Analysis of Approaches to Constructing a Plant Mathematical Model Based on Inaccurate Experimental Data

Abstract

The article analyzes the sources of uncertainty and peculiarities of using different models for describing inaccurate data, as well as approaches to constructing the forward and reverse static characteristics of a system from inaccurate data based on different uncertainty description models.

It is shown that application of the approach based on a probabilistic (statistical) uncertainty description model, which is most frequently used in practice, is expedient only in the case when the uncertainty is solely due to random variability. The description of other uncertainty sources within the framework of this model involves difficulties. In this case, interference is predominantly described by a normal distribution, the postulation of which serves as the basis for inferring the subsequent conclusions. As to the interference action mechanism, it is most frequently described using the model of additive interference at the output with a zero error value at the system input. Assuming that these prerequisites are fulfilled, the regression analysis techniques were taken to solve the problem of constructing the forward and reverse static characteristics. All optimal properties of the parameter estimates obtained in this case are only valid for the forward model, provided that the following sufficiently stringent system of assumptions is valid:

(i) The actual error distribution laws are various and often far from being normal.

(ii) To establish the actual form of the distribution function, it is necessary to carry out tests; the higher the confidence probability, the larger the number of tests should be.

(iii) The input quantities measurement errors are essential, which leads to biased estimates of the model parameters, and the output quantity measurement errors contain both random and systematic components, due to which the least-squares method is not optimal in the statistical sense. (iv) Random errors can be dependent, and their correlation matrix is unknown. Therefore, the usual least-squares estimates, which are frequently, although unreasonably, used in this case, will not provide the smallest variance values.

(v) The factor measurement and/or setting interference can be described not only by the additive model that is used in the majority of practical cases, but also by a multiplicative or an additive-multiplicative model.

In the regression analysis framework, there is no theoretically grounded method for constructing the reverse characteristic and its confidence interval. Therefore, attempts to apply the statistical approach to constructing reverse characteristics involve serious difficulties, and formal application of regression analysis yields results that are far from being true:

• The separation of variables into accurately measured input ones and output ones measured with errors is upset in constructing reverse characteristics.
• Analytical determination of the confidence interval for the predicted input variable value is not theoretically grounded and seems to be extremely difficult.
• A significant drawback of the error value confidence interval is the impossibility of summing up its components, because the confidence interval of the sum is not equal to the sum of the confidence intervals of the terms.
• Owing to essential nonlinearity of the reverse transformation, the error model is very complex and cannot be described in terms of absolute or relative errors.

In view of the above, the reverse function uncertainty interval seems to be most complete parameter characterizing the accuracy of solving the reverse function construction problem. In this regard, application of the approach based on the interval model that makes it possible to describe a wide class of uncertain, variable and inaccurate initial data is most promising.