High-order Volterra Model Predictive Control and its application to a nonlinear polymerisation process

Model Predictive Control (MPC) has recently found wide acceptance in the process industry, but existing design and implementation methods are restricted to linear process models. A chemical process, however, involves severe nonlinearity which cannot be ignored in practice. This paper aims to solve this nonlinear control problem by extending MPC to accommodate nonlinear models. It develops an analytical framework for nonlinear model predictive control (NMPC). It also offers a third-order Volterra series based nonparametric nonlinear modelling technique for NMPC design, which relieves practising engineers from the need for deriving a physical-principles based model first. An on-line realisation technique for implementing NMPC is then developed and applied to a Mitsubishi Chemicals polymerisation reaction process. Results show that this nonlinear MPC technique is feasible and very effective. It considerably outperforms linear and low-order Volterra model based methods. The advantages of the developed approach lie not only in control performance superior to existing NMPC methods, but also in eliminating the need for converting an analytical model and then convert it to a Volterra model obtainable only up to the second order.     

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.     

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.     

Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty

Model predictive control (MPC) has become one of the most popular control techniques in the process industry mainly because of its ability to deal with multiple-input–multiple-output plants and with constraints. However, in the presence of model uncertainties and disturbances its performance can deteriorate. Therefore, the development of robust MPC techniques has been widely discussed during the last years, but they were rarely, if at all, applied in practice due to the conservativeness or the computational complexity of the approaches. In this paper, we present multi-stage NMPC as a promising robust non-conservative nonlinear model predictive control scheme. The approach is based on the representation of the evolution of the uncertainty by a scenario tree, and leads to a non-conservative robust control of the uncertain plant because the adaptation of future inputs to new information is taken into account. Simulation results show that multi-stage NMPC outperforms standard and min–max NMPC under the presence of uncertainties for a semi-batch polymerization benchmark problem. In addition, the advantages of the approach are illustrated for the case where only noisy measurements are available and the unmeasured states and the uncertainties have to be estimated using an observer. It is shown that better performance can be achieved than by estimating the unknown parameters online and adapting the plant model.     

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.     

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.     

Design of observer based nonlinear model predictive controller for a continuous stirred tank reactor

Model predictive control (MPC) schemes are now widely used in process industries for the control of key unit operations. Linear model predictive control (LMPC) schemes which make use of linear dynamic model for prediction, limit their applicability to a narrow range of operation (or) to systems which exhibit mildly nonlinear dynamics.

In this paper, a nonlinear observer based model predictive controller (NMPC) for nonlinear system has been proposed. An approach to design NMPC based on fuzzy Kalman filter (FKF) and augmented state fuzzy Kalman filter (ASFKF) has been presented. The efficacy of the proposed NMPC schemes have been demonstrated by conducting simulation studies on the continuous stirred tank reactor (CSTR). The analysis of the extensive dynamic simulation studies revealed that, the NMPC schemes formulated produces satisfactory performance for both servo and regulatory problems. Simulation results also include an inferential control case, where the reactor concentration is not measured but estimated from temperature measurement and used in the NMPC based on FKF and ASFKF formulations.     

A convex approach for NMPC based on second order Volterra series models

In model predictive control (MPC), the input sequence is computed, minimizing a usually quadratic cost function based on the predicted evolution of the system output. In the case of nonlinear MPC (NMPC), the use of nonlinear prediction models frequently leads to non‐convex optimization problems with several minimums. This paper proposes a new NMPC strategy based on second order Volterra series models where the original performance index is approximated by quadratic functions, which represent a lower bound of the original performance index. Convexity of the approximating quadratic cost functions can be achieved easily by a suitable choice of the weighting of the control increments in the performance index. The approximating cost functions can be globally minimized by convex optimization techniques in order to compute the input sequence. The minimization of the performance index is carried out by an iterative optimization procedure, which guarantees convergence to the solution. Furthermore, for a nominal prediction model, asymptotic stability for the proposed NMPC strategy can be shown. In the case of considering an estimation error in the prediction model, input‐to‐state practical stability is assured. The control performance of the NMPC strategy is illustrated by experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

From linear to nonlinear MPC: bridging the gap via the real-time iteration

ABSTRACT

Linear model predictive control (MPC) can be currently deployed at outstanding speeds, thanks to recent progress in algorithms for solving online the underlying structured quadratic programs. In contrast, nonlinear MPC (NMPC) requires the deployment of more elaborate algorithms, which require longer computation times than linear MPC. Nonetheless, computational speeds for NMPC comparable to those of MPC are now regularly reported, provided that the adequate algorithms are used. In this paper, we aim at clarifying the similarities and differences between linear MPC and NMPC. In particular, we focus our analysis on NMPC based on the real-time iteration (RTI) scheme, as this technique has been successfully tested and, in some applications, requires computational times that are only marginally larger than linear MPC. The goal of the paper is to promote the understanding of RTI-based NMPC within the linear MPC community.     

A hybrid model predictive control strategy for nonlinear plant-wide control

A plant-wide control strategy based on integrating linear model predictive control (LMPC) and nonlinear model predictive control (NMPC) is proposed. The hybrid method is applicable to plants that can be decomposed into approximately linear subsystems and highly nonlinear subsystems that interact via mass and energy flows. LMPC is applied to the linear subsystems and NMPC is applied to the nonlinear subsystems. A simple controller coordination strategy that counteracts interaction effects is proposed for the case of one linear subsystem and one nonlinear subsystem. A reactor/separator process with recycle is used to compare the hybrid method to conventional LMPC and NMPC techniques.     

A RBF-ARX model-based robust MPC for tracking control without steady state knowledge

A RBF-ARX modeling and robust model predictive control (MPC) approach to achieving output-tracking control of the nonlinear system with unknown steady-state knowledge is proposed. On the basis of the RBF-ARX model with considering the system time delay, a local linearization state-space model is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system’s nonlinear behavior. Based on the two models, a quasi-min–max MPC algorithm with constraint is designed for output-tracking control of the nonlinear system with unknown steady state knowledge. The optimization problem of the quasi-min–max MPC algorithm is finally converted to the convex linear matrix inequalities (LMIs) optimization problem. Closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. Two examples, i.e. the modeling and control of a continuously stirred tank reactor (CSTR) and a two tank system demonstrate the effectiveness of the RBF-ARX modeling and robust MPC approach.     

Handling uncertainty in economic nonlinear model predictive control: A comparative case study

In the last years, the use of an economic cost function for model predictive control (MPC) has been widely discussed in the literature. The main motivation for this choice is that often the real goal of control is to maximize the profit or the efficiency of a certain system, rather than tracking a predefined set-point as done in the typical MPC approaches, which can be even counter-productive. Since the economic optimal operation of a system resulting from the application of an economic model predictive control approach drives the system to the constraints, the explicit consideration of the uncertainties becomes crucial in order to avoid constraint violations. Although robust MPC has been studied during the past years, little attention has yet been devoted to this topic in the context of economic nonlinear model predictive control, especially when analyzing the performance of the different MPC approaches. In this work, we present the use of multi-stage scenario-based nonlinear model predictive control as a promising strategy to deal with uncertainties in the context of economic NMPC. We make a comparison based on simulations of the advantages of the proposed approach with an open-loop NMPC controller in which no feedback is introduced in the prediction and with an NMPC controller which optimizes over affine control policies. The approach is efficiently implemented using CasADi, which makes it possible to achieve real-time computations for an industrial batch polymerization reactor model provided by BASF SE. Finally, a novel algorithm inspired by tube-based MPC is proposed in order to achieve a trade-off between the variability of the controlled system and the economic performance under uncertainty. Simulations results show that a closed-loop approach for robust NMPC increases the performance and that enforcing low variability under uncertainty of the controlled system might result in a big performance loss.     

Data-driven model reduction-based nonlinear MPC for large-scale distributed parameter systems

Model predictive control (MPC) has been effectively applied in process industries since the 1990s. Models in the form of closed equation sets are normally needed for MPC, but it is often difficult to obtain such formulations for large nonlinear systems. To extend nonlinear MPC (NMPC) application to nonlinear distributed parameter systems (DPS) with unknown dynamics, a data-driven model reduction-based approach is followed. The proper orthogonal decomposition (POD) method is first applied off-line to compute a set of basis functions. Then a series of artificial neural networks (ANNs) are trained to effectively compute POD time coefficients. NMPC, using sequential quadratic programming is then applied. The novelty of our methodology lies in the application of POD's highly efficient linear decomposition for the consequent conversion of any distributed multi-dimensional space-state model to a reduced 1-dimensional model, dependent only on time, which can be handled effectively as a black-box through ANNs. Hence we construct a paradigm, which allows the application of NMPC to complex nonlinear high-dimensional systems, even input/output systems, handled by black-box solvers, with significant computational efficiency. This paradigm combines elements of gain scheduling, NMPC, model reduction and ANN for effective control of nonlinear DPS. The stabilization/destabilization of a tubular reactor with recycle is used as an illustrative example to demonstrate the efficiency of our methodology. Case studies with inequality constraints are also presented.     

基于信赖域二次规划的非线性模型预测控制优化算法

针对非线性预测控制如何在有限时域内有效的求解非凸非线性规划这一关键问题, 本文采用序列二次规划方法, 将非线性规划转化为一系列二次子规划求解. 首先根据非线性规划联立方法将系统状态和控制量同时作为优化变量, 得到以控制量步长为优化变量, 只包含不等式约束的子二次规划问题, 并用它取代原SQP子规划, 减小了子问题的规模; 随后采用基于信赖域二次规划的方法求解子规划问题, 保证每次迭代的可行性; 同时采用一种能够保持SQP问题Hessian矩阵稀疏结构的更新方法, 也在一定程度上降低了算法的复杂程度.最后的仿真结果表明了该方法的有效性.     

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.     

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.     

Robust model predictive control of Wiener systems

Block-oriented models (BOMs) have shown to be appealing and efficient as nonlinear representations for many applications. They are at the same time valid and simple models in a more extensive region than time-invariant linear models. In this work, Wiener models are considered. They are one of the most diffused BOMs, and their structure consists in a linear dynamics in cascade with a nonlinear static block. Particularly, the problem of control of these systems in the presence of uncertainty is treated. The proposed methodology makes use of a robust identification procedure in order to obtain a robust model to represent the uncertain system. This model is then employed to design a model predictive controller. The mathematical problem involved in the controller design is formulated in the context of the existing linear matrix inequalities (LMI) theory. The main feature of this approach is that it takes advantage of the static nature of the nonlinearity, which allows to solve the control problem by focusing only in the linear dynamics. This formulation results in a simplified design procedure, because the original nonlinear model predictive control (MPC) problem turns into a linear one.     

基于T-S模糊模型的非线性预测控制策略

提出了一种新的基于T-S模糊模型的非线性预测控制策略. T-S模糊模型用于描述对象的非线性动态特性, 通过将模糊模型的输出反馈回来作为模型输入, 从而构成了模糊多步预报器. 由于T-S模糊模型每条规则的结论部分是一个线性模型, 因此整个模糊模型可以看作一个线性时变系统, 从而将模糊预测控制器中的非线性优化问题转化为一个线性二次寻优问题, 以方便求解. pH中和过程的仿真结果表明其性能优于传统的动态矩阵控制器.