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.

Update by Arkadiy Turevskiy at MathWorks on 6/18/2024

The answer above is valid for the software release that was current at the time the answer was posted. As of R2024a the doc link is still correct, but a different function should be used to compute controllable and observable forms.

Please use the function compreal, and set the argument type to "c" or "o" for controllable and observable forms respectively.

[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.