Transcript founderpresent

Spring 2003
CS-EE 481
The Inverted Pendulum
Authors
Jason Boyce
Jen Miller
Advisors
Dr. Robert Albright
Dr. Peter Osterberg
Industry Representative
Mr. Howard Voorheis
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University of Portland
School of Engineering
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CS-EE 481
Agenda
•
•
•
•
•
•
Introduction
Background
Methods
Results
Conclusions
Demonstration
Founder’s Day, 2003
University of Portland
Jen
Jen
Jen & Jason
Jason
Jason
Jen & Jason
School of Engineering
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Spring 2003
CS-EE 481
Introduction
•
•
•
•
Special Thanks
The Inverted Pendulum Problem
Analog Servo Mechanism
Purpose: Control System Demonstration
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University of Portland
School of Engineering
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CS-EE 481
Background
• Feedback Control
• Goal: Keep the pendulum from falling
• Functional Specifications
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CS-EE 481
Methods
•
•
•
•
•
•
•
•
Research
Mathematical Modeling
Feedback Diagrams
Transfer Functions
Stability Analysis
Circuit Design
Construction
Testing
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CS-EE 481
Research
– Kent Lundberg.
http://web.mit.edu/klund/www/pendulum.pdf
– William McC. Siebert. Circuits, Signals, and
Systems. MIT Press, Cambridge,
Massachusetts, 1986.
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University of Portland
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CS-EE 481
Mathematical Modeling
• Newtonian Physics
– Dynamics of Pendulum
• Transfer Functions
– Frequency Domain
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CS-EE 481
Feedback Diagram
c
Qc +
+
-1
-
Q
k´
M(s)
-
G(s)
K(s)
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University of Portland
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CS-EE 481
Transfer Functions
X (s)
M (s) 

V (s)
K (s) 
kM
s (  M s  1)
Ks 1
Ks
s / g
2
G (s) 
L 
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(  L s  1 )(  L s  1 )
l
g
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Closed Loop Transfer Function
x(s)
 (s)
k M (  K s  1 )(  L s  1 )
2

2


kM
2 2
s  K (  M s  1 )(  L s  1 )  ( )(  K s  1 ) 
g


2
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CS-EE 481
Stability Analysis
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CS-EE 481
Circuit Design
+15 V
R2
10k
C2
9.3 uF
+15V
10k
100
33
_
+
+
_
10k
+
_
10k
+
+
10k
_
Rotary Pot 1
_
10k
-15V
-15V
+15V
10k
+ _
Multi-turn
Potentiometer
10k
+
10k
+
_
10kW
24 V DC
MOTOR
_
Tachometer
_
+
_
+
10kW
10kW
-15 V
10k
10k
+
_
+
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University of Portland
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CS-EE 481
Cart-Track Schematic
Pendulum
(t)
V(t)
X(t)
Hinge
Motor
Motor Shaft
Pulley
Cart
Track
Belt
15
15
15
15
Potentiometer
Voltage proportional
to (t)
Ground
Multi-turn potentiometer
Voltage proportional to x(t)
Ground
From William McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Mass., 1986
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School of Engineering
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CS-EE 481
Conclusions
• Analog Servo Mechanism
• Control System Demonstration
– Aids in understanding control systems
– Easy problem to conceptualize
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University of Portland
School of Engineering
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Demonstration
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