matlab gives my answer inverted

my answer for A should be [0 1 -0.75,-0.300]
num=1;
den=[12.5 3.75 9.375];
[A,B,C,D]=tf2ss(num,den);
sys=tf(num, den);
step(sys)
grid
title('Step response', 'Color','blue')

[Mathieu NOE]

hello

The state space realization of a transfer function is not unique. In fact, there are infinitely many state space realizations to choose from, but each SS realization has the same transfer function

back to your code,seems to me A is matching your expectations

what result do you get ?

num=1;
den=[12.5 3.75 9.375];
[A,B,C,D]=tf2ss(num,den)

A = 2×2

-0.3000 -0.7500 1.0000 0

B = 2×1

1 0

C = 1×2

0 0.0800

D = 0

[Maria Cinthia]

Hi, thank you and Yes I do get that answer but according to my hand written work it should be 0 1 -0.75,-0.3 for A

[Maria Cinthia]

Thanks another question I have is if state space isn’t unique, does it make sense that Matlab would find the same numerical values but would flip both the rows and the columns for A and the rows on B.

And for C, you take the inverse laplace of 1, which should result in the Dirac delta function. Is there a reason Matlab returns 0.08?

Answer 1

[Sam Chak]

Hi @Maria Cinthia

It sounds like you're looking for a state-space realization in the Controllable Canonical Form. If that's the case, you can follow these simple steps using the compreal() command. It's also possible to achieve this by cleverly manipulating matrix operations without relying on the canon() or compreal() functions.

%% Plant Transfer Function
num  = 1;
den  = [12.5 3.75 9.375];
Gp   = minreal(tf(num, den))

Gp =
 
         0.08
  ------------------
  s^2 + 0.3 s + 0.75
 
Continuous-time transfer function.

%% Companion Canonical Form
csys = compreal(Gp, "c");   % introduced in R2023b release

%% Controllable Canonical Form transformation
A    = csys.A';
B    = csys.C';
C    = csys.B';
D    = 0*C*B;
sys  = ss(A, B, C, D)

sys =
 
  A = 
          x1     x2
   x1      0      1
   x2  -0.75   -0.3
 
  B = 
         u1
   x1     0
   x2  0.08
 
  C = 
       x1  x2
   y1   1   0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

%% Test if the transformed state-space yields the same transfer function
Gtf  = tf(sys)

Gtf =
 
         0.08
  ------------------
  s^2 + 0.3 s + 0.75
 
Continuous-time transfer function.

%% Step response
step(sys), grid on