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

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