Online identification for batch processes in closed loop incorporating priori controller knowledge

It is of great importance to develop an online modeling method for chemical processes operated in closed loop for better understanding, monitoring the process or other purposes without endangering the system. This paper intends to devise an online system identification method, particularly for the batch process, by fully exploiting its intrinsic repetitiveness. It properly uses the information from the time direction and the batch direction, thus leading to a gradual performance enhancement. In addition, the identification method formulates the priori controller knowledge such as closed-loop stability as optimization constraints to refine the parameter estimates. A trust region method is employed to overcome the significant computation burden of directly handling these constraints such as solving Lyapunov inequalities. An adaptive filter is introduced to further smooth the parameter estimates. Finally, the effectiveness of the approach is verified by three numerical examples including a two-tank system.     

Estimation of output measurement variances for error-in-variables model parameter estimation

Error-in-variables model (EVM) methods require information about variances of input and output measured variables when estimating the parameters in mathematical models for chemical processes. In EVM, using replicate experiments for estimating output measurement variances is complicated because true values of inputs may be different when multiple attempts are made to repeat an experiment. To address this issue, we categorize attempted replicate experiments as: (i) true replicates (TRs) when uncertain inputs are the same in replicated runs and (ii) pseudo replicates (PRs) when measured inputs are the same, but unknown true values of inputs are different. We propose methodologies to obtain output measurement variance estimates and associated parameter estimates for both situations. We also propose bootstrap methods for obtaining joint-confidence information for the resulting parameter estimates. A copolymerization case study is used to illustrate the proposed techniques. We show that different assumptions noticeably affect the uncertainties in the resulting reactivity-ratio estimates.     

用序贯法对反应速率模型中的参数作精确估算

对一个确定的模型,不仅要确定参数的估算值,还必须了解其可靠性。为了缩小参数的不定性,降低参数间的相关性,以扩大模型拟合的适定性和准确度,本文讨论了参数精确估算的序贯设计过程。 文内还详细讨论了精估参数的实验设计准则——最小联合置信容积准则和形状准则,以及序贯设计的实用价值。以氨合成为例,用序贯法精确估算了反应速度模型中的参数。这一模型是在序贯判别过程中选定的。把精估结果用于拟合大型生产过程的实测数据,得到了满意的结果。     

Sequential experimental design procedures for precise parameter estimation in ordinary differential equations

In this paper, it is shown how an experimental program for precise parameter estimation can be designed sequentially for the case that the mathematical model is given in the form of a set of ordinary differential equations. Two strategies are proposed. The first aims at minimizing the volume of the joint confidence region associated with the parameter estimates. The second attempts to alter as much as possible the shape towards a spherical region, by shortening the length of the longest principal axis of the confidence region to the maximum extent. The application of both criteria is illustrated by means of examples, representative for real problems in chemical reaction engineering. The techniques are easily applicable with our present day computing facilities. Qualitative indications are derived concerning the question when the use of an experimental design will result in an appreciable gain in significance for the parameter estimates.     

Large Sample Properties of Parameter Estimates for Periodic ARMA Models

This paper studies the asymptotic properties of parameter estimates for causal and invertible periodic autoregressive moving-average (PARMA) time series models. A general limit result for PARMA parameter estimates with a moving-average component is derived. The paper presents examples that explicitly identify the limiting covariance matrix for parameter estimates from a general periodic autoregression (PAR), a first-order periodic moving average (PMA(1)), and the mixed PARMA(1,1) model. Some comparisons and contrasts to univariate and vector autoregressive moving-average sequences are made.     

Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity

In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived. As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.     

Estimation for non‐negative time series with heavy‐tail innovations

For moving average processes where the coefficients are non‐negative and the innovations are positive random variables with a regularly varying tail at infinity, we provide estimates for the coefficients based on the ratio of two sample values chosen with respect to an extreme value criteria. We then apply this result to obtain estimates for the parameters of non‐negative ARMA models. Weak convergence results for the joint distribution of our estimates are established and a simulation study is provided to examine the small sample size behaviour of these estimates.     

Consistent estimation and order selection for nonstationary autoregressive processes with stable innovations

Abstract. A possibly nonstationary autoregressive process, of unknown finite order, with possibly infinite‐variance innovations is studied. The ordinary least squares autoregressive parameter estimates are shown to be consistent, and their rate of convergence, which depends on the index of stability, α, is established. We also establish consistency of lag‐order selection criteria in the nonstationary case. A small experiment illustrates the relative performance of different lag‐length selection criteria in finite samples.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

ESTIMATION FOR NONNEGATIVE AUTOREGRESSIVE PROCESSES WITH AN UNKNOWN LOCATION PARAMETER

Abstract. Consider an AR(1) process given by X ^t=γ+ø X _t+ Z _t≥ 1. where 0 ≤γ, 0 ≤ø 1 are unknown parameters and the innovations Z _t, ≥ 1, are independently and identically distributed positive random variables. We propose estimates of (γø) which are obtained as the solution to a linear programming problem and establish their strong consistency. When the Z _ts have the exponential distribution. our estimate becomes the conditional maximum likelihood estimate given X ₀. Under the assumption of regular variation of the innovation distribution at its left and right endpoints (assumed to be 0 and ∝ respectively), we establish asymptotic limit laws for the estimates. Consistent estimators for a class of moving-average processes with heavy-tailed innovation distribution are also presented.     

PARAMETER ESTIMATION IN EXPONENTIAL MODELS

Abstract. In this paper the problems of parameter estimation and order determination of an exponential (EX) model are studied in the time domain. In order to estimate the parameters, the parameter equations of an EX model are given in terms of the autocorrelation function, which is similar to the Yule-Walker equations of an autoregressive moving-average model. Estimates of parameters are obtained with the aid of the parameter equations and theorems are proved relating the convergence rate and asymptotic distribution of the estimates. We present two kinds of methods for estimating the order and prove that the estimates of the order are consistent.     

Maximum quasi-likelihood estimation for the near(2) model

Abstract.  Maximum quasi-likelihood estimation is investigated for the NEAR(2) model, an autoregressive time series model with marginal exponential distributions. In certain regions of the parameter space, simulations indicate that maximum quasi-likelihood estimators perform better than two-stage conditional least squares estimators in terms of the per cent of estimates falling in the parameter space. The problem of out-of-range estimates is shown to be caused by the lack of information in the data rather than the characteristics of the method of estimation.     

Resolving conflicting parameter estimates in multivariate population balance models

We present an extended methodology for parametric inference in complex population balance models. The aim is twofold. Firstly, it is assumed that the parameter distribution of the model is a multimodal Gaussian rather than a unimodal Gaussian. After projection of experimental data through a response surface approximation, estimates for the parameters and their uncertainties along with the associated weights of each mode are established. Secondly, the methodology is used to ask the following question—if n professors each have a ‘best’ estimate of a particular parameter, which of these estimates is more likely to be correct? A toy example is used to show the applicability of the methodology, aiding in the discrimination between a bimodal and trimodal parameter distribution. The identification of the ‘best’ model parameter among two conflicting estimates is demonstrated in an example from granulation modelling.     

FEASIBILITY OF ON-LINE REOPTIMIZATION IN BATCH PROCESSES

This work points out some serious drawbacks of the standard procedure for on-line optimization of batch processes. Commonly, at any given batch time, on-line measurement information is used to improve the estimate of the current state and the objective function is optimized again based on the estimated state. This strategy is useful when only the initial state is in error, but is unreliable in the more common case of an inaccurate dynamic model. Reoptimization based on the state estimates may then constitute a poor policy, even if perfect state estimates were available. Simulation examples of a semi-batch reactor are used to illustrate the associated pitfalls and to demonstrate what can be done to possibly detect and circumvent them.     

RECURSIVE GENERALIZED M ESTIMATES FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Outliers in time series seriously affect conventional parameter estimates. In this paper a robust recursive estimation procedure for the parameters of auto-regressve moving-average models with additive outliers is proposed. Using 'cleaned' residuals from an initial robust fit of an autoregression of high order as input, bounded influence regression is applied recursively. The proposal follows certain ideas of Hannan and Rissanen, who suggested a three-stage procedure for order and parameter estimation in a conventional setting.
A Monte Carlo study is performed to investigate the robustness properties of the proposed class of estimates and to compare them with various other suggestions, including least squares, M estimates, residual autocovariance and truncated residual autocovariance estimates. The results show that the recursive generalized M estimates compare favourably with them. Finally, possible modifications to master even vigourous situations are suggested.     

Parameter estimation and prediction uncertainties for multi-response kinetic models with uncertain inputs

Error-in-variables model (EVM) methods are used for parameter estimation when independent variables are uncertain. During EVM parameter estimation, output measurement variances are required as weighting factors in the objective function. These variances can be estimated based on data from replicate experiments. However, conducting replicates is complicated when independent variables are uncertain. Instead, pseudo-replicate runs may be performed where the target values of inputs for repeated runs are the same, but the true input values may be different. Here, we propose a method to estimate output-measurement variances for use in multivariate EVM estimation problems, based on pseudo-replicate data. We also propose a bootstrap technique for quantifying uncertainties in resulting parameter estimates and model predictions. The methods are illustrated using a case study involving n-hexane hydroisomerization in a well-mixed reactor. Case-study results reveal that assumptions about input uncertainties can have important influences on parameter estimates, model predictions and their confidence intervals.     

On Bartlett's Formula for Non-linear Processes

Bartlett's formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth-order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L ² and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non-negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study     

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.     

Robust batch‐to‐batch optimization in the presence of model‐plant mismatch and input uncertainty

In model‐based optimization in the presence of model‐plant mismatch, the set of model parameter estimates which satisfy an identification objective may not result in an accurate prediction of the gradients of the cost‐function and constraints. To ensure convergence to the optimum, the predicted gradients can be forced to match the measured gradients by adapting the model parameters. Since updating all available parameters is impractical due to estimability problems and overfitting, there is a motivation for adapting a subset of parameters for updating the predicted outputs and gradients. This article presents an approach to select a subset of parameters based on the sensitivities of the model outputs and of the cost function and constraint gradients. Furthermore, robustness to uncertainties in initial batch conditions is introduced using a robust formulation based on polynomial chaos expansions. The improvements in convergence to the process optimum and robustness are illustrated using a fed‐batch bioprocess. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2660–2670, 2017