How to convert a transfer function into state space representation?

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?

Answer 1

[Paul]

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.

Answer 2

[Walter Roberson]

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

At the moment I do not kow why the values do not match.