minreal tolerance not respected

minreal documentation says that it does a simple pole-zero comparsion for cancelation. However, simple examples seem to behave differently. Even the example in the Pole-Zero simplification document fails to behave as described:

G = zpk(3e-8,[-1,-3],1); 
C = pid(1,0.3);
T = feedback(G*C,1)

T =
 
    (s+0.3) (s-3e-08)
  ----------------------
  s (s+4.218) (s+0.7824)
 
Continuous-time zero/pole/gain model.

% compute minreal with a tolerance that ought to cancel
Tred = minreal(T, 1e-7)

Tred =
 
    (s+0.3) (s-3e-08)
  ----------------------
  s (s+4.218) (s+0.7824)
 
Continuous-time zero/pole/gain model.

As you can see, the zero at 3e-8 and 0 do not cancel, even though they are less than 1e-7 away. It eventually cancels at tolerance of 1e-3. It appears that the elimination criteria is the the square of the distance, rather than the distance, but that is not at all clear from the documentation or the example above.

Answer 1

[Paul]

Hi Daniel,

As best I can tell after sifting through the code, tol is generally compared to the distance between the zero and pole under test. However, the tol that is used is modified from what the user provides. This modification is very small, except when checking for cancellations of poles and zeros on (or perhaps very close to?) the imaginary axis, which includes the origin.

For example,

H = zpk(-3e-8,0,1);
minreal(H,1e-7)  % real pole/zero at/near the origin

ans =
 
  (s+3e-08)
  ---------
      s
 
Continuous-time zero/pole/gain model.

H = zpk(-1-3e-8,-1,1); % real pole/zero at -1 separated by 3e-8
minreal(H,1e-7)

ans =
 
  1
 
Static gain.

minreal(H,3e-8) % exact tol, cancellation occurs

ans =
 
  1
 
Static gain.

H = zpk(1j*(1-3e-8)*[1 -1] ,[-j,j],1)  % imaginary pole/zero at 1j separated by 3e-8

H =
 
  (s^2 + 1)
  ---------
  (s^2 + 1)
 
Continuous-time zero/pole/gain model.

minreal(H,3e-7)  % doesn't cancel

ans =
 
  (s^2 + 1)
  ---------
  (s^2 + 1)
 
Continuous-time zero/pole/gain model.

minreal(H,3e-4)  % need a larger tolerance

ans =
 
  1
 
Static gain.

H = zpk(-1+1j*(1-3e-8)*[1 -1] ,-1+[-j,j],1)  % complex poles and zeros separated by 3e-8

H =
 
  (s^2 + 2s + 2)
  --------------
  (s^2 + 2s + 2)
 
Continuous-time zero/pole/gain model.

minreal(H,3e-7)  % cancels

ans =
 
  1
 
Static gain.

I don't know if this helps, but it shows that poles/zeros around the origin and on the imaginary axis are treated a bit differently than those that aren't.

[Daniel Magree]

That is interesting analysis, but leaves me more confused as to how this function works. Someone from matlab needs to explaine why would it be implemented in that way

[Paul]

If I had to guess, I suspect that the idea might be that a pole/zero pair that are close to each other AND on or near the imaginary axis can have a larger effect on the frequency response than a pole/zero pair that are just as close to each other but further from the imaginary axis. If this is the case, then maybe they are trying to protect the user in some sense.

H = zpk(-2+1j*(1-3e-8)*[1,-1] ,-2+1j*[1,-1],1)  % imaginary pole/zero at -2+1j separated by 3e-8

H =
 
  (s^2 + 4s + 5)
  --------------
  (s^2 + 4s + 5)
 
Continuous-time zero/pole/gain model.

bode(H,logspace(-1,1,50000))

H = zpk(-0.01+1j*(1-3e-8)*[1,-1] ,-0.01+1j*[1,-1],1)  % imaginary pole/zero at -0.01+1j separated by 3e-8

H =
 
  (s^2 + 0.02s + 1)
  -----------------
  (s^2 + 0.02s + 1)
 
Continuous-time zero/pole/gain model.

bode(H,logspace(-1,1,50000))

The effect of the latter on the frequency response is three orders of magnitude higher than the former ( realize that this might not be the best example). Anyway, just a guess on my part. Whatever the reason, the doc should at least mention that tol is not always explicitly adhered to under certain condtions.