How to convert state space to transfer function

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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.

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Answer 1

Sam Chak 2023-7-26

1 个投票

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).

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Sam Chak 2023-7-26

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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 2023-8-1

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.

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