Transfer Function

Discussion in 'Homework Help' started by DJ.Rich, Oct 10, 2010.

  1. DJ.Rich

    Thread Starter New Member

    Oct 10, 2010
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    Hello,

    I am very new to Control Systems theory.

    I was just wondering would it be possible to get help with this question, and to maybe help me understand it more.

    Q. Given a system with the transfer function Gp(s) = 200/ ((s+20)(s+1)(s+10)) .

    If it is required to approximate the given Gp(s) using a 1st order system, please write down its transfer function (it should have the system gain as Gp(s)).

    Any help would be greatly appreciated.

    Thanks.
     
  2. shteii01

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  3. DJ.Rich

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    Oct 10, 2010
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    Thanks.

    Would I be right in thinking that the answer is:

    10 / (S^2 + 11s +10) ?

    If I remove the (s+20) bracket as it should affect the outcome of the transient response the least? Furthest away from the y axis?
     
  4. t_n_k

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    You've proposed a second order equivalent - rather than a first order.

    One approach would be to separate the total transfer function into its partial fraction equivalent and take the most persistent transient first order term as the approximation.
     
  5. Georacer

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    Nov 25, 2009
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    In order to linearize a TF you need to choose a point of operation Q, around which you will assume that your system will operate. This is a nescessary assumption in order to perform the linearization.

    The formula that linearizes the TF g(x_1,x_2,...,x_n) around the point (x_1o,x_2o,...,x_no) is the following:
    y=g(x_{1o},x_{2o},...,x_{no})+\left\   {\frac{\vartheta g}{\vartheta x_1}} \right|_{x=x_{1o}} \cdot (x-x_{1o})+\left\   {\frac{\vartheta g}{\vartheta x_2}} \right|_{x=x_{2o}}  \cdot (x-x_{2o})+...+\left\   {\frac{\vartheta g}{\vartheta x_n}} \right|_{x=x_{no}}  \cdot (x-x_{no})<br />
     
  6. DJ.Rich

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    Oct 10, 2010
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    Thanks for the replies. Would it be possible to show what the final answer should be, for :

    If it is required to approximate the given Gp(s) using a 1st order system, please write down its transfer function (it should have the system gain as Gp(s)).

    Would it be possible to maybe show workings at how to arrive at this answer for the 1st order system, so I can understand it better and possibly do other examples?

    Any help, greatly appeciated.

    Thanks.
     
  7. pramod_burnwal

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    Oct 10, 2010
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    i think answer is
    (20/19)/(s+20) + (200/171)/(s+1) + (-20/9)/(s+1)
     
  8. Georacer

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    The answer is definitely not the above. I will post it once I read more about linearization.
     
  9. Georacer

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    Ok, what I suggested doesn't apply in this case, as it is suitable only for equations in the t-domain. It can be used to linearize equations with non-linear time dependance in order to form the State Space linear system.

    So I used shteii01's second link (post #2) to go directly from the 3rd degree to the first and the results were remarkable!

    Step Response.png
     
  10. t_n_k

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    Some thoughts on what has been proposed so far.

    The unit step response of the original function

    G(s)=\frac{200}{(s+1)(s+10)(s+20)}

    will be of the form

    f(t)=A*exp(-t)+B*exp(-10*t)+C*exp(-20*t)

    By partial fraction decomposition the unknown terms are found as

    A=1.1695906
    B=- 2.2222222
    C=1.0526316

    Hence

    G(s)=\frac{1.1695906}{(s+1)}-\frac{2.2222222}{(s+10)}+\frac{1.0526316}{(s+20)}

    giving the unit step response

    f(t)=1.1695906e^{-t}-2.2222222e^{-10t}+1.0526316e^{-20t}

    It will be apparent that the final value (at t=∞) of the step response function will be zero and not 1 as proposed by 'shteii01' and developed by 'georacer'.

    While the goal is a first order approximation of the given transfer function I would think that the initial and final value time domain response conditions should (at least) be satisfied by the equivalent function.
     
  11. The Electrician

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    You're considering the impulse response, but it's the step response that doesn't go to zero at infinity; this had me going for a while!

    Your suggestion of using the most persistent part seems to have merit; see the attachment.

    The only problem is that it begins with a negative value. The 'shteii01' method adds a delay to help fix this problem.
     
  12. t_n_k

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    Yes - thanks Electrician - I have derived the impulse response rather than the step - my mistake on that one.
     
  13. t_n_k

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    Also apologies to 'Georacer' and 'shteii01' for the unnecessary distraction.

    Just one other point on the OP's question - the method suggested by 'shteii01' and elaborated by 'Georacer' works very well. I still wonder whether the pure delay factor falls within the scope of a first order system approximation.

    The pure delay term e^-Ts is often approximated (after Pade) by a ratio of polynomials of the form

    e^{-Ts}=\frac{1-\frac{Ts}{2}+\frac{(Ts)^2}{12} - ...}{1+\frac{Ts}{2}+\frac{(Ts)^2}{12} + ...}

    This form suggests the ideal pure delay has an infinity of frequency domain components of increasing order - which doesn't 'fit' with the first order approximation challenge.

    Just a thought which others might like to ponder ....
     
    Last edited: Oct 13, 2010
  14. Georacer

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    When analysing functions in the s-plane, a high-order TF can be a pain. Our aim is to reduce the order of the involved functions in order to combine, study and compensate for them more easilly. The Laplace Transform however is kind to the delay term, which can be easilly handled.

    Building a circuit with standard delay is also something manageable, I think.
     
  15. ELECTRONERD

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    May 26, 2009
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    Georacer,

    I've been plotting transfer functions myself, so I was curious as to what mathematics software you used?
     
  16. Georacer

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    It's Matlab R2009a. If you care for the code or any other help, let me know.
     
  17. ELECTRONERD

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    Unfortunately, I can't afford that at the moment. I appreciate your help nevertheless!
     
  18. Georacer

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    Unfortunately, at the time I don't know any other software that handles Transfer Functions and System Models effectively.
     
  19. t_n_k

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    You might consider Scilab or a similar Matlab clone. Scilab is gratis, reasonably documented / supported and handles transfer functions ['s' & 'z' domain], continuous and discrete time solutions, state space representation etc.

    See the link

    http://www.scilab.org/
     
    Last edited: Oct 15, 2010
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