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.     

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.     

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.     

Robust sampled‐data model predictive control for networked systems with time‐varying delay

In this paper, the problem of sampled‐data model predictive control (MPC) is investigated for linear networked control systems with both input delay and input saturation. The delay‐induced nonlinearity is overapproximatively modeled as a polytopic inclusion. The nonlinear behavior of input saturation is expressed as a convex polytope. The resulting closed‐loop systems are represented as linear systems with polytopic and additive norm‐bounded uncertainties. The aim is to determine a robust MPC controller that asymptotically stabilizes the uncertain system at the origin with a certain level of quadratic performance. The effectiveness of the proposed algorithm is demonstrated by a numerical example.     

Inherent robustness properties of quasi-infinite horizon nonlinear model predictive control

We consider inherent robustness properties of model predictive control (MPC) for continuous-time nonlinear systems with input constraints and terminal constraints. We show that MPC with a nominal prediction model and persistent but bounded disturbances has some degree of inherent robustness when the terminal control law and the terminal penalty matrix are chosen as the linear quadratic control law and the related Lyapunov matrix, respectively. We emphasize that the input constraint sets can be any compact set rather than convex sets, and our results do not depend on the continuity of the optimal cost function or of the control law in the interior of the feasible region.     

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

In this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.     

An improved robust model predictive control for linear parameter‐varying input‐output models

This paper describes a new robust model predictive control (MPC) scheme to control the discrete‐time linear parameter‐varying input‐output models subject to input and output constraints. Closed‐loop asymptotic stability is guaranteed by including a quadratic terminal cost and an ellipsoidal terminal set, which are solved offline, for the underlying online MPC optimization problem. The main attractive feature of the proposed scheme in comparison with previously published results is that all offline computations are now based on the convex optimization problem, which significantly reduces conservatism and computational complexity. Moreover, the proposed scheme can handle a wider class of linear parameter‐varying input‐output models than those considered by previous schemes without increasing the complexity. For an illustration, the predictive control of a continuously stirred tank reactor is provided with the proposed method.     

An iterative model predictive control algorithm for constrained nonlinear systems

An iterative model predictive control (MPC) scheme for constrained nonlinear systems is presented. The idea of the method is to detour from the solution of a non‐convex optimization problem using a time‐variant linearization of the nonlinear system model that is adjusted iteratively by solving an iterative quadratic programming optimization problem at each sampling time. The main advantage is the faster resolution of the optimization problem by using quadratic programming instead of non‐convex programming and yet, properly describing the nonlinear dynamics of the process being controlled. In this article, a general framework of the method is presented together with a discussion on the conditions under which the iterations converge and on the uncertainty of its results due to the linearization used, as well as some practical considerations about its implementation. The performance of the proposed controller is illustrated via two examples.     

Terrain Avoidance Nonlinear Model Predictive Control for Autonomous Rotorcraft

This paper describes a terrain avoidance control methodology for autonomous rotorcraft applied to low altitude flight. A simple nonlinear model predictive control (NMPC) formulation is used to adequately address the terrain avoidance problem, which involves stabilizing a nonlinear and highly coupled dynamic model of a helicopter, while avoiding collisions with the terrain as well as preventing input and state saturations. The physical input saturations are made intrinsic to the model, such that the control is always admissible and the MPC design is simplified. A comparison of several optimization approaches is provided, where the performance of the traditional gradient method with fixed step is compared with the quasi-Newton method and a line search algorithm. The simulation results show that the adopted strategy achieves good performance even when the desired path is on collision course with the terrain.     

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.     

A COMPARATIVE STUDY ON COMPUTATIONAL SCHEMES FOR NONLINEAR MODEL PREDICTIVE CONTROL

This paper briefly reviews development of nonlinear model predictive control (NMPC) schemes for finite horizon prediction and basic computational algorithms that can solve the stable real‐time implementation of NMPC in space state form with state and input constraints. In order to ensure stability within a finite prediction horizon, most NMPC schemes use a terminal region constraint at the end of the prediction horizon — a particular NMPC scheme using a terminal region constraint, namely quasi‐infinite horizon, that guarantees asymptotic closed‐loop stability with input constraints is presented. However, when nonlinear processes have both input and state constraints, difficulty arises from failure to satisfy the state constraints due to constraints on input. Therefore, a new NMPC scheme without a terminal region constraint is developed using soften state constraints. A brief comparative simulation study of two NMPC schemes: quasi‐infinite horizon and soften state constraints is done via simple nonlinear examples to demonstrate the ability of the soften state constraints scheme. Finally, some features of future research from this study are discussed.     

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.     

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.     

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。     

LMI-based model predictive control for switched nonlinear systems

This paper proposes an LMI approach to model predictive control of nonlinear systems with switching between multiple modes. In this approach, at each mode, the nonlinear system is divided to a linearized model in addition to a nonlinear term. A sum of squares (SOS) optimization problem is presented to find a quadratic bound for the nonlinear part. The stability condition of the switching system is obtained by using a discrete Lyapunov function and then the sufficient state feedback control law is achieved so that guarantees the stability of the system and also minimizes an infinite prediction horizon performance index. Moreover, two other LMI optimization problems are solved at each mode in order to find the maximum area region of convergence of the nonlinear system inscribed in the region of stability. The performance and effectiveness of the proposed MPC approach are illustrated by two case studies.     

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.     

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.     

一类输入受限不确定时滞系统的准Min-Max模型预测控制

针对一类输入受限离散不确定时滞系统,提出一种基于准Ｍｉｎ-Ｍａｘ的模型预测控制器设计方法．定义了时滞系统的鲁棒性能指标,给出了系统稳定的充分条件,通过求解ＬＭＩ凸优化获得控制器．准Ｍｉｎ-Ｍａｘ预测控制将当前控制量作为独立优化变量,与其他作为反馈控制的时域控制序列分开处理,有效地降低了算法的保守性,提高了可行性．仿真算例验证了所提出控制方法的有效性．     

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.