Put a series of compensators on a double integrator.

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See also TF2SS, LeadLag, FResp, Series
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Contents

Double integrator
Now we have more phase shift
Create a lead compensator with 40 deg of lead. Look at it alone.
Add the lead in series

%-------------------------------------------------------------------------------
%   Copyright (c) 2007 Princeton Satellite Systems, Inc.
%   All rights reserved.
%-------------------------------------------------------------------------------

Double integrator

%------------------
a = [0 1;0 0]; b = [0;1];
c = [1 0]; d = 0;
w = logspace(-5,1,1000);
FResp(a,b,c,d,1,1,w)
set(gcf,'name','Double Integrator')

% Add an integrator to the system and look at the frequency response

Now we have more phase shift

%-------------------------------------------------------------------
[aI,bI,cI,dI] = tf2ss( [1 0.1],[1 0] );
[a,b,c,d] = Series(a,b,c,d,aI,bI,cI,dI);
FResp(a,b,c,d,1,1,w)
set(gcf,'name','Open Loop with Integrator')
disp('Closed loop with integrator is unstable')
eig(a-b*c)

Closed loop with integrator is unstable

ans =

     0.049514 +     1.0037i
     0.049514 -     1.0037i
    -0.099029 +          0i

Create a lead compensator with 40 deg of lead. Look at it alone.

%-----------------------------------------------------------------
[aL,bL,cL,dL] = LeadLag(0.6,40,1);
FResp(aL,bL,cL,dL,1,1,w)
set(gcf,'name','Lead Compensator')

Add the lead in series

%-----------------------
[a,b,c,d] = Series(a,b,c,d,aL,bL,cL,dL);
FResp(a,b,c,d,1,1,w,'unwrap');
set(gcf,'name','Open Loop with Lead')
disp('Closed loop with lead is stable')
eig(a-b*c)


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% PSS internal file version information
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% $Date$
% $Id: c3336befd7391503710fa877643ff1ad504d2d01 $

Closed loop with lead is stable

ans =

     -0.44084 +     2.1397i
     -0.44084 -     2.1397i
     -0.30625 +          0i
    -0.098774 +          0i