CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 101
D. G. MacMynowski
Fall 2009
Problem Set #8
Issued:
23 Nov 09
Due:
2 Dec 09
Note: In the upper left hand corner of the
second
page of your homework set, please
put the number of hours that you spent on this homework set (including reading).
1. Consider a secondorder process of the form
P
(
s
) =
k
s
2
+ 2
ζω
0
s
+
ω
2
0
k, ζ, ω
0
>
0
.
In this problem we will explore various methods for designing a PID controller for the system.
(a) (Eigenvalue assignment) Suppose that we want the closed loop dynamics of the system
to have a characteristic polynomial given by
p
(
s
) =
s
3
+
a
1
s
2
+
a
2
s
+
a
3
.
Compute a formula for the controller parameters of a PID controller (
k
p
,
k
i
and
k
d
) that
gives the desired closed loop response.
(b) (Eigenvalue assignment) Let the process parameters be given by
k
= 1,
ζ
= 0
.
5 and
ω
0
= 2. Using the formulas from part (a), compute a feedback control law that places
the closed loop poles of the system at
λ
=
{
1
,

2
±
j
}
.
Plot the step response and
frequency response for the closed loop systems, and compute the gain and phase margins
for your design.
(c)
Optional:
(Ziegler–Nichols step response) Using the same process parameters as above,
plot the step response for the system and use the Ziegler–Nichols rules to design PID
controllers.
Plot the step response and frequency response for each of your controller
designs, and compute the gain and phase margins for each design.
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 Fall '09
 MacMynowski
 Closed loop, closed loop poles, D. G., California Institute, TECHNOLOGY Control and Dynamical Systems

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