frqlead.m

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function [numopen, denopen, denclsd] = frqlead(num, den)
% Hadi Saadat, 1998

%discr=[
%'                                                                           '
%'  The function [numopen,denopen,denclsd]=frqlead(num, den) is used for the '
%'  frequency response design of a phase-lead compensator.  num & den are    '
%'  row vectors of polynomial coefficients of the uncompensated open-loop    '
%'  plant transfer function. Design is based on the phase margin criterion.  '
%'  The user is prompted to enter the desired Phase Margin & the controller  '
%'  DC gain Gc(0) = Kc*Z0/P0.  The program finds and displays a compensated  '
%'  gain crossover frequency range for an stable controller.  The user is    '
%'  then prompted to enter the gain crossover frequency in this range.  The  '
%'  controller transfer function and the frequency-domain specifications     '
%'  before and after compensation are found.  The function returns the open- '
%'  loop and closed-loop numerator and denominators of the compensated system'
%'  transfer function.                                                       '];
%disp(discr);

r=abs(roots(den));
i=find(r>0); rp=r(i);
rmx=max(rp); rmn=min(rp); wst=0.1*round(rmn); wf=20*round(rmx);;dw=wf/800;
w=wst:dw:wf;

clc
a0=input('Enter the compensator DC Gain -> ');
pm=input('Enter desired Phase Margin -> ');
fprintf('\n')

[mag, phase] = bode(num, den, w);
phase=180/pi*unwrap(phase*pi/180);

if phase(1) > (-180+pm)
     i = find(phase < (-180 + pm));
else
     i = find(phase > (-180 + pm));
end
if length(i)==0
     disp('Phase does not cross (-180 + P.M.). ')
return, else
     i2=i(1); i1 = i2 -1;
     if i1==0 wpm=w(i2); else
     wa = w(i1); wb = w(i2); p1 = phase(i1); p2 = phase(i2);
     wpm = wa + (-180+pm - p1)/(p2-p1)*(wb-wa);
     end
end

i=find(mag<1/a0);
if length(i)==0
     disp('Gain does not cross 1/a0. ')
return, else
     i2=i(1); i1=i2 -1;
     if i1==0 wga=w(i2); else
     wa = w(i1); wb = w(i2); m1 = mag(i1); m2 = mag(i2);
     wga =wa+(1/a0-m1)/(m2-m1)*(wb-wa);
     end
end

w1=max(wga, wpm);

[M,ph]=bode(num, den, w1);  % Returns the mag. and phase of G(w)H(w1)
thta=-180 + pm - ph;
thtar=thta*pi/180;
[a1,b1]=frcntlr(num,den,w1,a0,pm);

stab=0;
wmx=10*w1; dw1=wmx/100;
while a1 < 0 | b1 < 0  & w1 < wmx
  w1=w1+dw1;
  [M,ph]=bode(num, den, w1);  % Returns the mag. and phase of G(w)H(w1)
  thta=-180 + pm - ph;
  thtar=thta*pi/180;
  [a1,b1]=frcntlr(num,den,w1,a0,pm);
end
w1mn=w1;
k=0;
while a1 > 0 & b1 > 0  & w1 < wmx
  k=k+1;
  stab=stab+1;
  w1=w1+dw1;
  [M,ph]=bode(num, den, w1);  % Returns the mag. and phase of G(w)H(w1)
  thta=-180 + pm - ph;
  thtar=thta*pi/180;
  [a1,b1]=frcntlr(num,den,w1,a0,pm);
    if w1<wmx
    wm=sqrt(a0/a1*1/b1); wdd = w1-wm;
    wddm(k)=abs(wdd);  wk(k)=w1;
    end
end
  [wmin,i]=min(wddm); wcal=wk(i);
  w1mx=w1- dw1;
if stab==0
  fprintf('Unstable controller. Reduce the compensator DC Gain or Phase Margin and repeat.\n'),
  return
  else
  fprintf('For a stable controller select a compensated gain crossover\n')
  fprintf('frequency wgc between %7.3g',w1mn),fprintf(' and %7.3g\n',w1mx)
    if wcal < wmx
    fprintf('Suggested wgc for max. phase lead is %7.3g\n',wcal)
    else,end
end

w1=input('Enter wgc -> ')
clc
fprintf('Uncompensated control system \n')
[Gm1, Pm1, wpc1, wgc1]=margin(mag, phase, w);
fprintf('Gain Margin  = %7.3g',Gm1),fprintf('    Gain crossover w = %7.3g\n',wgc1)
fprintf('Phase Margin = %7.3g',Pm1),fprintf('   Phase crossover w = %7.3g\n',wpc1)
fprintf('\n')
[a1,b1]=frcntlr(num,den,w1,a0,pm);
KKc=a1/b1; Z0=a0/a1; P0=1/b1;

fprintf('Controller transfer function \n')
fprintf('   Gc(0) = %g',a0),fprintf(',     Gc = %g',KKc),fprintf('(s + %g',Z0),
fprintf(')/(s + %g',P0), fprintf(')\n\n')
% the following statements will form the characteristic Equation
% of the compensated system.
m=length(num); n=length(den);
if n > m
o=zeros(1,n-m); mk=[o,1]; num1=conv(num,mk);
else, num1=num, end
numgc=[KKc,KKc*Z0]; numopen=conv(numgc,num1);
dengc=[1,P0];       denopen=conv(dengc,den);
denclsd=denopen+numopen;
%fprintf('Row vectors of polynomial coefficients of the compensated system:\n')
%fprintf('Open-loop num. '),disp(numopen)
%fprintf('Open-loop den. '),disp(denopen)
%fprintf('Closed-loop den'),disp(denclsd)
fprintf('Compensated open-loop ')
GH = tf(numopen, denopen)
fprintf('Compensated closed-loop ')
T = tf(numopen, denclsd)

[magp,phasep]=bode(numopen,denopen,w);
[Gm, Pm, wpc,wgc]=margin(magp,phasep,w);
if Pm>360 Pm=Pm-360; else, end
fprintf('Gain Margin  = %7.3g',Gm),fprintf('    Gain crossover w = %7.3g\n',wgc)
fprintf('Phase Margin = %7.3g',Pm),fprintf('   Phase crossover w = %7.3g\n',wpc)
fprintf('\n')
[M,ph]=bode(numopen,denclsd,w);
frqspec(w,M)

discr2=[
'Roots of the compensated characteristic equation:       '];
disp(discr2)
r=roots(denclsd);
disp(r)
rreal=real(r);
   for l=1:n
   if rreal(l) >=0
  fprintf('   Root on the RHP, system is unstable.\n')
  else,end
  end