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Paper presentation code no. 086-eei-17
1. STABILIZATION OF THE INVERTED
PENDULUM SYSTEM USING THREE
ZONE CONTROL STRATEGY
1. Sandip Joardar
2. Somnath Garai
Master of Electrical Engineering
Assistant Professor
Electrical Measurement and Instrumentation
Dept. of Instrumentation and
Dept. of Electrical Engineering
Control Engineering
Jadavpur University
CIEM, WBUT
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SOMNATH GARAI, CIEM
2. CONTENTS
• INTRODUCTION
• PHYSICAL MODELLING OF THE SYSTEM
• IMPLEMENTATION OF THE THREE ZONE
CONTROL STRATEGY
• SIMULATION RESULTS
• APPLICATION
• CONCLUSION
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3. INTRODUCTION
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4. INTRODUCTION
The stabilization of an
system is a classical control problem. The
inverted pendulum (IP) is among the
systems to control in the field of
control engineering. The design and
implementation of a control strategy for the
and equally as it is
an .
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9. EQUATIONS OF MOTION
The system dynamics is represented by the
following equations :-
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10. EQUATIONS OF MOTION (contd.)
Linearization:-
The system dynamics, after linearization, is
represented by the following equations :-
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11. OUTPUT RESPONSE
The angular displacement of the PENDULUM ROD when
subjected to an Impulse input is shown in the figure below.
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12. OUTPUT RESPONSE(contd.)
The linear displacement of the cart when subjected to an
Impulse input is shown in the figure below.
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13. IMPLEMENTATION OF
THE THREE ZONE
CONTROL STRATEGY
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14. THREE ZONE CONTROL STRATEGY
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15. SWING UP CONTROL STRATEGY
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17. STATE FEEDBACK CONTROL STRATEGY
(contd.)
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18. STATE FEEDBACK CONTROL STRATEGY
(contd.)
psi(x1) vs. time psi-dot(x2) vs. time
0.3 1
0.2
Psi-dot(x2)
0
Psi(x1)
0.1
-1
0
-0.1 -2
0 2 4 6 0 2 4 6
t ( in seconds t ( in seconds
X-displacement(x3) vs. time X-dot-velocity(x4) vs. time
0.6 1
X-displacement(x4)
0.4
0.5
X-dot(x4)
0.2
0
0
-0.2 -0.5
0 2 4 6 0 2 4 6
t ( in seconds t ( in seconds
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19. ZONE OF SMOOTH SWITCHING
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27. APPLICATION
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28. AREAS OF APPLICATION
• Attitude Control of Space Boosters
• Automatic Aircraft Landing System
• Balancing of a ship on a tide
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29. CONCLUSION
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30. CONCLUSION
Therefore, total operational region, has been
.
In the only the linear state feedback
controller acts, in the
both the linear state feedback control strategy and the
swing – up control strategy acts simultaneously, and
finally, in the only the swing – up
control law acts. Therefore,
, to stabilize the pendulum rod at its
vertically upright position, is
and the obtained experimental results
were very .
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31. REFERENCES
[1] Khalil Sultan, “INVERTED PENDULUM, ANALYSIS, DESIGN AND IMPLEMENTATION”, IIEE Visionaries, Document Version 1.0.
[2] Bugeja M. (2003), Non – linear swing up and stabilizing control of an inverted pendulum system, EUROCON 2003, Ljubljana, Slovenia.
[3] Udhayakumar K. and Lakshmi P. (2007), Design of Robust Energy control for cart – inverted pendulum, International Journal of
Engineering and Technology, Vol. 4, No. 1, 2007, pp. 66-76.
[4] A. Stephenson, “On a new type of dynamical stability”, Manchester Memoirs 8, 1–10 (1908).
[5] D.J. Acheson, “A pendulum theorem”, Proc. R.Soc. Lond. A (443), 239–245 (1993).
[6] D.J. Acheson, “Upside-down pendulums”, Nature 336, 215– 216 (1993).
[7] J. Baillieul and B. Lehman, “Open-loop control using oscillatory input”, in CRC Control Handbook, edited by W.S.Levine, 967–980 (1996).
[8] Z. Minglian, H. Jiankang, and H. Weidong, “Human imitating intelligent control and triple inverted pendulum”, Chinese J. of Aeronautics
2, 135–146 (1996).
[9] K.G. Eltohamy and C.Y. Kuo, “Real time stabilization of a triple link inverted pendulum using single control input”, IEE Proc- Control
Theory Appl. 5, 498–504 (1997).
[10] K.G. Eltohamy and C.Y. Kuo, “Nonlinear optimal control of a triple link inverted pendulum with single control input”, Int. J. Control 2,
239–256 (1998).
[11] S. Mori, H. Nishihara, and K. Furuta, “Control of unstable mechanical systems: Control of pendulum”, Internat. J. Control 23, 673–692
(1976).
[12] K. Furuta, M. Yamakita, and S. Kobayashi, “Swing-up control of inverted pendulum using pseudo-state feedback”, Proc. Instn. Mech.
Engrs., 206, 263–269 (1993).
[13] Aström K.J. and Furuta K., (1996), Swinging up a pendulum by energy control, presented at 13th IFAC World Congress.
[14] M. Yamakita and K. Furuta, “Toward robust state transfer control of titech double pendulum”, The Aström Symposium on Control, ed.
by Wittenmark and Rantzer, 73–269 (1999).
[15] Fradkov A.L., “Speed – gradient laws of control and evolution”, Proc. 1st European Control Conference, Grenoble, pp. 1861 – 1865,
1991.
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32. THANK YOU
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