convert a transfer function to controllable and observable canonical form

Hi, I want to convert a transfer function to controllable and observable canonical form for the

num = [4];
den = [1 0.8 4];
Gp = tf (num , den)

Gp =
 
         4
  ---------------
  s^2 + 0.8 s + 4

Answer 1

[Star Strider]

num = [4];
den = [1 0.8 4];
Gp = tf (num , den);

The canon function requesting the 'companion' canonical form directly produces the observable canonical form:

GpssObs = canon(Gp,'companion')
GpssObsA = GpssObs.A
GpssObsB = GpssObs.B
GpssObsC = GpssObs.C
GpssObsD = GpssObs.D

producing:

GpssObsA =
            0           -4
            1         -0.8

GpssObsB =
     1
     0

GpssObsC =
     0     4

GpssObsD =
     0

The controllable canonical form is then:

GpssConA = GpssObsA.'
GpssConB = GpssObsC.'
GpssConC = GpssObsB.'
GpssConD = GpssObsD

producing:

GpssConA =
            0            1
           -4         -0.8

GpssConB =
     0
     4

GpssConC =
     1     0

GpssConD =
     0

Se the documentation section on: Canonical State-Space Realizations for a full discussion.

[Star Strider]

My pleasure!

If my Answer helped you solve your problem, please Accept it!

[Bradley Beck]

Hello there,

The documentation on observable canonical form states that the B matrix should contain the values from the transfer function numerator while the C matrix should be a standard basis vector. However using the "canon(...,'companion')" command produces B and C matrices that are swapped to what is expected per the documentation, both in the given example above and my own experiences.

Although turning this state space back into a transfer function produces the correct result I cannot figure out why this discrepancy exists (when I try to do the math by hand it produced some equation with time derivatives of u which I didn't bother to solve).

If you have any insight as to why this is the case I would greatly appreciate it. Thanks in advance.

[Bill Tubbs]

The outputs in this answer (for observable and controllable forms) do not match my class notes or other documentation I found online, for example here and here. The differences are in the B and C matrices.

[Sean Doherty]

Star Strider. This MATLAB example contradicts the documentation (https://uk.mathworks.com/help/control/ug/canonical-state-space-realizations.html)

documentaion says observable canonical form has:

in example: n = 2, b0 = 0, bn1 = 0; b2 = 4, a0 = 1, a1 = 0.8, a0 = 0.4 should give:

B0 = [4 0]'

C0 = [0 1]

MATLAB example gives:

B0 = [1 0]'

C0 = [0 4]

Documentation is correct, MATLAB's canon() is wrong?

Answer 2

[TALAL alghattami]

for the given matric how can i designe observer and pole placment and then implement the designe in simulink

apreicate your support

deno=[1 0.8 4];

num=[4];

Gp=(num,deno)

A =[0 1 ; -4 - -0.8];

B =[0;4];

C =[1 0];

D =[0];

[Star Strider]

I have very little experience with Simulink. I cannot help you with it.

Use the place function to do pole placement. (This is a Control System Toolbox function. I have no idea how to do it in Simulink.)

[TALAL alghattami]

thanks

[Star Strider]

As always, my pleasure!

Answer 3

[Branislav Hatala]

I would like to ask how can I convert SIMO system to controllable form. I did not find anything about SIMO or MIMO systems and this cannot be applied since C and B matrices will result in frong dimensions.