模式〖DK〗(ARX)结构和高斯径向基神经网络串联的非线性预测控制器。利用BP神经网络训练预测控制输入序列和拟牛顿算法求解非线性预测控制律,从而实现一种基于支持向量机Hammerstein-Wiener辨识模型的非线性神经网络预测控制算法。对CSTR的仿真结果表明,该方法能够更有效地跟踪控制反应物浓度。     

Simultaneous optimization and control for polypropylene grade transition with two-layer hierarchical structure

In this paper, a two-layer hierarchical structure of optimization and control for polypropylene grade transition was raised to overcome process uncertain disturbances that led to the large deviation between the open-loop reference trajectory and the actual process. In the upper layer, the variant time scale based control vector parametric methods (VTS-CVP) was used for dynamic optimization of transition reference trajectory, while nonlinear model predictive controller (NMPC) based on closed-loop subspace and piece-wise linear (SSARX-PWL) model in the lower layer was tracking to the reference trajectory from the upper layer for overcoming high-frequency disturbances. Besides, mechanism about trajectory deviation detection and optimal trajectory updating onlinewere introduced to ensure a smooth transition for the entire process. The proposed method was validated with the real data from an industrial double-loop propylene polymerization reaction process with developed dynamic mechanismmathematicalmodel.     

State‐Space Digital PI Controller Design for Linear Stochastic Multivariable Systems with Input Delay

In this paper, a centralized digital PI control scheme is proposed for linear stochastic multivariable systems with input delay. The discrete linear quadratic regulator (LQR) approach with pole placement is used to achieve satisfactory set‐point tracking with guaranteed closed‐loop stability. In addition, the innovation form of Kalman gain is employed for state estimation with no prior knowledge of noise properties. Compared with existing designs, the proposed scheme provides an optimal closed‐loop design via the digitally implementable PI controller for linear stochastic multivariable systems with input delay. Its effectiveness will be demonstrated by the simulation study on examples from both industrial process control and aircraft control.     

Feasibility analysis and online adjustment of constraints in model predictive control integrated with soft sensor

Feasibility analysis of soft constraints for input and output variables is critical for model predictive control (MPC).When encountering the infeasible situation,some way should be found to adjust the constraints to guarantee that the optimal control law exists.For MPC integrated with soft sensor,considering the soft constraints for critical variables additionally makes it more complicated and difficult for feasibility analysis and constraint adjustment.Therefore,the main contributions are that a linear programming approach is proposed for feasibility analysis,and the corresponding constraint adjustment method and procedure are given as well.The feasibility analysis gives considerations to the manipulated,secondary and critical variables,and the increment of manipulated variables as well.The feasibility analysis and the constraint adjustment are conducted in the entire control process and guarantee the existence of optimal control.In final,a simulation case confirms the contributions in this paper.     

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015     

混杂系统基于参数规划的状态反馈预测控制

In this article, state feedback predictive controller for hybrid system via parametric programming is proposed. First, mixed logic dynamic （MLD） modeling mechanism for hybrid system is analyzed, which has a distinguished advantage to deal with the logic rules and constraints of a plant. Model predictive control algorithm with moving horizon state estimator （MHE） is presented. The estimator is adopted to estimate the current state of the plant with process disturbance and measurement noise, and the state estimated are utilized in the predictive controller for both regulation and tracking problems of the hybrid system based on MLD model. Off-line parametric programming is adopted and then on-line mixed integer programming problem can be treated as the parameter programming with estimated state as the parameters. A three tank system is used for computer simulation, results show that the proposed MHE based predictive control via parametric programming is effective for hybrid system with model/olant mismatch, and has a potential for the engineering applications.     

Control of a nonadiabatic packed bed reactor under periodic flow reversal

A full parametric study of the open-loop behavior of a packed bed reactor-recuperator system operating under periodic flow reversal produced a series of parametric maps which slow regions of operating conditions for which the system exhibits runaway, stable operation or extinction of the reaction. The reaction is CO oxidation over a Pt/alumina catalyst. A set of optimal operating conditions in terms of cycle time and heat transfer coefficient can be directly extracted from the parametric maps. A preliminary study on the control of periodic flow reversal tested and compared two strategies. 1) feedback PID control of the exit CO concentration and 2) model based feedforward control.     

Blood glucose regulation with stochastic optimal control for insulin-dependent diabetic patients

Diabetes is a disease resulting from the impaired mechanism of insulin secretion from the pancreas, which prevents glucose from entering the cells and being utilized and leads to wide swings of blood sugar and many complications such as heart disease and stroke, kidney disease and amputations. In order to prevent these complications and achieve a better quality of life for diabetic patients, effective regulation of blood glucose is essential. This study aims to achieve a better blood glucose control profile by incorporating the time-dependent uncertainties in diabetic patient parameters into formulations of optimal control using a novel approach which originates from finance literature. The time-dependent uncertainties are represented using stochastic processes called Ito processes and the mathematical formulation for this problem is presented. The usefulness of this approach is shown using experimental data from a diabetic patient and stochastic and deterministic optimal control profiles are computed. The stochastic profile results in fewer variations in blood glucose from the reference value of 4.5 mmol/L as compared to the deterministic profile in the presence of parametric uncertainty. This method holds a lot of promise in reducing the wide swings of blood glucose observed in diabetic patients and preventing possible complications of diabetes.     

基于PDE降阶模型的最优控制

针对带有约束条件的偏微分方程（PDE）模型最优控制的实时性要求和巨大的内存开销问题,提出了基于降阶模型的输入/状态约束的最优实时控制方法。采用特征正交分解和奇异值分解以及Galerkin投影方法导出了动态PDE具有高精度离散形式的状态空间低阶模型,提出了一定输入/状态约束条件下的基于二次规划单步滚动最优控制并与基于线性二次调节器的极值验证最优控制策略相互验证。通过对流-扩散-反应过程的控制仿真结果证明了所提方法的高效性和准确性。     

Integrating data‐based modeling and nonlinear control tools for batch process control

A data‐based multimodel approach is developed in this work for modeling batch systems in which multiple local linear models are identified using latent variable regression and combined using an appropriate weighting function that arises from fuzzy c‐means clustering. The resulting model is used to generate empirical reverse‐time reachability regions (RTRRs) (defined as the set of states from where the data‐based model can be driven inside a desired end‐point neighborhood of the system), which are subsequently incorporated in a predictive control design. Simulation results of a fed‐batch reactor system under proportional‐integral (PI) control and the proposed RTRR‐based design demonstrate the superior performance of the RTRR‐based design in both a fault‐free and faulty environment. The data‐based modeling methodology is then applied on a nylon‐6,6 batch polymerization process to design a trajectory tracking predictive controller. Closed‐loop simulation results illustrate the superior tracking performance of the proposed predictive controller over PI control. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

An optimal filter based MPC for systems with arbitrary disturbances

In this study, a linear model predictive control (MPC) approach with optimal filters is proposed for handling unmeasured disturbances with arbitrary statistics. Two types of optimal filters are introduced into the framework of MPC to relax the assumption of integrated white noise model in existing approaches. The introduced filters are globally optimal for linear systems with unmeasured disturbances that have unknown statistics. This enables the proposed MPC to better handle disturbances without access to disturbance statistics. As a result, the effort required for disturbance modeling can be alleviated. The proposed MPC can achieve offset-free control in the presence of asymptotically constant unmeasured disturbances. Simulation results demonstrate that the proposed approach can provide an improved disturbance õrejection performance over conventional approaches when applied to the control of systems with unmeasured disturbances that have arbitrary statistics.     

A New Optimal Control System Design for Chemical Processes

Based on frequency response and convex optimization, a novel optimal control system was developed for chemical processes. The feedforward control is designed to improve the tracking performance of closed loop chemical systems. The parametric model is not required because the system directly utilizes the frequency response of the loop transfer function, which can be measured accurately. In particular, the extremal values of magnitude and phase can be solved according to constrained quadratic programming optimizer and convex optimization. Simulation examples show the effectiveness of the method. The design method is simple and easily adopted in chemical industry.