- #1

gfd43tg

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## Homework Statement

## Homework Equations

## The Attempt at a Solution

I'm working on parts (c) and (d) now, but I will show my work for (a) and (b) for completeness.

(a)

Starting with the process transfer function

$$ g_{p} = \frac {-1.43}{s^{2}+3.687s-7.177} $$

The denominator is factorized,

$$ g_{p} = \frac {-1.43}{(s-1.399)(s+5.086)} $$

$$ = \frac {-1.43}{-1.399(\frac {-1}{1.399}s+1)5.086(\frac

{1}{5.086}s+1)} $$

$$ g_{p} = \frac {0.201}{(-0.715s+1)(0.197s+1)} $$

where ##\tau_{1}= -0.715## and ##\tau_{2} = 0.197##. The first time constant makes the process unstable. A singularity occurs at

##s = \frac {1}{\tau_{1}} = \frac {1}{0.715} = 1.399##

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(b)

$$ f(s) = \frac {\gamma s + 1}{(\lambda s + 1)^{n}} $$

The controller must be semi-proper with an unstable system.

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(c)

The invertible part of the process transfer function is found,

$$ \tilde {g_{p-}} = \frac {0.201}{0.197s+1} $$

with a filter

$$ f = \frac {\gamma s+1}{5s+1} $$

$$ q = \tilde {g_{p-}}^{-1}f = \frac {0.197s+1}{0.201} \cdot \frac {\gamma s+1}{5s+1} $$

The controller transfer function is determined,

$$ g_{c} = \frac {q}{1-q \tilde {g_{p-}}} $$

$$ = \frac { \frac {0.197s+1}{0.201} \frac {\gamma s+1}{5s+1}}{1 - \frac

{0.201}{0.197s+1} \cdot \frac {0.197s+1}{0.201} \cdot \frac {\gamma s+1}{5s+1} } $$

$$ = \frac {\frac {0.197s+1}{0.201} \cdot \gamma s+1} {(5- \gamma)s} $$

$$ = 4.975 \bigg [ \frac {0.197 \gamma s^{2}+(0.197+ \gamma)s + 1}{(5- \gamma)s} \bigg ] $$

Which is of the form of an ideal PID controller,

$$ g_{c,PID} = k_{c} \bigg [ \frac {\tau_{I} \tau_{D} s^{2} + \tau_{I}s + 1}{\tau_{I}s} \bigg ] $$

Therefore, for the ##\tau_{I}## term to be equal, ##5- \gamma = 0.197+ \gamma##, so ##\gamma = 2.402##.

$$ g_{c} = 4.975 \bigg [ \frac {0.473s^{2}+2.599s+1}{2.599s} \bigg ] $$

##k_{c} = 4.975##, ##\tau_{I} = 2.599##, ##\tau_{D} = 0.473/2.599 = 0.182##

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(d)

Here is my simulink model

And the inputs to the PID controller

However, my output and manipulated inputs look totally wrong, this controller is not working at all!

So I am wondering if I may have done part (c) incorrectly.