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Unformatted text preview: 16.06 Lecture 10 The Eﬀect of Zeroes Karen Willcox
September 25, 2003 Today’s Topics
1. Zero near a dominant quadratic mode
2. General observations on the eﬀect of a zero
3. Other examples of the eﬀects of zeroes
Reading: 5.3, l.n. 1 1 Zero near a dominant quadratic mode This situation usually results when we add a PD controller. 2
s + z1
ωn
G(s) =
2 + 2ζω s + ω 2
z1 s
n
n c(t) = If A ≈ z1 and α ≈ 0, then c(t) is the standard secondorder response.
A far away zero has negligible eﬀect.
As the zero is moved to the right, it has a greater eﬀect as follows: 2 Tp = P.O. =
Example: •
• 3 2 General observations on the eﬀect of a zero Given G(s) = 4
s+a
a (s+1)(s+4) (a) Step response c(t) =
Note: (b) Observations:
(i) (ii) 4 (iii) (iv) 3 Other examples of the eﬀects of zeroes
• Another way to look at the eﬀect of a zero
Suppose we have the closedloop transfer function C (s)
N (s) =
R(s)
D(s) Consider a single zero on the real axis C (s) =
R(s) and a step input C (s) = Then the time response can be written as c(t) = c1 (t) + c2 (t)
c1 (t) =
c2 (t) =
5 Now c2 (t) is the c1 (t) is the c1 (t) is the c(t) =
• Discussion of attached ﬁgures. 6 ...
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 Fall '03
 willcox
 Aeronautics

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