Estimation of time-varying time delay and parameters of a class of jump Markov nonlinear stochastic systems

It is a challenging problem to estimate time-varying time delay and parameters, especially for systems subject to disturbances with unknown statistics in measurements. The desirable filter should be sensitive to unmodeled dynamics caused by random changes in time delay and parameters, and also be robust to disturbances. Recently, we proposed a finite-horizon robust Kalman filter (RKF) through designing and simultaneously minimizing the upper bounds of unknown covariances of prediction errors, filtering residuals and estimation errors. Unfortunately, unmodeled dynamics and disturbances must be hypothesized to be zero-mean white noises in the RKF. To cope with more general unmodeled dynamics and/or disturbances, a class of jump Markov stochastic systems (JMSS) subject to unmodeled dynamics and disturbances is considered in this article so that a priori system information, such as the value range of unknown and/or randomly changing parameters, can be introduced. Through combining the RKF with the interacting multiple model (IMM) estimation technique, a RKF/IMM algorithm is proposed for such JMSS. Then it is applied to estimate time-varying time delay and parameters of a continuous stirred tank reactor (CSTR) with sensors subject to Gaussian disturbances with unknown means and/or covariances. The RKF/IMM algorithm is compared with the extended Kalman filter (EKF), the strong tracking filter (STF) and the RKF through computer simulations. The results show that, in the case that measurement disturbances are zero-mean noise with unknown covariances, the RKF/IMM and RKF achieve almost the same accurate estimates, which are superior to those of the STF and EKF; in the case that such disturbances have unknown covariances and time-varying means, only the RKF/IMM remains the ability to estimate time-varying time delay and parameters. Furthermore the RKF/IMM has unique ability to identify the disturbance mean no matter whether it is constant or time-varying. Moreover, the RKF/IMM algorithm is shown having strong robustness against the a priori filter parameters, this may be attractive in practical applications.     

On-line monitoring of a CSTR polymerization reactor using multivariable filters

Three multivariable filters are evaluated for on-line monitoring of a CSTR polymerization reactor. The first filtering algorithm is the Kaiman filter. This linear filter is simple to implementation, but cannot exactly estimate the dynamic behavior of the polymerization reactor. To compensate the state model inadequacies, nonlinear models can be considered in the filtering algorithm. The precise state estimation can be guaranteed by the extended Kaiman filter (EKF). Finally, the auto-regressive exogenous inputs model based filter (ARXF) is developed to reduce the modeling cost. These different filters are applied to the continuous solution polymerization of a MMA-AIBN-EA system as a case study. The ARXF is easy to implement and shows satisfactory results.     

EXTENDED KALMAN FILTER CONTROLLER: FIRST PRINCIPLES MODELS TO NEURAL NETWORKS

Extended Kalman filters (EKF) have been widely employed for state and parameter estimation in chemical engineering systems. Gao et al. [Gao, F., Wang, F. and Li, M. (1999). Ind. Eng. Chem. Res., 38, 2345-2349] have proposed the use of EKF for control computation using a neural network representation of the system in a discrete-time framework. In the present study, an EKF controller is proposed in a continuous time framework with models incorporating different levels of process knowledge. The problem of process-model mismatch is handled by incorporating EKF-based state and/or parameter estimation along with control computation. A batch reactor temperature control problem for a highly exothermic reaction between maleic anhydride and hexanol to form hexyl monoester of maleic acid is considered as a test bed to evaluate the performance of the proposed control schemes. Three different models are considered, namely the first principles model, a reduced-order process model, and an artificial neural network (ANN) model for formulation of the control schemes. The performance of the proposed control scheme using first principles model is compared to that of generic model control, and a similar performance is achieved. The present study illustrates the usefulness of the proposed control schemes and can be easily extended to general chemical engineering systems.     

A particle filter approach to identification of nonlinear processes under missing observations

A novel maximum likelihood solution to the problem of identifying parameters of a nonlinear model under missing observations is presented. If the observations are missing, then it is difficult to build a partial likelihood function consisting of only the available observations. Hence, an expectation–maximization (EM) algorithm, which uses the expected value of the complete log‐likelihood function including the missing observations, is developed. The expected value of the complete log‐likelihood (E‐step) in the EM algorithm is approximated using particle filters and smoothers. New expressions for particle filters and smoothers under missing observations are derived. In order to reduce the variance on the smoothed states, a point‐wise (as opposed to path‐based) state estimation procedure is used. The maximization step (M‐step) in the EM algorithm is performed using standard optimization routines. The proposed nonlinear identification approach is illustrated through numerical examples.     

Fitting Stochastic Volatility Models in the Presence of Irregular Sampling via Particle Methods and the EM Algorithm

Abstract. Stochastic volatility (SV) models have become increasingly popular for explaining the behaviour of financial variables such as stock prices and exchange rates, and their popularity has resulted in several different proposed approaches to estimating the parameters of the model. An important feature of financial data, which is commonly ignored, is the occurrence of irregular sampling because of holidays or unexpected events. We present a method that can handle the estimation problem of SV models when the sampling is somewhat irregular. The basic idea of our approach is to combine the expectation‐maximization (EM) algorithm with particle filters and smoothers in order to estimate parameters of the model. In addition, we expand the scope of application of SV models by adopting a normal mixture, with unknown parameters, for the observational error term rather than assuming a log‐chi‐squared distribution. We address the problems by using state–space models and imputation. Finally, we present simulation studies and real data analyses to establish the viability of the proposed method.     

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.     

Simultaneous estimation of the heat of reaction and the heat transfer coefficient by calorimetry: estimation problems due to model simplification and high jacket flow rates—theoretical development

In batch and semi-batch reactors, the heat of reaction is normally estimated using calorimetry. If all temperatures and volumes are measured correctly and the measurements are filtered sensibly, the results are usually very good. For many applications, the overall heat transfer coefficient k also needs to be known. In heat balance calorimetry k and can be calculated simultaneously, if the correct model is used. It is shown that the commonly used model simplifications pose serious problems for large reactors. Subsequently a sensible model extension is discussed. For this extension we propose the application of an Extended Kalman Filter (EKF) to estimate the heat of reaction () and the heat transfer coefficient (k) simultaneously, as the EKF can handle the model extension well. Our emphasis lies on three important factors.Firstly and mainly, the jacket of a jacketed reactor is generally modelled as a stirred tank. When looking at real jacketed reactors, the jacket behaves more like a plug-flow reactor. We propose a model extension to overcome this problem. Secondly, the heat transfer coefficient (k) is for many reactions strongly dependent on the batch time and should therefore also be estimated. With the usual models, errors may result which can be corrected by the model extension. Thirdly, the flow rate through the jacket and the hold-up in the reactor strongly influence the estimation quality. With a lower jacket flow rate estimation quality increases but cooling decreases, a trade-off has to be made. Using an EKF, good estimation quality can still be achieved for high flow rates. However, the trade-off is considered and the tuning is adjusted to the flow rate. An optimal flow rate calculation is suggested. Finally, it will be shown that adding measurements in the jacket rather than in the reactor will improve calorimetric estimation for the proposed model extension.     

Consensus-based approach for parameter and state estimation of agro-hydrological systems

The development of advanced closed-loop irrigation systems requires accurate soil moisture information. In this work, we address the problem of soil moisture estimation for the agro-hydrological systems in a robust and reliable manner. A nonlinear state-space model is established based on the discretization of the Richards equation to describe the dynamics of the agro-hydrological systems. We consider that model parameters are unknown and need to be estimated together with the states simultaneously. We propose a consensus-based estimation mechanism, which comprises two main parts: (a) a distributed extended Kalman filtering algorithm used to estimate several model parameters; and (b) a distributed moving horizon estimation algorithm used to estimate the state variables and one remaining model parameter. Extensive simulations are conducted, and comparisons with existing methods are made to demonstrate the effectiveness and superiority of the proposed approach. In particular, the proposed approach can provide accurate soil moisture estimate even when poor initial guesses of the parameters and the states are used, which can be challenging to be handled using existing algorithms.     

基于热扩散核密度确定密度峰值法的历史工况识别

在工业生产过程中,生产决策的调整或生产状况的变化会导致生产过程多模态化,常用的数据聚类方法进行工况识别时存在参数选取困难或需要先验知识等限制。基于此,提出一种将人工智能领域的热扩散核密度确定密度峰的技术与高斯混合模型相结合的方法,可有效克服目前方法的缺点。该方法首先利用热扩散核密度确定密度峰的技术估算每个数据点的密度及其与局部密度较大点的距离,获取数据集的聚类中心并完成聚类;其次,利用高斯混合模型获取不同工况的特征参数：平均值、协方差和先验概率,从而对多工况历史过程进行准确的描述;最后,利用文献中仿真数据和Tennessee Eastman过程两个案例进行验证,并与K-均值法和F-J改进的高斯混合模型进行比较,证明了本文提出方法可更加方便、有效地对历史工况进行准确识别。     

Dynamic data rectification using particle filters

The basis of dynamic data rectification is a dynamic process model. The successful application of the model requires the fulfilling of a number of objectives that are as wide-ranging as the estimation of the process states, process signal denoising and outlier detection and removal. Current approaches to dynamic data rectification include the conjunction of the Extended Kalman Filter (EKF) and the expectation-maximization algorithm. However, this approach is limited due to the EKF being less applicable where the state and measurement functions are highly non-linear or where the posterior distribution of the states is non-Gaussian. This paper proposes an alternative approach whereby particle filters, based on the sequential Monte Carlo method, are utilized for dynamic data rectification. By formulating the rectification problem within a probabilistic framework, the particle filters generate Monte Carlo samples from the posterior distribution of the system states, and thus provide the basis for rectifying the process measurements. Furthermore, the proposed technique is capable of detecting changes in process operation and thus complements the task of process fault diagnosis. The appropriateness of particle filters for dynamic data rectification is demonstrated through their application to an illustrative non-linear dynamic system, and a benchmark pH neutralization process.     

基于PSO-LSSVR的环状流截面含气率软测量方法

由于测量条件高,环状流的截面含气率直接测量往往比较困难。软测量技术的关键在于建立优良的数学模型,在分析了微粒群优化算法(PSO)和最小支持向量回归机(LS-SVR)原理的基础上,利用粒子群算法优化最小二乘支持向量回归机参数的算法,建立了软测量模型,实现了环状流的截面含气率的软测量。实验表明:该模型泛化能力强,测试精度比较高,为环状流截面含气率的测量提供了一种新的测量途径。     

基于混合微分进化算法的生化过程动态参数估计(英文)

Determination of the optimal model parameters for biochemical systems is a time consuming iterative process. In this study, a novel hybrid differential evolution (DE) algorithm based on the differential evolution technique and a local search strategy is developed for solving kinetic parameter estimation problems. By combining the merits of DE with Gauss-Newton method, the proposed hybrid approach employs a DE algorithm for identifying promising regions of the solution space followed by use of Gauss-Newton method to determine the optimum in the identified regions. Some well-known benchmark estimation problems are utilized to test the efficiency and the robustness of the proposed algorithm compared to other methods in literature. The comparison indicates that the present hybrid algorithm outperforms other estimation techniques in terms of the global searching ability and the convergence speed. Additionally, the estimation of kinetic model parameters for a feed batch fermentor is carried out to test the applicability of the proposed algorithm. The result suggests that the method can be used to estimate suitable values of model parameters for a complex mathematical model.     

Maximum likelihood estimation of noise covariance matrices for state estimation of autonomous hybrid systems

A critical aspect of developing Bayesian state estimators for hybrid systems, that involve a combination of continuous and discrete state variables, is to have a reasonably accurate characterization of the stochastic disturbances affecting their dynamics. Recently, Bavdekar et al. (2011) have proposed a maximum likelihood (ML) based framework for estimation of the noise covariance matrices from operating input–output data when an EKF is used for state estimation. In this work, the ML framework is extended to estimation of the noise covariance matrices associated with autonomous hybrid systems, and, to a wider class of recursive Bayesian filters. Under the assumption that the innovations generated by an estimator form a white noise sequence, the proposed ML framework computes the noise covariance matrices such that they maximize the log-likelihood function of the estimator innovations. The efficacy of the proposed scheme is demonstrated through the simulation and experimental studies on the benchmark three-tank system.     

Extended kalman filter‐based nonlinear model predictive control of a continuous KCl‐NaCl crystallizer

An extended Kalman filter (EKF)‐based nonlinear quadratic dynamic matrix control (EQDMC) for an evaporative cooling draft‐tube baffled (DTB) KCl crystallizer is developed. The controller is used to maintain the productivity, crystal mean size and impurity of crystals. Since these controlled variables are not directly measurable, the EKF is used to estimate them. The nonlinear controller is a combination of an extended linear dynamic matrix control (EDMC) and the quadratic dynamic matrix control (QDMC). This combination provided good control of the system despite the process nonlinearity, constraints, and inadequate reliable online measurement of the controlled variables. The performance of the controller in the presence of plant/model mismatch, disturbance, wrong estimation and simultaneous step changes in the controller setpoints is discussed.     

Model predictive control for the reactant concentration control of a reactor

The reactant concentration control of a reactor using Model Predictive Control (MPC) is presented in this paper. Two major difficulties in the control of reactant concentration are that the measurement of concentration is not available for the control point of view and it is not possible to control the concentration without considering the reactor temperature. Therefore, MIMO control techniques and state and parameter estimation are needed. One of the MIMO control techniques widely studied recently is MPC. The basic concept of MPC is that it computes a control trajectory for a whole horizon time minimising a cost function of a plant subject to a dynamic plant model and an end point constraint. However, only the initial value of controls is then applied. Feedback is incorporated by using the measurements/estimates to reconstruct the calculation for the next time step. Since MPC is a model based controller, it requires the measurement of the states of an appropriate process model. However, in most industrial processes, the state variables are not all measurable. Therefore, an extended Kalman filter (EKF), one of estimation techniques, is also utilised to estimate unknown/uncertain parameters of the system. Simulation results have demonstrated that without the reactor temperature constraint, the MPC with EKF can control the reactant concentration at a desired set point but the reactor temperator is raised over a maximum allowable value. On the other hand, when the maximun allowable value is added as a constraint, the MPC with EKF can control the reactant concentration at the desired set point with less drastic control action and within the reactor temperature constraint. This shows that the MPC with EKF is applicable to control the reactant concentration of chemical reactors.     

Blocking effect of twin boundaries on partial dislocation emission from void surfaces

Recent discovery that nanoscale twin boundaries can be introduced in ultrafine-grained metals to improve strength and ductility has renewed interest in the mechanical behavior and deformation mechanisms of these nanostructured materials. By controlling twin boundary spacing, the effect of twin boundaries on void growth is investigated by using atomistic simulation method. The strength is significantly enhanced due to the discontinuous slip system associated with these coherent interfaces. Atomic-scale mechanisms underlying void growth, as well as the interaction between twin boundaries and the void, are revealed in details.     

Gas emission source term estimation with 1-step nonlinear partial swarm optimization-Tikhonov regularization hybrid method

Source term identification is very important for the contaminant gas emission event.Thus,it is necessary to study the source parameter estimation method with high computation efficiency,high estimation accuracy and reasonable confidence interval.Tikhonov regularization method is a potential good tool to identify the source parameters.However,it is invalid for nonlinear inverse problem like gas emission process.2-step nonlinear and linear PSO (partial swarm optimization)-Tikhonov regularization method proposed previously have estimated the emission source parameters successfully.But there are still some problems in computation efficiency and confidence interval.Hence,a new 1-step nonlinear method combined Tikhonov regularization and PSO algorithm with nonlinear forward dispersion model was proposed.First,the method was tested with simulation and experiment cases.The test results showed that 1-step nonlinear hybrid method is able to estimate multiple source parameters with reasonable confidence interval.Then,the estimation performances of different methods were compared with different cases.The estimation values with 1-step nonlinear method were close to that with 2-step nonlinear and linear PSO-Tikhonov regularization method.1-step nonlinear method even performs better than other two methods in some cases,especially for source strength and downwind distance estimation.Compared with 2-step nonlinear method,1-step method has higher computation efficiency.On the other hand,the confidence intervals with the method proposed in this paper seem more reasonable than that with other two methods.Finally,single PSO algorithm was compared with 1-step nonlinear PSO-Tikhonov hybrid regularization method.The results showed that the skill scores of 1-step nonlinear hybrid method to estimate source parameters were close to that of single PSO method and even better in some cases.One more important property of 1-step nonlinear PSO-Tikhonov regularization method is its reasonable confidence interval,which is not obtained by single PSO algorithm.Therefore,1-step nonlinear hybrid regularization method proposed in this paper is a potential good method to estimate contaminant gas emission source term.     

Parameter Estimation and Subset Selection for Separable lower Triangular Bilinear Models

Abstract. Parameter estimation and subset selection for separable lower triangular bilinear (SLTBL) models are considered. Under a flat prior, we present an expectation–maximization (EM) algorithm to obtain the maximum likelihood estimation. Furthermore, two sub‐procedures are designed to select the best subset model after an initial fitting. Example with two simulated and one real data set illustrate the feasibility and validity of the proposed methods.     

Estimation of time-dependent reaction coefficient with extended Kalman filter approach

This work introduces an extended Kalman filter (EKF) to the estimation of the unknown time-dependent reaction coefficient based on the concentration measurement data. An autocatalytic reaction pathway is chosen as a model problem. This inverse estimation algorithm does not assume any functional form of the reaction coefficient. The performance of the proposed algorithm is verified through the numerical experiments with the exact and the contaminated concentration measurement data.     

Estimation of the quantity of product within cryogenic storage vessels

Cryogenic products, such as oxygen, nitrogen and argon, can be delivered to customers either within vessels (cryogenic liquids) or cylinders (gaseous state), depending on the requested quantity.Cryogenic applications are numerous, from hospitals to glass furnaces and electronics manufacturers.Accurate estimation of the quantity of product stored at any time in cryogenic vessels is critical to avoid run-outs for the customer and to optimize the product deliveries, which means saving money and reducing pollution due to trucks fuel consumption.Our paper first details a method to estimate the vessel parameters and the mass stored within using measurement of the pressure at the top of the vessel and the differential pressure between the top and the bottom of the vessel. It is based upon a physical model of the vessel and estimates the liquid mass inside the vessel and the deliverable mass at each time-step.The added-value of the proposed algorithm has been assessed comparing to simple algorithm currently used in operations. The weaknesses of this simple algorithm are analyzed at the beginning of this paper.The method described is applicable for vertical, horizontal and spherical vessels. It has been validated on 40 vessels storing nitrogen, argon and oxygen (around 6 months of data involving 1000 fillings for all tanks) in reference to delivery bills (for deliverable mass estimation).Instantaneous mass estimation has been validated on vessels specially equipped with load cells measuring the weight of the vessel every hour.Validation showed this method to be at least twice as accurate as the simple algorithm previously used to estimate the mass. For high pressure vessels (above 15 barg), it reaches three times as accurate as the simple algorithm.Such results led to the industrialization of the algorithm, including simplifications to fit with operational constraints (especially calculation time).