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Nonlinear modeling, estimation and predictive control in APMonitor

http://www.nt.ntnu.no/users/skoge/prost/proceedings/cpc8-focapo-2012/data/papers/007.pdf

Advances and selected recent developments in state and parameter estimation

Chemical engineers who develop fundamental models often have difficulties estimating all model parameters due to problems with parameter identifiability and estimability. These two concepts are reviewed, as are techniques for assessing identifiability and estimability. When some parameters are not estimable from the data, modellers must decide whether to conduct new experiments, change the model structure, or to estimate only a subset of the parameters and leave the others at fixed values. Estimating a reduced number of parameters can lead to better model predictions with lower mean squared error (MSE). MSE-based techniques for parameter subset selection are discussed and compared. © 2011 Canadian Society for Chemical Engineering

Mathematical modelling of chemical processes—obtaining the best model predictions and parameter estimates using identifiability and estimability procedures

Background Kinetic models of biochemical systems usually consist of ordinary differential equations that have many unknown parameters. Some of these parameters are often practically unidentifiable, that is, their values cannot be uniquely determined from the available data. Possible causes are lack of influence on the measured outputs, interdependence among parameters, and poor data quality. Uncorrelated parameters can be seen as the key tuning knobs of a predictive model. Therefore, before attempting to perform parameter estimation (model calibration) it is important to characterize the subset(s) of identifiable parameters and their interplay. Once this is achieved, it is still necessary to perform parameter estimation, which poses additional challenges. Methods We present a methodology that (i) detects high-order relationships among parameters, and (ii) visualizes the results to facilitate further analysis. We use a collinearity index to quantify the correlation between parameters in a group in a computationally efficient way. Then we apply integer optimization to find the largest groups of uncorrelated parameters. We also use the collinearity index to identify small groups of highly correlated parameters. The results files can be visualized using Cytoscape, showing the identifiable and non-identifiable groups of parameters together with the model structure in the same graph. Results Our contributions alleviate the difficulties that appear at different stages of the identifiability analysis and parameter estimation process. We show how to combine global optimization and regularization techniques for calibrating medium and large scale biological models with moderate computation times. Then we evaluate the practical identifiability of the estimated parameters using the proposed methodology. The identifiability analysis techniques are implemented as a MATLAB toolbox called VisId, which is freely available as open source from GitHub ( https://github.com/gabora/visid ). Conclusions Our approach is geared towards scalability. It enables the practical identifiability analysis of dynamic models of large size, and accelerates their calibration. The visualization tool allows modellers to detect parts that are problematic and need refinement or reformulation, and provides experimentalists with information that can be helpful in the design of new experiments.

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Parameter identifiability analysis and visualization in large-scale kinetic models of biosystems

Nonlinear ill-posed problem analysis in model-based parameter estimation and experimental design

Parameter estimation in complex mathematical models is difficult, especially when there are too many unknown parameters to estimate, and the available data for parameter estimation are limited. Estimability analysis ranks parameters from most estimable to least estimable based on the model structure, uncertainties in initial parameter guesses, measurement uncertainties, and experimental settings. Difficulties associated with poor numerical conditioning are avoided by only estimating those parameters that are most estimable. The remaining parameters are left at their initial values or can be removed from the model via simplification. In this paper, a mean squared error (MSE)-based model-selection criterion is used to determine the optimal number of parameters to estimate from the ranked parameter list, so that the most reliable model predictions can be obtained. This methodology is illustrated using a dynamic chemical reactor model.

Selection of optimal parameter set using estimability analysis and MSE-based model-selection criterion

Engineers who develop fundamental models for chemical processes are often unable to estimate all of the parameters, especially when available data are limited or noisy. In these situations, modelers may decide to select only a subset of the parameters for estimation. An orthogonalization algorithm combined with a mean squared error (MSE) based selection criterion has been used to rank parameters from most to least estimable and to determine the parameter subset that should be estimated to obtain the best predictions. A robustness test is proposed and applied to a batch reactor model to assess the sensitivity of the selected parameter subset to initial parameter guesses. A new ranking and selection technique is also developed based on the MSE criterion and is compared with existing techniques in the literature. Results obtained using the proposed ranking and selection techniques agree with those from leave-one-out cross-validation but are more computationally attractive.

Mean-Squared-Error Methods for Selecting Optimal Parameter Subsets for Estimation

The effects of cathode materials and electrolyte additives on magnesium wetting were studied with the goal of improving current efficiency in a magnesium electrolysis cell. The study consisted of static wetting and electrolysis tests, both conducted in a visual cell with a molten salt electrolyte of MgCl2-CaCl2-NaCl-KCl-CaF2. The wetting conditions were tested using high resolution photography and contact angle software. The electrolysis tests were completed to qualitatively assess the effect of additives to the melt and were recorded with a digital video camcorder. Results from the static wetting tests showed a significant variation in wetting depending on the material used for the cathode. Mo and a Mo-W alloy, with contact angles of 60° and 52° respectively, demonstrated excellent wetting. The contact angle for steel was 132° and it ranged from 142°–154° for graphite depending on the type. Improvements to the cathode wetting were observed with tungsten and molybdenum oxide additives.

Kevin A. P. McLean

Papers

Mathematical modelling of chemical processes—obtaining the best model predictions and parameter estimates using identifiability and estimability procedures

Selection of optimal parameter set using estimability analysis and MSE-based model-selection criterion

Mean-Squared-Error Methods for Selecting Optimal Parameter Subsets for Estimation

Cathode Wetting Studies in Magnesium Electrolysis