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STABILIZATION OF THE INVERTED PENDULUM SYSTEM USING THREE
ZONE CONTROL STRATEGY
1. Sandip JoardarMaster of Electrical EngineeringElectrical Measurement and InstrumentationDept. of Electrical EngineeringJadavpur University
2. Somnath GaraiAssistant Professor
Dept. of Instrumentation and Control Engineering
CIEM, WBUT
3/31/2013 1SANDIP JOARDAR, JU
SOMNATH GARAI, CIEM
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CONTENTS
• INTRODUCTION
• PHYSICAL MODELLING OF THE SYSTEM
• IMPLEMENTATION OF THE THREE ZONE CONTROL STRATEGY
• SIMULATION RESULTS
• APPLICATION
• CONCLUSION
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SOMNATH GARAI, CIEM
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INTRODUCTION
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SOMNATH GARAI, CIEM3
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INTRODUCTION
The stabilization of ansystem is a classical control problem. Theinverted pendulum (IP) is among the
systems to control in the field ofcontrol engineering. The design andimplementation of a control strategy for the
and equally as it isan .
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INVERTED PENDULUM
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POSITIONS OF THE I.P.
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PHYSICAL MODELLING OF THE SYSTEM
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FREE BODY DIAGRAMS
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EQUATIONS OF MOTION
The system dynamics is represented by thefollowing equations :-
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EQUATIONS OF MOTION (contd.)
Linearization:-
The system dynamics, after linearization, isrepresented by the following equations :-
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OUTPUT RESPONSE
The angular displacement of the PENDULUM ROD whensubjected to an Impulse input is shown in the figure below.
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OUTPUT RESPONSE(contd.)
The linear displacement of the cart when subjected to anImpulse input is shown in the figure below.
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IMPLEMENTATION OF THE THREE ZONE
CONTROL STRATEGY
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THREE ZONE CONTROL STRATEGY
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SWING UP CONTROL STRATEGY
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STATE FEEDBACK CONTROL STRATEGY
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STATE FEEDBACK CONTROL STRATEGY (contd.)
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STATE FEEDBACK CONTROL STRATEGY (contd.)
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0 2 4 6-0.1
0
0.1
0.2
0.3psi(x1) vs. time
t ( in seconds
Psi
(x1)
0 2 4 6-2
-1
0
1psi-dot(x2) vs. time
t ( in seconds
Psi
-dot
(x2)
0 2 4 6-0.2
0
0.2
0.4
0.6X-displacement(x3) vs. time
t ( in seconds
X-d
ispl
acem
ent(
x4)
0 2 4 6-0.5
0
0.5
1X-dot-velocity(x4) vs. time
t ( in seconds
X-d
ot(x
4)
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ZONE OF SMOOTH SWITCHING
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SIMULATION RESULTS
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OUTPUT RESPONSES
ANGULAR DISPLACEMENT
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OUTPUT RESPONSES (contd.)
ANGULAR VELOCITY
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OUTPUT RESPONSES (contd.)
ANGULAR ACCELERATION
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OUTPUT RESPONSES (contd.)
LINEAR DISPLACEMENT
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OUTPUT RESPONSES (contd.)
LINEAR VELOCITY
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OUTPUT RESPONSES (contd.)
LINEAR ACCELERATION
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APPLICATION
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AREAS OF APPLICATION
• Attitude Control of Space Boosters
• Automatic Aircraft Landing System
• Balancing of a ship on a tide
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CONCLUSION
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CONCLUSION
Therefore, total operational region, has been
.In the only the linear state feedbackcontroller acts, in theboth the linear state feedback control strategy and theswing – up control strategy acts simultaneously, andfinally, in the only the swing – upcontrol law acts. Therefore,
, to stabilize the pendulum rod at itsvertically upright position, is
and the obtained experimental resultswere very .
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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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THANK YOU
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