TIME DELAYED CONTROL OF A PENDULUM Analysis

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TIME DELAYED CONTROL OF A PENDULUM
(AnalysisExperiments)

• Objective To force a pendulum through a desired
  trajectory using time delayed full state
  feedback.
• Highlights of exercise
• Stability analysis of the dynamics. The Direct
  
• Method (IEEE-TAC Vol47, No.5,
  793-797,2002, Olgac Sipahi)
• 2) Validation of analytical findings

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Pendulum Control System
Amplifier(Driver)
Computer (Controller)
Pendulum(System)
DC Motor(Actuator) Encoder(Feedback)
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System Overview
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1) Stability Analysis

• There are stability switches for the system at
  hand (system
• details are suppressed). They dictate the
  stability outlook

Note1 Delay (t)0, stable control system
Note2 Listed delays cause resonance at
corresponding frequencies
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Experimental validation of stability switching
points (using impulse responses)
0.25
0.2

0.2
0.15
0.15
0.1
0.1
0.05
0.05
q rad
q rad
0
0
-0.05
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
-0.2
-0.25
-0.25
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Time
Time
t 63.5 ms, w7.46 rad/s
t 425ms, w5.88 rad/s
Note All traces are in the steady state
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Experimental validation of stability switching
points (using impulse responses) Contd
q rad
q rad
t 928 ms, w7.42 rad/s
t 1426 ms, w5.96 rad/s
7

Experimental validation of stability switching
points (using impulse responses) Contd
q rad
t 1849 ms, w7.28 rad/s
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Achieving Optimum Rightmost Pole Using Time
Delay
Method The stability charts given in
Mechatronics, Vol.12, 393-413,2002,
Filipovic-Olgac. Claim Optimum rightmost
poles can be achieved by adding time delay to
system feedback without any other change in
control. Therefore performance of system can be
improved.
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Rightmost Pole Influence on Impulse Response
Stable interval 1 0 lt t lt 63.5 ,
optimum t1 is 0 ms
2 432.4 lt t lt 911.4 , optimum t2 is 706
ms 3 1523 lt t lt
1759 , optimum t3 is 1666 ms
Impulse
Impulse
Stable, t2 706 ms Settling time (5) 1.71
sec
Stable, t1 0 ms Settling time (5) 3.45 sec

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Rightmost Pole Influence on Impulse
Response Contd
Impulse
Impulse
Stable, t3 1666 ms Settling time (5) 4.1
sec
Unstable, t 200 ms
Conclusion t2 performs better than both t1
and t3
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2) Trajectory Tracking I / Experimental
verificationDesired Trajectory
qd0.1sin(12.56 t)
Time
t1 0 ms , gives best transient performance
within interval 1 Figure shows the steady state.
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Trajectory Tracking I (Contd)
t2 706 ms, gives best transient performance
within interval 2 Figure shows the steady state
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Trajectory Tracking I (Contd)
t3 1666 ms, gives best transient performance
within interval 3 Figure shows the steady state.
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Conclusion of Trajectory Tracking I
All optimal stable operations for t1 , t2 ,
t3 show similar tracking errors (at the steady
state). Their respective transient
characteristics are on pages 9,10 .
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Trajectory Tracking II, Dual frequencyDesired
Trajectory qd0.1sin(12.56 t)0.02sin(9.4 t)
t1 0 ms, gives best transient performance
within interval 1 Figure shows the steady state.
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Trajectory Tracking II (Contd)
t2 706 ms, gives best transient performance
within interval 2 Figure shows the steady state.

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Trajectory Tracking II (Contd)
t3 1666 ms, gives best transient performance
within interval 3 Figure shows the steady state
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Conclusion of Trajectory Tracking II
Tracking for dual frequency trajectory exhibit
similar tracking errors (at the steady state)
for all three stable pockets.
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Results

• Stability pockets are analytically determined
  and experimentally verified.
• Time delay on feedback can be used as a parameter
  which improves the control performance.