Different MIMO zeros, poles when using tf and ss system representation

Suppose that we initialize the A,B,C,D arrays of the steady state representation as follows:

A = [-1, 2, 0; 2, 1, 0; -2, -1, 0];
B = [1, 1; -1, 2; 1, -2];
C = [1,-1, 0; -1, 0, 0];
D = zeros(2,2);

If we try to find the zeros and poles of the MIMO system above we end up with different results when using the state space representation and the transfer function:

fprintf('ss representation:\n\n');
fprintf('zeros:\n');
tzero(ss_sys)
fprintf('poles:\n');
pole(ss_sys)
fprintf('tf representation:\n\n');
fprintf('zeros:\n')
tzero(tf_sys)
fprintf('poles:\n');
pole(tf_sys)

ss representation:

zeros:

ans =

       0

poles:

ans =

     -2.2361
      2.2361
           0

tf representation:

zeros:

ans =

     -2.2361
      2.2361

poles:

ans =

      2.2361
     -2.2361
      2.2361
     -2.2361

Why is this happening? Which representation should we choose each time we want to calculate the properties of the system?

Thanks in advance,

Answer 1

[Arkadiy Turevskiy]

Your post does not include the code for creating ss_sys and tf_sys, but I assume you created tf_sys by doing:

tf_sys=tf(ss_sys);

The reason why you are seeing the difference in poles and zeros is due to numerical issues. Please take a look here . As the doc states, it is recommended to use state space models; transfer function representation is not recommended for working with MIMO systems.

HTH.

[Amine]

Hello sir! I am using 'ss' and 'tf' commands in order to get the transfer function form of an H infinity controller. Does it means that my controller in its system matrix form is NOT the same in its transfer function form? Thank you!

[Arkadiy Turevskiy]

did you have a chance to read the doc I linked to? Again, it is recommended to use state space representation for MIMO systems- it is better behaved numerically.

[Amine]

Yes sir I did, thank you so much it was realy helpful! thank you!