transfer function realization from frequency response
[h [,err]]=frep2tf(frq,repf,dg [,dom,tols,weight])
:frq vector of frequencies in Hz. : :repf vector of frequency response : :dg degree of linear system : :dom time domain ( ‘c’ or ‘d’ or dt) : :tols a vector of size 3 giving the relative and absolute tolerance
and the maximum number of iterations (default values are rtol=1.e-2; atol=1.e-4, N=10).
: :weight vector of weights on frequencies : :h SISO transfer function : :err error (for example if dom=’c’ `sum(abs(h(2i*pi*frq) -
rep)^2)/size(frq,*)`)
:
Frequency response to transfer function conversion. The order of h is a priori given in dg which must be provided. The following linear system is solved in the least square sense.
weight(k)*(n( phi_k) - d(phi_k)*rep_k)=0, k=1,..,n
where phi_k= 2*%i*%pi*frq when dom=’c’ and phi_k=exp(2*%i*%pi*dom*frq if not. If the weight vector is not given a default penalization is used (when dom=’c’).
A stable and minimum phase system can be obtained by using function factors.
s=`poly`_(0,'s');
h=`syslin`_('c',(s-1)/(s^3+5*s+20))
frq=0:0.05:3;repf=`repfreq`_(h,frq);
`clean`_(frep2tf(frq,repf,3))
Sys=`ssrand`_(1,1,10);
frq=`logspace`_(-3,2,200);
[frq,rep]=`repfreq`_(Sys,frq); //Frequency response of Sys
[Sys2,err]=frep2tf(frq,rep,10);Sys2=`clean`_(Sys2)//Sys2 obtained from freq. resp of Sys
[frq,rep2]=`repfreq`_(Sys2,frq); //Frequency response of Sys2
`clf`_();`bode`_(frq,[rep;rep2]) //Responses of Sys and Sys2
[`gsort`_(`spec`_(Sys('A'))),`gsort`_(`roots`_(Sys2('den')))] //poles
dom=1/1000; // Sampling time
z=`poly`_(0,'z');
h=`syslin`_(dom,(z^2+0.5)/(z^3+0.1*z^2-0.5*z+0.08))
frq=(0:0.01:0.5)/dom;repf=`repfreq`_(h,frq);
[Sys2,err]=frep2tf(frq,repf,3,dom);
[frq,rep2]=`repfreq`_(Sys2,frq); //Frequency response of Sys2
`clf`_();`plot2d1`_("onn",frq',`abs`_([repf;rep2])');