Controllable and observable canonical form

Hi, I want to convert a transfer function to controllable and observable canonical form. Tried with tf2ss but it did not work. I am sharing a part of my code. Is there any way to get those A,B,C,D matrices by any Matlab functions??

My code:

clc; clear all;

Den=[0 1 1]; Num=[1 5 6];

s=tf(Den,Num)

[A B C D]=tf2ss(s)

Answer 1

[Star Strider]

The tf2ss function wants a transfer function as input, not a system object.

Try this:

[A B C D]=tf2ss(Den,Num)

A =
    -5    -6
     1     0

B =
     1
     0

C =
     1     1

D =
     0

EDIT —

To get the state space representation from a system object, just use the ss funciton:

[A B C D] = ss(s);

[M. M. Farhad]

But does tf2ss convert to controllable canonical form or observable canonical form?? I did the same but it does not match those forms.

[Star Strider]

It does, but not directly. It produces the ‘[A B C D]’ matrices.

You must use the canon function with the 'companion' option to get the controllability canonical form, that it produces by default.

The observability canonical form requires that you transpose the ‘A’ matrix it produces, and then the ‘B’ vector (or matrix) becomes the transposed ‘C’, and the ‘C’ vector (or matrix) becomes the transposed ‘B’.

Example with your system:

Den = [0 1 1];
Num = [1 5 6];
s = tf(Den,Num);
s_ss = ss(s);

s_can_c = canon(s_ss, 'companion');

Controllability —

Amtx_c = s_can_c.A
Bmtx_c = s_can_c.B
Cmtx_c = s_can_c.C

Amtx_c =
     0    -6
     1    -5

Bmtx_c =
     1
     0

Cmtx_c =
     1    -4

Observability —

Amtx_o = s_can_c.A'
Bmtx_o = s_can_c.C'
Cmtx_o = s_can_c.B'

Amtx_o =
     0     1
    -6    -5

Bmtx_o =
     1
    -4

Cmtx_o =
     1     0

I went back to my textbooks to be certain I got this correct. It would help if MATLAB made these a bit easier to find and interpret in the documentation, but then understanding the Jordan-form and companion matrices are essential to understanding controllability and observability.

I apologise for the delay. Life intrudes.

[Christopher Rösel]

Isn´t it supposed to be the other way around? Your controllable canonical form is your observable canonical form? I checked your code with another transfer function and it doesn´t provide me the results I calculated.

[Mohamed Ibrahim]

It is, yes.

[Shady Hassan]

its the other way around, controllability and observability matrices are reversed in zour explanation above..

Answer 2

[Mackyle Naidoo]

i would like to obtain the state space repsentation for controllable , observable and diagonal canonical form

using the following transfer function of the 𝑌𝑌(𝑠𝑠) 𝑈𝑈(𝑠𝑠) = 𝑠 + 4 /𝑠^2 + 13s + 42. Using matlab code to get the desired outcome can anyone help?

[Sam Chak]

@Mackyle Naidoo

Your transfer function looks strange when interpreted according to standard order of operations in inline math mode and computer programming.

If you this is not, please enter it in MATLAB code. I'll show an example:

If , then type:

G = tf([2],[3 5])

G =
 
     2
  -------
  3 s + 5
 
Continuous-time transfer function.

[Star Strider]

This should be posted as a new Question.

[Mackyle Naidoo]

@Sam Chak

[Mackyle Naidoo]

@Sam Chak transfer function in matlab is as follows

g = tf ([1,4],[1^2 13 42])

@star strider how do i post this as a new question ?

[Star Strider]

@Mackyle Naidoo —

Start here.