th th th SEAMS - GMU 2011 Yogyakarta - Indonesia, 12 - 15 July 2011 th th Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications th Mathematics and Its Applications in the Development of Sciences and Technology. Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications th Department of Mathematics Faculty of Mathematics & Natural Sciences Universitas Gadjah Mada Sekip Utara Yogyakarta - INDONESIA 55281 Phone : +62 - 274 - 552243 ; 7104933 Fax. : +62 - 274 555131 MATHEMATICS AND ITS APPLICATIONS IN THE DEVELOPMENT OF SCIENCES AND TECHNOLOGY International Conference on Mathematics and Its Applications The ISBN 978-979-17979-3-1
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ththth
SEAMS - GMU 2011
Yogyakarta - Indonesia, 12 - 15 July 2011th th
Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications
th
Mathem
atics and Its Applications in the
Developm
ent of Sciences and Technology.
Proceedings of the 6 S
EA
MS
-GM
U International
Conference on M
athematics and Its A
pplications
th
Department of Mathematics
Faculty of Mathematics & Natural Sciences
Universitas Gadjah Mada
Sekip Utara Yogyakarta - INDONESIA 55281
Phone : +62 - 274 - 552243 ; 7104933
Fax. : +62 - 274 555131
MATHEMATICS AND ITS APPLICATIONS IN THE DEVELOPMENT OF SCIENCES AND TECHNOLOGY
International Conference on Mathematics and Its Applications
TheISBN 978-979-17979-3-1
PROCEEDINGS OF THE 6TH
SOUTHEAST ASIAN MATHEMATICAL SOCIETY
GADJAH MADA UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS
AND ITS APPLICATIONS 2011
Yogyakarta, Indonesia, 12th – 15th July 2011
DEPARTMENT OF MATHEMATICS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
UNIVERSITAS GADJAH MADA
YOGYAKARTA, INDONESIA
2012
Published by
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Gadjah Mada
Sekip Utara, Yogyakarta, Indonesia
Telp. +62 (274) 7104933, 552243
Fax. +62 (274) 555131
PROCEEDINGS OF
THE 6TH SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS 2011
Copyright @ 2012 by Department of Mathematics, Faculty of Mathematics and
Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia
ISBN 978-979-17979-3-1
PROCEEDINGS OF THE 6TH
SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA
UNIVERSITY
INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS
APPLICATIONS 2011
Chief Editor:
Sri Wahyuni
Managing Editor :
Indah Emilia Wijayanti Dedi Rosadi
Managing Team :
Ch. Rini Indrati Irwan Endrayanto A. Herni Utami Dewi Kartika Sari
Nur Khusnussa’adah Indarsih Noorma Yulia Megawati Rianti Siswi Utami
Soeharyadi (Indonesia), Subanar (Indonesia) Supama (Indonesia), Asep K. Supriatna
(Indonesia) and Indah Emilia Wijayanti (Indonesia). Most of the contributed papers
were delivered by mathematicians from Asia.
We would like to sincerely thank all plenary and invited speakers who
warmly accepted our invitation to come to the Conference and the paper
contributors for their overwhelming response to our call for short presentations.
Moreover, we are very grateful for the financial assistance and support that we
received from Universitas Gadjah Mada, the Faculty of Mathematics and Natural
Sciences, the Department of Mathematics, the Southeast Asian Mathematical
Society, and UNESCO.
We would like also to extend our appreciation and deepest gratitude to all
invited speakers, all participants, and referees for the wonderful cooperation, the
great coordination, and the fascinating efforts. Appreciation and special thanks are
addressed to our colleagues and staffs who help in editing process. Finally, we
acknowledge and express our thanks to all friends, colleagues, and staffs of the
Department of Mathematics UGM for their help and support in the preparation
during the conference.
The Editors
October, 2012
CONTENTS
Title i Publisher and Copyright ii Managerial Boards iii Editorial Boards iv List of Reviewers v Preface vii Paper of Invited Speakers On Things You Can’t Find : Retrievability Measures and What to do with Them ……............... 1 Andreas Rauber and Shariq Bashir
A Quasi-Stochastic Diffusion-Reaction Dynamic Model for Tumour Growth ..……................... 9 Ang Keng Cheng
*-Rings in Radical Theory ...………………………………………………..................................................... 19 H. France-Jackson
Clean Rings and Clean Modules ...………………………………………................................................... 29 Indah Emilia Wijayanti
Research on Nakayama Algebras ……...…………………………………................................................. 41 Intan Muchtadi-Alamsyah
Mathematics in Medical Image Analysis: A Focus on Mammography ...…............................... 51 Murk J. Bottema, Mariusz Bajger, Kenny MA, Simon Williams
The Order of Phase-Type Distributions ..………………………………….............................................. 65 Reza Pulungan
The Linear Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System… 79 Salmah
Chaotic Dynamics and Bifurcations in Impact Systems ………………........................................... 89 Sergey Kryzhevich
Contribution of Fuzzy Systems for Time Series Analysis ……………….......................................... 121 Subanar and Agus Maman Abadi
Contributed Papers Algebra
Degenerations for Finite Dimensional Representations of Quivers ……................................... 137 Darmajid and Intan Muchtadi-Alamsyah
On Sets Related to Clones of Quasilinear Operations …...……………………………………………………. 145 Denecke, K. and Susanti, Y.
Normalized H Coprime Factorization for Infinite-Dimensional Systems …………………………… 159 Fatmawati, Roberd Saragih, Yudi Soeharyadi
Construction of a Complete Heyting Algebra for Any Lattice ………………………………………………. 169 Harina O.L. Monim, Indah Emilia Wijayanti, Sri Wahyuni
The Fuzzy Regularity of Bilinear Form Semigroups …………………………………………………..…………. 175 Karyati, Sri Wahyuni, Budi Surodjo, Setiadji
The Cuntz-Krieger Uniqueness Theorem of Leavitt Path Algebras ………………………………………. 183 Khurul Wardati, Indah Emilia Wijayanti, Sri Wahyuni
Application of Fuzzy Number Max-Plus Algebra to Closed Serial Queuing Network with
Fuzzy Activitiy Time ………………………………………………………………………………………………..…………… 193 M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo
Enumerating of Star-Magic Coverings and Critical Sets on Complete Bipartite Graphs………… 205 M. Roswitha, E. T. Baskoro, H. Assiyatun, T. S. Martini, N. A. Sudibyo
Construction of Rate s/2s Convolutional Codes with Large Free Distance via Linear System
Approach ………………………………………………………………………………………………........…………………….. 213 Ricky Aditya and Ari Suparwanto
Characteristics of IBN, Rank Condition, and Stably Finite Rings ………....................................... 223 Samsul Arifin and Indah Emilia Wijayanti
The Eccentric Digraph of n mP P Graph ………………………………………………………………….….……… 233 Sri Kuntarti and Tri Atmojo Kusmayadi
On M -Linearly Independent Modules ……………………………………………………………………….......... 241 Suprapto, Sri Wahyuni, Indah Emilia Wijayanti, Irawati
The Existence of Moore Penrose Inverse in Rings with Involution …........................................ 249 Titi Udjiani SRRM, Sri Wahyuni, Budi Surodjo
Analysis
An Application of Zero Index to Sequences of Baire-1 Functions ………….................................. 259 Atok Zulijanto
Regulated Functions in the n-Dimensional Space ……………………....................….....................… 267 Ch. Rini Indrati
Compactness Space Which is Induced by Symmetric Gauge ……..........................………………... 275 Dewi Kartika Sari and Ch. Rini Indrati
A Continuous Linear Representation of a Topological Quotient Group …................................ 281 Diah Junia Eksi Palupi, Soeparna Darmawijaya, Setiadji, Ch. Rini Indrati
On Necessary and Sufficient Conditions for L into 1 Superposition Operator ……...………... 289 Elvina Herawaty, Supama, Indah Emilia Wijayanti
A DRBEM for Steady Infiltration from Periodic Flat Channels with Root Water Uptake ……….. 297 Imam Solekhudin and Keng-Cheng Ang
Boundedness of the Bimaximal Operator and Bifractional Integral Operators in Generalized
Morrey Spaces …………………………………...................................................................................... 309 Wono Setya Budhi and Janny Lindiarni
Applied Mathematics
A Lepskij-Type Stopping-Rule for Simplified Iteratively Regularized Gauss-Newton Method.. 317 Agah D. Garnadi
Asymptotically Autonomous Subsystems Applied to the Analysis of a Two-Predator One-
Prey Population Model ............................................................................................................. 323 Alexis Erich S. Almocera, Lorna S. Almocera, Polly W.Sy
Sequence Analysis of DNA H1N1 Virus Using Super Pair Wise Alignment .............................. 331 Alfi Yusrotis Zakiyyah, M. Isa Irawan, Maya Shovitri
Optimization Problem in Inverted Pendulum System with Oblique Track ……………………………. 339 Bambang Edisusanto, Toni Bakhtiar, Ali Kusnanto
Existence of Traveling Wave Solutions for Time-Delayed Lattice Reaction-Diffusion Systems 347 Cheng-Hsiung Hsu, Jian-Jhong Lin, Ting-Hui Yang
Effect of Rainfall and Global Radiation on Oil Palm Yield in Two Contrasted Regions of Sumatera, Riau and Lampung, Using Transfer Function ..........................................................
Expected Value Approach for Solving Multi-Objective Linear Programming with Fuzzy
Random Parameters …..................................................................................................... ......... 427 Indarsih, Widodo, Ch. Rini Indrati
Chaotic S-Box with Piecewise Linear Chaotic Map (PLCM) ...................................................... 435 Jenny Irna Eva Sari and Bety Hayat Susanti
Model of Predator-Prey with Infected Prey in Toxic Environment .......................................... 449 Lina Aryati and Zenith Purisha
On the Mechanical Systems with Nonholonomic Constraints: The Motion of a Snakeboard
on a Spherical Arena ………………………………………..……………………................................................ 459 Muharani Asnal and Muhammad Farchani Rosyid
Safety Analysis of Timed Automata Hybrid Systems with SOS for Complex Eigenvalues …….. 471 Noorma Yulia Megawati, Salmah, Indah Emilia Wijayanti
Global Asymptotic Stability of Virus Dynamics Models and the Effects of CTL and Antibody Responses ………………………………………………………………………………….……………………………………….. 481
Nughtoth Arfawi Kurdhi and Lina Aryati
A Simple Diffusion Model of Plasma Leakage in Dengue Infection …………………………..………… 499 Nuning Nuraini, Dinnar Rachmi Pasya, Edy Soewono
The Sequences Comparison of DNA H5N1 Virus on Human and Avian Host Using Tree
Diagram Method …………………………………..……………………………………………………………..…………….. 505 Siti Fauziyah, M. Isa Irawan, Maya Shovitri
Fuzzy Controller Design on Model of Motion System of the Satellite Based on Linear Matrix
Unified Structural Models and Reduced-Form Models in Credit Risk by the Yield Spreads …. 697 Di Asih I Maruddani, Dedi Rosadi, Gunardi, Abdurakhman
The Effect of Changing Measure in Interest Rate Models …….…....................…....................... 705 Dina Indarti, Bevina D. Handari, Ias Sri Wahyuni
New Weighted High Order Fuzzy Time Seriesfor Inflation Prediction ……................................ 715 Dwi Ayu Lusia and Suhartono
Detecting Outlier in Hyperspectral Imaging UsingMultivariate Statistical Modeling and
Numerical Optimization ………………………………………...........................................……………......... 729 Edisanter Lo
Prediction the Cause of Network Congestion Using Bayesian Probabilities ............................. 737 Erwin Harapap, M. Yusuf Fajar, Hiroaki Nishi
Solving Black-Scholes Equation by Using Interpolation Method with Estimated Volatility……… 751 F. Dastmalchisaei, M. Jahangir Hossein Pour, S. Yaghoubi
Artificial Ensemble Forecasts: A New Perspective of Weather Forecast in Indonesia ............... 763 Heri Kuswanto
Second Order Least Square for ARCH Model …………………………....................………………............. 773 Herni Utami, Subanar, Dedi Rosadi, Liqun Wang
Two Dimensional Weibull Failure Modeling ……………………..…..................……….......................... 781 Indira P. Kinasih and Udjianna S. Pasaribu
Simulation Study of MLE on Multivariate Probit Models …......................................................... 791 Jaka Nugraha
Clustering of Dichotomous Variables and Its Application for Simplifying Dimension of
Quality Variables of Building Reconstruction Process ............................................................. 801 Kariyam
Valuing Employee Stock Options Using Monte Carlo Method ……………................................. 813 Kuntjoro Adji Sidarto and Dila Puspita
Classification of Epileptic Data Using Fuzzy Clustering .......................................................... 821 Nazihah Ahmad, Sharmila Karim, Hawa Ibrahim, Azizan Saaban, Kamarun Hizam
Mansor
Recommendation Analysis Based on Soft Set for Purchasing Products ................................. 831 R.B. Fajriya Hakim, Subanar, Edi Winarko
Heteroscedastic Time Series Model by Wavelet Transform ................................................. 849 Rukun Santoso, Subanar, Dedi Rosadi, Suhartono
Parallel Nonparametric Regression Curves ............................................................................ 859 Sri Haryatmi Kartiko
Ordering Dually in Triangles (Ordit) and Hotspot Detection in Generalized Linear Model for
Poverty and Infant Health in East Java ……................................…………………………………………. 865 Yekti Widyaningsih, Asep Saefuddin, Khairil Anwar Notodiputro, Aji Hamim Wigena
Empirical Properties and Mixture of Distributions: Evidence from Bursa Malaysia Stock
Market Indices …………………....................................................................................................... 879 Zetty Ain Kamaruzzaman, Zaidi Isa, Mohd Tahir Ismail An Improved Model of Tumour-Immune System Interactions …………………………………………….. 895 Trisilowati, Scott W. Mccue, Dann Mallet
515
Proceedings of ”The 6th
SEAMS-UGM Conference 2011” Applied Mathematics, pp. 515 – 528.
REGULA FUZZY CONTROLLER DESIGN ON MODEL OF
MOTION SYSTEM OF THE SATELLITE BASED ON LINEAR
MATRIX INEQUALITY
SOLIKHATUN AND SALMAH
Abstract. In this paper, the linear state feedback controller for fuzzy model is designed by using
the linear matrix inequalities (LMIs). Fuzzy model are described as sum of weighting of r
subsystems. The controller design is guarantees the stability of system and satisfies the desired transient responses of system. If the r approach to infinity, the existence of controller that stabilize
the system will be difficult to obtain. It is caused by find the solution of set of LMIs that satisfying
some the conditions. By relaxing the stability conditions we will formulate the problem of controller design in LMIs feasibility problem. Keywords and Phrases : fuzzy control, feedback, linear matrix inequalities, Lyapunov stability,
pole placement, Takagi-Sugeno model, relaxed stability conditions .
1. INTRODUCTION
In recent years, there have been many research efforts on these issues based on the
Takagi-Sugeno (TS) model based fuzzy control. For this TS model based fuzzy control
system, Wang et all [14] proved the stability by finding a common symmetric positive
definite matrix P for the r subsystems in general and suggested the idea of using Linear
Matrix Inequalities (LMIs). The process of controller design are involves an iterative process,
that is, for each rule a controller is designed based on consideration of local performance
only, then LMI-based stability analysis is carried out to check the global stability condition.
In the case that the stability conditions are not satisfied, the controller for each rule should be
redesigned.
Olsder [9] and Ogata, K, [8] presented about the basic theories of systems and controls.
Boyd, et all [1] discussed about the linear matrix inequality in system and control theory.
Lam, H.K et all [4] designed the controller to fuzzy system by linear matrix inequality
approach. Mastorakis, N.E. [5] discussed about the modeling of dynamic system by TS fuzzy
model. Messousi, W.E. et all [6] discussed a bout how the pole placement on fuzzy model by
516 SOLIKHATUN AND SALMAH
LMI-approach. Tanaka, K., dan Sugeno, M.[12] analyzed the stability and designed the fuzzy
control system.
Motivated by the LMI formulation of pole placement constraint of the conventional
state feedback in Chilali [2] and Hong, S.K. & Nam, Y [3], Solikhatun [11] modify the
formulation and apply to the multi-objective TS model based fuzzy logic controller design
problem. The design the fuzzy controller system for the way of simultaneously guaranteeing
global stability and adequate transient behavior (pre-specified transient performance) are
formulated. Tanaka and Wang [13] wrote that if the r approach to infinity then the existence
of controller that stabilize the system will be difficult to obtain. It is caused by find the
solution of set of LMIs that satisfying some the conditions. By relaxing the stability
conditions we will formulate the problem of controller design in LMIs feasibility problem. Polderman [10] presented about the dynamic of satellite of motion system and the
factors that affect it and derive the equations of satellite motion of dynamic without
disturbance. Last, we will present the simulation results by apply the proposed methodology
to model of motion system of the satellite. The followed definitions, lemmas and theorems
are used in the main results.
Definition 1. The matrix nxnP R is called positive definite matrix if 0,t nu Pu u R .
Lemma 2. Suppose , nxnQ R R are symmetric matrix and matrix nxnS R . The condition
0t
Q S
S R
is equivalent to
10, 0R Q SR S .
Lemma 2 is known as Schur complement.
Definition 3. Consider the linear system 0)0(,,, xxRxRAAxx nnxn .
Equilibrium point x is stable if for all 0 there 0)( such that for each solution
),( 0xtx
If xx0 then 00 ,),( ttxxtx .
Equilibrium point x is asymptotically stable if x stable and there 00 such that for each
solution ),( 0xtx
if 00 xx then 0),(lim 0
xxtxt
.
The stability of linear system can be formulated by LMI. It is known by Lyapunov
theorem about stability. In stabilization, we need a controller such that the state feedback
fuzzy control system is asymptotically stable, i.e.
lim ( ) 0t
x t
for initial condition (0)x and 0x .
Theorem 4. Consider the linear system nnxn RxRAAxx ,, . Matrix A is called
stable if there exist positive definite matrix P such that
0tA P PA .
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 517
Theorem 5. Consider set nRS invariant and 0x is equilibrium point of system
nnxn RxRAAxx ,, . If the Lyapunov function RSV : is differentiable
continuously then
1. 0)0( V and }0{\,0)( SxxV .
2. If SxxV ,0)( then 0x is stable.
3. If }0{\,0)( SxxV then 0x is asymptotically stable.
Definition 6. Consider the set X and set XM . Consider function M is defined as
function of
]1,0[: XM
that corresponding the each element of Xx with the real number )(xM on ]1,0[ , with
the )(xM represent membership function grades for x on M . The fuzzy set XM is
defined by Xxxx M ,)(, .
Definition 7. A subset of D of the complex plane is called an LMI-D region if there exist a
symmetric matrix [ ] mxm
kl R and [ ] mxm
kl R such that
( ) 0DD z C f z
where the characteristic function Df is given by
1 ,( ) [ ]D kl kl kl k l mf z z z .
Example 8. Consider a circle LMI region D
2 2 2( )D x iy C x q y r
centered at (-q, 0) and has radius 0r , where the characteristic function is given by
( )D
r z qf z
z q r
.
As shown in Figure 2, we can chose the poles in region D such that desired transient
responses of system.
518 SOLIKHATUN AND SALMAH
Fig. 2. Circular region D for pole placement
Definition 9. Consider the circular region D of the left half complex plane. The system nxnn RARxAxx ,, is said D-stable if all the poles lies on LMI-D region.
1.1 Affine Fuzzy Model. The problem of LMI-based fuzzy state feedback controller becomes
yet more complex if some of model parameters are unknown. By using a Takagi-Sugeno (TS)
fuzzy model, a nonlinear model can be expressed as a weighted sum of r simple subsystem.
The inference performed via the Takagi-Sugeno model is an interpolation of all the relevant
linear models. Takagi and Sugeno define the inference in the rule base as the weighted
average of each rule’s consequents :
1
1
( ( ) ( ) )
( )
r
i i i i
i
r
i
i
A x t B u t d
x t
. (1)
The TS fuzzy model consists of an if-then rule base. The rule antecedents partition a subset of
the model variables into fuzzy sets. The consequent of each rule is a simple functional
expression. The i-th rule of the Takagi-Sugeno fuzzy model is of the following form :
If 1( )x t is 1iL and ( )nx t is inL and ( )u t is iM
then ( ) ( ) ( )i i ix t A x t Bu t d
where 1,2,...,i r and r is the number of rules and , 1,2,...,ijL j n and iM are fuzzy
sets centered at the i-th operating point. The categories of the fuzzy sets are expressed as
N_Left, Z_Equal and P_Right where N_Left represents negative, Z_Equal zero and P_Right
positive (Figure 1).
max d
Re
Im max n
(-q,0)
r
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 519
The truth value of the i-th rule in the set i is obtained as the product of the
membership function grades :
1 1( , ) ( )... ( ). ( )i in ii L L n Mx u x x u
with ( )ijL jx represent membership function grades for ijL at jx . Consider the linearized
state space form with the bias term d induced from the model linearization is follow:
( ) ( ) ( )x t Ax t Bu t d (2)
with 1 2( , ,..., ) n
nx x x x R , nxnA R ,
nxmB R , mu R and
nd R . The model
(2) is known an affine model. When 0d , this model is called a linear model.
1.2 Design Controller. The linear control theory can be used to design the consequent parts
of the fuzzy control rules because they are described by linear state equations. Suppose that
the control input u is
0( ) ( )i i iu t u t k
in order to cancel the bias term id . Then the Takagi-Sugeno fuzzy model is described by
( ) ( ) ( ), 1,2,...,i i ix t A x t Bu t i r .
Hence, the state feedback controller described by
( ) ( )i iu t K x t
where n
iK R is vector of feedback gains to be chosen for i-th operating point. Therefore a
set of r control rules takes the following form:
If 1( )x t is 1iL and ( )nx t is inL and ( )u t is iM
then ( 1) ( )i i oiu t K x t k
where the index 1t in the consequent part is introduced to distinguish the previous control
action in the antecedent part in order to avoid algebraic loops.
The resulting total control action is
-1 0 1
N_Left Z_Equal P_Right
Fig. 1. membership function
520 SOLIKHATUN AND SALMAH
0
1
1
( )r
i i i
i
r
i
i
K x k
u
. (3)
Substituting (3) into (1), the state feedback fuzzy control system can be represented by
1 1
1 1
( )
( )
r r
i j i i j
i j
r r
i j
i j
A B K x
x t
. (4)
Define
r
i
i
ii
z
zh
1
)(
)(
then 1)(
1
r
i
i zh . The system (4) can be written by
ji
ijji
r
i
iii GhhGhtx 2)(1
2 ,
riKBAG iiiii ,...2,1, and rjiKBAKBA
Gijjjii
ij
,2
)()(.
Corollary 10. 0)()(21
1)(
11
2
r
i ji
ji
r
i
i zhzhr
zh where 0)(,1)(1
zhzh i
r
i
i
for all i.
Proof. It holds since
0))()((1
1)()(2
1
1)( 2
111
2
r
i ji
ji
r
i ji
ji
r
i
i zhzhr
zhzhr
zh .■
Corollary 11. If the number of rules that fire for all t is less than or equal to s, where
rs 1 , then
0)()(21
1)(
11
2
r
i ji
ji
r
i
i zhzhs
zh
where 0)(,1)(1
zhzh i
r
i
i for all i.
Proof. It follows directly from Corollary 10.■
LMI Formulation for Stability Requirement
The stability of feedback system can be formulated by LMI according to Theorem 4. A
sufficient quadratic stability condition derived by Tanaka [12] for ensuring stability of (4) is
given by Theorem 12 as follow :
Theorem 12. The fuzzy control system (4) is asymptotically stable for some stable feedback
jK if there exists a common positive definite matrix P such that
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 521
( ) ( ) 0, , 1,2,...,t
i i j i i jA B K P P A B K i j r . (5)
LMI Formulation for Pole Placement Requirement
In the synthesis of control system, meeting some desired performances should be
considered a long with stability. Generally, stability condition doesn’t directly deal with the
transient responses of the closed loop system. In contrast, a satisfactory transient response of
a system can be guaranteed by confining its poles in a prescribed region. To this purpose, we
introduce the following LMI-based representation of stability region. The pole placement
problem design a controller such that the state feedback fuzzy control system are located in a
prescribed sub region D in the left half plane to prevent too fast controller dynamics and
achieve desired transient behavior, i. e.
( )i i jA B K D
for initial condition (0)x .
Motivated by Chilali [2] an extended Lyapunov theorem for system (4) is develop with
the above definition of an LMI-based circular pole region as below.
Theorem 13. The fuzzy control system (4) is D-stable (all the complex poles lying in LMI
region D) if only if there exists a positive definite matrix Q such that
rjirQQKBAqQ
KBAQqQrQ
jii
t
jii,...,2,1,,0
)(
)(
.
Proof. Let D as eigenvalue of rjiKBA jii ,...,2,1,, and nCv is eigenvector
that corresponding with . Then ** )( vKBAv jii . For each
nCv then
00
0
)(
)(
0
0*
*
v
v
rQQKBAqQ
QKBAqQrQ
v
vt
jii
jii
0)(
)(***
***
QvrvvKBAQvQvqv
QvKBAvQvqvQvrvt
jii
jii
0*
rq
qrQvv
Because 0Q then we obtain
0)(
rq
qrfD
.
In other word the fuzzy control system (4) is D-stable.
Contrary. Consider the fuzzy control system (4) is D-stable. We divide into two cases. For the
case rjiKBA jii ,...,2,1,, is diagonal matrix DnlDiag ll ,,...,2,1),( .
Suppose that 0
rIIqI
IqIrI t
then according to the Lemma 2, we obtain
522 SOLIKHATUN AND SALMAH
0))((1
0)())((
0
1
IqIIqIr
rIIqIrIIqIrI
rI
tt
Because 0r then 0rI . It is contradiction with assumption. For the case
rjiKBA jii ,...,2,1,, is diagonalizable matrix then there exists an invertible matrix T
such that
rjiTKBAT jii ,...,2,1,,)(1
with is a diagonal matrix that elements of are eigenvalues of
rjiKBA jii ,...,2,1,, . Then
.0
))((
))((1
1
rIIqI
IqIrI
rIITKBATqI
TKBATIqIrI
t
jii
t
jii
For the case rjiKBA jii ,...,2,1,, is not diagonalizable matrix then there exists an
invertible matrix T such that
rjiJTKBAT jii ,...,2,1,,)(1
with J is a Jordan matrix. Then
.0
))((
))((1
1
rIJIqI
IJqIrI
rIITKBATqI
TKBATIqIrI
t
jii
t
jii
Let *TTQ such that 0Q . Then
00
0
))((
))((
0
0*
*
1
1
T
T
rIITKBATqI
TKBATIqIrI
T
T
jii
t
jii
0))((
))((**1*
1***
rTTTTTKBATqTT
TKBATTTqTTrTT
jii
t
jii.
Furthermore if we take Re( )Q is matrix that its element is real part of Q then
0))((
))((1
1
rQQTKBATqQ
TKBATQqQrQ
jii
t
jii
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 523
0)Re()Re())(()Re(
))()(Re()Re()Re(1
1
QrQTKBATQq
TKBATQQqQr
jii
t
jii
Because nxnRQ and TKBAT jii )(1
similar with jii KBA then
rjirQQKBAqQ
KBAQqQrQ
jii
t
jii,...,2,1,,0
)(
)(
.■
2.THE MAIN RESULTS
Formulation for synthesis
We will formulate a problem for the design of fuzzy state feedback control system
that guarantees stability and satisfies desired transient responses by using the LMIS
constraints. The LMIs formulations of fuzzy state feedback synthesis problem are followed:
Theorem 14. The fuzzy control system (4) can be stabilized in the LMI-D region if there
exists a common positive definite matrix Q and iY such that the following conditions hold
(6)
rjirQYBQAqQ
BYQAqQrQ
iii
t
i
t
i
t
i ,...,2,1,,0
.
Given solution ( , )iQ Y , the fuzzy state feedback gain is obtained by 1
i iK YQ .
Proof. System (4) can be presented as
1
1( ) 2
r
i i ii i j ij
i i j
x t G G xW
,
1( ) ( ),
2ij i i j j j iG A B K A B K i j ,
1( ) ( )
2ii i i i i i iG A B K A B K and
1 1
r r
i j
i j
W
. By define 1Q P then LMI of (5) of stability Lyapunov can be rewrite
as
0, 1,2,...,
0, .
t
ii ii
t
ij ij
QG G Q i r
QG G Q i j r
It is equivalent to
022
0
t
j
t
iij
t
jj
t
i
t
jji
t
ii
t
i
t
iii
t
ii
BYYBQAQABYYBQAQA
BYYBQAQA
524 SOLIKHATUN AND SALMAH
( ) ( ) 0
( ) ( ) ( ) ( )0
2 2
t
i i i i i i
t t
i i j i i j j j i j j i
Q A B K A B K Q
Q A B K A B K Q Q A B K A B K Q
( ) ( ) 0
( ) ( ) ( ) ( ) 02 2
t
i i i i i i
t t
i i j i i j j j i j j i
Q A B K A B K Q
Q QA B K A B K Q A B K A B K Q
By define i iY K Q then we have
0
0.2 2
t t t
i i i i i i
t t t t t t
i i i j j i j j j i i j
AQ QA BY Y B
AQ QA BY Y B A Q QA B Y Y B
The last sufficient can be derived immediately from Theorem 13.■
Fuzzy controller model (4) is described as sum of weighting of r subsystems. The controller
design is guarantees the stability of system and satisfies the desired transient responses of
system. If the r approach to infinity, the existence of controller that stabilize the system will
be difficult to obtain. It is caused by find the solution of set of LMIs that satisfying some the
conditions.
There are two approach to relax the stability conditions according [7], namely, the global and
the regional Membership Function Shape Dependent (MFSD). In this paper we used the
regional MFSD. The operating regions of membership functions is divided into r region. Each
of region have has individual constraints that brings regional information to relaxation of
stability condition by some slack matrices T. We will formulate the problem of controller
design in LMIs feasibility problem as followed:
Theorem 15. Assume that the number of rules that fire for all t is less than or equal to s,
where 1 s r . The fuzzy control system (4) can be stabilized in the specified region D if
there exists a common positive definite matrix Q , iY and a common positive semi definite
matrix T such that the following conditions hold
(7)
ji
iii
t
i
t
i
t
i hhrjirQYBQAqQ
BYQAqQrQ,,...,2,1,0
where 1s . Given solution ( , )iQ Y , the fuzzy state feedback gain is obtained by
1
i iK YQ .
Proof. Consider a candidate of Lyapunov function 0),()())(( PtPxtxtxV t . Then
02
)1(
22
0)1(
TsBYYBQAQABYYBQAQA
TsBYYBQAQA
t
j
t
iij
t
jj
t
i
t
jji
t
ii
t
i
t
iii
t
ii
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 525
)(2
)(
2
)()()()(2
)())(()())((
1
1
2
txGG
PPGG
txzhzh
txPGPGtxzhtxV
jiij
t
jiijr
i ji
t
ji
iiii
tr
i
i
1( ) ( ),
2ij i i j j j iG A B K A B K i j and
1( ) ( )
2ii i i i i i iG A B K A B K .
From condition of second LMI (7) and Corollary 10, we have
).())1()(()(
)()()()1()())(()(
)()()()(2)())(()())((
1
2
1
2
1
2
11
2
txTsPGPGtxzh
tTxtxzhstxPGPGtxzh
tTxtxzhzhtxPGPGtxzhtxV
iiii
tr
i
i
tr
i
iiiii
tr
i
i
r
i ji
t
jiiiii
tr
i
i
Because condition of the first LMI (7) holds by 1Q P and i iY K Q ,
0)(,0))(( txtxV .■
Simulation Results on Motion System of Satellite
Last, simulation results are presented by application of the proposed methodology to
model of motion system of the satellite. Consider the model in state space form is followed:
2
0 0
0 1 0 0 10
3 0 0 2( ) ( ) ( )
0 0 0 1 0 0
0 2 0 0 10
mx t x t u t
m
Where ( )( )
( )
r
rx t
t
and ( )
( )( )
ru tu t
u t
.
The )(tr is the distance between the surface earth and the satellite that is affected by
time. The tt )( is the different between angle and angle position that is affected by time.
Let the 1( )x t as variable fuzzy, then exist three power one rules as followed:
526 SOLIKHATUN AND SALMAH
If 1( )x t is ZE then 1 1( ) ( ) ( )x t A x t B u t ,
If 1( )x t is PO (or NE) and u(t) is ZE then 2 2( ) ( ) ( )x t A x t B u t ,
If 1( )x t is PO (or NE) and u(t) is NE (or PO) then
3 3( ) ( ) ( )x t A x t B u t .
The fuzzy rules for controller are given
If 1( )x t is ZE then ( 1) ( )i iu t K x t
If 1( )x t is PO (or NE) and u(t) is ZE then ( 1) ( )i iu t K x t ,
If 1( )x t is PO (or NE) and u(t) is NE (or PO) then
( 1) ( )i iu t K x t .
Supposed the matrices are followed
0020
1000
2003
0010
1A ,
0040
1000
40012
0010
2A ,
0080
1000
80048
0010
3A
and
1620
10
00
01620
100
321 BBB . The results of simulation are expressed in Graphics
of response impulse are followed:
-20
-15
-10
-5
0
5x 10
9
To: O
ut(1
)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5
0
5
10
15x 10
5
To: O
ut(2
)
Respon impulse before are given controller
Time (sec)
Ampl
itude
-0.5
0
0.5
1
1.5
2
2.5
To: O
ut(
1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
x 10-4
To: O
ut(
2)
Respon impulse after are given controller
Time (sec)
Am
plit
ude
Regula Fuzzy Cont ro ller Design on Model of Mot ion System. . . 527
-0.5
0
0.5
1
1.5
2
2.5T
o: O
ut(
1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
x 10-4
To: O
ut(
2)
Respon impulse after are given controller by relaxing the stability
Time (sec)
Am
plit
ude
3. CONCLUSIONS
In this paper, the linear state feedback fuzzy controller with guaranteed stability and pre-
specified transient performance is presented. By formulate the system into TS-fuzzy model
and recasting these constraints into LMIs, we formulate an LMI feasibility problem for the
design of the fuzzy state feedback control system. If the r approach to infinity then the
existence of controller that stabilizes the system will be difficult to obtain. It is caused by find
the solution of set of LMIs that satisfying some the conditions. By relaxing the stability
conditions we have formulated the problem of controller design in LMIs feasibility problem.
Acknowledgement. This project was supported by Research Grant of Mathematics
Department for 2010-2011, Faculty of Mathematics and Natural Sciences, Gadjah Mada
University.
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