Structural Identification of Static Systems with Distributed Lags

The approach to structural identification of static systems with the distributed lags is offered. The criterion of an estimation of linearity of system in parametrical space is introduced. The criterion is based on construction of set of secants for system. The special space for an estimation of structural parameters of system is selected. The concept of level of nonlinearity of system is introduced and the method of h is estimation is reduced. The analogue of criterion of Darbin-Watson is reduced. Criteria o f decision-making in the presence of a lag as in an output variable, and input variables are offered. It is shown that as magnitude of a lag performance on an output variable one can to use an estimat ion of parameters of coefficient of structural properties on the specified variable. Estimations of parameters of coefficient of structural properties are based on an evaluation of Lyapunov characteristic indicators.


Introduction
With the d istributed delays the set of publicat ions is devoted problems of identification of static plants. Such mo d e ls are w id ely ap p l ied in eco n o met r ics an d econo my [1][2][3][4], to th e techn ician [5][6][7], med icin e [8][9][10]. Delay can have both independent, and dependent variables. The account of the distributed lags leads to autocorrelation b et ween v ariab les [1,2,4]. A u t o co rrelat io n h amp ers identificat ion of parameters of p lant. Various models of approximation o f parameters are applied to a abatement of influence of autocorrelation on identification process at the distributed lags. It allo ws to reduce nu mber of estimated parameters of plant. The Koyck scheme [2,12], based on a change of factors of model on a decreasing geo met rical progression is most widely applied. The model I. Fisher [1,11] is based on a change of facto rs of model on the set d ecreas ing arith met ical p rog ress io n. A rith met ical and geo met rical models are app lied in that case when p lant parameters decrease from the first members of a progression. Considering it, S. Almon [13] has added more flexib ilit ies of model of I. Fisher, having applied polyno mial the law of a modification of factors. In [11] the law o f a change o f factors is offered at the distributed lags according to logarith mic normal distribution. The same idea based on application the exponential law of distribution, is considered in [14]. In [15] Pascal distribution which leads to the simp le form for factors is applied.
The a priori representation of correlation between the distributed lags in the form of some rat ional polyno mial is considered in [16]. Other approaches to description of factors at the distributed lags are studied in [4].
The considered models of factors min imise number of unknown parameters. To an estimat ion of parameters apply a method least-squares method or its modifications [1][2][3][4]. In these works the model structure is postulated a priori and the problem of paramet rical identification is considered. In [17] the interactive algorith m of an estimation of parameters of static plant with the distributed lags is offered. The length of a lag is set and does not become any suppositions about correlation between plant parameters. The case of a piecewise monotonic change of parameters of plant is considered. In a number of works methods of a choice of maximu m length of a lag are offered. At the heart of applied approaches the statistics which are based on the analysis of residuals [1,15,18] lie. In [19,20] various methods of an estimation of parameters of model with the distributed lag in a case of a priori set structure of model are considered. In [21] influence of the a priori in formation received fro m the analysis of the empirical data, on a choice of structure of model with the distributed lag on an output variable is analyzed. In work [22] are described algorithm of an estimation of parameters of model with the distributed lag. On the basis of results of modelling the structure of model which exp lains a mis match of model applied now in the American interest rate on federal funds is selected. In [23] process of inflation with the help autoregressive models is researched. The choice of length of a lag is carried out on the basis of consecutive magnification of delays and an estimation of adequacy of the received model. Then the criterion of A kaike and Bayesian Information Criteria are applied and the solution is made on model structure. The case of a priori uncertainty concerning structure and plant parameters was not study.
In the given work the approach to structural identification of static plants in the conditions of uncertainty is offered. It is based on application of the static structures describing properties of plant [24] in special space. The criterion of linearity (autocorrelation) of variab les of plant is introduced. Algorith ms of decision-making on maximu m length of the distributed lag are offered. They do not demand an evaluation of the statistics.  Then the field of structures (4). The theorem 1 gives linearity sufficient conditions (nonlinearities, collinearity) systems (1) on the set field of structures S S . If the condition (4) is fulfilled, that field S S is full. Hence S S is a linear span of an exit of system (1).
Otherwise make a solution on presence of nonlinearity or collinearity (autocorrelation) in system (1). Let

Set for an Estimation of Structure of the Distributed Lag of Plant
Following [7,24], generate au xiliary set for an estimat ion of structure (1)  On N J will order , , S . Such approach well works at an estimat ion of structure of nonlinear static systems [24]. For systems with the d istributed lag he demands modificat ion. In work the mult iple -functional approach to structural identification is applied. Therefore known methods of a choice of length of the lag, based on statistical criteria (section 1 see), are inapplicab le.

Decision-Making on Length of a Lag in
Considering it, to an estimation of independence of elements of vector n X apply the theorem 1.

Conclusions
The is functional-mu ltiple approach to structural identification of static systems with the distributed lag is offered. So lution about structure of the distributed lag of system receive on the basis of the analysis of special static structures. The criterion of an estimation of degree of linearity of system in parametrical space is introduced. It is considered variants of identification of system in the availability of a lag on dependent and independent variables. The approach to an