How to convert transfer function to state space

[kanazi.1]

Hi,how can i convert the following transfer function to state space form by hand.

H(s)=(s^2+27s+50)/(2s^2+27s+50)

Any help will be appreciated!

 

[fact0ry]

Well, first you have get a proper transfer function. You do that by dividing the numerator to the denominator to find the direct part of the sistem (D) : y(s) = D*u(s). So in your case that would be 1/2.
Now you have a proper transfer function (a first degree polynomial on a second degree polynomial). Now there are two well-known forms of the state space:

If your proper H(s) = N(s)/D(s),
where N(s) = s^n+ a1*s^(n-1) +...+an
D(s) = c1*s^(n-1)+...+cn

Standard reachability form:
Ar =
[ -a1 -a2 ... -an
1 0 ... 0
0 1 ... 0
...
0 0 .. 1 0 ]
Br= [1 0 ...0]'
Cr=[c1 ... cn]
(also Ar1 = the rows upside down ; br1=[0...1]; cr1=[cn...c1] )
Standard observability form:

Ao =
[ -a1 1 0 ... 0
-a2 0 1 ... 0
-a3 0 0 ... 0
....
-an 0 0 ... 0 ]
Co= [1 ... 0]
Bo= [c1...cn]
(also there is an equivalent ). If you need that I'll write it.

And of course there is D, which is 1/2 for your H(s).

 

[berkeman]

Thread moved from EE to Homework Help forums.

Welcome to the PF, guys. Homework and coursework questions need to be posted here in the Homework Help forums, and when you post a HH question, you need to use the HH Template that is provided, and show us your work so that we can provide tutorial assistance.

And fact0ry, on homework/coursework questions like this (it may not have been obvious to you, since it was posted incorrectly in the EE forum), we are not allowed to do the original poster's (OP's) work for them. You can offer hints and point out mistakes in the OP's work, but the OP needs to do the bulk of the work.

 

[kanazi.1]

well, first you have get a proper transfer function. You do that by dividing the numerator to the denominator to find the direct part of the sistem (d) : Y(s) = d*u(s). So in your case that would be 1/2.
Now you have a proper transfer function (a first degree polynomial on a second degree polynomial). Now there are two well-known forms of the state space:

If your proper h(s) = n(s)/d(s),
where n(s) = s^n+ a1*s^(n-1) +...+an
d(s) = c1*s^(n-1)+...+cn

standard reachability form:
Ar =
[ -a1 -a2 ... -an
1 0 ... 0
0 1 ... 0
...
0 0 .. 1 0 ]
br= [1 0 ...0]'
cr=[c1 ... Cn]
(also ar1 = the rows upside down ; br1=[0...1]; cr1=[cn...c1] )
standard observability form:

Ao =
[ -a1 1 0 ... 0
-a2 0 1 ... 0
-a3 0 0 ... 0
....
-an 0 0 ... 0 ]
co= [1 ... 0]
bo= [c1...cn]
(also there is an equivalent ). If you need that i'll write it.

And of course there is d, which is 1/2 for your h(s).

you are a lifesaver!

 

[fact0ry]

Thread moved from EE to Homework Help forums.

Welcome to the PF, guys. Homework and coursework questions need to be posted here in the Homework Help forums, and when you post a HH question, you need to use the HH Template that is provided, and show us your work so that we can provide tutorial assistance.

And fact0ry, on homework/coursework questions like this (it may not have been obvious to you, since it was posted incorrectly in the EE forum), we are not allowed to do the original poster's (OP's) work for them. You can offer hints and point out mistakes in the OP's work, but the OP needs to do the bulk of the work.

Yeah, I thought he asked for a general solution. Or is that a homework/coursework question too?
Anyway, I like the hints idea :).

 

[sausage]

im sorry but I am confused as how you would apply your general solution to this problem.. i don't know how to convert a transfer function with an equal degree numerator and denominator into state space and i went looking on google and came across this page. I have the issue where my C matrix is one column too big when I use your method..

like how can you use the general method you proposed the specific example given?