Geometric Method of Reconstructing Evolution Equations from Experimental Data (pp.373-400)
Authors: (E. Nikulchev, Moscow State University of Printing Arts, Moscow, Russia)
Abstract: One of the most important tasks in solving problems of control technical systems is the
evolutionary behavior formalization of the object. These models are required for the
formation control algorithms. The engineers operate only with the basic linear models,
leaving the solution of problems of regulation of nonlinear phenomena in industrial
controllers. As a result, there are actually functioning control systems that work not only
optimal, but, even in manual mode set up control processes.
We can distinguish three classical approaches to modeling of nonlinear phenomena in
technical systems. The first is to use stochastic models; it is assumed that the nonlinear
oscillations are realization of a random process whose characteristics are to be found.
Obviously, in this case can not be rigorously justified assumptions, and assumptions that are
not explicitly probabilistic nature, and the observed behavior is possible for a whole class of
nonlinear objects. Assessing the adequacy of the construction of stochastic models is to test
the hypotheses match the original assumptions. The second approach - the construction of
models based on the physical properties of all occurring phenomena in technical systems and
the withdrawal of evolution equations. Despite the evidence of such a decision, it is almost
impossible for technical systems, since closed and structural complexity of structures defines
a knowledge-intensive and cumbersome nature of the physical simulation. Obtaining
necessary to control the evolution equations of the models is a separate complex challenge.
The third - for the use of an approach to modeling is to choose the kind of mathematical
model as an evolution equation and the subsequent identification of parameters or
nonparametric model identification. Model is considered adequate if the evaluation of the
adequacy of a given criterion, calculated as the residual model dependence of the
experimental data is within acceptable limit.