rpem

Recursive Prediction-Error Minimization estimation

Calling Sequence

[w1,[v]]=rpem(w0,u0,y0,[lambda,[k,[c]]])

Arguments

w0 list(theta,p,l,phi,psi) where:  

:theta [a,b,c] is a real vector of order 3*n

:a,b,c a=[a(1),...,a(n)], b=[b(1),...,b(n)], c=[c(1),...,c(n)] :

: :p (3*n x 3*n) real matrix. : :phi,psi,l real vector of dimension 3*n :

Applicable values for the first call:

theta=phi=psi=l=0*`ones`_(1,3*n). p=`eye`_(3*n,3*n)

: :u0 real vector of inputs (arbitrary size). ( u0($) is not used by

rpem)

: :y0 vector of outputs (same dimension as u0). ( y0(1) is not

used by rpem). If the time domain is (t0,t0+k-1) the u0 vector contains the inputs u(t0),u(t0+1),..,u(t0+k-1) and y0 the outputs y(t0),y(t0+1),..,y(t0+k-1)

:

Optional arguments

:lambda optional argument (forgetting constant) choosed close to 1 as convergence occur: lambda=[lambda0,alfa,beta] evolves according to :

lambda(t)=alfa*lambda(t-1)+`beta`_

with lambda(0)=lambda0 : :k contraction factor to be chosen close to 1 as convergence occurs. k=[k0,mu,nu] evolves according to:

k(t)=mu*k(t-1)+nu

with k(0)=k0. : :c Large argument.( c=1000 is the default value). :

Outputs:

:w1 Update for w0. : :v sum of squared prediction errors on u0, y0.(optional). In

particular w1(1) is the new estimate of theta. If a new sample u1, y1 is available the update is obtained by: [w2,[v]]=rpem(w1,u1,y1,[lambda,[k,[c]]]). Arbitrary large series can thus be treated.

:

Description

Recursive estimation of arguments in an ARMAX model. Uses Ljung- Soderstrom recursive prediction error method. Model considered is the following:

y(t)+a(1)*y(t-1)+...+a(n)*y(t-n)=
b(1)*u(t-1)+...+b(n)*u(t-n)+e(t)+c(1)*e(t-1)+...+c(n)*e(t-n)

The effect of this command is to update the estimation of unknown argument theta=[a,b,c] with

a=[a(1),...,a(n)], b=[b(1),...,b(n)], c=[c(1),...,c(n)].