INTRODUCTION:
This project is based on Project 1.
The DC motor model obtained as
G(s) = b / (s
4
+ a
1
s
3
+ a
2
s
2
+ a
3
s + a
4
)
where b = 5*10
8
, a
1
= 139.4615, a
2
= 17884.6154, a
3
= 1.29385*10
6
, and a
4
= 32*10
6
.
In
this project we would like to design a unity negative feedback control system as outlined
in the following parts.
Part 1:
Design a PID compensator using one of two following methods:
(a)
Pole/Zero cancellation and approximation
(b)
Ultimate gain method as in ZieglerNichols tuning
to satisfy the 12% overshoot requirement, while reducing both the settling time
with 4% error), and the steadystate error to unit ramp input.
Because pure PID
compensators can not be implemented physically, an extra stable pole far away
from the imaginary axis should be added in.
Part 2:
Analyze the performance of the control system you designed in terms of
(a)
Transient performance, and steadystate performance
(b)
Sensitivity requirement based on the results of Project 1
(c)
Disturbance rejection, if the control input to the plant involves an
additive disturbance ω(t) in the low frequency range
(d)
Noise rejection, if the output measurement involves noise in the high
frequency range.
1
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Main Body:
Solving for the roots of the transfer function obtained in Project 1 yields:
G(s) = 5*10
8
/ [(s + 17.44 ± j105.52) (s + 52.29 ± j7.96)].
Canceling two poles that are far on the real axis for an approximation yields:
G(s) = 5*10
8
/ [2797.6057 (s + 17.44 ± j105.52)].
A more simplified system found by setting s to zero for one of the poles yields:
G(s) = 178724.257 / (s
2
+ 34.88s + 11438.624).
Matlab Code for Figure 1 (The Original System):
%Computing the poles of the system
num=[5e8];
den=[1 139.4615 17884.6154 1.29385e6 32e6];
y=tf(num,den);
[z,p,k]=tf2zp(num,den);
%Step Response of the Original System
t=0:0.0001:0.3;
step(y,'r',t);
title('Figure 1: Step Response of the Original Function');
2
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 Fall '07
 Chen
 Steady State

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