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Inverted
Pendulum
Analysis, Design and
Implementation

IIEE Visionaries

Document Version 1.0

Reference:
The work included in this document has been carried out in the Instrumentation
and Control Lab at the Institute of Industrial Electronics Engineering, Karachi,
Pakistan.

CONTENTS

4

W HAT'S INSIDE T HIS REPORT
(CONTENTS IN DETAIL)

4

CONTENTS IN DETAIL

…

3

THE AUTHORS

…

6

PREFACE

…

9

INTRODUCTION

…

12

…

19

…

26

…

34

…

36

ü
ü

ABOUT THE AUTHOR
TECHNICAL ADVISOR

ü
ü

INTRODUCTION TO INVERTED PENDULUM
APPLICATIONS OF INVERTED PENDULUM
o SIMULATION OF DYNAMICS OF A ROCKET VEHICLE
o MODEL OF A HUMAN STANDING STILL
ü PROBLEM DESCRIPTION

MATHEMATICAL WORK
ü

MATHEMATICAL ANALYSIS
o SETUP DESCRIPTION
o INVERTED PENDULUM SYSTEM EQUATIONS
o ACTUATION MECHANISM
o TRANSFER FUNCTION OF THE W HOLE SYSTEM
ü SYSTEM P ARAMETERS

ANALYSIS OF UNCOMPENSATED SYSTEM
ü
ü
ü
ü
ü
ü

POLE ZERO MAP OF UNCOMPENSATED OPEN LOOP S YSTEM
IMPULSE RESPONSE OF UNCOMPENSATED OPEN LOOP S YSTEM
ROOT LOCUS OF THE UNCOMPENSATED S YSTEM
STEP RESPONSE OF UNCOMPENSATED OPEN LOOP S YSTEM
SIMULINK M ODEL FOR THE OPEN LOOP IMPULSE RESPONSE
SIMULINK M ODEL FOR THE OPEN LOOP STEP RESPONSE

COMPENSATION DESIGN
ü HOW CAN THE COMPENSATION BE DESIGNED?
(POSSIBLE OPTIONS)

ROOT LOCUS SYSTEM DESIGN
ü W HY COMPENSATION IS REQUIRED?
ü COMPENSATION GOALS
ü COMPENSATION DESIGN
T HE SISO DESIGN T OOL
ü W HAT IS THE SISO DESIGN TOOL
ü IMPORTING MODELS INTO THE SISO DESIGN TOOL
ü OPENING THE SISO DESIGN TOOL
ü DESIGN SPECIFICATIONS
ü ROOT LOCUS DESIGN WITH SISO DESIGN TOOL
ü ADDING POLES AND ZEROS TO THE COMPENSATOR
ü PROCEDURE

ANALYSIS OF COMPENSATED SYSTEM

…

43

…

56

…

61

CONCLUSION

…

63

APPENDIX

…

65

ü
ü
ü
ü
ü
ü
ü

POLE-ZERO MAP OF COMPENSATED OPEN LOOP S YSTEM
ROOT LOCUS OF THE COMPENSATED S YSTEM
POLE-ZERO MAP OF COMPENSATED CLOSED-LOOP S YSTEM
IMPULSE RESPONSE OF PID COMPENSATED SYSTEM
STEP RESPONSE OF PID COMPENSATED S YSTEM
CONCLUSION OF COMPENSATION ANALYSIS
SIMULINK M ODEL FOR CLOSED-LOOP STEP RESPONSE OF COMPENSATED
SYSTEM
ü SIMULINK M ODEL FOR CLOSED-LOOP IMPULSE RESPONSE OF COMPENSATED
SYSTEM
ü SIMULINK M ODEL FOR RESPONSE TO DISTURBANCE IN THE FORCE ON THE
CART OF COMPENSATED S YSTEM
ü SIMULINK M ODEL FOR RESPONSE TO DISTURBANCE IN THE POSITION OF
INVERTED BROOM OF COMPENSATED S YSTEM

PRACTICAL IMPLEMENTATION
ü CONTROLLER IMPLEMENTATION
ANALOG PID CONTROLLER DESIGNS
ü DESIGN 1: IDEAL PID ALGORITHM
ü DESIGN 2: P ARALLEL PID ALGORITHM
ü DESIGN 3: SERIES PID A...
Inverted
Pendulum
Analysis, Design and
Implementation
IIEE Visionaries
Document Version 1.0
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