Feedback Systems & Applications
Examples from
MIT 6.003
Signals & Systems
Spring 2011 (Prof. Q. Hsu) &
Spring
2010 (Prof. D. Freeman)
2
Why use Feedback?
•
Reducing Nonlinearities
•
Reducing Sensitivity to Uncertainties and Variability
•
Stabilizing Unstable Systems
•
Reducing Effects of Disturbances
•
Tracking
•
Shaping System Response Characteristics (bandwidth/speed)
A Typical
Feedback
System
feed forward
feed backward
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, EE 6.003 S11
Controller
3
One Motivating Example
–– pointing a telescope
Open-loop System
Aim –– shoot
Closed-loop
Feedback
System
Not done until it is pointed
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, EE 6.003 S11
4
Another example ––
A robot car
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, EE 6.003 S11

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System Function of a Closed-loop System
Example:
A basic Feedback System — By its nature, we
are dealing with real physical systems.
⇒
They are all
causal
.
E
(
s
)
=
X
(
s
)
"
R
(
s
)
=
X
(
s
)
"
G
(
s
)
Y
(
s
)
"
Q
(
s
)
=
Y
(
s
)
X
(
s
)
=
H
(
s
)
1
+
G
(
s
)
H
(
s
)
Y
(
s
)
=
H
(
s
)
E
(
S
)
=
H
(
s
)[
X
(
s
)
"
G
(
s
)
Y
(
s
)]
G
(
s
)
Y
(
s
)
–– System function
of the close-loop
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, EE 6.003 S11
6
Can show for any closed-loop systems, the system function is given by
Black
±
s formula
(H. S. Black in the 1920
±
s, along with Nyquist and Bode):
Forward gain — total gain along the forward path from the
input
to the
output
the gain of an adder is
≡
1
Loop gain — total gain along the closed loop — shared by all the system functions
General formula for a closed-loop system:
Black
±
s Formula
Closed - loop system function
=
forward gain
1 - loop gain
Q
(
s
)
=
Y
(
s
)
X
(
s
)
=
H
(
s
)
1
+
G
(
s
)
H
(
s
)
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, EE 6.003 S11
Feedback and Control
Using feedback to enhance performance.
Examples:
•
improve performance of an op amp circuit.
•
control position of a motor.
•
reduce sensitivity to unwanted parameter variation.
•
reduce distortions.
•
stabilize unstable systems
≠
magnetic levitation
≠
inverted pendulum
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, EE 6.003 S10
13
•
P
(
s
) — unstable
•
Design
C
(
s
),
G
(
s
) so that the closed-loop system
is stable
⇒
poles
of
Q
(
s
) =
roots
of 1+
C
(
s
)
P
(
s
)
G
(
s
) on
LHP
(Do not use pole/zero cancellation.)
)
(
)
(
)
(
1
)
(
)
(
)
(
s
G
s
P
s
C
s
P
s
C
s
Q
+
=
Stabilization of Unstable Systems
http://ocw.mit.edu/courses/
, EE 6.003 S11