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.     

Real-time energy management for electric arc furnace operation

Electric arc furnaces are used extensively in the steel industry for steel production. Development of energy savings strategies for the highly energy-intensive batch process is extremely challenging due to the complexity of the process and lack of measurements due to the harsh operating conditions. Here we introduce a new energy management approach that effectively curtails the energy cost in real-time through the implementation of economically optimal operating decisions. An economics- oriented shrinking horizon nonlinear model predictive control (NMPC) algorithm that exploits time-varying electricity prices is coupled with a multi-rate moving horizon estimator (MHE) to form an integrated decision- making framework. With a detailed first-principles dynamic model functioning at the core, the multi-variable interactions and plant variations are successfully incorporated into the control strategy to achieve reliable performance. We also present a novel initialization scheme for obtaining fast on-line solutions of the economic NMPC and multi-rate MHE dynamic optimization problems. Using this initialization algorithm, we show that the optimal input decisions are obtained with sufficient computational speed for real-time implementation. The energy usage optimization results indicate a significant reduction in the operating cost and peak electricity demand compared to the case where the electricity price profile is not updated.     

Adaptive peak seeking control of a proton exchange membrane fuel cell

The primary aim of operating any fuel cell (PEMFC) system is to produce the power/electricity at maximum efficiency. The cell voltage/current manipulation appear to be the most suitable choice for controlling the power density. However, the power density exhibits a highly nonlinear and complex dynamic relationship with respect to the cell voltage. Since the process output variable (i.e. power density) itself is the objective function for the optimization, there exists a singularity at the optimum operating condition. In addition, the location of the optimum operating point changes with time due to the occurrence of variety of disturbances and/or changes in the operating conditions. Thus, the need to operate the PEMFC at its peak power density and track the shifting optimum turns out to be a challenging control problem. The task of on-line optimizing control of PEMFC poses difficulties in real time control due to its fast dynamics and it is impractical to employ a mechanistic model for locating the changing optimum on-line. In this context the adaptive optimizing control scheme developed by Bamberger and Isermann (1978) [1] appears interesting. Their scheme is based on on-line adaptation of a nonlinear black box time series models and facilitates analytical computation of changing optimum. Recently, Bedi et al. (2007) [2] have developed a closed form multi-step predictive control law under nonlinear internal model control framework using a black-box nonlinear model and employed it for peak power control in PEMFC. From the viewpoint of PEMFC operation, this nonlinear IMC controller meets the demand on the fast computations as a closed form solution is obtained for the nonlinear control problem at each time step. In this work, we propose to develop an adaptive optimizing control scheme, which combines the attractive features of the on-line optimization approach proposed by Bamberger and Isermann (1978) [1] and closed form control law developed by Bedi et al. (2007) [2]. We demonstrate the effectiveness of the proposed adaptive optimizing scheme by conducting simulation studies on the distributed an along-the-channel model of PEMFC. Analysis of the simulation results indicate that the proposed adaptive optimizing control scheme satisfactorily tracks the shifting optimum operating point in the face of changing unmeasured disturbances     

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.     

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.     

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.     

Dual-mode Nonlinear MPC via Terminal Control Laws With Free-parameters

The paper presents a new dual-mode nonlinear model predictive control (NMPC) scheme for continuous-time nonlinear systems subject to constraints on the state and control. The idea of control Lyapunov functions for nonlinear systems is used to compute the terminal regions and terminal control laws with some free-parameters in the dual-mode NMPC framework. The parameters of the terminal controller are selected offline to estimate the terminal region as large as possible; and the parameters are optimized online to gain optimality of the terminal controller with respect to given cost functions. Then a dual-mode NMPC algorithm with varying time-horizon is formulated for the constrained system. Recursive feasibility and closed-loop stability of this NMPC are established. The example of a spring-cart is used to demonstrate the advantages of the presented scheme by comparing to the dual-mode NMPC via the linear quadratic regulator (LQR) method.      

Identification and predictive control of a multistage evaporator

A recurrent neural network-based nonlinear model predictive control (NMPC) scheme in parallel with PI control loops is developed for a simulation model of an industrial-scale five-stage evaporator. Input–output data from system identification experiments are used in training the network using the Levenberg–Marquardt algorithm with automatic differentiation. The same optimization algorithm is used in predictive control of the plant. The scheme is tested with set-point tracking and disturbance rejection problems on the plant while control performance is compared with that of PI controllers, a simplified mechanistic model-based NMPC developed in previous work and a linear model predictive controller (LMPC). Results show significant improvements in control performance by the new parallel NMPC–PI control scheme.     

Latent‐variable Nonlinear Model Predictive Control Strategy for a pH Neutralization Process

Linear model predictive control (MPC) is a widely‐used control strategy in chemical processes. Its extension to nonlinear MPC (NMPC) has drawn increasing attention since many process systems are inherently nonlinear. When implementing the NMPC based on a nonlinear predictive model, a nonlinear dynamic optimization problem must be calculated. For the sake of solving this optimization problem efficiently, a latent‐variable dynamic optimization approach is proposed. Two kinds of constraint formulations, original variable constraint and Hotelling T² statistic constraint, are also discussed. The proposed method is illustrated in a pH neutralization process. The results demonstrate that the latent‐variable dynamic optimization based the NMPC strategy is efficient and has good control performance.     

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.     

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 general NMPC framework for a diesel engine air path

This paper presents the formulation of a parameterised nonlinear model predictive control (NMPC) scheme to be applied on a diesel engine air path. The most important feature of the proposed controller is that it uses no structural properties of the system model. Therefore, the proposed NMPC scheme can be applied to any nonlinear system, leading to a general framework for a diesel engine air path. Moreover, the computational burden is substantially reduced due to an optimisation problem of low dimension obtained by means of the parameterised approach. Simulation results and an experimental validation are presented in order to emphasise the controller's efficiency and the real-time implementability.     

Nonlinear model predictive control of fed-batch fermentations using dynamic flux balance models

Fed-batch fermentation is an important production technology in the biochemical industry. Using fed-batch Saccharomyces cerevisiae fermentation as a prototypical example, we developed a general methodology for nonlinear model predictive control of fed-batch bioreactors described by dynamic flux balance models. The control objective was to maximize ethanol production at a fixed final batch time by adjusting the glucose feeding rate and the aerobic–anaerobic switching time. Effectiveness of the closed-loop implementation was evaluated by comparing the relative performance of NMPC and the open-loop optimal controller. NMPC was able to compensate for structural errors in the intracellular model and parametric errors in the substrate uptake kinetics and cellular energetics by increasing ethanol production between 8.0% and 14.7% compared with the open-loop operating policy. Minimal degradation in NMPC performance was observed when the biomass, glucose, and ethanol concentration and liquid volume measurements were corrupted with Gaussian white noise. NMPC based on the dynamic flux balance model was shown to improve ethanol production compared to the same NMPC formulation based on a simpler unstructured model. To our knowledge, this study represents the first attempt to utilize a dynamic flux balance model within a nonlinear model-based control scheme.     

Hybrid NMPC of supermarket display cases

This paper presents an NMPC of a supermarket refrigeration system. This is a hybrid process involving switched nonlinear dynamics and discrete events, on/off manipulated variables (valves and compressors), continuous controlled variables (goods temperatures) and, finally, several operation constraints. The hybrid controller is based on a parameterization of the on/off control signals in terms of time of occurrence of events instead of using directly binary values; in this way, the optimization problem can be re-formulated as an NLP problem. A rigorous model of a real supermarket refrigeration system is presented, as well as results of the hybrid controller operating on it.     

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.     

A two-layer architecture for economically optimal process control and operation

A two-layer architecture for dynamic real-time optimization (or nonlinear modelpredictive control (NMPC) with an economic objective) is presented, where the solution of the dynamic optimization problem is computed on two time-scales. On the upper layer, a rigorous optimization problem is solved with an economic objective function at a slow time-scale, which captures slow trends in process uncertainties. On the lower layer, a fast neighboring-extremal controller is tracking the trajectory in order to deal with fast disturbances acting on the process. Compared to a single-layer architecture, the two-layer architecture is able to address control systems with complex models leading to high computational load, since the rigorous optimization problem can be solved at a slower rate than the process sampling time. Furthermore, solving a new rigorous optimization problem is not necessary at each sampling time if the process has rather slow dynamics compared to the disturbance dynamics. The two-layer control strategy is illustrated with a simulated case study of an industrial polymerization process.     

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。     

Computation delay compensation for real time implementation of robust model predictive control

The implementation of model predictive control (MPC) requires to solve an optimization problem online. The computation time, often not negligible especially for nonlinear MPC (NMPC), introduces a delay in the feedback loop. Moreover, it impedes fast sampling rate setting for the controller to react to uncertainties quickly. In this paper, a dual time scale control scheme is proposed for linear/nonlinear systems with external disturbances. A pre-compensator works at fast sampling rate to suppress uncertainty, while the outer MPC controller updates the open loop input sequence at a slower rate. The computation delay is explicitly considered and compensated in the MPC design. Four robust MPC algorithms for linear/nonlinear systems in the literature are adopted and tailored for the proposed control scheme. The recursive feasibility and stability are rigorously analysed. Three simulation examples are provided to validate the proposed approaches.     

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.     

A novel adaptive fuzzy predictive control for hybrid systems with mixed inputs

This paper proposes a new adaptive nonlinear model predictive control (NMPC) methodology for a class of hybrid systems with mixed inputs. For this purpose, an online fuzzy identification approach is presented to recursively estimate an evolving Takagi–Sugeno (eTS) model for the hybrid systems based on a potential clustering scheme. A receding horizon adaptive NMPC is then devised on the basis of the online identified eTS fuzzy model. The nonlinear MPC optimization problem is solved by a genetic algorithm (GA). Diverse sets of test scenarios have been conducted to comparatively demonstrate the robust performance of the proposed adaptive NMPC methodology on the challenging start-up operation of a hybrid continuous stirred tank reactor (CSTR) benchmark problem.