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.TH armax 3 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
armax - armax identification
.SH CALLING SEQUENCE
.nf
[arc,la,lb,sig,resid]=armax(r,s,y,u,[b0f,prf])
.fi
.SH PARAMETERS
.TP 15
y
: output process y(ny,n); ( ny: dimension of y , n : sample size)
.TP
u
: input process u(nu,n); ( nu: dimension of u , n : sample size)
.TP
r and s
: auto-regression orders r >=0 et s >=-1
.TP
b0f
: optional parameter. Its default value is 0 and it means that the coefficient b0 must be identified. if bof=1 the b0 is supposed to be zero and is not identified
.TP
prf
: optional parameter for display control. If prf =1, the default value,
a display of the identified Arma is given.
.TP
arc
: a Scilab arma object (see armac)
.TP
la
: is the list(a,a+eta,a-eta) ( la = a in dimension 1)
; where eta is the estimated standard deviation.
, a=[Id,a1,a2,...,ar] where each ai is a matrix of size (ny,ny)
.TP
lb
: is the list(b,b+etb,b-etb) (lb =b in dimension 1)
; where etb is the estimated standard deviation.
b=[b0,.....,b_s] where each bi is a matrix of size (nu,nu)
.TP
sig
: is the estimated standard deviation of the noise and resid=[ sig*e(t0),....] (
.SH DESCRIPTION
armax is used to identify the coefficients of a n-dimensional
ARX process
.nf
A(z^-1)y= B(z^-1)u + sig*e(t)
.fi
where e(t) is a n-dimensional white noise with variance I.
sig an nxn matrix and A(z) and B(z):
.nf
A(z) = 1+a1*z+...+a_r*z^r; ( r=0 => A(z)=1)
B(z) = b0+b1*z+...+b_s z^s ( s=-1 => B(z)=0)
.fi
for the method see Eykhoff in trends and progress in system identification, page 96.
with
z(t)=[y(t-1),..,y(t-r),u(t),...,u(t-s)]
and
coef= [-a1,..,-ar,b0,...,b_s]
we can write
y(t)= coef* z(t) + sig*e(t) and the algorithm minimises
sum_{t=1}^N ( [y(t)- coef'z(t)]^2)
where t0=maxi(maxi(r,s)+1,1))).
.SH EXAMPLE
.nf
//-Ex1- Arma model : y(t) = 0.2*u(t-1)+0.01*e(t-1)
ny=1,nu=1,sig=0.01;
Arma=armac(1,[0,0.2],[0,1],ny,nu,sig) //defining the above arma model
u=rand(1,1000,'normal'); //a random input sequence u
y=arsimul(Arma,u); //simulation of a y output sequence associated with u.
Armaest=armax(0,1,y,u); //Identified model given u and y.
Acoeff=Armaest('a'); //Coefficients of the polynomial A(x)
Bcoeff=Armaest('b') //Coefficients of the polynomial B(x)
Dcoeff=Armaest('d'); //Coefficients of the polynomial D(x)
[Ax,Bx,Dx]=arma2p(Armaest) //Results in polynomial form.
//-Ex2- Arma1: y_t -0.8*y_{t-1} + 0.2*y_{t-2} = sig*e(t)
ny=1,nu=1;sig=0.001;
// First step: simulation the Arma1 model, for that we define
// Arma2: y_t -0.8*y_{t-1} + 0.2*y_{t-2} = sig*u(t)
// with normal deviates for u(t).
Arma2=armac([1,-0.8,0.2],sig,0,ny,nu,0);
//Definition of the Arma2 arma model (a model with B=sig and without noise!)
u=rand(1,10000,'normal'); // An input sequence for Arma2
y=arsimul(Arma2,u); // y = output of Arma2 with input u
// can be seen as output of Arma1.
// Second step: identification. We look for an Arma model
// y(t) + a1*y(t-1) + a2 *y(t-2) = sig*e(t)
Arma1est=armax(2,-1,y,[]);
[A,B,D]=arma2p(Arma1est)
.fi
.SH AUTHOR
J-Ph. Chancelier.
.SH SEE ALSO
imrep2ss, time_id, arl2, armax, frep2tf
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