Bilinear system identification by Volterra kernels estimation

Abstract

N1)-lV, respectively, and using &,V = AV, cl V = 0, we get AV*(N;'-Nl)-lV+XV*(N~l-Nl)-lV =V*(N,-'-Nl)-lNlc~lNl(N;'-Nl)-lV -V * ( N ; ' -N l ) -l N ; ' c~l N ; ' ( N ; ' -N l ) -l V . (12) Since Nl is diagonal, we can write Nl(N;'-Nl)-lV=(N;'-N,)-lNIV=[N;'(I-N:)]-lNIV = ( Z -N ; ) -l N ; V . Thus, c1Nl(N;' -Nl)-'V=el(I-N;)-'N;V= HV, say. Then the first term on the right-hand side of (12) becomes ( H V ) * ( H V ) . Now consider clNC1(N<' -N1)-'V. We can write c l N~l ( N ; l -N l ) -l V =~l ( z -N f ) -l V . (13) But clV=O; hence c l ( 1 -N f ) ( I -N ; ) -' V = O , and further, cl(Z-N~)~l V =~l N~( Z -N~)~l V =~l ( Z -N~)~l N~V . Therefore, (13) implies that c1Nr1(N;' -Nl)-'V= cl(Z-N:)-'N;V= HV. The secondterm on the right-hand side of (12) is thus ( H V ) * ( H V ) , and hence the right-hand side of (12) is 0. But the left-hand side of (12) isSince the right-hand side of (12) is 0, we get 2Re(A)V*(N;'-N1)-'V = 0, which since (N;'-N1)-' > 0 implies Re@)= 0. This is a contradiction, since (&, GI, cl) is asymptotically stable. Thus, there is no nonzero V such that &V= AV and cl V = 0, and hence (&, el) is observable.That (&,cl1) is controllable can be shown in a similar manner by considering (8) and (9). This proves the minimality of (Al, cl, el). Corulhry 1: Under the conditions of Theorem 2, (&, Gl, el, Z/2) is positive real, and (&l,E+l,cl) and (&l,cl,E-l) are minimal. Proof: Defiie L. Pemebo and L. M. Silverman, "Model reduction ria b a l a n d state space representation," IEEE Tram Automat. Contr., voL AC-27, pp. 382-381.1982. P. Faurre, "Stochastic realization algorithms" in Sysrem Idoltifiicotion: Adfiances and A. Lmdquist and G. P i a "On the stochastic realization problem," S I A M J . Conzr.Abstract -"his note considers the identification of bilinear discretetime dynamic systems from sequences of input and noise cormpted output measurements. In contrast to other approaches, the proposed algorithm is simple and does not require knowledge of the noise statistics. It is also shown that the obtained estimates are unbiased and consistent, which is not shorn in the previous papers. I.A S. Goldberger, Econometric Theory. New Yo&: Wiley, 1964. M. I~g a k i and R. Kamiya, ''Bilinear system identification by estimated Volterra kernels," Elect. Eng. Japan, Abstract --This paper presents a method of reducing a two-dimensional (2D) rational function to an irreducible one. It is achieved by searching the first linearly dependent row, in order from top to bottom, of a

A computational intelligence technique for the identification of non-linear non-stationary systems

“…. , y (N ) of y, which can be put in matrix form as X = [x (1) x (2) · · · x (N ) ] and Y = [y (1) y (2) · · · y (N ) ]. A currently used estimation of the autocorrelation matrix is given by…”

“…The system to be identified is governed by a nonstationary version of Narendra's difference equation: (33) where t = n Δt with Δt = 0.05, [·] denotes the real part, and α is an r.v. uniformly distributed in the interval [1,50]. Figure 2 shows the identification capabilities of the proposed approach for this system, with N = 50, L = 240, and polynomial order p = 5.…”

“…uniformly distributed in the interval [1,50]. Figure 3 shows the identification results with N = 50, L = 240, and a polynomial order p = 5.…”

“…A large class of problems in automatic control [1], communications [2], [3], but also in image processing [4] and signal processing [5] is described by nonlinear nonstationary models. System Identification (SI) is the discipline interested in building mathematical models of physical systems starting from experimental data, measurements, or observations.…”

Modelagem Estocástica do Algoritmo LMS para Formas Bilineares

“…Tais representações buscam caracterizar o comportamento desses sistemas através de sistemas lineares, fornecendo assim um adequado compromisso entre a precisão da representação obtida e a simplicidade do tratamento matemático. Para tal, diferentes algoritmos adaptativos envolvendo formas bilineares vêm sendo desenvolvidos visando aplicações de identificação de sistema [2]- [4], equalização de canal [5], cancelamento de eco [6], controle ativo de ruído [7], [8], redes neurais [9], dentre outras. Contudo, os algoritmos discutidos até então têm considerado que o termo bilinear seja definido com respeito aos dados, isto é, com vistas à relação de entradas e saídas do sistema.…”