rpem(1)

NAME
rpem - RPEM estimation

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

PARAMETERS

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

w0  : list(theta,p,phi,psi,l) where:

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

    p   : (3*n x 3*n) real matrix.

    phi,psi,l
        : real vector of dimension 3*n
During the first call on can take:
theta=phi=psi=l=0*ones(1,3*n). p=eye(3*n,3*n)

u0  : real vector of inputs (arbitrary size) (if no input take u0=[ ]).

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

DESCRIPTION
Recursive estimation of parameters 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 parameter theta=[a,b,c] with

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

Optional parameters

lambda
    : optional parameter (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 parameter.(c=1000 is the default value).

Output parameters:

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.