Nonlinear observer design for two-time-scale systems

A two-time-scale system involves both fast and slow dynamics. This article studies observer design for general nonlinear two-time-scale systems and presents two alternative nonlinear observer design approaches, one full-order and one reduced-order. The full-order observer is designed by following a scheme to systematically select design parameters, so that the fast and slow observer dynamics are assigned to estimate the corresponding system modes. The reduced-order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow-motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that the reduced-order observer is capable of providing accurate estimation of the states for the detailed system with an exponentially decaying estimation error. In the last part of the article, the two proposed observers are designed for an anaerobic digestion process, as an illustrative example to evaluate their performance and convergence properties.     

Robust modeling of a microalgal heterotrophic fed-batch bioreactor

Microalgal feedstocks have shown potential for the production of biofuels and fine chemicals. Recently, an optimal experimental input profile for the identification of parameters of a microalgal bioreactor, containing 6 states and 12 unknown parameters has been proposed. In this work, the proposed design is implemented and parameters are estimated. It was found that the parameter estimation procedure can be made more computational efficient by the use of a novel iterative non-linear model reparameterization algorithm. By applying the proposed algorithm to experimental data, a good degree of output prediction was achieved along with bounds on the parameter values. The final, validated, model can be used for optimal control and process simulation.     

Parameter estimation in batch process using EM algorithm with particle filter

This paper investigates a parameter estimation problem for batch processes through the maximum likelihood method. In batch processes, the initial state usually relates to the states of previous batches. The proposed algorithm takes batch-to-batch correlations into account by employing an initial state transition equation to model the dynamics along the batch dimension. By treating the unmeasured states and the parameters as hidden variables, the maximum likelihood estimation is accomplished through the expectation–maximization (EM) algorithm, where the smoothing for the terminal state and the filtering for the initial state are specially considered. Due to the nonlinearity and non-Gaussianity in the state space model, particle filtering methods are employed for the implementation of filtering and smoothing. Through alternating between the expectation step and the maximization step, the unknown parameters along with states are estimated. Simulation examples demonstrate the proposed estimation approach.     

非线性观测器及其应用(上)一类高阶非线性系统的观测器

针对子系统的不可测状态之间呈下三角阵式关联的一类主阶非线性系统,本文给出了其观测器型式和反馈增益阵的设计方法,同时进一步讲座了存在不可测输入时系统的观测,这种观测器设计简单,便于实施和调变量进行的处理,对于化工生产过程中经常遇到的这类高阶非线性系统,这种观测器设计简单,便于实施和调整。     

化工过程基于广义非线性状态观测器的不可测输入和不确定模型参数软仪表(英文)

Chemical processes are usually nonlinear singular systems. In this study, a soft sensor using nonlinear singular state observer is established for unknown inputs and uncertain model parameters in chemical processes, which are augmented as state variables. Based on the observability of the singular system, this paper presents a simplified observability criterion under certain conditions for unknown inputs and uncertain model parameters. When the observability is satisfied, the unknown inputs and the uncertain model parameters are estimated online by the soft sensor using augmented nonlinear singular state observer. The riser reactor of fluid catalytic cracking unit is used as an example for analysis and simulation. With the catalyst circulation rate as the only unknown input without model error, one temperature sensor at the riser reactor outlet will ensure the correct estimation for the catalyst cir- culation rate. However, when uncertain model parameters also exist, additional temperature sensors must be used to ensure correct estimation for unknown inputs and uncertain model parameters of chemical processes.     

基于渐进黑箱理论的多变量辨识方法

为克服传统单变量开环阶跃测试法测试时间长和误差大的缺点,提出一种基于渐进黑箱理论的多变量辨识方法.针对辨识的几个基本问题:测试信号的设计、模型结构的选择、模型阶次的判别和参数估计,进行了全新的设计.采用平移的方法,把一个周期较长的伪随机二进制序列平移若干次,从而得到若干个近似两两互不相关的伪随机二进制序列作为多变量测试信号.选取高ARX模型作为参数模型,并用输出误差(OE)模型进行降阶模型的参数估计,降阶模型的阶次由最小描述长度(MDL)准则来判别.实例仿真的结果表明,用该方法解决多变量辨识问题,能减少测试时间,降低测试期间对设备产生的干扰,辨识的结果也优于常规辨识法.     

Advances and selected recent developments in state and parameter estimation

This paper deals with two topics from state and parameter estimation. The first contribution of this work provides an overview of techniques used for determining which parameters of a model should be estimated. This is a question that commonly arises when fundamental models are used as these models often contain more parameters than can be reliably estimated from data. The decision of which parameters to estimate is independent of the observer/estimator design, however, it is directly affected by the structure of the model as well as the available data. The second contribution is an overview of recent developments regarding the design of nonlinear Luenberger observers, with special emphasis on exact error linearization techniques, but also discussing more general issues, including observer discretization, sampled data observers and the use of delayed measurements.     

基于机理模型的2-KGA混菌发酵过程关键状态变量预报

维生素C生产的前体2-KGA是巨大芽孢杆菌和普通生酮古龙酸菌混合发酵的产物。利用先前建立的2-KGA混菌发酵动力学模型对背景厂80个批次的实测数据进行分析,结果表明该模型能够很好地符合工业生产的实际情况。在模型参数灵敏度分析的基础上固定了部分模型参数,并选取具有代表性的3个罐批(劣等、普通、优势),利用移动数据窗口技术和滚动参数辨识方法成功地进行了2-KGA浓度和底物浓度的超前4 h和8 h拟在线预报,预报误差均在5%以内。同时还比较了固定长度时间窗口和变长度时间窗口的预报结果,并根据现场实际数据特点分析了二者的优劣。     

ESTIMATION AND BLIND DECONVOLUTION OF AUTOREGRESSIVE SYSTEMS WITH NONSTATIONARY BINARY INPUTS

Abstract. The problem of parameter estimation and blind deconvolution of auto-regressive (AR) systems with independent nonstationary binary inputs is considered. The estimation procedure consists of applying a moving-average filter (equalizer) to the observed data and adjusting the parameters of the filter so as to minimize a criterion that measures the binariness of its output. The output sequence itself serves as an estimate of the unobservable binary input of the AR system. Without assuming stationarity of the inputs, it is shown that the proposed method produces a consistent estimator of the AR system not only in the sense of converging to the true parameter as the sample size increases, but also in the sense of attaining the true parameter of the AR system for a sufficiently large sample size. For noisy data, the estimation criterion is modified on the basis of an asymptotic analysis of the effect of the noise. It is shown that the modified criterion is also consistent (in the usual sense) and its variability depends upon the filtered noise. Some simulation results are presented to demonstrate the performance of the proposed method for parameter estimation as well as for blind deconvolution.     

Subsystem decomposition and configuration for distributed state estimation

Distributed state estimation plays a very important role in process control. Improper subsystem decomposition for distributed state estimation may increase the computational burdens, degrade the estimation performance, or even deteriorate the observability of the entire system. The subsystem decomposition problem for distributed state estimation of nonlinear systems is investigated. A systematic procedure for subsystem decomposition for distributed state estimation is proposed. Key steps in the procedure include observability test of the entire system, observable states identification for each output measurement, relative degree analysis and sensitivity analysis between measured outputs and states. Considerations with respect to time‐scale multiplicity and direct graph are discussed. A few examples are used to illustrate the applicability of the methods used in different steps. The effectiveness of the entire distributed state estimation configuration procedure is also demonstrated via an application to a chemical process example used in coal handling and preparation plants. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1995–2003, 2016     

An experimental evaluation of Kalman filtering

The effectiveness of the stationary form of the discrete Kalman filter for state estimation in noisy process systems was demonstrated by simulated and experimental tests on a pilot plant evaporator. The filter was incorporated into a multivariable, computer control system and resulted in good control despite process and/or measurement noise levels of 10%. The results were significantly better than those obtained when the Kalman filter was omitted or replaced by conventional exponential filters. In this application the standard Kalman filter was reasonably insensitive to incorrect estimates of initial conditions or noise statistics and to errors in model parameters. The filter estimates were sensitive to unmeasured process disturbances. However this sensitivity could be reduced by treating the noise covariance matrices R and Q as design parameters rather than noise statistics and selecting values which result in increased weighting of the process measurements relative to the calculated model states.     

Nonlinear inferential control

Due to unmeasured distrubances and nonlinearities typical for chemical processes the performance of state reconstruction schemes based on linearized system is often unsatisfactory. Unmeasured disturbances yield biased state estimates because generally only proportional feedback is used in the estimators. Set point changes or large disturbances make linear estimators invalid for most chemical processes. Static and dynamic nonlinear estimation schemes are derived in this paper when persistent or slowly varying disturbances (nonstationary noise) affect the system. Because an analytical solution for general nonlinear systems is impossible, approximate methods are suggested to reduce the computational effort necessary for evaluating the estimate. The method is applied to two CSTR's in series where concentration estimates are obtained from temperature measurements. The results are significantly better than those obtained by linear estimation techniques. A convenient measurement selection criterion is also derived which aims at minimizing the sensitivity of the estimate to unmeasured disturbances.     

Sequential experimental design based on multiobjective optimization procedures

Model-based sequential experimental designs are frequently applied for discrimination of rival models and/or estimation of precise model parameters. Although the development and use of a single design criterion to perform the simultaneous model discrimination and precise parameter estimation seem appealing, published material indicates that previous attempts to develop such a single design criterion have not been successful. Despite that, this problem has rarely been analyzed with the help of multiobjective optimization procedures. In this work, a multiobjective optimization method based on the particle swarm optimization procedure is used to build the Pareto fronts in experimental design problems where distinct design criteria used for discrimination of rival models and/or estimation of precise model parameters are considered simultaneously. It is shown through the rigorous analysis of the Pareto sets that both design objectives are frequently conflicting, which means that optimum discrimination of rival models and estimation of precise model parameters cannot be performed simultaneously in many cases. However, it is also shown that the use of the posterior covariance matrix of estimated model parameters for model discrimination makes the design of experiments for the simultaneous optimum model discrimination and estimation of model parameters possible in many experimental design problems.     

Iterative model-based experimental design for spherical agglomeration processes

Spherical agglomeration (SA) is a process intensification strategy, which can reduce the number of unit operations in pharmaceutical manufacturing. SA merges drug substance crystallization with drug product wet granulation, reducing capital, and operating costs. However, SA is a highly nonlinear process, thus for its efficient operation model-based design and control strategies are beneficial. These require the development of a high-fidelity process model with appropriately estimated parameters. There are two major problems associated with the development of a high-fidelity process models—(i) selection of the appropriate model corresponding to the underlying process mechanisms, and (ii) accurate estimation of the parameters. This work focuses on the identification of the best fitting model that correlates with experimental observations using cross-validation experiments. Further, an iterative model-based experimental design strategy is developed, which uses D-optimal experimental design criterion to minimize the number of experiments necessary to obtain accurate parameter estimates.     

A systematic approach to SMB processes model identification from batch experiments

The optimization, monitoring and control of simulated moving bed (SMB) units require the use of a process model and the estimation of the model parameters. A systematic numerical procedure for determining parameters of SMB models from batch experiments is presented and evaluated. The unknown parameters are estimated by minimizing a cost function measuring the difference between experimental and simulated concentration profiles. In contrast with previous studies, parameter identifiability is studied and errors on the estimated parameters are calculated. A sensitivity analysis is used to design the experiments and to compare the identifiability of different chromatographic models. Then, the influence of local minima is evaluated by applying the numerical procedure on fictitious measurements generated from a model with known parameters.     

Selection of simplified models: II. Development of a model selection criterion based on mean squared error

Simplified models (SMs) with a reduced set of parameters are used in many practical situations, especially when the available data for parameter estimation are limited. A variety of candidate models are often considered during the model formulation, simplification, and parameter estimation processes. We propose a new criterion to help modellers select the best SM, so that predictions with lowest expected mean squared error can be obtained. The effectiveness of the proposed criterion for selecting simplified nonlinear univariate and multivariate models is demonstrated using Monte‐Carlo simulations and is compared with the effectiveness of the Bayesian Information Criterion (BIC).     

Application of state estimation based NMPC to an unstable nonlinear process

Model predictive control (MPC) has become very popular both in process industry and academia due to its effectiveness in dealing with nonlinear, multivariable and/or hard-constrained plants.Although linear MPC can be applied for controlling nonlinear processes by obtaining a linearized model of the plant, this is only valid in a limited region. Therefore, a substantial improvement can be achieved by using the whole knowledge of the process dynamics, specially in the presence of marked nonlinearities. This effect can be strong if the process to control is open-loop unstable.The purpose of this paper is to introduce a nonlinear model predictive controller (NMPC) based on nonlinear state estimation, in order to exploit the knowledge of the nonlinear dynamics and to avoid modeling simplifications or linearization.A state-space formulation is proposed to achieve the control objective. To update the optimization involved in NMPC strategy, state estimation based on the measured outputs is proposed.As a particular application, we consider an open-loop unstable jacketed exothermic chemical reactor. This CSTR is widely recognized as a difficult problem for the purpose of control. In order to achieve the control goal, a NMPController coupled with a state observer are designed. The observer is also used to estimate some unmeasured disturbances. Finally, computer simulations are developed for showing the performance of both the nonlinear observer and the control strategy.     

KINETIC PARAMETER ESTIMATION BY STANDARD OPTIMIZATION METHODS IN CATALYTIC CONVERTER MODELING

The application of mathematical models to the prediction of the performance of automotive catalytic converters is gaining increasing interest, both for gasoline and diesel engined-vehicles. This article addresses converter modeling in the transient state under realistic experimental conditions. The model employed in this study relies on Langmuir-Hinshelwood kinetics, and a number of apparent kinetic parameters must be tuned to match the behavior of each different catalyst formulation. The previously applied procedure of manually tuning kinetics parameters requires significant manpower. This article presents a methodology for kinetic parameter estimation that is based on standard optimization methods. The methodology is being applied in the exploitation of synthetic gas experiments and legislated driving cycle tests and the assessment of the quality of information contained in the test results. Although the optimization technique employed for parameter estimation is well known, the development of the specific parameter estimation methodology that employs the results of the available types of experiments is novel and required significant development. Application of this refined tuning methodology increases the quality and reliability of prediction and also greatly reduces the required manpower, which is important in the specific engineering design process. The parameter estimation procedure is applied to the example of modeling of a diesel catalytic converter with adsorption capabilities, based on laboratory experiments and vehicle driving cycle tests.