Change out a PI controller Question

Discussion in 'Homework Help' started by Kayne, Jun 26, 2010.

  1. Kayne

    Thread Starter Active Member

    Mar 19, 2009
    105
    0
    The Controller transfer function is given by  G_c(s) = k(1+\frac{1}{T_rs})


    The Analoge PI Controller is given by M_n=k(e_n + \frac {T}{T_r} \sum^{n}_{j=1}   e_j_-_1)

    where k=0.2, Tr=1 and Sampling Time T=0.1

     G_c(s) = k(1+\frac{1}{T_rs})

     G_c(s) = 0.2(1+\frac{1}{s})

    = 0.2+\frac{0.2}{s} \Rightarrow = \frac{0.2s+0.2}{s}

    To change from the s to z domain use  \frac{1}{s} \Rightarrow \frac{z}{z-1}

    I am having trouble with the transformation as the answer I am getting is

    \frac{0.2z^2+0.2z}{z-1}

    The transfer function of the control object with a ZOH is

    G_H_P(z) = \frac {0.1}{z-0.9} and I have been trying to work out the closed loop system transfer function but it becomes complicated which makes me think that I am incorrect with \frac{0.2z^2+0.2z}{z-1}:confused:

    Any suggestion??

    Thanks
     
  2. tskaggs

    New Member

    Jun 17, 2010
    26
    3
    If you are just trying to transform using \frac{1}{s}\Rightarrow\frac{1}{z-1}

    wouldn't that imply that

    0.2+\frac{0.2}{s}\Rightarrow\frac{0.2z-0.2}{z-1}+\frac{0.2z}{z-1}

    ?
     
  3. Kayne

    Thread Starter Active Member

    Mar 19, 2009
    105
    0
    Yes something simple which I didnt see. Thank you.


    so the Closed Loop TF is T(z)=\frac{G_c(z)G_H_p(z)}{1+G_c(z)G_H_p(z)}

    So I have done the following

     G_H_P(z) = \frac{0.1}{z-0.9} , G_c(z) = \frac{0.4z-0.2}{z-1}

    Multplying these to together I get

    = \frac{0.1}{z-0.9} \times \frac{0.4z-0.2}{z-1}

    = \frac{0.4z-0.2}{(z-0.9)(z-1)}

    Now putting into the Closed Loop TF this is the answer I have

    T(z) = \frac{\frac{0.4z-0.2}{(z-0.9)(z-1)}}{{1+\frac{0.4z-0.2}{(z-0.9)(z-1)}}

    = \frac{\frac{0.4z-0.2}{(z-0.9)(z-1)}}{{1+\frac{0.4z-0.2}{(z-0.9)(z-1)}}

    = {\frac{0.4z-0.2}{(z-0.9)(z-1)}} \times {\frac{(z-0.9)(z-1)}{{z^2 -1.5z-0.7 }


    T(z)= {\frac{0.4z-0.2}{z^2 -1.5z-0.7}

    I know that I have showed all the steps which is probably not needed but if I have made a mistake then it should be found easier.

    Thanks
     
  4. tskaggs

    New Member

    Jun 17, 2010
    26
    3
    I see a quick error which is carried through the whole problem. When you are calculating the loop transfer function G_c*G_hp, the numerator is calculated incorrectly. You should have 0.1(0.4z-0.2)=0.04z-0.02
     
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