Identification of a Nonlinear System with the Modified GMDH

Abstract

In this paper, I propose the modified GMDH algorithm. The GMDH was developed in USSR, and recently regarded as a very useful method. The reason is that it is very easy to use and applicable to model complex systems and with insufficient data. On the other hand, the obtained model with GMDH has usually too high a degree of nonlinearity, and the termination of modeling is vague. So the result is usually unstable. For example, when the input signal is varied in a wide frequency band or made to have a large amplitude, the simulation results become unreliable.
The modified GMDH preserves the desirable features of GMDH, and institutes the termination criterion in the modeling. The main concepts of this algorithm are the structural stability and the parametric stability which are defined as follows.
Structural stability: At first, we divide the data to two sets. In one data set, we make the regression analysis. With the obtained model, we make the prediction for the other data set, and get the mean square of the residual. Then we exchange the roles of the data sets. We get the mean square of the residual in the same manner. We consider the obtained model as structurally stable when the sum of the mean square of the residual is small.
Parametric stability: When the regression coefficients of some term are similar for each data set, the term is considered as parametrically stable.
The hypothesis which is used in this paper is that the essence of the system is brought by structural stable polynomials which consist of parametrically stable terms. The results of the modified GMDH are presented in computer simulations and in the convergency of identification. I am sure that this method is very practical in analyzing various systems.