Models with One Unknown Parameter

The identification of the parameters of a process can be examined from two completely different viewpoints. The former is given by laboratory researchers, who consider the identification of parameters together with a deep experimental analysis it is then frequently difficult to criticize the experimental working methods, the quality and quantity of the experimental data. The latter is given by researchers specialized in mathematical modelling and simulation. These researchers consider that the mathematical aspects in the identification of parameters are prevailing. Nevertheless, this last consideration has some limits because, in all cases, a similar number of parameters and independent experimental data are necessary for a correct identification. 

It is important to notice that, from both viewpoints, as well as in all working procedures, experimental data are required and that, at the same time, mathematical models are absolutely needed for data processing. Generally, when the mathematical model of a process is relatively complex, a good accuracy and an important volume of experimental data are simultaneously required. Therefore, in these cases the quality of the determination of parameters is the most important factor to ensure model relevance. The strategy adopted in these cases is very simple for all the parameters of the process that accept an indirect identification, the research procedure of identification is carried out separately from the real process whereas for the very specific process parameters that are difficult to identify indirectly, experiments are carried out with the actual process. 

When we have N measures for the exit variables in a process, the technical problem of identification of the unknown parameter resides in solving the equation l (p) = 0. From the theoretical viewpoint, all the methods recommended for the solution of the transcendent equation can be used to determine parameter p. The majority of these methods are of iterative type and require an expression or an evaluation of the (p) derivate. When we evaluate the derivate numerically, as in the case of a complex process model, then important deviations can be introduced into the iteration chain. Indeed, the deviation propagation usually results in an increasing and non-realistic value of the parameter. This problem can be avoided by solving the equation (p) =0 by integral methods such as the method of minimal function value (MFV). When (p) values are only obtained in the area of influence of parameter p, the MFV method is reduced to a dialogue with the mathematical model of the process and then the smallest (p) value gives the best value for the parameter. 

The following example details how the MFV method is used to identify the diffusion coefficient of species with respect to their motion in a particle of activated carbon. 

Diffusion of Species Inside a Particle of Activated Carbon 

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Some Aspects of Parameters Identification in Mathematical Modelling 167 3.5.4.2 Complex Models with One Unknown Parameter... 

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