How can we represent this transfer function into state space representation in MATLAB?

First, find the state space form on the transfer function using the “tf2ss” function. This output will be in the form of [A, B, C, D].
After finding the state space form use the “canon” function to find the state space form.

[A_c, B_c, C_c, D_c] = canon(A, B, C, D, 'companion') %transform linear model into the canonical realization

Refer the below MATLAB documentation for state space Realizations:

https://in.mathworks.com/help/control/ug/canonical-state-space-realizations.html

You can refer to the following MATLAB documentation page for more information about “tf2ss”

https://in.mathworks.com/help/signal/ref/tf2ss.html

Refer to this MATLAB page for the “canon” function

https://in.mathworks.com/help/control/ref/dynamicsystem.canon.html

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Answer 2

Sam Chak 2023-5-11

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1955984-how-can-we-represent-this-transfer-function-into-state-space-representation-in-matlab#answer_1232979

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Hi @Hind Aljallaf

As far as I know, there is no MATLAB command in the Control System Toolbox for directly computing the controllable canonical form. The canon() command only convert the identified state-space model to either the modal or companion canonical form.

G    = tf([1 3], [1 3 2])
G =
 
      s + 3
  -------------
  s^2 + 3 s + 2
 
Continuous-time transfer function.
sys = ss(G);                       % convert from TF to random SS (non-unique)

You need to write a code to get the Observable Canonical form first and then transform it to Controllable Canonical form

obsys = ss(Aa, Bb, Cc, Dd)          % Observable Canonical form
obsys =
 
  A = 
       x1  x2
   x1   0  -2
   x2   1  -3
 
  B = 
       u1
   x1   3
   x2   1
 
  C = 
       x1  x2
   y1   0   1
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.
%% State-space in Conventional Controllable Canonical form
A = Aa';
B = Cc';
C = Bb';
D = Dd;
ctsys = ss(A, B, C, D)              % Controllable Canonical form
ctsys =
 
  A = 
       x1  x2
   x1   0   1
   x2  -2  -3
 
  B = 
       u1
   x1   0
   x2   1
 
  C = 
       x1  x2
   y1   3   1
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.
cttf = tf(ctsys)
cttf =
 
      s + 3
  -------------
  s^2 + 3 s + 2
 
Continuous-time transfer function.