How to convert state space to transfer function

I have the following state space:

Where x is 12 by 1, A is 12 by 12, B is 12 by 3, w is 12 by 1, y is 6 by 1, H is 6 by 12, and v is 6 by 1. How can I convert this to transfer functions in MATLAB? I wanted to use the ss2tf command but I realized that it takes the state space in the format of:

Which is not what I have.

Answer 1

[Sam Chak]

Hi @Ali Almakhmari

I think you can rearrange the matrices manually because

can be expressed as

.

So, if , then in compact form, it looks like

.

Now you should be able to use ss2tf(A, B, C, D).

[Sam Chak]

There will be a total of 126 transfer functions.

6*21

ans = 126

[Ali Almakhmari]

Are you sure the conversion you mentioned is correct, especially the new B and D matrices you mentioned?

[Sam Chak]

Hi @Ali Almakhmari

According to the theoretical mathematics presented, and based on the info you provided, It should work. Have you tried and show us the result? Here is an example:

A  = [0  1;             % state matrix
     -1 -2];
G1 = [0;                % input matrix of u
      1]

G1 = 2×1

0 1

G2 = eye(2)             % input matrix of w

G2 = 2×2

1 0 0 1

G3 = zeros(2, 3)        % input matrix of v

G3 = 2×3

0 0 0 0 0 0

B  = [G1 G2 G3];        % augmented input matrix B
C  = [1 0];             % state matrix
D  = [0, 0 0, 1 1 1];   % augmented direct feedforward matrix D

% State-Space Model
sys = ss(A, B, C, D)

sys =
 
  A = 
       x1  x2
   x1   0   1
   x2  -1  -2
 
  B = 
       u1  u2  u3  u4  u5  u6
   x1   0   1   0   0   0   0
   x2   1   0   1   0   0   0
 
  C = 
       x1  x2
   y1   1   0
 
  D = 
       u1  u2  u3  u4  u5  u6
   y1   0   0   0   1   1   1
 
Continuous-time state-space model.

% Transfer functions 
P  = tf(sys)            % predicted number of TFs: 1*6 = 6 Tfs

P =
 
  From input 1 to output:
        1
  -------------
  s^2 + 2 s + 1
 
  From input 2 to output:
      s + 2
  -------------
  s^2 + 2 s + 1
 
  From input 3 to output:
        1
  -------------
  s^2 + 2 s + 1
 
  From input 4 to output:
  1
 
  From input 5 to output:
  1
 
  From input 6 to output:
  1
 
Continuous-time transfer function.

[Ali Almakhmari]

Thank you so much. You have been very helpful. One last question I have, what if in the new system you told me to use, I only care about the first three rows of transfer functions: I care about the transfer functions in terms of u only in the [u; v; w] vector. So a total of 6*3 transfer functions. How can I tell ss2tf to do that for me? Because I dont want it to waste so much time solving for all the other ones.