in Wiley InterScience (www.interscience.wiley.com).Mathematical models of physical systems often have parameters that must be estimated from measured data. Inestimable models have more parameters than can be estimated from available data. In this work, a method for identifying inestimable parameters or parameter combinations is proposed. The method is based on partitioning the parameter space into estimable and inestimable subspaces. Parameter combinations in the inestimable subspace have little effect on measured values and can therefore be fixed at a nominal value. The number of effective parameters is thereby reduced to the dimension of the estimable subspace. The proposed method is applicable over a range of experimental conditions. Detailed examples, including a batch bioreactor and a three-phase reactor system, are included for illustration. Keywords: process control, control, parameter estimation
IntroductionMathematical models of physical systems are used in almost every branch of science and engineering. Such models are, generally, functions of unknown parameters that must be estimated from available data. Often, models contain more parameters than can be estimated from a given data set. In this case the model parameters are said to be inestimable. Inestimability implies that there are several possible parameter values that yield statistically indistinguishable predictions.Estimability in models is a very useful property. If a model is estimable (from a given data set) then the value of each parameter in the model can be accurately determined. As a result, the model, or parameters, can be used for design, analysis, and scale-up. Furthermore, accurate parameter estimates allow system behavior to be extrapolated beyond the region where data is collected. This is especially useful for systems whose behavior cannot be studied under all conditions. For example, accurate parameter estimation may allow a clinician to choose an appropriate drug dosage for a human patient.The problem of inestimability has been well documented in the literature. [1][2][3][4] For every model that is inestimable there exists a simpler, estimable, model (i.e., one with fewer parameters) that provides near identical predictions over the region for which data is available. Usually, a sensitivitybased approach 1,3,4 is used to identify model parameters that have little or no effect on model predictions and can therefore be either lumped, discarded, or held at a fixed value for the purpose of estimation.The sensitivity-based framework for model simplification holds that a simpler identifiable model can be obtained by effectively removing some of the model parameters. However, the sensitivities of the model predictions to parameter values are functions of estimated parameter values and the experimental conditions. As a result, sensitivity analysis can suggest reparameterizations that only hold locally (i.e., near a nominal parameter value).In this work, a method is proposed for reparameterizing inestimable models. The proposed approach al...