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[thm,yhat,p,phi] = rarx(z,nn,adm,adg,th0,p0,phi)
RARX Computes estimates recursively for an ARX model.
[THM,YHAT] = RARX(Z,NN,adm,adg)
Z: The output-input data z=[y u]
NN : NN=[na nb nk], The orders and delay of an ARX model (see also ARX)
adm: Adaptation mechanism. adg: Adaptation gain
adm='ff', adg=lam: Forgetting factor algorithm, with forg factor lam
adm='kf', adg=R1: The Kalman filter algorithm with R1 as covariance
matrix of the parameter changes per time step
adm='ng', adg=gam: A normalized gradient algorithm, with gain gam
adm='ug', adg=gam: An Unnormalized gradient algorithm with gain gam
THM: The resulting estimates. Row k contains the estimates "in alpha-
betic order" corresponding to data up to time k (row k in Z)
YHAT: The predicted values of the outputs. Row k corresponds to time k.
Initial value of parameters(TH0) and of "P-matrix" (P0) can be given by
[THM,YHAT,P] = RARX(Z,NN,adm,adg,TH0,P0)
Initial and last values of auxiliary data vector phi are
obtained by [THM,YHAT,P,phi]=RARX(Z,NN,adm,adg,TH0,P0,phi0).
See also RARMAX, ROE, RBJ, RPEM and RPLR.
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function [thm,yhat,p,phi] = rarx(z,nn,adm,adg,th0,p0,phi)
% L. Ljung 10-1-89
% Copyright (c) 1986-98 by The MathWorks, Inc.
% $Revision: 2.3 $ $Date: 1997/12/02 03:43:50 $
if nargin < 4
disp('Usage: MODEL_PARS = RARX(DATA,ORDERS,ADM,ADG)')
disp(' [MODEL_PARS,YHAT,COV,PHI] = RARX(DATA,ORDERS,ADM,ADG,TH0,COV0,PHI)')
disp(' ADM is one of ''ff'', ''kf'', ''ng'', ''ug''.')
return
end
adm=lower(adm(1:2));
if ~(adm=='ff'|adm=='kf'|adm=='ng'|adm=='ug')
error('The argument ADM should be one of ''ff'', ''kf'', ''ng'', or ''ug''.')
end
[nz,ns]=size(z);
if ns==1,if length(nn)~=1,
error('For a time series nn should be a scalar nn=na!')
end,end
if 2*ns-1~=length(nn)
error('Incorrect number of orders specified in nn. nn=[na nb nk]')
end
na=nn(1);if ns>1,nb=nn(2:ns);nk=nn(ns+1:2*ns-1);else nk=1;nb=0;end
nu=1;
if any(nk<1)
error('Sorry, this routine requires nk>0; Shift input sequence if necessary!')
end
d=na+sum(nb);
nbm=nb+nk-1;ncbm=na+cumsum([0 nbm]);
ii=[1:na+sum(nbm)];
i=[1:na];
for ku=1:ns-1,i=[i ncbm(ku)+nk(ku):ncbm(ku+1)];end
dm=na+sum(nbm);
if nargin<7, phi=zeros(dm,1);end
if nargin<6, p0=10000*eye(d);end
if nargin<5, th0=eps*ones(d,1);end
if isempty(phi),phi=zeros(dm,1);end
if isempty(p0),p0=10000*eye(d);end
if isempty(th0),th0=eps*ones(d,1);end
if length(th0)~=d, error('The length of th0 must equal the number of estimated parameters!'),end
[th0nr,th0nc]=size(th0);if th0nr<th0nc, th0=th0';end
p=p0;th=th0;
if adm(1)=='f', R1=zeros(d,d);lam=adg;end
if adm(1)=='k', [sR1,SR1]=size(adg);
if sR1~=d | SR1~=d
error(['The R1 matrix should be a square matrix with dimension ',...
'equal to number of parameters.'])
end
R1=adg;lam=1;
end
if adm(2)=='g', grad=1;else grad=0;end
for kcou=1:nz
yh=phi(i)'*th;
epsi=z(kcou,1)-yh;
if ~grad,K=p*phi(i)/(lam + phi(i)'*p*phi(i));
p=(p-K*phi(i)'*p)/lam+R1;
else K=adg*phi(i);end
if adm(1)=='n', K=K/(eps+phi(i)'*phi(i));end
th=th+K*epsi;
epsilon=z(kcou,1)-th'*phi(i);
phi(ii+1)=phi(ii);
if na>0,phi(1)=-z(kcou,1);end
if any(ncbm>0),phi(ncbm(1:ns-1)+1)=z(kcou,2:ns)';end
thm(kcou,:)=th';yhat(kcou)=yh;
end
yhat=yhat';