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// OE Model parameter estimation
///////////////////////////////////////
/////////// ARX Model /////////////////
function [thetaN_arx,covt_arx,nvar,res] = arxc(data,na,nb,nk)
az = max(na,nb+nk-1);
zer = zeros(az,1);
zd = data;
// Zeros appended
zd1(:,1) = [zer; zd(:,1)];
zd1(:,2) = [zer; zd(:,2)];
[r,c] = size(zd1);
t = az+1:r;
yt = zd1(:,1); ut = zd1(:,2);
yt1 = yt'; ut1 = ut'; // row vector
len1 = length(yt1);
yt2 = zeros(1,len1-az); ut2 = zeros(1,len1-az);
// arx(Data,[na nb nk])
for i=1:na
yt2 = [yt2; -yt1(t-i)];
end;
for i=nk:nb+nk-1
ut2 = [ut2; ut1(t-i)];
end;
[r1,c1] = size(yt2); [r2,c2] = size(ut2);
phit = [yt2(2:r1,:); ut2(2:r2,:)];
m1 = phit*phit';
[qm,rm] = qr(m1);
m2 = phit*zd(:,1);
thetaN_arx = inv(rm)*qm'*m2;
// thetaN_arx = inv(m1)*m2;
// thetaN_arx = m1\m2;
[r11,c11] = size(thetaN_arx);
a = thetaN_arx(1:na); b = thetaN_arx(na+1:r11);
// Sum of squared residuals
yhat = phit'*thetaN_arx;
res = zd(:,1) - yhat;
N = length(res);
q = rank(phit);
ssr = res'*res;
sig2 = ssr/(N-q);
nvar = sqrt(sig2);
cov_arx = inv(m1);
covt_arx = diag(cov_arx);
endfunction;
///////////////////////////////
//////////////////////////////
///////////////////////////////////////
////////// Model Display /////////////
function disp_mod(N1,covN1)
len = length(covN1);
B1 = pol2str(N1);
ind = strindex(B1,['+','-']);
ind = ind - 1;
B2 = strsplit(B1,ind);
covB = string(covN1);
if ascii(B2(1)) == 32
B2 = B2(2:len+1);
end;
B3(1) = ' ';
for i=1:len
B3(i) = strsubst(B2(i),'*x','(+-' + covB(i) + ')*x');
end;
B4 = B3(1);
for i=2:len
B4 = B4 + ' ' + B3(i);
end;
disp(B4);
endfunction;
///////////////////////////////////////
///////////////////////////////////////
function [thetaN_oe,covN_oe,nvar,resid] = oe(zd,nb,nf,nk)
[thetaN,covfN,nvar,res] = arxc(zd,nf,nb,nk);
[r1,c1] = size(thetaN);
yt = zd(:,1);
m=50;
if nf==0
thetaN_oe = thetaN;
covN_oe = covfN;
else
for k=1:m
a = thetaN(1:nf);
b = thetaN(nf+1:r1);
A = [1 a']; // Filter
y = yt(1:length(u))';
yf = deconvol(y,A);
uf = deconvol(u,A);
zf = [yf(1:length(uf))' uf'];
zdf = detrend(zf,'constant');
[thetaNf,covf_a,nvar,resid] = arxc(zdf,nf,nb,nk)
thetaN = thetaNf;
a1 = thetaN(1:nf);
b = (norm(a-a1))/norm(a1);
if b<0.005
break;
end;
end;
thetaN_oe = thetaN;
covN_oe = covf_a;
end;
[rt,ct] = size(thetaN_oe);
f_oe = [1 thetaN_oe(1:nf)']; b1 = zeros(1,nk);
b_oe = [b1 thetaN_oe(nf+1:rt)'];
cov_f = covN_oe(1:nf); cov_b = covN_oe(nf+1:rt);
x = poly(0,'x');
if nf ==0
disp('Discrete time model: y(t) = B(x)u(t) + e(t)');
else
disp('Discrete time model: y(t) = [B(x)/F(x)]u(t) + e(t)');
end;
F = poly(f_oe,'x','coeff');
cov_f1 = [0 cov_f'];
B = poly( b_oe,'x','coeff');
cov_b1 = cov_b';
if nb==0
error('All B parameters are zero');
else
disp('B(x) = ');
disp_mod(B,cov_b);
end;
if nf~=0
disp('F(x) = ');
disp_mod(F,cov_f1);
end;
endfunction;
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