Evaluating the order of connectivity and the degree of polynomial phenomenological equations from experimental data

Abstract

The authors of [1] proposed a method for constructing a phenomenological equation for a dynamic system that is one of the constructive forms of analytic mathematical descriptions of complex (multifactor) objects (systems) that are difficult to formalize. However, the constructed equation was validated under the assumption that the connectivity and the degree of the polynomial equation are known (found), and the problem is thus reduced to simply finding coefficients.Below we develop a solution to the problem for the case of a priori unknown connectivity and degree for the desired phenomenological equation.Statement of the Problem. We will study some generally nonlinear dynamic (time varying) system, assuming that measurements (observations) are made only of the output signal X k, which is recorded at discrete times t = tk:X k = X(tk), k = 1, 2 ..... N. We neglect the measurement error and external noise, assuming that tt.e measurements of k k are exact. Analysis of the stability of our algorithm in the presence of measurement error and external noise is an independent problem whose solution is beyond the scope of the present article.We assume that the operation of the system can be described by some phenomenological equation, generally stochastic, i.e., )-.=f(~-~_,,L._ 2, ,~-k_.)+{.;fE F ,