APPLICATIONS OF
NONLINEAR CONTROL
Edited by Meral Altınay
Applications of Nonlinear Control
Edited by Meral Altınay
Published by InTech
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Copyright © 2012 InTech
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First published June, 2012
Printed in Croatia
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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
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.
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