Determine the output of the given Block diagram

Hi,

I have calculated by hand the output of the given block diagram Y(z), in Z-Domain. T=1 sec.

I get as result: Y(z)=4z/[3(z-1)^2]

I want to calculate the same result with matlab. Therefore I have written the following matlab script:

clear all
clc
s = tf('s');
% model Gp(s)
Gp = 2/s;
H = s;
% close-loop transfer function in continuous domain
Tc = feedback(Gp,H);
% model Y(s)
Y = Tc*(2/s);
%% continuous to discrete transformation
T = 1; % sampling time 1 sec
Y_z = c2d(Y,T,'zoh')

But with matlab I receive as result:

Y_z = (0.6667 z + 0.6667)/(z^2 - 2 z + 1)

So my question is, is my matlab code wrong? Can I define with the matlab command c2d also the location where the sampler is placed?

Thank you!

Answer 1

[Paul]

Not clear how that Matlab code would mimic the by-hand solution without seeing the by-hand solution. But I really doubt that it does. Did your by-hand solution solve for the Y(s) by ignoring the sampler and then discretizing the result?

I solved the problem by hand and then evaluated it in Matlab and got the same result as your by-hand solution. If you post your by-hand solution we can probably figure out how to implement it in Matlab.

[Mark S]

I have calculated this by hand like this picture.

I have not specified every single step here. Is your Matlab Code the same as mine?

When you have the same by hand result as me, then my matlab code is not ok.

[Paul]

Excellent (but replace the 5 with an s!).

Let's use the zpk forms instead of tf forms to make it easier to follow.

First define the transform variables:

s = zpk('s');
T = 1; % sample time is T = 1 as in the notes
z = zpk('z',T); 

% define the continuous plant
Gs = 2/s

Gs =
 
  2
  -
  s
 
Continuous-time zero/pole/gain model.

Now to discretize the combination of the plant and the ZOH, we do exactly as in your notes, and use the 'impulse' method of c2d on G(s)/s

Gz = (1 - inv(z))*c2d(Gs/s,T,'impulse')

Gz =
 
  2 z (z-1)
  ---------
  z (z-1)^2
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.

Gz(z) has some pole-zero cancellations, so we can get rid of those

Gz = minreal(Gz)

Gz =
 
    2
  -----
  (z-1)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.

Because G(s) is preceded by a ZOH, it's easier (and clearer?) to do

Gz = c2d(Gs,T,'zoh')

Gz =
 
    2
  -----
  (z-1)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.

Now we have to operate on the product G(s)*H(s). Again, this product is preceded by the ZOH, so

Hs = s;
GHz = c2d(Gs*Hs,T,'zoh')

GHz =
 
  2 (z-1)
  -------
   (z-1)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.

In this case we still have a pole-zero cancellation to deal with because nothing in the above operations "knew" to cancel the s in the numerator of Hs with the s in the denominator of Gs. So we can apply minreal to the output, but it's probaby cleaner to apply minreal to the input

GHz = c2d(minreal(Gs*Hs),T,'zoh')

GHz =
 
  2
 
Static gain.

Now compute the transfer function:

YoverR = Gz/(1+GHz)

YoverR =
 
  0.66667
  -------
   (z-1)
 
Sample time: 1 seconds
Discrete-time zero/pole/gain model.

Define the reference input

Rs = 2/s;

At this point, all that's left to do is compute Rz using c2d, keeping in mind that Rz is NOT preceded by a ZOH, and mulitply it by the transfer function to yield Yz.

[Mark S]

"Excellent (but replace the 5 with an s!)." Yes you are right.

I got it now, thank you very much! It is not so easy to do this calculation with matlab. But now I have understood it.