Advanced step nonlinear model predictive control for air separation units

Cryogenic air separation units constitute an integral part of many industrial processes and next generation power plants. These units are characterized by fluctuating operating conditions to respond to changing product demands. The dynamics of these transitions are highly nonlinear and energy-intensive. Consequently, nonlinear model predictive control (NMPC) based on rigorous dynamic models is essential for high performance in these applications. Currently, the implementation of NMPC controllers is limited by the computational complexity of the associated on-line optimization problems. In this work, we make use of the so-called advanced step NMPC controller to overcome these limitations. We demonstrate that this sensitivity-based strategy reduces the on-line computational time to just a single CPU second, while incorporating a highly detailed dynamic air separation unit model. Finally, we demonstrate that the controller can handle nonlinear dynamics over a wide range of operating conditions.     

A reduced regularization strategy for economic NMPC

Nonlinear Model Predictive Control (NMPC) enables the incorporation of detailed dynamic process models for nonlinear, multivariable control with constraints. This optimization-based framework also leads to on-line dynamic optimization with performance-based and so-called economic objectives. Nevertheless, economic NMPC (eNMPC) still requires careful formulation of the nonlinear programming (NLP) subproblem to guarantee stability. In this study, we derive a novel reduced regularization eNMPC approach with stability guarantees. Compared with full state regularization, the proposed strategy is less conservative and easier to implement. The resulting eNMPC framework is firstly demonstrated on a nonlinear continuous stirred-tank reactor (CSTR) example and a large-scale double distillation system example. Then the proposed strategy is applied to a challenging nonlinear CO₂ capture model, where bubbling fluidized bed models comprise a solid-sorbent post-combustion carbon capture system. Our results indicate the benefits of this improved eNMPC approach over tracking to the setpoint, and better stability over eNMPC without regularization.     

Theoretical analysis of unconstrained nonlinear model predictive control

Model predictive control (MPC) is a well-established controller design strategy for linear process models. Because many chemical and biological processes exhibit significant nonlinear behaviour, several MPC techniques based on nonlinear process models have recently been proposed. The most significant difference between these techniques is the computational approach used to solve the nonlinear model predictive control (NMPC) optimization problem. Consequently, analysis of NMPC techniques is often connected to the computational approach employed. In this paper, a theoretical analysis of unconstrained NMPC is presented that is independent of the computational approach. A nonlinear discrete-time, state-space model is used to predict the effects of future inputs on future process outputs. It is shown that model inverse, pole-placement, and steady-state controllers can be obtained by suitable selection of the control and prediction horizons. Moreover, the NMPC optimization problem can be modified to yield nonlinear internal model control (NIMC). The computational requirements of NIMC are considerably less than NMPC, but the NIMC approach is currently restricted to nonlinear models with well-defined and stable inverses. The NIMC controller is shown to provide superior servo and regulatory performance to a linear IMC controller for a continuous stirred tank reactor.     

Advanced-multi-step nonlinear model predictive control

Nonlinear model predictive control (NMPC) has gained widespread attention due to its ability to handle variable bounds and deal with multi-input, multi-output systems. However, it is susceptible to computational delay, especially when the solution time of the nonlinear programming (NLP) problem exceeds the sampling time. In this paper we propose a fast NMPC method based on NLP sensitivity, called advanced-multi-step NMPC (amsNMPC). Two variants of this method are developed, the parallel approach and the serial approach. For the amsNMPC method, NLP problems are solved in background multiple sampling times in advance, and manipulated variables are updated on-line when the actual states are available. We present case studies about a continuous stirred tank reactor (CSTR) and a distillation column to show the performance of amsNMPC. Nominal stability properties are also analyzed.     

Robust stability of economically oriented infinite horizon NMPC that include cyclic processes

Nonlinear model predictive control (NMPC) with economic objective attracts growing interest. In our previous work [1], nominal stability of economically oriented NMPC for cyclic processes was proved by introducing a transformed system, and an infinite horizon NMPC formulation with discount factors was proposed. Moreover, the nominal stability property for economically oriented NMPC was analyzed in [2] for a class of systems satisfying strong duality. In this study, we extend the previous stability analysis in [1] to a general infinite horizon NMPC formulation with economic objectives. Instead of the strong duality assumption, we require the stage cost to be strongly convex, which is easier to check for a general nonlinear system. In addition, robust stability of this NMPC controller is also analyzed based on the Input-to-State Stability (ISS) framework. A simulated nonlinear double tank system subject to periodic change in electricity price is presented to illustrate the stability property. Finally, an industrial size air separation unit case study with periodic electricity cost is presented.     

基于FPGA的非线性预测控制器设计

对于非线性程度较高的复杂对象,非线性模型预测控制（NonlinearModelPredictiveControl,NMPC）是一种有效的控制策略。为了实现对这类对象的有效控制,设计了一种基于FPGA（FieldProgrammableGateArray）的非线性预测控制器,该嵌入式控制器具有灵活性和高适应性等特点,能够应用于工业现场控制。为了满足工业控制的可行性和实时性要求,提出了一种序贯二次规划（SQP）算法的改进算法,在FPGA有限的计算资源下,保证每个采样间隔内都能得到NMPC优化问题的可行解。经仿真实验证明,采用非线性预测控制器在计算速度和精度上都能达到较好的性能。     

Explicit output-feedback nonlinear predictive control based on black-box models

Nonlinear model predictive control (NMPC) algorithms are based on various nonlinear models. A number of on-line optimization approaches for output-feedback NMPC based on various black-box models can be found in the literature. However, NMPC involving on-line optimization is computationally very demanding. On the other hand, an explicit solution to the NMPC problem would allow efficient on-line computations as well as verifiability of the implementation. This paper applies an approximate multi-parametric nonlinear programming approach to explicitly solve output-feedback NMPC problems for constrained nonlinear systems described by black-box models. In particular, neural network models are used and the optimal regulation problem is considered. A dual-mode control strategy is employed in order to achieve an offset-free closed-loop response in the presence of bounded disturbances and/or model errors. The approach is applied to design an explicit NMPC for regulation of a pH maintaining system. The verification of the NMPC controller performance is based on simulation experiments.     

基于扩张状态观测器的车辆横摆稳定模型预测控制器设计

针对车辆横摆稳定性控制问题,本文提出一种基于扩张状态观测器的线性模型预测控制器设计方法.首先,将非线性车辆模型线性化,建立带有模型误差干扰项的线性模型,其中线性化导致的模型误差采用扩张状态观测器估计得到,并证明了观测器的稳定性.然后基于此模型设计线性预测控制器,近似实现了非线性预测控制器的控制效果,同时降低了计算量.最后,通过不同路况下的仿真实验结果,验证了所提方法的计算性能和控制效果.     

Serial advanced-multi-step nonlinear model predictive control using an extended sensitivity method

Nonlinear model predictive control (NMPC) can directly handle multi-input multi-output nonlinear systems and explicitly consider input and state constraints. However, the computational load for nonlinear programming (NLP) of large-scale systems limits the range of possible applications and degrades NMPC performance. An NLP sensitivity based approach, advanced-step NMPC, has been developed to address the online computational load. In addition, for cases where the NLP solving time exceeds one sampling time, two types of advanced-multi-step NMPC (amsNMPC), parallel and serial, have been proposed. However, in previous studies, a serial amsNMPC could not be applied to large-scale problems because of the size of extended Karush–Kuhn–Tucker matrix and its Schur complement decomposition, and the robustness was analyzed under a conservative assumption for memory effects. In this paper, we propose a serial amsNMPC using an extended sensitivity method to increase the online computation speed further. We successfully apply it to a large-scale air separation unit using the sparse matrix handling packages of Python, Pyomo, and k_aug tools. Furthermore, an auxiliary NLP formulation is defined to analyze the robustness. Using this with the key properties of an extended sensitivity matrix, we can prove robustness while avoiding the memory effects term.     

Evaluation study of an efficient output feedback nonlinear model predictive control for temperature tracking in an industrial batch reactor

The paper illustrates the benefits of nonlinear model predictive control (NMPC) for the setpoint tracking control of an industrial batch polymerization reactor. Real-time feasibility of the on-line optimization problem from the NMPC is achieved using an efficient multiple shooting algorithm. A real-time formulation of the NMPC that takes computational delay into account is described. The control relevant model for the NMPC is derived from the complex-first principles model and is fitted to the experimental data using maximum likelihood estimation. A parameter adaptive extended Kalman filter (PAEKF) is used for state estimation and on-line model adaptation. The performance of the NMPC implementation is assessed via simulation and experimental results.     

Diesel engine air path control based on neural approximation of nonlinear MPC

This paper deals with a control design problem for a diesel engine air path system that has strong nonlinearity and requires multi-input and multi-output control to satisfy requirements and constraints. We focus on a neural network based approximation of nonlinear model predictive control (NMPC) for high-speed computation. Most neural approximation methods are verified only through simulation; further, the influence of approximation on the closed-loop performance has been not sufficiently discussed. In this study, we discuss this influence, and propose a new method to improve stability against degradation due to an approximation error. The control system is assembled using a neural network based controller, obtained by the proposed method, and an unscented Kalman filter. This system is verified both numerically and experimentally; the results demonstrate the capability of the proposed method to track the boost pressure, EGR rate, and pumping loss according to the reference values, and satisfy the constraints of compressor surge and choke. The high computation speed that can be achieved using a standard on-board ECU is also demonstrated using the approximated controller.     

Comparative evaluation of nonlinear model predictive and flatness-based two-degree-of-freedom control design in view of industrial application

Two advanced nonlinear model-based control design methods – nonlinear model predictive control (NMPC) and a two-degree-of-freedom control-scheme with flatness-based feedforward control design and decentralised PI-controllers (FB-2DOF) – are compared in view of industrial application. The comparative evaluation is carried out on a setpoint-transition of the Klatt–Engell reactor model. Based on an analysis of simulation scenarios, the controllers are compared with respect to controller performance, robustness criteria, and implementation issues. Thereby, the choice of the control task and the comparison methodology are oriented on industrial practice.In the considered comparative evaluation, NMPC exhibits performance advantages when it comes to time-efficient setpoint-transitions in the nominal case, in which FB-2DOF control design is restricted by the existing input constraints. In return, robustness of stability of the FB-2DOF controller is determined only by the feedback controll part; it is therefore independent from the setpoint-transition performance – determined by the feedforward controll part – whereas the NMPC suffers from degradation of robustness properties if it is tuned for time-efficiency only. NMPC allows direct incorporation of process models and constraints, but, as it employs computationally expensive online optimisation, has to be connected to the digital control system (DCS) via some standard interface. The FB-2DOF controller in contrast can be directly implemented in a DCS, whereby the feedforward-part can be realised as an extension of an already existing feedback-part.     

Multirate sliding mode disturbance compensation for model predictive control

In this paper, a novel hierarchical multirate control scheme for nonlinear discrete‐time systems is presented, consisting of a robust nonlinear model predictive controller (NMPC) and a multirate sliding mode disturbance compensator (MSMDC). The proposed MSMDC acts at a faster rate than the NMPC in order to keep the system as close as possible to the nominal trajectory predicted by NMPC despite model uncertainties and external disturbances. The a priori disturbance compensation turns out to be very useful in order to improve the robustness of the NMPC controller. A dynamic input allocation between MSMDC and NMPC allows to maximize the benefits of the proposed scheme that unites the advantages of sliding mode control (strong reduction of matched disturbances, low computational burden) to those of NMPC (optimality, constraints handling). Sufficient conditions required to guarantee input‐to‐state stability and constraints satisfaction by the overall scheme are also provided. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on stability, robustness and performance of output feedback nonlinear model predictive control

In recent years, nonlinear model predictive control (NMPC) schemes have been derived that guarantee stability of the closed loop under the assumption of full state information. However, only limited advances have been made with respect to output feedback in the framework of nonlinear predictive control. This paper combines stabilizing instantaneous state feedback NMPC schemes with high-gain observers to achieve output feedback stabilization. For a uniformly observable MIMO system class it is shown that the resulting closed loop is asymptotically stable. Furthermore, the output feedback NMPC scheme recovers the performance of the state feedback in the sense that the region of attraction and the trajectories of the state feedback scheme can be recovered to any degree of accuracy for large enough observer gains, thus leading to semi-regional results. Additionally, it is shown that the output feedback controller is robust with respect to static sector bounded nonlinear input uncertainties.     

Nonlinear system identification for predictive control using continuous time recurrent neural networks and automatic differentiation

In this paper, a continuous time recurrent neural network (CTRNN) is developed to be used in nonlinear model predictive control (NMPC) context. The neural network represented in a general nonlinear state-space form is used to predict the future dynamic behavior of the nonlinear process in real time. An efficient training algorithm for the proposed network is developed using automatic differentiation (AD) techniques. By automatically generating Taylor coefficients, the algorithm not only solves the differentiation equations of the network but also produces the sensitivity for the training problem. The same approach is also used to solve the online optimization problem in the predictive controller. The proposed neural network and the nonlinear predictive controller were tested on an evaporation case study. A good model fitting for the nonlinear plant is obtained using the new method. A comparison with other approaches shows that the new algorithm can considerably reduce network training time and improve solution accuracy. The CTRNN trained is used as an internal model in a predictive controller and results in good performance under different operating conditions.     

时变非线性不确定系统H∞鲁棒模型预测控制

针对输入受限的时变不确定非线性系统,提出一种H∞鲁棒模型预测控制策略。假设线性化系统矩阵一致有界,将非凸的无穷时域优化问题转化为带有单个线性矩阵不等式(LMI)约束的凸优化问题,降低控制量求解难度。结合滚动优化原理与H∞控制方法在线极小化性能指标,使得闭环系统满足控制性能和约束。在LMI框架下给出H∞NMPC的求解方法及其鲁棒稳定性充分条件。仿真实验对比验证了该策略的有效性。     

Advanced-step multistage nonlinear model predictive control: Robustness and stability

Nonlinear model predictive control (NMPC) has been popular in many applications, especially when constraint satisfaction is critical. However, due to plant-model mismatch and disturbances, robust NMPC generally faces three challenges: robust performance, real-time implementation, and stability. In this paper, we propose a parallelizable advanced-step multistage NMPC (as-msNMPC), which provides a non-conservative robust control solution that explicitly addresses two types of uncertainty: model parameters and unmeasured noise. The first type is attended to by incorporating scenario trees and the second by applying nonlinear programming (NLP) sensitivity. In addition, robust stability concepts have been discussed for both ideal multistage NMPC (ideal-msNMPC) and as-msNMPC. Under suitable assumptions, as-msNMPC has demonstrated input-to-state practical stability properties with the presence of two types of uncertainty. Lastly, the as-msNMPC framework has been applied to continuous stirred tank reactor (CSTR) and quad-tank case studies for tracking setpoints to demonstrate its performance in robustness and efficiency.     

Unscented Kalman filter based nonlinear model predictive control of a LDPE autoclave reactor

This paper presents a multivariable nonlinear model predictive control (NMPC) scheme for the regulation of a low-density polyethylene (LDPE) autoclave reactor. A detailed mechanistic process model developed previously was used to describe the dynamics of the LDPE reactor and the properties of the polymer product. Closed-loop simulations are used to demonstrate the disturbance rejection and tracking performance of the NMPC algorithm for control of reactor temperature and weight-averaged molecular weight (WAMW). In addition, the effect of parametric uncertainty in the kinetic rate constants of the LDPE reactor model on closed-loop performance is discussed. The unscented Kalman filtering (UKF) algorithm is employed to estimate plant states and disturbances. All control simulations were performed under conditions of noisy process measurements and structural plant–model mismatch. Where appropriate, the performance of the NMPC algorithm is contrasted with that of linear model predictive control (LMPC). It is shown that for this application the closed-loop performance of the UKF based NMPC scheme is very good and is superior to that of the linear predictive controller.     

Predictive optimal control for thermo-mechanical pulping processes with multi-stage low consistency refining

In this paper, we present a nonlinear model based dynamic optimization approach for optimal control for thermo-mechanical pulping (TMP) processes. The method is based on nonlinear model predictive control (NMPC) technique with an economic objective function. In our previous work [1], an economically oriented NMPC (econNMPC) was presented for a two-stage TMP process with high consistency (HC) refiners. The two-stage TMP process was simulated up to the secondary refiner. In this study, we extend the previous study to multiple stages that include low consistency (LC) refining. Here we can exploit the complexity of interactions of multiple stage TMP for both performance and optimal operations. The first process is a conventional refining process with two stages of HC refining with a primary and a secondary refiner, followed by a latency chest and a third-stage LC refiner. The second TMP process has only a HC primary refiner, and the secondary stage HC refiner is replaced by multiple stages of LC refiners after the latency chest. Both TMP schemes are dynamically optimized against disturbances to produce the same target final pulp qualities while minimizing specific energy consumption of the processes. The simulation results show the potential economic benefits of the proposed method, through a reduction in total specific energy, about 24%, for the second TMP process.