Improving prediction capabilities of complex dynamic models via parameter selection and estimation

Many mathematical models describing complex (bio-)chemical reaction networks with a high level of detail are available in the literature. While such detailed models are desirable for investigating the dynamics of a particular component, it is often questionable if it is possible to verify these models due to the limited amount of quantitative data that can be generated. Even if it is not possible to estimate all parameters of these models then one would still be interested in determining which parameters should be estimated.This paper addresses this point and introduces a technique for determining the parameters of a model that should be estimated from experimental data. The focus of this work is on ensuring that the model has good prediction capability as over-fitting the model to noisy experimental data is avoided. Towards this goal, a forward selection approach for selecting a subset of parameters for estimation while taking uncertainty into account is developed to minimize the mean squared prediction error of the model. It is shown that the developed technique is closely related to the often-used orthogonalization method. The technique is applied to a model of the NF-κB signal transduction pathway. The presented method is able to generate a smaller mean squared error than estimation of all parameters and also outperforms the orthogonalization method.     

Systematic procedure for the reduction of complex biological reaction pathways and the generation of macroscopic equivalents

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows equivalent `input-output' representations of the system to be derived. The procedure is illustrated with a few examples, and a comparison is made with another recently published method for generating and evaluating macroscopic reaction schemes.     

State-preserving nonlinear model reduction procedure

Model reduction has proven to be a powerful tool to deal with challenges arising from large-scale models. This is also reflected in the large number of different reduction techniques that have been developed. Most of these methods focus on minimizing the approximation error; however, they usually result in a loss of physical interpretability of the reduced model. A new reduction technique, which preserves a non-prescribed subset of the original state variables in the reduced model, is presented in this work. The technique is derived from the Petrov–Galerkin projection by adding constraints on the projection matrix. This results in a combinatorial problem of which states need to be selected. A sequential algorithm has been developed based on the modified Gram–Schmidt orthogonalization procedure. The presented technique is applied to two examples where the reduction error is found to be comparable to the traditional POD method. At the same time, the technique has the advantage that the physical interpretation of the remaining states is retained.     

Experimental design for the joint model discrimination and precise parameter estimation through information measures

Experimental design procedures for model discrimination and for estimation of precise model parameters are usually treated as independent techniques. In order to conciliate the objectives of both experimental design procedures, the present paper proposes the use of experimental design criteria that are based on measures of the information gain when new experiments are carried out. The proposed criterion depends on the volumes of the confidence regions of the model parameters and presents a number of advantageous aspects, such as the conciliation of the usual experimental design objectives and the fact that the obtained criterion values can be easily interpreted in terms of the information eliminated after carrying out additional experiments. Besides, the proposed design criterion can easily accommodate multiobjective experimental design approaches, as shown in the examples.     

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.     

A new approach for sequential experimental design for model discrimination

Model discrimination procedures are useful tools for selection of the best mathematical models to be used to represent a specific chemical process. The present paper presents and discusses a new sequential discrimination procedure, which makes use of model probabilities and concentrates the efforts on models with higher probabilities. Model probabilities are determined based on simple statistical arguments. Four numerical examples illustrate the application of the proposed discrimination procedure. The obtained results indicate that the new procedure is able to discriminate kinetic models with fewer experiments when compared to other procedures and also indicates when model discrimination is not possible and, thus, when the sequential design must be halted. Furthermore, the speed of the proposed discrimination procedure can be controlled by tuning a design parameter which reflects the analyst's mood (confidence) towards the discrimination problem and allows for increase or decrease of the number of experiments required for model discrimination during the sequential procedure.     

Closed loop model reduction applied to a tank reactor process

The subject of this article is the direct assessment of model simplification from a feedback control perspective. Normally, dynamic systems are simplified dimensionally and structurally from an open loop perspective. In spite of the intentions, the resulting models still often tend to be unnecessary complex for controller design and synthesis. Here, a four step method is proposed that incorporates a feedback controller, and makes use of the closed loop sensitivity functions to indicate significant impact on the closed loop behaviour from a performed model simplification. The method is applied to a first order reaction in an ideally stirred tank reactor with a cooling system. For this system a general and detailed model is derived. The model includes temperature dependent parameters such as specific heat capacities and densities. This reference model is locally unstable in most operating points, making open loop simplification impractical. The proposed closed loop simplification method makes it possible to evaluate which approximations of the system that can be justified.     

Scaling and sensitivity analysis of a reverse flow reactor

Scaling analysis is presented as a systematic procedure to analyze and understand the operation of a complex process such as the autothermal reverse flow reactor (RFR). The reactor is complex from an operational point of view due to its hybrid and periodic nature. An adequate model of the RFR involves highly nonlinear equations. Using simple mathematical operations, these model equations are non-dimensionalized, scaled to order 1 and used to determine the contributions of the controlling physical phenomena taking place in it. The scale factors lead to several analytical expressions useful for suggesting efficient operational strategies for the RFR. Based on a specified error tolerance, we also illustrate how model approximation can be carried out and justified. The sensitivity of important operational parameters that determine sustainability (i.e., maximum temperature and overall conversion) to variables such as reactor length, switching time and mass transfer rate are also analyzed for the pseudo-steady-state condition. The results obtained prove that prudent ways of operating an RFR can be determined through scaling and sensitivity analysis.     

Kinetic model for the hydrolysis of sterilized palm press fibre

In this study, a kinetic model for enzymatic hydrolysis of oil palm residues considering the effect of sterilization was proposed and validated. Experiments were performed in batch reactor using palm oil mill effluent (POME) supplemented with sterilized and non-sterilized pressed pericarp fibers (PPF) as substrates. Kinetic parameters were estimated by fitting the experimental data to the models. It was found that the sterilization process as well as the variety in substrate particle size exerted an effect on the apparent rate constant (k), but no effect on the apparent Michaelis constant (K_M) and apparent competitive inhibition constant (K_I). When compared with the experimental data, the kinetic model provided good prediction to the oil palm residues hydrolysis with mean square error less than 10%.     

Correction procedures for extra-column effects in dynamic column breakthrough experiments

Dynamic column breakthrough experiments, routinely used to complement adsorption and diffusion studies at the particle scale, constitute an important step in the development and verification of dynamic models for simulation of adsorption processes. Various parts of the experimental setup contribute to the retention time and band broadening of the experimental breakthrough curve. However, the effect of the extra-column contributions have to be properly accounted for in order to compare the experimental results with theoretical calculations. A common practice is to measure a blank response under the same flow rate, pressure and temperature conditions as the actual experiment by simply bypassing the adsorption column with a tube (or a connector) of negligible volume. This blank response is then subtracted point-by-point from the composite response (i.e., including the adsorption column) to account for extra-column contributions. The underlying assumption here is that blank and column responses are linearly additive, both in terms of mean residence time and band broadening. It is shown that this method of correction can, under certain operating conditions, lead to erroneous results. An alternative procedure based on linear regression is introduced and the improvements achieved by this method are illustrated using simulation examples.     

An alternative method for parameter identification from temperature programmed reduction (TPR) data

The experimental method of temperature programmed reduction (TPR) for the investigation of gas-solid reactions is well established and widely used since the 1970s. With regard to the temporal and financial effort for TPR measurements, the quantitative model-based analysis of the data is not adequately developed. The TPR data analysis is comprised of two aspects: a discrete model identification and a continuous parameter optimisation. In this contribution, a general model for TPR experiments is introduced and a strategy for the model identification is proposed. This results in a large number of continuous optimisation problems which can be solved very efficiently by a proposed optimisation algorithm that is especially tailored for the problem at hand. The applicability of the method is demonstrated using TPR measurements of iron oxide in hydrogen gas.     

Sequential experimental design for model discrimination: Taking into account the posterior covariance matrix of differences between model predictions

Techniques for experimental design of experiments for model discrimination constitute important tools for scientists and engineers, as analyzed phenomena can very often be described fairly well by different mathematical models. As interpretation and use of available experimental data depend on the model structure, techniques for design of experiments for selection of the best model are of fundamental importance. Besides, experiments must often be designed for estimation of model parameters and reduction of variances of model predictions (or parameter estimates). These two classes of experimental design techniques generally lead to different experimental designs, although model discrimination and reduction of variances of parameter estimates are closely related to each other. In this work the posterior covariance matrix of difference between model predictions is taken into account during the design for model discrimination for the first time. The obtained results show that the model discrimination power becomes much higher when the posterior covariance matrix of difference between model predictions are considered during the experimental design, increasing the capability of model discrimination and simultaneously leading to improved parameter estimates.     

Nonlinear system identification and model reduction using artificial neural networks

We present a technique for nonlinear system identification and model reduction using artificial neural networks (ANNs). The ANN is used to model plant input–output data, with the states of the model being represented by the outputs of an intermediate hidden layer of the ANN. Model reduction is achieved by applying a singular value decomposition (SVD)-based technique to the weight matrices of the ANN. The sequence of state values is used to convert the model to a form that is useful for state and parameter estimation. Examples of chemical systems (batch and continuous reactors and distillation columns) are presented to demonstrate the performance of the ANN-based system identification and model reduction technique.     

Asymptotic analysis of a complex reaction scheme in solid-liquid system

Asymptotic analysis of a complex reaction, Claisen condensation, is discussed. Under certain assumptions related to experimental observations the kinetics scheme of the reaction can be considerably reduced. Additional complications in the problem arise due to the fact that one of the substances is present in a sparingly soluble solid phase. We show how reductions for increasingly complex systems with kinetics, mass transfer and feed can be systematically handled by employing the methodology of asymptotic analysis. Generalizations of the reduction approach for the case of an arbitrary number of fast intermediate reactions are also considered.The identification of model parameters is considered with aid of the reduced systems. Questions of parameter estimation and optimal design of experiments are discussed using the MCMC methods.     

Nonlinear system identification using Wiener type Laguerre-Wavelet network model

Compact nonlinear black box models, capable of representing highly nonlinear systems, are of high demand both in industry and academia. In this paper it has been shown that the use of Laguerre basis filters coupled with a wavelet network in Wiener type model structure are capable of modeling highly nonlinear systems with acceptable accuracy. Although individual merits of orthonormal basis functions (especially Laguerre filters) and multiresolution wavelet decompositions and/or wavelet network have been well documented in various literatures on system identification in the past, their combinational advantages in nonlinear system identification has never been explored. Laguerre filter models have the ability to approximate linear systems (even with time delay) with a model order lower than the traditional ARX (e.g., FIR, AR) modeling. Use of Laguerre models for mildly nonlinear system is possible only with a piece-wise linear models. Wavelet basis functions have the property of localization in both time and frequency and they can approximate even severe nonlinearities with appreciable accuracy with fewer model terms. But wavelet approximations fail miserably, in terms of model parsimony, if used for approximating linear or mildly nonlinear systems. In the present work a Laguerre-Wavelet network model has been proposed which combines the Laguerre and Wavelet approximations. In the said model, merits of both these approximations are retained whereas their demerits are suppressed.     

Thermal model validation of plate heat exchangers with generalized configurations

Thermal models of plate heat exchangers rely on correlations for the evaluation of the convective heat transfer coefficients inside the channels. It is usual to configure the exchanger with one countercurrent single-pass arrangement for acquiring heat transfer experimental data. This type of configuration approaches the ideal case of pure countercurrent flow conditions, and therefore a simplified mathematical model can be used for parameter estimation. However, it is known that the results of parameter estimation depend on the selected exchanger configuration because the effects of flow maldistribution inside its channels are incorporated into the heat transfer coefficients. This work presents a parameter estimation procedure for plate heat exchangers that handles experimental data from multiple configurations. The procedure is tested with an Armfield FT-43 heat exchanger with flat plates and the parameter estimation results are compared to those obtained from the usual method of single-pass arrangements. It can be observed that the heat transfer correlations obtained for plate heat exchangers are intimately associated with the configuration(s) experimentally tested and the corresponding flow distribution pattern(s).