Transform a MIMO system to state space.

I'm trying to transform the following MIMO system to state space with the following code:

nums = {[1 -1] [1 7.5];[1 0] 6.5};
dens = [1 1 6.5];
sys = tf(nums,dens)

sys =
 
  From input 1 to output...
           s - 1
   1:  -------------
       s^2 + s + 6.5
 
             s
   2:  -------------
       s^2 + s + 6.5
 
  From input 2 to output...
          s + 7.5
   1:  -------------
       s^2 + s + 6.5
 
            6.5
   2:  -------------
       s^2 + s + 6.5
 
Continuous-time transfer function.

[A,B,C,D] = tf2ss(sys)

Error using tf2ss (line 33)
Not enough input arguments.

I wanted to know a way to transform transfer functions from a MIMO system to state space.

Answer 1

[Paul]

Therer really isn't a need to use tf2ss anymore. The functions tf(), ss(),and zpk() can be used to convert from one form to another. In this case:

nums = {[1 -1] [1 7.5];[1 0] 6.5};
dens = [1 1 6.5];
systf = tf(nums,dens)

systf =
 
  From input 1 to output...
           s - 1
   1:  -------------
       s^2 + s + 6.5
 
             s
   2:  -------------
       s^2 + s + 6.5
 
  From input 2 to output...
          s + 7.5
   1:  -------------
       s^2 + s + 6.5
 
            6.5
   2:  -------------
       s^2 + s + 6.5
 
Continuous-time transfer function.

sysss = ss(systf)

sysss =
 
  A = 
          x1     x2     x3     x4
   x1     -1  -3.25      0      0
   x2      2      0      0      0
   x3      0      0     -1  -3.25
   x4      0      0      2      0
 
  B = 
       u1  u2
   x1   2   0
   x2   0   0
   x3   0   2
   x4   0   0
 
  C = 
          x1     x2     x3     x4
   y1    0.5  -0.25    0.5  1.875
   y2    0.5      0      0  1.625
 
  D = 
       u1  u2
   y1   0   0
   y2   0   0
 
Continuous-time state-space model.

The matrices in the state space model can be accessed with dot indexing

sysss.A

ans = 4×4

-1.0000 -3.2500 0 0 2.0000 0 0 0 0 0 -1.0000 -3.2500 0 0 2.0000 0

Note that the result of ss() when coverting a MIMO tranfer function matrix will typically be non-minimal. In this case sysss has four states, when only two are needed.

[Vagner Martinelli]

there is an explanation why the tf2ss function does not solve this problem?

[Paul]

tf2ss (Signal Processing Toolbox) takes the num and den as inputs, not the tf object (Control System Toolbox). And, as far as I know, tf2ss only works for single input systems. And the num input is a numeric array, not a cell array. So using the first input, for example

% nums = {[1 -1] [1 7.5];[1 0] 6.5};
nums = [1 -1; 1 0];
dens = [1 1 6.5];
[A,B,C,D] = tf2ss(nums,dens)

A = 2×2

-1.0000 -6.5000 1.0000 0

B = 2×1

1 0

C = 2×2

1 -1 1 0

D = 2×1

0 0

Then one can do the same for the second input to come up with a second state space realization, and then use other functions to combine the two realizations to form the 2-input/2-output system.

[Vagner Martinelli]

Thank you very much. helped me a lot.

[savan]

thank you very much...