Change out a PI controller Question

Thread Starter

Kayne

Joined Mar 19, 2009
105
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
 

tskaggs

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

wouldn't that imply that

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

?
 

Thread Starter

Kayne

Joined Mar 19, 2009
105
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
 

tskaggs

Joined Jun 17, 2010
26
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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