Scilab Function

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.