Estimation of parameters of linear fractional order ARX systems with noise in the input signal

The paper deals with the problem of parameter estimation of linear ARX systems with observation noise in the input signals, described by the equations m=\ z = b0 Д ^ -\ + Za0 A *i i, W = x + C m=\ where x. and w. are unobservable and observable input variables; z is the observable output variable; . is a noise; Г. is the noise observation of the input signal. We proposed the criterion for estimating the parameters of linear fractional order ARX systems, which is a generalization of the method of least squares: • Z (y -yT b) in Z^-jM-, ВеВ i=\ а| + a Mr a where Поступила в редакцию 14 февраля 2014 г. y. =f(yZ ) i(yww J) , y Tp 1 ^ \ z-1-\ 1 г [P\ 1 v У n(i) - Z (-\) 1 =0 z -1 -\, ..., Z (-\) 1=0 Tp 1 1 f (4 V ) Bo = (bf j af f, bo 4(U b0 f, ao = (, ..., ( J yw = Z (-\) 1=o w '-1 '-1 1=0 J J у 1 N а = lim - Z^ 2, ^ N N = ' hp r Z y£° ) Mr = lim - E 'r N N hf^ ... hp J · tp 11 1 к n V J Уу Tp 1 h( = lim - E Z Z (-\) 1=0 C-1 -Z (-\) N ^да N i=0 1=0 N-\ N-\ С R Л fa Л 1 NN-\1 R = lim - ZZ P a?(i -1), m = r , n = r J It is proved that under non-restrictive conditions on the signal and noise, the proposed algorithm provides strongly consistent estimates. Note that there does not require knowledge of the laws of distribution of noise to obtain the strongly consistent estimates. The results can be used to create new highly automated systems of control by technological processes.

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