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function sys = arx(varargin)
// Parameters Estimation of ARX model using Input Output time-domain data
//
// Calling Sequence
// sys = arx(ioData,[na nb nk])
//
// Parameters
// ioData : iddata or [outputData inputData] ,matrix of nx2 dimensions, type plant data
// na : non-negative integer number specified as order of the polynomial A(z^-1)
// nb : non-negative integer number specified as order of the polynomial B(z^-1)+1
// nk : non-negative integer number specified as input output delay, Default value is 1
// sys : idpoly type polynomial have estimated coefficients of A(z^-1) and B(z^-1) polynomials
//
// Description
// Fit ARX model on given input output data
// The mathematical equation of the ARX model
// <latex>
// begin{eqnarray}
// A(q)y(n) = B(q)u(n) + e(t)
// end{eqnarray}
// </latex>
// It is SISO type model. It minimizes the sum of the squares of nonlinear functions using Levenberg-Marquardt algorithm.
//
// sys ,idpoly type, have different fields that contains estimated coefficients, sampling time, time unit and other estimated data in Report object.
//
// Examples
// u = idinput(1024,'PRBS',[0 1/20],[-1 1])
// a = [1 0.5];b = [0 2 3];
// model = idpoly(a,b,'Ts',0.1)
// y = sim(u,model) + rand(length(u),1)
// plantData = iddata(y,u,0.1)
// sys = arx(plantData,[2,2,1])
//
// Examples
// u = idinput(1024,'PRBS',[0 1/20],[-1 1])
// a = [1 0.5];b = [0 2 3];
// model = idpoly(a,b,'Ts',0.1)
// y = sim(u,model) + rand(length(u),1)
// plantData = [y,u]
// sys = arx(plantData,[2,2,1])
//
// Authors
// Ashutosh Kumar Bhargava, Harpreet, Inderpreet
[lhs , rhs] = argn();
if ( rhs < 2 ) then
errmsg = msprintf(gettext("%s: Unexpected number of input arguments : %d provided while should be 2"), "arx", rhs);
error(errmsg)
end
z = varargin(1)
if typeof(z) == 'iddata' then
Ts = z.Ts;unit = z.TimeUnit
z = [z.OutputData z.InputData]
elseif typeof(z) == 'constant' then
Ts = 1;unit = 'seconds'
end
if ((~size(z,2)==2) & (~size(z,1)==2)) then
errmsg = msprintf(gettext("%s: input and output data matrix should be of size (number of data)*2"), "arx");
error(errmsg);
end
if (~isreal(z)) then
errmsg = msprintf(gettext("%s: input and output data matrix should be a real matrix"), "arx");
error(errmsg);
end
n = varargin(2)
if (size(n,"*")<2| size(n,"*")>3) then
errmsg = msprintf(gettext("%s: The order and delay matrix [na nb nk] should be of size [2 3]"), "arx");
error(errmsg);
end
if (size(find(n<0),"*") | size(find(((n-floor(n))<%eps)== %f))) then
errmsg = msprintf(gettext("%s: values of order and delay matrix [na nb nk] should be nonnegative integer number "), "arx");
error(errmsg);
end
na = n(1); nb = n(2); // nk = n(3); // nf = n(4);
//
if (size(n,"*") == 2) then
nk = 1
else
nk = n(3);
end
// storing U(k) , y(k) and n data in UDATA,YDATA and NDATA respectively
YDATA = z(:,1);
UDATA = z(:,2);
NDATA = size(UDATA,"*");
function e = G(p,m)
e = YDATA - _objfunarx(UDATA,YDATA,p,na,nb,nk);
endfunction
tempSum = na+nb
p0 = linspace(0.1,0.9,tempSum)';
[var,errl] = lsqrsolve(p0,G,size(UDATA,"*"));
err = (norm(errl)^2);
opt_err = err;
resid = G(var,[]);
a = 1-poly([var(nb+1:nb+na)]',"q","coeff");
b = poly([repmat(0,nk,1);var(1:nb)]',"q","coeff");
a = (poly([1,-coeff(a)],'q','coeff'))
t = idpoly(coeff(a),coeff(b),1,1,1,Ts)
// estimating the other parameters
[temp1,temp2,temp3] = predict(z,t)
[temp11,temp22,temp33] = pe(z,t)
estData = calModelPara(temp1,temp1,n(1)+n(2))
// pause
t.Report.Fit.MSE = estData.MSE
t.Report.Fit.FPE = estData.FPE
t.Report.Fit.FitPer = estData.FitPer
t.Report.Fit.AIC = estData.AIC
t.Report.Fit.AICc = estData.AICc
t.Report.Fit.nAIC = estData.nAIC
t.Report.Fit.BIC = estData.BIC
t.TimeUnit = unit
sys = t
endfunction
function yhat = _objfunarx(UDATA,YDATA,x,na,nb,nk)
x=x(:)
q = poly(0,'q')
tempSum = nb+na
// making polynomials
b = poly([repmat(0,nk,1);x(1:nb)]',"q","coeff");
a = 1 - poly([x(nb+1:nb+na)]',"q","coeff")
aSize = coeff(a);bSize = coeff(b)
maxDelay = max([length(aSize) length(bSize)])
yhat = [YDATA(1:maxDelay)]
for k=maxDelay+1:size(UDATA,"*")
tempB = 0
for ii = 1:size(bSize,'*')
tempB = tempB + bSize(ii)*UDATA(k-ii+1)
end
tempA = 0
for ii = 1:size(aSize,"*")
tempA = tempA + aSize(ii)*YDATA(k-ii)
end
yhat = [yhat; [ tempA+tempB ]];
end
endfunction
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