Nonlinear identification and control of a high-purity distillation column: a case study

Identification and control of ill-conditioned, interactive and highly nonlinear processes pose a challenging problem to the process industry. In the absence of a reasonably accurate model, these processes are fairly difficult to control. Using a high-purity distillation column as an example, model identification and control issues are addressed in this paper. The structure of the identified models is that of the polynomial type nonlinear autoregressive models with exogenous inputs (NARX). While most of the work in this area has concentrated on linear models (one-time scale and two-time scale models), this work is aimed at identifying the inherent nonlinearities. Comparisons are drawn between the identified models based on statistical criteria (AIC etc.) and other validation tests. Simulation results are provided to demonstrate the closed-loop performance of the nonlinear ARX models in the control of the distillation column. The controller employed is based on a nonlinear model predictive scheme with state and parameter estimation.     

多自由度非线性系统动态参数化模型建模方法研究

针对多自由度非线性系统的动态模型辨识问题,基于NARX(Non-linear Autoregressive with Exogenous inputs)模型的建模方法,考虑系统的物理设计参数,建立非线性系统动态参数化模型.首先,根据系统输入、输出数据建立系统不同参数下的NARX模型,并通过EFOR(Extended Forward Orthogonal Regression)算法对不同参数下NARX模型进行修正,以统一辨识得到的系统模型结构.随后,建立NARX模型系数与物理设计参数间的函数关系,得到多自由度非线性系统的动态参数化模型.以单输入、单输出两自由度非线性系统为例,根据数值仿真结果,对系统的动态参数化模型建模过程进行说明.最后,以带非线性涂层阻尼的悬臂梁作为试验对象,建立其动态参数化模型以反映其动力学特性.试验结果表明,非线性系统动态参数化模型能准确预测多自由度非线性系统的输出响应,为非线性系统的分析与优化设计提供了理论基础.     

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.     

Experimental evaluation of a multivariable adaptive nonlinear predictive controller

A new multivariable adaptive nonlinear predictive controller is designed using a general nonlinear input-output model and variable transformations. The controller is similar in form to typical linear predictive controllers can be tuned analogously or by specifying a single parameters for each controlled variable. In addition, the design procedure is computationally efficient. The new controller is compared to a multi-loop proportional-integral (PI) controller with one-way static decoupling and to an adaptive linear predictive controller through tests on a simulated nonlinear distillation column. The new controller performed well in an experimental application to a multicomponent distillation column.     

Determination of model order for NARX models directly from input-output data

Nonlinear auto-regressive models with exogenous inputs (NARX models) have proved to be versatile and useful empirical models for industrial processes. There are a wide variety of identification methods and model structures from which to choose; in all methods, however, key parameters are the model orders, which are the number of past outputs and inputs used in the model. In this paper the methods of Lipschitz numbers and false nearest neighbors are evaluated as a means of estimating the model orders of dynamic, discrete-time NARX models. No specific model structure is assumed and the model orders are estimated directly from input-output data using only geometric concerns and the continuity property. The two methods are applied to several dynamic systems, including realistic process simulations and experimental data from the UCSB pH neutralization process, and the consistency and accuracy of these methods are examined. The usefulness of these methods of model order determination for radial basis function network (RBFN) identification is examined.     

Order and delay selection for neural network modelling by identification of linearized models

An approach to selecting the order and delay for neural network modelling of nonlinear dynamic systems is proposed by identifying local linear models at points spanning the system operating range. The method is based on relationships between linear and nonlinear models and is developed for three popular nonlinear model structures; nonlinear autoregressive with exogenous inputs (NARX), NARX with a linear noise model and nonlinear autoregressive moving-average with exogenous inputs (NARMAX). Simulation results illustrate the application of the method, and the suitability of the orders and delays selected are demonstrated by nonlinear system identification using radial basis function neural networks. The method is also shown to indicate the suitability of a particular nonlinear model structure for representing a system.     

On robustness in control and LTI identification: Near-linearity and non-conic uncertainty

Robustness issues are studied in the context of linear models in linear controller design and in system identification when the true system is nonlinear. The notion of nearly linear system is generalized to include time-varying and open-loop unstable systems. This class of systems, although it presents the simplest possible (global) generalization of linear systems, is in many ways nontrivial for the purpose of linear controller design and linear model identification from input-output data. This is mainly due to the presence of non-conic uncertainty, which is not included in standard treatments of robust control theory and stochastic system identification theory. Signal distribution theory, a realistic non-stochastic signal analysis tool, is used to study the limiting least squares estimates of linear finite impulse response (FIR) and autoregressive with external input (ARX) model parametrizations for two classes of nonlinear systems. Some connections to worst-case analysis of linear model identification are also discussed.     

Hybrid control of a pneumatic artificial muscle (PAM) robot arm using an inverse NARX fuzzy model

We investigated the possibility of applying a hybrid feed-forward inverse nonlinear autoregressive with exogenous input (NARX) fuzzy model-PID controller to a nonlinear pneumatic artificial muscle (PAM) robot arm to improve its joint angle position output performance. The proposed hybrid inverse NARX fuzzy-PID controller is implemented to control a PAM robot arm that is subjected to nonlinear systematic features and load variations in real time. First the inverse NARX fuzzy model is modeled and identified by a modified genetic algorithm (MGA) based on input/output training data gathered experimentally from the PAM system. Second the performance of the optimized inverse NARX fuzzy model is experimentally demonstrated in a novel hybrid inverse NARX fuzzy-PID position controller of the PAM robot arm. The results of these experiments demonstrate the feasibility and benefits of the proposed control approach compared to traditional PID control strategies. Consequently, the good performance of the MGA-based inverse NARX fuzzy model in the proposed hybrid inverse NARX fuzzy-PID position control of the PAM robot arm is demonstrated. These results are also applied to model and to control other highly nonlinear systems.     

Wavelet based non-parametric NARX models for nonlinear input–output system identification

Wavelet based non-parametric additive NARX models are proposed for nonlinear input–output system identification. By expanding each functional component of the non-parametric NARX model into wavelet multiresolution expansions, the non-parametric estimation problem becomes a linear-in-the-parameters problem, and least-squares-based methods such as the orthogonal forward regression (OFR) approach, coupled with model size determination criteria, can be used to select the model terms and estimate the parameters. Wavelet based additive models, combined with model order determination and variable selection approaches, are capable of handling problems of high dimensionality.     

Globally optimal nonlinear model predictive control based on multi-parametric disaggregation

Nonlinear model predictive control is appropriate for controlling highly nonlinear processes, particularly when operating conditions change frequently. If the problem is nonconvex, the controller must lead the process to a global, rather than a local optimum. This work deals with computation of the control actions which lead to the global optimum via the normalized multi-parametric disaggregation technique. The continuous process model is transformed into a nonlinear programming (NLP) problem via discretization which uses an implicit integration method. The NLP problem is relaxed into a mixed integer linear programming (MILP) model. Iterations between solving MILP (lower bound) and using its solution as a starting point for a local nonlinear optimizer (which computes the upper bound) continue until the gap is closed (an l₁-norm objective function is used). Controller performance is illustrated by several examples. Relative simplicity of the algorithm makes it possible to be implemented by a wide audience.     

基于Volterra模型的预测控制及应用

由于工业实践的需要,非线性预测控制近年来受到广泛地关注.Volterra模型是一类特殊的非线性模型,非常适合描述工业过程中的无记忆非线性对象.传统的基于Volterra模型的控制器合成法及迭代计算预测控制器法计算量大,且不便于处理控制约束.非线性模型预测控制求解是典型的非线性规划问题,序列二次规划(sequential quadratic program,SQP)算法是求解非线性规划问题常用方法之一.针对Volterra非线性模型预测控制求解问题,本文将滤子法与一种信赖域SQP算法相结合,提出一种改进SQP算法用于基于非线性Volterra模型的带控制约束的多步预测控制求解,并分析了所提方法的收敛性.工业实例仿真结果证实了所提方法的可行性与有效性.     

永磁同步电机高效非线性模型预测控制

永磁电机控制器要求电机有很强的转速跟踪能力,并且要保证系统参数变化及负荷扰动下系统的鲁棒性. 永磁电机包含很多不确定因素,是强耦合的非线性系统,传统的线性控制器很难对其进行控制. 针对永磁电机的转速控制构造非线性模型预测控制方法. 非线性永磁电机模型通过输入-输出反馈线性化策略解耦成为新的线性系统. 为保证可行解的收敛性,提出一种迭代二次规划方法来处理由输入-输出反馈线性化导致的非线性约束. 仿真结果表明,控制器能有效降低计算负担,具有很好的动态控制性能,能抑制转矩脉动,并保证在参数变化和负荷扰动下控制系统的鲁棒性.     

Data-driven predictive control for continuous-time linear parameter varying systems with application to wind turbine

A new data-driven predictive control method based on subspace identification for continuous-time linear parameter varying (LPV) systems is presented in this paper. It is developed by reformulating the continuous-time LPV system which utilizes Laguerre filters to obtain the subspace prediction of output. The subspace predictors are derived by QR decomposition of input-output and Laguerre matrices obtained by input-output data. The predictors are then applied to design the model predictive controller. It is shown that the integrated action is incorporated in the control effect to eliminate the steady-state offset. We control the continuous-time LPV systems to obtain the attractive performance with the proposed data-driven predictive control method. The proposed controller is applied to a wind turbine to verify its effectiveness and feasibility.     

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.     

Forecasting peak air pollution levels using NARX models

Air pollution has a negative impact on human health. For this reason, it is important to correctly forecast over-threshold events to give timely warnings to the population. Nonlinear models of the nonlinear autoregressive with exogenous variable (NARX) class have been extensively used to forecast air pollution time series, mainly using artificial neural networks (NNs) to model the nonlinearities. This work discusses the possible advantages of using polynomial NARX instead, in combination with suitable model structure selection methods. Furthermore, a suitably weighted mean square error (MSE) (one-step-ahead prediction) cost function is used in the identification/learning process to enhance the model performance in peak estimation, which is the final purpose of this application. The proposed approach is applied to ground-level ozone concentration time series. An extended simulation analysis is provided to compare the two classes of models on a selected case study (Milan metropolitan area) and to investigate the effect of different weighting functions in the identification performance index. Results show that polynomial NARX are able to correctly reconstruct ozone concentrations, with performances similar to NN-based NARX models, but providing additional information, as, e.g., the best set of regressors to describe the studied phenomena. The simulation analysis also demonstrates the potential benefits of using the weighted cost function, especially in increasing the reliability in peak estimation.     

Identification of the rainfall–runoff relationship in urban drainage networks

The calibration of conceptual models for the design of urban drainage networks is an important and well-known problem in hydraulic engineering. In this paper the problem is analysed and the use of black-box identification methods is proposed and applied to experimental data. Both linear (ARX and state space) and nonlinear (polynomial and neural NARX) models are considered and their performance in the simulation and prediction of the network flow from rainfall measurements is evaluated.     

Regressor and structure selection in NARX models using a structured ANOVA approach

Regressor selection can be viewed as the first step in the system identification process. The benefits of finding good regressors before estimating complex models are especially clear for nonlinear systems, where the class of possible models is huge. In this article, a structured way of using the tool analysis of variance (ANOVA) is presented and used for NARX model (nonlinear autoregressive model with exogenous input) identification with many candidate regressors.     

Simple recursive algorithm for linear-in-theparameters nonlinear model identification

This paper introduces a simple recursive algorithm for nonlinear dynamic system identification using linear-in-the-parameters models for NARX or RBF network where both the structure and parameters can be obtained simultaneously and recursively. The main objective is to improve the numerical stability when the model terms are highly correlated. This is based on the "innovation" idea and net contribution criteria. Using the recursive formulae for the computation of the Moore-Penrose inverse of matrices and th...     

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.