1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
|
.TH rpem 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
rpem - RPEM estimation
.SH CALLING SEQUENCE
.nf
[w1,[v]]=rpem(w0,u0,y0,[lambda,[k,[c]]])
.fi
.SH PARAMETERS
.TP 15
a,b,c
:
\fVa=[a(1),...,a(n)], b=[b(1),...,b(n)], c=[c(1),...,c(n)]\fR
.TP
w0
: \fVlist(theta,p,phi,psi,l)\fR where:
.RS
.TP
theta
: [a,b,c] is a real vector of order \fV3*n\fR
.TP
p
: (3*n x 3*n) real matrix.
.TP
phi,psi,l
: real vector of dimension \fV3*n\fR
.RE
During the first call on can take:
.nf
theta=phi=psi=l=0*ones(1,3*n). p=eye(3*n,3*n)
.fi
.TP
u0
: real vector of inputs (arbitrary size)
(if no input take \fVu0=[ ]\fR).
.TP
y0
: vector of outputs (same dimension as \fVu0\fR if \fVu0\fR is not empty).
(\fVy0(1)\fR is not used by rpem).
.LP
If the time domain is \fV(t0,t0+k-1)\fR the \fVu0\fR vector contains
the inputs
.LP
\fVu(t0),u(t0+1),..,u(t0+k-1)\fR and \fVy0\fR the outputs
.LP
\fVy(t0),y(t0+1),..,y(t0+k-1)\fR
.SH DESCRIPTION
Recursive estimation of parameters in an ARMAX model.
Uses Ljung-Soderstrom recursive prediction error method.
Model considered is the following:
.nf
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)
.fi
The effect of this command is to update the estimation of
unknown parameter \fVtheta=[a,b,c]\fR with
.LP
\fVa=[a(1),...,a(n)], b=[b(1),...,b(n)], c=[c(1),...,c(n)]\fR.
.SH Optional parameters
.TP 10
lambda
: optional parameter (forgetting constant) choosed close to 1 as
convergence occur:
.LP
\fVlambda=[lambda0,alfa,beta]\fR evolves according to :
.nf
lambda(t)=alfa*lambda(t-1)+beta
.fi
with \fVlambda(0)=lambda0\fR
.LP
\fVk\fR : contraction factor to be chosen close to 1 as convergence occurs.
.LP
\fVk=[k0,mu,nu]\fR evolves according to:
.nf
k(t)=mu*k(t-1)+nu
.fi
with \fVk(0)=k0\fR.
.LP
\fVc\fR : large parameter.(\fVc=1000\fR is the default value).
.SH Output parameters:
.LP
\fVw1\fR: update for \fVw0\fR.
.LP
\fVv\fR: sum of squared prediction errors on \fVu0, y0\fR.(optional).
.LP
In particular \fVw1(1)\fR is the new estimate of \fVtheta\fR.
If a new sample \fVu1, y1\fR is available the update is obtained by:
.LP
\fV[w2,[v]]=rpem(w1,u1,y1,[lambda,[k,[c]]])\fR.
Arbitrary large series can thus be treated.
.LP
|
