Distributed Parameter Model Oriented Identification

Abstract

By means of mathematically defined digital filters, the disturbance affecting the system response may be eliminated and the real response recovered. Thereby a multivariable exponential test signal is assumed, but any other type of input is also pertinent. This achievement involves both a functional and a stochastic component. The former consists in the proved property that a multivariable function given over a finite interval may be approximately expressed by a product of functions of only one variable. Further, each such factor is expandable in a finite sum of exponential terms. The latter component consists in the low probability of coincidence regarding the test signal exponent and any exponent of the disturbance (approximate) spectrum. The developed procedure enables to estimate the constant coefficients included in a distributed parameter model of the process. Short observation time, test signal / noise small ration, the good accuracy of the model estimation and simple identification algorithm will be achieved. These features are due to the method peculiarity, to operate with any incipient segment of the response, and to the simple structure and good selectivity of the proposed digital filters. One may give up the test signal use, by having recoursed to exponential decomposition of the actual system input. INTRODUCTION Let us consider the dynamic process of a distributed parameter system, a process expressed by the partial differential eq. ) , , ( ) ( ) ( ) ( 0 0 0 0 y x t b t,x,y a ijk N k k i,j ijk ijk N k k i,j ijk D D (1) Above ijk D denotes the partial derivate operator j i k j i k ijk y x t D (2) and ijk a , ijk b are some constant coefficients. This eq. relates the system response ) , , ( y x t to the external action ) , , ( y x t . The following considerations may be easily extended to the functions of arguments z y x t , , , . We aim at producing an alternative to stochastic methods (Unbehauen 1990), (Chen and Wahlberg, Eds. 1997) in order to achieve a shorter observation time of the process. The developed procedure is based on the exponential decomposition of a continuous function, providing an individual exponential spectrum (Cehan-Racovita 1999, 2002). The resulting sum is a generalization of the Fourier’s finite sum. In contrast to current methods, a higher efficiency of disturbance filtering is performed. EXPONENTIAL DECOMPOSITION The Approximate Let ) (t f be a process and ) (t fn an approximate of ) (t f , over a finite interval [0,T], with the structure t A t f i m i i m exp ) ( 1 , const Ai (3) where i A and i are complex constants; exp(.) means (.) e . This approximate has to fulfill the conditions ) ( ) ( 0 0 kt f kt fm , const t0 (4) for M k k , 0 , T t kM 0 . The approximate (3), (4) may be considered as an extension of a Fourier’s finite sum. The frequency spectrum of the last sum does not characterize ) (t f . On the contrary, the complex spectrum m i i 1 } { expresses the individual character of ) (t f over the given interval. Consequently this spectrum may separate components from a combined signal. Spectrum Determination We define the shifting operator q by its effect on any function ) (t g , i.e. ) ( ) ( 0 t t g t g q , (5) 0 0 t . One may prove that polynomials of q may be handled like algebraic polynomials. Note that ) exp( ) ( ) exp( ) ( 0 t q q t q i iq q , (6) where ) exp( 0 t q i i . (7) Further observe that (6) vanishes if i q q . Consequently, denoting