Computer-aided nonlinear control system design

Computer-aided Nonlinear Control System Design

Amir Nassirharand

Computer-aided Nonlinear

Control System Design

Using Describing Function Models

123

Amir Nassirharand

Faculty of Engineering

The University of Nottingham

Malaysia Campus

Jalan Broga

43500 Semenyih

Malaysia

ISBN 978-1-4471-2148-0 e-ISBN 978-1-4471-2149-7

DOI 10.1007/978-1-4471-2149-7

Springer London Dordrecht Heidelberg New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2011942925

© Springer-Verlag London Limited 2012

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To Shabnam, Sam, and Daniel who suffer as

I do my research

Of course to Nicole, Jessie, Kaydence, and

Kiera

To my mother, Shokat Vahdat, and my late

father, Hassan, who are proud of this work

Preface

The aim of this book is to give an in-depth treatment of computer-aided design

of nonlinear control systems using describing function models. The approach

is systematic, and it is based on a describing function platform. A large class

of nonlinear systems is covered with no restrictions on the number, type, and

arrangement of nonlinear terms, system order, and the number of inputs and/or

the outputs. Unlike other books in the area of nonlinear control that are highly

mathematically oriented, this book is software oriented.

It is believed that today’s tight control system performance requirements do not

allow a linear frame of mind for the designer. The designer’s first choice to linearize

a nonlinear model places so many unrealistic restrictions on the real model that

drives up the costs of final product.

A full nonlinear frame of mind for design of nonlinear feedback control systems

requires a computer-aided design approach. Many experts argue that such an

approach would not guarantee stability. Here, without loss of generality, stability is

demonstrated using the fact that if the sinusoidal input describing function models

of a nonlinear system is obtainable, then that nonlinear system is stable around the

operating regimes of interest.

I like to emphasize that having a full nonlinear frame of mind does not mean that

designers should throw away other available linear and more restricted nonlinear

control literature. On the contrary, one should capitalize on others’ intellectual

developments to bring good low-cost products for mankind. This book is a vehicle to

account for nonlinear effects, and it simultaneously enables the designer to interface

with other control system design works.

A person with a thorough knowledge of calculus and an elementary knowledge

of numerical analysis and software engineering would most benefit from this book.

This book should be of interest to aerospace, chemical, electrical, mechanical, and

mechatronics students and practicing control engineers as well as other practical

engineers and managers who are simply interested in design of low-cost real-life

nonlinear feedback systems.

vii

viii Preface

Many key MATLAB®

functions are developed and presented to allow easy design

of nonlinear feedback systems. The algorithms and the corresponding MATLAB®

functions are demonstrated by solving example problems of the sort encountered in

mechatronics and aerospace. A versatile controller design software based on the

presented describing function design platform (using MATLAB®

Graphics User

Interface (GUI)) is under development which will be commercially available by

early 2013.

Semenyih, Malaysia Amir Nassirharand

Acknowledgments

First, I like to sincerely thank my mentor, teacher, friend, counselor, adviser,

and academic father, Professor James H. Taylor, for his continued support and

encouragement throughout my academic and professional life. He has always been

there to answer my questions.

I also like to thank my students. Special thanks to Professor Abhijit Patwardhan

whose contributions on search of the parameter in simultaneous stabilization theory

shed light on this important problem while he was pursuing his Master’s degree at

the University of Kentucky; my apologies if I was too hard on you! Mr. Li See Chew,

my Ph.D. student, who is working on extension of some of the works presented here

to the unstable case in a direct way and who is a pioneer in experimental work among

my group members, deserves special thanks for putting up with my supervision.

I sincerely thank Mr. Sze Hong Teh (University of Nottingham – Malaysia Campus,

(UNMC)) for extending my Ph.D. work on dual-range linear controller design from

the single-variable nonlinear case to the multi-variable nonlinear case. His other

works on idle speed control of automobile engines, bank-angle control of aerospace

vehicles with hard nonlinearities, and initiation of projects in chemical engineering

and continuing to play with the spacecraft attitude control problem have contributed

to preparation of this monograph. His collaboration in developing a versatile

software package to support this monograph is acknowledged. I certainly have been

lucky to have the opportunity to work with you. Support of Brandon (my Master’s

degree student at UNMC) is appreciated; I appreciate his aggressiveness to do his

thesis on application of describing function techniques to nonlinear multivariable

problem of satellite attitude control. Finally many thanks to all those undergraduate

Mechanical Engineering and Mechatronics students at UNMC (Ahmed, Ajitpal,

Aimen, Arun, Athira, Awaii, Bartholomew, Bushra, Chan, Chin, Gift, Hammad,

Khawja, Lee, Lim, Mohd, Muhammad, Prem, Saeid, Sameer, Sanz, Sean, Shola,

Tan, Tee, Yuvanraj) who selected my projects in the dynamic systems and control

field as their final year theses, final year design projects, and final year group

development projects.

Many thanks to Dr. Karimi and his research group (Norouz, Seyed Mehdi,

Seyed Reza, Ali, Davood, ::: ) at K.N.T. University of Technology who taught me

ix

x Acknowledgments

the practical aspects of modeling, simulation, and control of engineering systems;

they substantially contributed to my professional growth that is reflected in this

monograph. Contributions of Seyed Reza in solving the linear and nonlinear H1

control problems and nonlinear lead-lag design problem given in this monograph

are greatly appreciated. Contributions of Norouz in application of algebraic linear

multivariable controller design approach of C. T. Chen to describing function-based

single range controllers are also acknowledged.

I like to express my sincere appreciation to Mr. Hadjirasouliha (former President

of the Farab Company) and other personnel (Ardavan, Ata, Farshid, Farzan,

Massoud, Mehdi, Mohamad Reza, Pouria, Samira, ::: ) at Farab for supporting me

and allowing me to do my research while I was doing my engineering tasks.

Special thanks to Professor Seamus Garvey and Professor Michael Wood of

University of Nottingham (UK Campus) for their invaluable support of my teaching

in terms of providing their lecture notes and powerpoint slides as well as answering

my detailed questions on Systems Engineering, Design of Experiments, Dynamic

Systems, Reliability, and Robustness; this helped me allocate the extra time required

to do my research and complete this monograph.

I appreciate the help, guidance, and supervision of Dr. Stewart McWilliam and

Andrew Spowage while at UNMC. Special thanks to Professor Mike Cloke (former

Dean) and Professor Ian Pashby (CEO) for giving me the opportunity to continue

my academic career at UNMC.

The support, guidance, direction, and critique of Professor James Rowland

(University of Kansas) of my initial proposal for my Ph.D. studies, which led

to my pursuit of controller design for nonlinear systems, are acknowledged. His

undergraduate text on control systems inspired me to pursue and develop closed￾form solutions for design of linear lead-lag, linear PID, and other classical linear

controllers (that are given in the Appendices) in an algorithmic and systematic way.

I also wish to express my gratitude to Mr. Oliver Jackson (Springer-Verlag

London) for keeping an open mind in making the decision to go ahead with

publication of this monograph. His understanding and support for the need for

a software-oriented approach to design of nonlinear feedback control systems is

noteworthy. It is noted that without the help and support of Ms. Charlotte Cross

(Springer-Verlag London), this manuscript would not have been completed in time;

special thanks to Ms. Cross for her help in obtaining the required permissions for

use of copy righted material.

Last but not least, supports of my sisters Zohreh and Gita are appreciated. I also

thank Mr. Hassan Bojnordiazad and Nasrin Rayat as they have touched my life in a

positive way. Their encouragement keeps me going. I like to thank John and Patsy

Hiser for contributing to my spiritual growth.

Thank you all!

Contents

1 Introduction .................................................................. 1

1.1 What’s New? ........................................................... 1

1.2 Design Platform ........................................................ 1

1.3 Objectives............................................................... 3

1.4 Software................................................................. 4

1.5 Organization of the Monograph ....................................... 4

2 Frequency Domain Modeling ............................................... 5

2.1 Early Formulation ...................................................... 5

2.1.1 Example: Limiter .............................................. 6

2.2 Modern Formulation ................................................... 9

2.2.1 Single-Input Single-Output Case ............................. 9

2.2.2 Multi-input Multi-output Case ................................ 13

3 Single-Range Controller Design ............................................ 23

3.1 Procedure ............................................................... 23

3.2 Example – Single-Variable Case: A Decoupled Liquid

Propellant Engine....................................................... 25

3.3 Example – Multivariable Case: Idle Speed Control Problem ........ 34

4 Dual-Range Controller Design ............................................. 43

4.1 Background ............................................................. 43

4.2 Controller Synthesis.................................................... 44

4.3 Software................................................................. 46

4.4 Example 1: A Single-Variable Servo System ......................... 46

4.5 Example 2: Multivariable Bank Angle Control Problem ............. 48

5 Multirange Nonlinear Controller Design.................................. 57

5.1 Background ............................................................. 57

5.2 Controller Synthesis.................................................... 57

5.2.1 Describing Function Inversion (Nassirharand 2009a)* ...... 60

5.3 Software................................................................. 66

5.3.1 Software for Step 5 ............................................ 66

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xii Contents

5.3.2 Software for Describing Function Inversion

(Nassirharand 2009a).......................................... 69

5.4 Example: Single-Variable Case: Servo System ....................... 72

5.4.1 Nonlinear PID Controller Design ............................. 72

5.4.2 Nonlinear Lead-Lag Controller Design

(Nassirharand and Mousavi Firdeh 2008) .................... 75

5.4.3 Nonlinear H1 Controller Design............................. 78

5.5 Example: Multivariable Case: Bank Angle Control of a UAV ....... 82

6 Experimental Study: Unstable SISO Systems ............................ 87

6.1 Background ............................................................. 87

6.2 Experimental Rig ....................................................... 87

6.3 Nonlinear PID Synthesis ............................................... 88

6.4 Conclusions............................................................. 95

Appendix A Closed-Form Solution to Linear Classical Controllers...... 97

A.1 Linear Lead-Lag Controller Design ................................... 97

A.1.1 Software........................................................ 99

A.1.2 Demonstration Example Problem............................. 100

A.2 Linear PID Controller Design.......................................... 103

A.2.1 Software........................................................ 104

A.2.2 Demonstration Example Problem............................. 104

A.3 Proportional Plus Rate Feedback Controller Design.................. 105

A.3.1 Software........................................................ 109

A.3.2 Demonstration Example Problem............................. 111

A.4 Proportional-Integral Plus Rate Feedback Controller Design ........ 112

A.4.1 Software........................................................ 116

A.4.2 Demonstration Example Problem............................. 116

Appendix B Algebraic Linear Multivariable Controller Design .......... 121

B.1 Theory................................................................... 121

B.1.1 Discussion (Nassirharand and Karimi 2004c)**.............. 125

B.1.2 Systematizing the Design Approach.......................... 127

B.2 Software................................................................. 127

Appendix C Additional Example Problems for Multivariable

Single-Range Controller Design ............................... 129

C.1 Example Problem: One Combustion Chamber Liquid

Propellant Engine....................................................... 129

C.2 Example Problem: Five Combustion Chamber Liquid

Propellant Engine....................................................... 137

C.3 Example Problem: An Unstable Multivariable Robot Arm .......... 146

Appendix D Dual-Range Controllers and Simultaneous

Stabilization Theory............................................. 153

D.1 Bezout’s Identity ´ ....................................................... 153

D.1.1 Multivariable Case............................................. 154

Contents xiii

D.1.2 Single-Variable Case .......................................... 155

D.1.3 Software........................................................ 159

D.2 DRLCD Using Simultaneous Stabilization Theory ................... 159

D.2.1 Theory and Algorithm......................................... 159

D.2.2 Software........................................................ 165

D.2.3 Example Problems............................................. 165

D.2.4 Discussion ..................................................... 171

References......................................................................... 175

Index ............................................................................... 179