Table Of ContentLecture Notes
in Control and Information Sciences 247
Editor: .M Thoma
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London
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Series Advisory Board
A. Bensoussan • M.J. Grimble • P. Kokotovic • A.B. Kurzhanski •
H. Kwakernaak • J.L. Massey . M. Morari
Editors
K.D. Young, PhD
Applied Technology Center, Hong Kong University of Science and Technology, Clear
Water Bay, Kowloon, Hong Kong
O. Ozgfiner, PhD
Department of Electrical Engineering, Ohio State University, Columbus,
43210 USA
ISBN 1-85233-197-6 Springer-Vedag London Berlin Heidelberg
British Library Cataloguing in Publication Data
Variable structure systems, sliding mode and nonlinear
control. - (Lectures notes in control and information
sciences ; )742
1.Automatic control - Congresses
LYoung, K.D. ILOzguner, U.
8.926
NBSI 6791332581
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This volume of work has been selected from papers presented at VSS'98 (5th
International Workshop on Variable Structure Systems) held in Sarasota, Florida.
This workshop was the fifth in a series of SSV international workshops, and the first
tboe held in the United States.
The initiator of this workshop series, Dr. Asif Sabanovic first organized and chaired
VSS'90 in Sarajevo amidst the warmth of the Bosnian people and the spectacular
surroundings. Since then an international workshop has been organized and held
biennially. The second workshop was in (Sheffield, 1992 )KU and the third in 4991
(Benevento, Italy). VSS'96 (Tokyo, Japan) was the first in the series to be held
outside Europe.
Work presented herein on theoretical developments and applications on SSV and
Sliding Mode, reflects how trends have advanced beyond the original ideas that are
now well documented in a number of books and research monographs. In particular,
the concepts of Sliding Sector and Second Order Sliding Mode introduced in this
volume, will stimulate discussions and invite further extensions. Also, the focus on
Sampled Data systems represents a positive trend towards practical industrial
implementations of sliding mode controllers.
.K David Young, Hong Kong
Omit 0zgiiner, Columbus, Ohio, ASU
stnetnoC
Sliding Sector for Variable Structure System
oK Furuta and Y. Pan ................................................................................................................ 1
On Discrete Variable Structure Control with Switching Sector
C. Y. Tang and E. A. Misawa ................................................................................................. 27
Variable Structure Control of Nonlinear Sampled Data Systems by Second
Order Sliding Modes
.3( Bartolini, A. Pisano and E. Usai ...................................................................................... 43
On Sampled Data Variable Structure Control Systems
.-X. Xu, .:1 Zheng and T. H. Lee. ........................................................................................... 69
Sliding Mode Control of Systems with Delayed States and Controls
X. Li and S. Yurkovich ............................................................................................................ 39
On Global Stabilization of Nonlinear Dynamical Systems
X. Yu, Y. Wu and M. Zhihong ............................................................................................ 901
A General Canonical Form for Sliding Mode Control of Nonlinear Systems
H. Sira-Ramfrez .................................................................................................................... 321
Handling Stiction with Variable Structure Control
C. Hatipoglu and (). (Jzgiiner ............................................................................................. 341
Equivalent Value Filters in Disturbance Estimation and State Observation
L Haskara and O. Ozgiiner ................................................................................................. 761
Sliding Mode Control of a Car-Like Mobile Robot Using Single-Track
Dynamic Model
A. Stotsky and X. Hu ............................................................................................................ 181
Park Vector Based Sliding Mode Control of UPS With Unbalanced and
Nonlinear Load
P. Korondi and H. Hashimoto ............................................................................................ 391
iiiv
Dynamical Adaptive Sliding Mode Control of Observable Minimum-Phase
Uncertain Nonlinear Systems
M. Rios-Bolfvar and A. S. L Zinober ................................................................................. 112
Symbolic Computing Tools for Nonsmooth Dynamics and Control
.C Teolis, H.G. Kwatny and M. Mattice ............................................................................ 237
Sliding Mode Control with Gain Scheduled Hyperplane for LPV Plant
K. Nonami and S. Sivrioglu ................................................................................................ 263
A Study on Parameterized Output Feedback Sliding Mode Controller
.(1 Nonaka and K. Furuta ................................................................................................... 182
Sliding Mode Tracking Control of Systems with Unstable Zero Dynamics
H.-S. ]eong and V. L Utkin .................................................................................................. 303
On Second Order Sliding Mode Controllers
G. Bartolini, A. Ferrara, A. Levant and E. Usai .............................................................. 329
A Tale of Two Discontinuities
K. D. Young .......................................................................................................................... 153
The Problem of Chattering: an Averaging Approach
L. Fridman ............................................................................................................................. 363
Sliding Sector for Variable Structure System
Katsuhisa l~lruta* Yaodong Pan**
* Department of Mechanical and Environmental Informatics
Tokyo Institute of Technology
2-12-10h-okayama, Meguro-ku, Tokyo 152, Japan
** Department of Computers and Systems Engineering
Tokyo Denki University
Hiki-gun, Saitama 350-0394, Japan
Abstract. This paper presents VS controllers with sliding sector. The
sliding sector is designed so that some norm of the state is decreasing
inside it without any control action. VS control law ensures the norm
decreases in the state space and thus yields a quadratic stable VS control
system. The discrete-time system described by transfer function is also
taken into consideration.
1 Introduction
The variable structure control has started in Russia by many researchers, like
Barbashin1, Utkin2, Emelyanov3. The variable structure control has been
mainly considered for continuous time systems in the form of sliding mode con-
trol. But when the continuous-time sliding mode is implemented by the digital
controller, the unstable phenomena is known to be possibly happened as follows:
The stability on the sliding mode
(1)
8 ~- cTx
for the continuous-time linear system represented by
~: = Ax + Bu (2)
is determined by the zeros of the transfer function between the input u and s.
The zeros of the transfer function are those of
But when the continuous system is realized by the digital control, not only the
chattering around the sliding mode may be generated, but also the stable zeros
may be happened unstable as/~strom pointed out 4.
To overcome this problem discrete-time VS controllers have been proposed in
literatures 5, 6, 7, 8 and 9. Drakunov and Utkin 5 proposed a discrete-
time variable structure control based on the contraction mapping. Furuta 6
proposed to use the conventional Lyapunov like approach and has shown that
2
the sliding sector should be considered for the discrete-time system. He also con-
sidered the variable structure control using the transfer function for the unknown
parameter system 10.
The close observation of the sliding mode for a simple linear system
01 0 ~f
x= 0 x+ u
s=alx, a>0
Decompose the state space into three sets by
s+ = {zls > o} (4)
So = {~Js = o} (5)
s_ = ~1:~{ < 0} (6)
and the control law is given by
u_ Vx E S+
u= o vxcSo (7)
u+ Vx E S_
For this system, the state can not be stay inside the sliding mode for a sampling
interval even if the sampling mode is small. So instated of the sliding mode, the
set of the state in which the state has the desirable properties without control
will be considered. One of such sets may be one in which the norm of the state
decreases with zero control input. To define such set, the following P-norm of
the state is introduced.
~11~1I = (xTpx) 98(cid:1)( (8)
Letting
P= 2I
then
and the derivative is
L = xT(ATp -4 PA)x = 2xlx2
where/2 is the 2 (cid:141) 2 identity matrix. This tells that the norm of the zero input
system is deceasing inside the sector
S = {~lxlx2 < 0}
i.e., inside the second and fourth quadrants, the P-norm of the state is decreasing
for zero input. If we restrict the condition in that the derivative of L is less than
-xTRx for S, then the following set can be defined.
Sn = (zlL < -xr Rz}
1
e.O
6.0
0.4
0.2
o
-0.2
-0.4
-0.6
-0.8
-1
--1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X 1
Fig. 1. Sectors S, ,Sin, Sn2, and 8R3 for R = 02, R = 0.2. I2, R = 0.412, and
It -- 0.512, respectively
Inside this set the P-norm is decreasing fa.stcr and
SacS
Figure 1 shows the sector where R is chosen to be 02, 0.2 * I2, 0.4 */2, and
0.5 */2, respectively where 02 is the 2 (cid:141) 2 zero matrix. The sector is reduced
to a hyperplane if R -- 0.5 */2 and is the origin if R > 0.5 */2. This sector is
named PR-sliding sector and we take the control be zero when the state is in
the sector. And when the state is outside of the sector, the control to transfer
the state into the sector will be considered. The sector has been shown to be
useful in the control design not only for the continuous-time system but also for
the discrete-time system [15] and the sampled-data system [14].
In the paper, the definition and design of the PR-sliding sector for continuous-
time system is presented and the correspom|ing VS controller is designed. Then
parameter uncertainty and external disturbance are considered, a robust VS
controller and an adaptive VS controller are proposed.
For a discrete-time system described by transfer function, another kind of
sliding sector is also proposed. Inside the sliding sector, the closed loop system
