Design a PD controller for pitch for the ComStar satellite.

If requested saves the matrices to a file PD.txt in the ComStar folder.

------------------------------------------------------------------------- See also PDDesign, C2DelZOH, C2DZOH, Series, AC, ComStar, FResp, Plot2D, WriteCM -------------------------------------------------------------------------

Constants
System Parameters
Design the pitch loops
If requested save and dump the controller matrices
Append the ESA Noise filter
Look at the closed loop eigenvalues
Simulate the loop

%--------------------------------------------------------------------------
%	Copyright 1996 Princeton Satellite Systems, Inc. All rights reserved.
%--------------------------------------------------------------------------
%   Since version 2.
%--------------------------------------------------------------------------

Constants

%----------
degToRad = pi/180;
radToDeg = 180/pi;

System Parameters

%------------------
inr        = ComStar('MO Inertia');
cMatrices  = 'yes';
dT         = 0.25; % This is the loop sampling time

Design the pitch loops

%-----------------------
zeta          = 1.0; % This must be higher than 0.7071 because of the noise filter lag
wN            = 0.1;
wD            = 1.0;
[aC,bC,cC,dC] = PDDesign(zeta,wN,wD,inr(2,2));

If requested save and dump the controller matrices

%---------------------------------------------------
if( strcmp(cMatrices,'yes') )
  fPath = fileparts(which(mfilename));
  fID           = fopen(fullfile(fPath,'PD.txt'),'w');
  [aD,bD]       = C2DelZOH(aC,bC,dT);
  cD = cC;
  dD = dC;
  WriteCM(fID,'A Matrix','fPLAMatrix',reshape(aD',1,length(aD)^2),12,20,3)
  WriteCM(fID,'B Matrix','fPLBMatrix',bD,12,20,3)
  WriteCM(fID,'C Matrix','fPLCMatrix',cD,12,20,3)
  WriteCM(fID,'D Matrix','fRPLDMatrix',dD,12,20,3)
  fclose(fID);
  disp('Pitch Loop PD Matrices')
  disp(aD);
  disp(bD);
  disp(cD);
  disp(dD);
end

Pitch Loop PD Matrices
     -0.25918
      0.25918
      -191.45
       199.36

Append the ESA Noise filter

%----------------------------
sN = load('NoiseF');
[aC,bC,cC,dC] = Series(aC,bC,cC,dC,sN.aN,sN.bN,sN.cN,sN.dN);

w = logspace(-2,1,400);
FResp(aC,bC,cC,dC,1,1,w,'unwrap');

Look at the closed loop eigenvalues

%------------------------------------
aP = [0,1;0 0];
bP = [0;1]/inr(2,2);
cP = [1,0];
dP = 0;

[a,b,c,d] = Series(aP,bP,cP,dP,aC,bC,cC,dC);

aCL = a - b*c;
disp(eig(aCL));

     -0.88768 +     0.4804i
     -0.88768 -     0.4804i
     -0.50017 +          0i
     -0.26449 +          0i
    -0.074199 +          0i

Simulate the loop

%------------------
nSim    = 300;
yPlot   = zeros(1,nSim);
uStep   = 0.1;
n       = length(aCL);
x       = zeros(n,1); % Rest conditions at the start
c       = [1 zeros(1,n-1)];

[aCL, b] = C2DZOH( aCL, b, dT );

for k = 1:nSim
   yPlot(k) = c*x;
   x        = aCL*x + b*uStep;
end

titleStr = sprintf('Pitch PD response to a %6.2f Nm step',uStep);

Plot2D((0:(nSim-1))*dT,yPlot*radToDeg,'Time (sec)','Pitch (deg)',titleStr);


%--------------------------------------