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

CONTENTS

• INTRODUCTION

• PHYSICAL MODELLING OF THE SYSTEM

• IMPLEMENTATION OF THE THREE ZONE CONTROL STRATEGY

• SIMULATION RESULTS

• APPLICATION

• CONCLUSION

3/31/2013 2SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM

INTRODUCTION

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM3

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 .

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM4

INVERTED PENDULUM

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM5

POSITIONS OF THE I.P.

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM6

PHYSICAL MODELLING OF THE SYSTEM

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM7

FREE BODY DIAGRAMS

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM8

EQUATIONS OF MOTION

The system dynamics is represented by thefollowing equations :-

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM9

EQUATIONS OF MOTION (contd.)

Linearization:-

The system dynamics, after linearization, isrepresented by the following equations :-

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM10

OUTPUT RESPONSE

The angular displacement of the PENDULUM ROD whensubjected to an Impulse input is shown in the figure below.

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM11

OUTPUT RESPONSE(contd.)

The linear displacement of the cart when subjected to anImpulse input is shown in the figure below.

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM12

IMPLEMENTATION OF THE THREE ZONE

CONTROL STRATEGY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM13

THREE ZONE CONTROL STRATEGY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM14

SWING UP CONTROL STRATEGY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM15

STATE FEEDBACK CONTROL STRATEGY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM16

STATE FEEDBACK CONTROL STRATEGY (contd.)

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM17

STATE FEEDBACK CONTROL STRATEGY (contd.)

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM18

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)

ZONE OF SMOOTH SWITCHING

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM19

SIMULATION RESULTS

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM20

OUTPUT RESPONSES

ANGULAR DISPLACEMENT

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM21

OUTPUT RESPONSES (contd.)

ANGULAR VELOCITY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM22

OUTPUT RESPONSES (contd.)

ANGULAR ACCELERATION

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM23

OUTPUT RESPONSES (contd.)

LINEAR DISPLACEMENT

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM24

OUTPUT RESPONSES (contd.)

LINEAR VELOCITY

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM25

OUTPUT RESPONSES (contd.)

LINEAR ACCELERATION

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM26

APPLICATION

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM27

AREAS OF APPLICATION

• Attitude Control of Space Boosters

• Automatic Aircraft Landing System

• Balancing of a ship on a tide

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM28

CONCLUSION

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM29

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 .

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM30

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.

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM31

THANK YOU

3/31/2013SANDIP JOARDAR, JU

SOMNATH GARAI, CIEM32