rpem

rpem — RPEM estimation

Calling sequence

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

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:

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)

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)].

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).

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.