MATLAB: C2d function ‘zoh’ method formula

c2ddiscreteMATLABpi controllertransfer functionzzohztransform

Hi!
My following question would be, say i have a continuous time PI controller:
and i apply the function c2d(,,'zoh'), then it appears that matlab uses he substitution:
.
Which is completly fine, however if i have a transfer function like:
then when applying the c2d function c2d(,,'zoh'), it appears that the previous substitution in this case changes (s!=(z-1)/Ts)depending on the value of . I would like to know why Matlab does this, and what is its algebraic formula if possible?
Thank you!

Best Answer

I suspect that in your first case for C the substitution you cite is only applicable because of the form of C. In general, the ZOH approximation does not use that substitution. Though probably not implemented this way, the genaral form for the ZOH approximation can be implemented as shown below, and compared to what Matlab produces
P=5;Ti=6;Ts=.1;  % example data
C=tf(P*[Ti 1],[Ti 0]);
Cz=c2d(C,Ts,'zoh');
Cznew = minreal(c2d(C*tf(1,[1 0]),Ts,'impulse')*tf([1 -1],[1 0],Ts)/Ts);
[Cz Cznew]
ans =
 
  From input 1 to output:
  5 z - 4.917
  -----------
     z - 1
 
  From input 2 to output:
  5 z - 4.917
  -----------
     z - 1
 
Sample time: 0.1 seconds
Discrete-time transfer function.
R=10;Te = 5; W=tf(1,[R*Te 1]);
Wz=c2d(W,Ts,'zoh');
Wznew = minreal(c2d(W*tf(1,[1 0]),Ts,'impulse')*tf([1 -1],[1 0],Ts)/Ts);
[Wz Wznew]
ans =
 
  From input 1 to output:
  0.001998
  ---------
  z - 0.998
 
  From input 2 to output:
  0.001998
  ---------
  z - 0.998
 
Sample time: 0.1 seconds
Discrete-time transfer function.
Th substitution you cite, s = (z -1)/Ts, the forward rectangular rule, which appears to be an allowable, albeit undocumented, method input to c2d.
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