How to convert a transfer function into state space representation?

Question

Hind Aljallaf 2023-5-1

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1956039-how-to-convert-a-transfer-function-into-state-space-representation

编辑： Paul 2023-5-2

I was trying to convert a transfer function into state space representation, but the matrices in the output are not quiet correct. The numbers are flipped like how in B 0 should be up and 1 should be down, or in C where 2 should be right and 3 should be left. How can I fix this issue?

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Answer 1

Paul 2023-5-1

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1956039-how-to-convert-a-transfer-function-into-state-space-representation#answer_1226864

编辑：Paul 2023-5-2

在 MATLAB Online 中打开

The state space realization of a transfer function is not unique. In fact, there are infinitely many state space realizations to choose from.

num = [0 3 2];
den = [1 4 4];

The Control System Toolbox uses one methodology

G = ss(tf(num, den))
G =
 
  A = 
       x1  x2
   x1  -4  -2
   x2   2   0
 
  B = 
       u1
   x1   2
   x2   0
 
  C = 
        x1   x2
   y1  1.5  0.5
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

and the Signal Processing Toolbox uses another

[A,B,C,D] = tf2ss(num,den)
A = 2×2
    -4    -4
     1     0
B = 2×1
     1
     0
C = 1×2
     3     2
D = 0

Each realization has the same transfer function

tf(G)
ans =
 
     3 s + 2
  -------------
  s^2 + 4 s + 4
 
Continuous-time transfer function.
[b,a] = ss2tf(A,B,C,D)
b = 1×3
         0    3.0000    2.0000
a = 1×3
     1     4     4

Any other realization can be obtained via similarity transformation. The CST provides a function ss2ss to do that

ss2ss(ss(A,B,C,D),[0 1;1 0])
ans =
 
  A = 
       x1  x2
   x1   0   1
   x2  -4  -4
 
  B = 
       u1
   x1   0
   x2   1
 
  C = 
       x1  x2
   y1   2   3
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.
tf(ans)
ans =
 
     3 s + 2
  -------------
  s^2 + 4 s + 4
 
Continuous-time transfer function.

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Answer 2

Walter Roberson 2023-5-1

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1956039-how-to-convert-a-transfer-function-into-state-space-representation#answer_1226849

在 MATLAB Online 中打开

num = [0 3 2];
den = [1 4 4];
G = tf(num, den);
S = ss(G)
S =
 
  A = 
       x1  x2
   x1  -4  -2
   x2   2   0
 
  B = 
       u1
   x1   2
   x2   0
 
  C = 
        x1   x2
   y1  1.5  0.5
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.
S.A
ans = 2×2
    -4    -2
     2     0
S.B
ans = 2×1
     2
     0
S.C
ans = 1×2
    1.5000    0.5000
S.D
ans = 0
[A, B, C, D] = tf2ss(num, den)
A = 2×2
    -4    -4
     1     0
B = 2×1
     1
     0
C = 1×2
     3     2
D = 0