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APPLICATIONS OF

NONLINEAR CONTROL

Edited by Meral Altınay

Applications of Nonlinear Control

Edited by Meral Altınay

Published by InTech

Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2012 InTech

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Statements and opinions expressed in the chapters are these of the individual contributors

and not necessarily those of the editors or publisher. No responsibility is accepted for the

accuracy of information contained in the published chapters. The publisher assumes no

responsibility for any damage or injury to persons or property arising out of the use of any

materials, instructions, methods or ideas contained in the book.

Publishing Process Manager Anja Filipovic

Technical Editor Teodora Smiljanic

Cover Designer InTech Design Team

First published June, 2012

Printed in Croatia

A free online edition of this book is available at www.intechopen.com

Additional hard copies can be obtained from orders@intechopen.com

Applications of Nonlinear Control, Edited by Meral Altınay

p. cm.

ISBN 978-953-51-0656-2

Contents

Preface IX

Chapter 1

Application of Input-Output Linearization 1

Erdal Şehirli and Meral Altinay

Chapter 2

Lyapunov-Based Robust and Nonlinear Control

for Two-Stage Power Factor Correction Converter 21

Seigo Sasaki

Chapter 3

Nonlinear Control Applied to the Rheology of

Drops in Elongational Flows with Vorticity 37

Israel Y. Rosas, Marco A. H. Reyes, A. A. Minzoni and E. Geffroy

Chapter 4

Robust Control Research of Chaos Phenomenon

for Diesel-Generator Set on Parallel Connection 57

Man-lei Huang

Chapter 5

A Robust State Feedback Adaptive

Controller with Improved Transient Tracking Error

Bounds for Plants with Unknown Varying Control Gain 79

A. Rincon, F. Angulo and G. Osorio

Chapter 6

A Robust Motion Tracking

Control of Piezo-Positioning

Mechanism with Hysteresis Estimation 99

Amir Farrokh Payam,

Mohammad Javad Yazdanpanah and Morteza Fathipour

Chapter 7

Nonlinear Observer-Based Control Allocation 115

Fang Liao, Jian Liang Wang and Kai-Yew Lum

Chapter 8

Predictive Function Control of the

Single-Link Manipulator with Flexible Joint 129

Zhihuan Zhang and Chao Hu

VI

Contents

Chapter 9

On Optimization Techniques for a

Class of Hybrid Mechanical Systems 147

Vadim Azhmyakov and Arturo Enrique Gil García

Chapter 10

Optimized Method for Real Time Nonlinear Control 163

Younes Rafic, Omran Rabih and Rachid Outbib

Chapter 11

Nonlinear Phenomena and Stability

Analysis for Discrete Control Systems 187

Yoshifumi Okuyama

Preface

All practical systems contain nonlinear dynamics. Control system development for

these systems has traditionally been based on linearized system dynamics in

conjunction with linear control techniques. Sometimes it is possible to describe the

operation of systems by a linear model around its operating points. Linearized system

can provide approximate behavior of the system. But in analyzing the overall system

behavior, the resulting system model is inadequate or inaccurate. Moreover, the

stability of the system cannot be guaranteed. However, nonlinear control techniques

take advantage of the given nonlinear dynamics to produce high‐performance designs.

Nonlinear Control Systems represent a new trend of investigation during the last few

decades. There has been great excitement over the development of new mathematical

techniques for the control of nonlinear systems. Methods for the analysis and design of

nonlinear control systems have improved rapidly. A number of new approaches, ideas

and results have emerged during this time. These developments have been motivated

by comprehensive applications such as mechatronic, robotics, automotive and air‐craft

control systems.

The book is organized into eleven chapters that include nonlinear design topics such

as Feedback Linearization, Lyapunov Based Control, Adaptive Control, Optimal

Control and Robust Control. All chapters discuss different applications that are

basically independent of each other. The book will provide the reader with

information on modern control techniques and results which cover a very wide

application area. Each chapter attempts to demonstrate how one would apply these

techniques to real‐world systems through both simulations and experimental settings.

Lastly, I would like to thank all the authors for their excellent contributions in different

applications of Nonlinear Control Techniques. Despite the rapid advances in the field,

I believe that the examples provided here allow us to look through some main

research tendencies in the upcoming years. I hope the book will be a worthy

contribution to the field of Nonlinear Control, and hopefully it will provide the

readers with different points of view on this interesting branch of Control Engineering.

Dr. Meral Altınay

Kocaeli University,

Turkey

1

Application of Input-Output Linearization

Erdal Şehirli and Meral Altinay

Kastamonu University & Kocaeli University

Turkey

1. Introduction

In nature, most of the systems are nonlinear. But, most of them are thought as linear and the

control structures are realized with linear approach. Because, linear control methods are so

strong to define the stability of the systems. However, linear control gives poor results in

large operation range and the effects of hard nonlinearities cannot be derived from linear

methods. Furthermore, designing linear controller, there must not be uncertainties on the

parameters of system model because this causes performance degradation or instability. For

that reasons, the nonlinear control are chosen. Nonlinear control methods also provide

simplicity of the controller (Slotine & Li, 1991).

There are lots of machine in industry. One of the basic one is dc machine. There are two

kinds of dc machines which are brushless and brushed. Brushed type of dc machine needs

more maintenance than the other type due to its brush and commutator. However, the

control of brushless dc motor is more complicated. Whereas, the control of brushed dc

machine is easier than all the other kind of machines. Furthermore, dc machines need to dc

current. This dc current can be supplied by dc source or rectified ac source. Three – phase ac

source can provide higher voltage than one phase ac source. When the rectified dc current is

used, the dc machine can generate harmonic distortion and reactive power on grid side.

Also for the speed control, the dc source must be variable. In this paper, dc machine is fed

by three – phase voltage source pulse width modulation (PWM) rectifier. This kind of

rectifiers compared to phase controlled rectifiers have lots of advantages such as lower line

currents harmonics, sinusoidal line currents, controllable power factor and dc – link voltage.

To make use of these advantages, the filters that are used for grid connection and the control

algorithm must be chosen carefully.

In the literature there are lots of control methods for both voltage source rectifier and dc

machine. References (Ooi et al., 1987; Dixon&Ooi, 1988; Dixon, 1990; Wu et al., 1988, 1991)

realize current control of L filtered PWM rectifier at three – phase system. Reference (Blasko

& Kaura, 1997) derives mathematical model of Voltage Source Converter (VSC) in d-q and

α-β frames and also controlled it in d-q frames, as in (Bose, 2002; Kazmierkowski et al.,

2002). Reference (Dai et al., 2001) realizes control of L filtered VSC with different decoupling

structures. The design and control of LCL filtered VSC are carried out in d-q frames, as in

(Lindgren, 1998; Liserre et al., 2005; Dannehl et al., 2007). Reference (Lee et al., 2000; Lee,

2003) realize input-output nonlinear control of L filtered VSC, and also in reference

(Kömürcügil & Kükrer, 1998) Lyapunov based controller is designed for VSC. The feedback

linearization technique for LCL filtered VSC is also presented, as in (Kim & Lee, 2007; Sehirli

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2

Applications of Nonlinear Control

& Altınay, 2010). Reference (Holtz, 1994) compares the performance of pulse width

modulation (PWM) techniques which are used for VSC. In (Krishnan, 2001) the control

algorithms, theories and the structure of machines are described. The fuzzy and neural

network controls are applied to dc machine, as in (Bates et al., 1993; Sousa & Bose, 1994).

In this chapter, simulation of dc machine speed control which is fed by three – phase voltage

source rectifier under input – output linearization nonlinear control, is realized. The speed

control loop is combined with input-output linearization nonlinear control. By means of the

simulation, power factor, line currents harmonic distortions and dc machine speed are

presented.

2. Main configuration of VSC

In many industrial applications, it is desired that the rectifiers have the following features;

high-unity power factor, low input current harmonic distortion, variable dc output voltage

and occasionally, reversibility. Rectifiers with diodes and thyristors cannot meet most of

these requirements. However, PWM rectifiers can provide these specifications in

comparison with phase-controlled rectifiers that include diodes and thyristors.

The power circuit of VSC topology shown in Fig.1 is composed of six controlled switches

and input L filters. Ac-side inputs are ideal three-phase symmetrical voltage source, which

are filtered by inductor L and parasitic resistance R, then connected to three-phase rectifier

consist of six insulated gate bipolar transistors (IGBTs) and diodes in reversed parallel. The

output is composed of capacitance and resistance.

Fig. 1. L filtered VSC

3. Mathematical model of the VSC

3.1 Model of the VSC in the three-phase reference frame

Considering state variables on the circuit of Fig.1 and applying Kirchhoff laws, model of

VSC in the three-phase reference frame can be obtained, as in (Wu et al., 1988, 1991; Blasko

& Kaura, 1997).

The model of VSC is carried out under the following assumptions.

The power switches are ideal devices.

All circuit elements are LTI (Linear Time Invariant)

The input AC voltage is a balanced three-phase supply.

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Application of Input-Output Linearization

3

For the three-phase voltage source rectifier, the phase duty cycles are defined as the duty

cycle of the top switch in that phase, i.e., da= d(S1), db= d(S3), dc= d(S5) with d representing

duty cycle.

1

(1)

3

1

(2)

3

1

(3)

3

1

1

(4)

This model in equations (1) – (4) is nonlinear and time variant. Using Park Transformation,

the ac-side quantities can be transformed into rotating d-q frame. Therefore, it is possible to

obtain a time-invariant system model with a lower order.

3.2 Coordinate transformation

On the control of VSC, to make a transformation, there are three coordinates whose relations

are shown by Fig 2, that are a-b-c, -β and d-q. a-b-c is three phase coordinate, -β is

stationary coordinate and d-q is rotating coordinate which rotates ω speed.

Fig. 2. Coordinates diagram of a-b-c, -β and d-q

4

Applications of Nonlinear Control

The d-q transformation is a transformation of coordinates from the three-phase stationary

coordinate system to the d-q rotating coordinate system. A representation of a vector in any

n-dimensional space is accomplished through the product of a transpose n-dimensional

vector (base) of coordinate units and a vector representation of the vector, whose elements

are corresponding projections on each coordinate axis, normalized by their unit values. In

three phase (three dimensional) space, it looks like as in (5).

(5)

Assuming a balanced three-phase system, a three-phase vector representation transforms to

d-q vector representation (zero-axis component is 0) through the transformation matrix T,

defined as in (6).

cos

cos

cos

(6)

sin

sin

sin

In (6), ω is the fundamental frequency of three-phase variables. The transformation from

(three-phase coordinates) to

(d-q rotating coordinates), called Park Transformation, is

obtained through the multiplication of the vector

by the matrix T, as in (7).

.

(7)

The inverse transformation matrix (from d-q to a-b-c) is defined in (8).

cos

sin

cos

sin

(8)

cos

sin

The inverse transformation is calculated as in (9).

′.

(9)

3.3 Model of the VSC in the rotating frame

Let x and u be the phase state variable vector and phase input vector in one phase of a

balanced three-phase system with the state equation in one phase as in (10).

(10)

Where A and B are identical for the three phases. Applying d-q transformation to the three-

phase system, d-q subsystem with d and q variables is obtained (xd-xq and ud-uq ). The

system equation in (10) becomes as in (11) (Mao et al., 1998; Mihailovic, 1998).

0