mathematics and its applications in the development … · 2015-03-14 · proceedings of the 6th...

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SEAMS - GMU 2011

Yogyakarta - Indonesia, 12 - 15 July 2011th th

Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications

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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


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

Hadrian Andradi

Supporting Team :

Parjilan Warjinah Siti Aisyah Emiliana Sunaryani Yuniastuti

Susiana Karyati Tutik Kristiastuti Sudarmanto

Tri Wiyanto Wira Kurniawan Sukir Widodo Sumardi

EDITORIAL BOARDS

Algebra, Graph and Combinatorics

Budi Surodjo

Ari Suparwanto

Analysis

Supama

Atok Zulijanto

Applied Mathematics

Fajar Adi Kusumo

Salmah

Computer Science

Edi Winarko

MHD. Reza M.I. Pulungan

Statistics and Finance

Subanar

Abdurakhman

LIST OF REVIEWERS

Abdurakhman

Universitas Gadjah Mada, Indonesia

Insap Santosa


Achmad Muchlis

Institut Teknologi Bandung, Indonesia

Intan Muchtadi-Alamsyah


Adhitya Ronnie Effendi


Irawati


Agus Buwono

Institut Pertanian Bogor, Indonesia

Irwan Endrayanto A.


Agus Maman Abadi

Universitas Negeri Yogyakarta, Indonesia

Jailani


Agus Yodi Gunawan


Janson Naiborhu


Ari Suparwanto


Joko Lianto Buliali

Institut Teknologi Sepuluh Nopember,

Indonesia

Asep. K. Supriatna

Universitas Padjadjaran, Indonesia

Khreshna Imaduddin Ahmad S.


Atok Zulijanto


Kiki Ariyanti Sugeng

Universitas Indonesia

Azhari SN


Lina Aryati


Budi Nurani


M. Farchani Rosyid


Budi Santosa

Institut Teknologi Sepuluh Nopember, Indonesia

Mardiyana

Universitas Negeri Surakarta, Indonesia

Budi Surodjo


MHD. Reza M. I. Pulungan


Cecilia Esti Nugraheni

Universitas Katolik Parahyangan, Indonesia

Miswanto

Universitas Airlangga, Indonesia

Ch. Rini Indrati


Netty Hernawati

Universitas Lampung, Indonesia

Chan Basarrudin


Noor Akhmad Setiawan


Danardono


Nuning Nuraini


Dedi Rosadi


Rieske Hadianti


Deni Saepudin

Institut Teknologi Telkom, Indonesia

Roberd Saragih


Diah Chaerani


Salmah


Edy Soewono


Siti Fatimah

Universitas Pendidikan Indonesia

Edy Tri Baskoro


Soeparna Darmawijaya


Edi Winarko


Sri Haryatmi


Endar H Nugrahani


Sri Wahyuni


Endra Joelianto


Subanar


Eridani

Universitas Airlangga, Indonesia

Supama


Fajar Adi Kusumo


Suryanto


Frans Susilo

Universitas Sanata Dharma, Indonesia

Suyono

Universitas Negeri Jakarta, Indonesia

Gunardi


Tony Bahtiar


Hani Garminia


Wayan Somayasa

Universitas Haluoleo, Indonesia

Hartono


Widodo Priyodiprojo


Hengki Tasman


Wono Setyo Budhi


I Wayan Mangku


Yudi Soeharyadi


Indah Emilia Wijayanti


PREFACE

It is an honor and great pleasure for the Department of Mathematics –

Universitas Gadjah Mada, Yogyakarta – INDONESIA, to be entrusted by the

Southeast Asian Mathematical Society (SEAMS) to organize an international

conference every four years. Appreciation goes to those who have developed and

established this tradition of the successful series of conferences. The SEAMS -

Gadjah Mada University (SEAMS-GMU) 2011 International Conference on

Mathematics and Its Applications took place in the Faculty of Mathematics and

Natural Sciences of Universitas Gadjah Mada on July 12th – 15th, 2011. The

conference was the follow up of the successful series of events which have been

held in 1989, 1995, 1999, 2003 and 2007.

The conference has achieved its main purposes of promoting the exchange

of ideas and presentation of recent development, particularly in the areas of pure,

applied, and computational mathematics which are represented in Southeast Asian

Countries. The conference has also provided a forum of researchers, developers,

and practitioners to exchange ideas and to discuss future direction of research.

Moreover, it has enhanced collaboration between researchers from countries in the

region and those from outside.

More than 250 participants from over the world attended the conference.

They come from USA, Austria, The Netherlands, Australia, Russia, South Africa,

Taiwan, Iran, Singapore, The Philippines, Thailand, Malaysia, India, Pakistan,

Mongolia, Saudi Arabia, Nigeria, Mexico and Indonesia. During the four days

conference, there were 16 plenary lectures and 217 contributed short

communication papers. The plenary lectures were delivered by Halina France-

Jackson (South Africa), Jawad Y. Abuihlail (Saudi Arabia), Andreas Rauber (Austria),

Svetlana Borovkova (The Netherlands), Murk J. Bottema (Australia), Ang Keng Cheng

(Singapore), Peter Filzmoser (Austria), Sergey Kryzhevich (Russia), Intan Muchtadi-

Alamsyah (Indonesia), Reza Pulungan (Indonesia), Salmah (Indonesia), Yudi

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 ..........................................................

365

Divo D. Silalahi, J.P. Caliman, Yong Yit Yuan

Continuously Translated Framelet............................................................................................ 379 Dylmoon Hidayat

Multilane Kinetic Model of Vehicular Traffic System ............................................................. 386 Endar H. Nugrahani

Analysis of a Higher Dimensional Singularly Perturbed Conservative System: the Basic

Properties ….............................................................................................................................. 395 Fajar Adi Kusumo

A Mathematical Model of Periodic Maintence Policy based on the Number of Failures for

Two-Dimensional Warranted Product ..…................................................................................. 403 Hennie Husniah, Udjianna S. Pasaribu, A.H. Halim

The Existence of Periodic Solution on STN Neuron Model in Basal Ganglia …………………………. 413 I Made Eka Dwipayana

Optimum Locations of Multi-Providers Joint Base Station by Using Set-Covering Integer

Programming: Modeling & Simulation ….................................................................................. 419 I Wayan Suletra, Widodo, Subanar

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

Inequality …………………………………………………………………….…………….…………..…….……………………..

515 Solikhatun and Salmah

Unsteady Heat and Mass Transfer from a Stretching Suface Embedded in a Porous Medium

with Suction/injection and Thermal Radiation Effects…………………………………………………………. 529 Stanford Shateyi and Sandile S Motsa

Level-Set-Like Method for Computing Multi-Valued Solutions to Nonlinear Two Channels

Dissipation Model ……………………………………………………………………………….....………………………….. 547 Sumardi, Soeparna Darmawijaya, Lina Aryati, F.P.H. Van Beckum

Nonhomogeneous Abstract Degenerate Cauchy Problem: The Bounded Operator on the

Nonhomogen Term ………………………………………………………………………..…………………..……………… 559 Susilo Hariyanto, Lina Aryati,Widodo

Stability Analysis and Optimal Harvesting of Predator-Prey Population Model with Time

Delay and Constant Effort of Harvesting ………………………………………………………..……………………. 567 Syamsuddin Toaha

Dynamic Analysis of Ethanol, Glucose, and Saccharomyces for Batch Fermentation ………….. 579 Widowati, Nurhayati, Sutimin, Laylatusysyarifah

Computer Science, Graph and Combinatorics

Survey of Methods for Monitoring Association Rule Behav ior ............................. 589 Ani Dijah Rahajoe and Edi Winarko

A Comparison Framework for F ingerprint Recognition Methods . . . . . . . . . . . . . . . . . . . . 601 Ary Noviyanto and Reza Pulungan

The Global Behav ior of Certa in Turing System …………………. ..................................... 615 Janpou Nee

Logic Approach Towards Formal Verif ication of Cryptographic Protocol . . . . . . . . . 621 D.L. Crispina Pardede, Maukar, Sulistyo Puspitodjati

A Framework for an LTS Semantics for Promela ….. . . . . . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . 631 Suprapto and Reza Pulungan

Mathematics Education

Modelling On Lecturers’ Performance with Hotteling-Harmonic-Fuzzy …................................ 647 H. A. Parhusip and A. Setiawan

Differences in Creativity Qualities Between Reflective and Impulsive Students in Solving

Mathematics …………………......................................……...................…………………........................ 659 Warli

Statistics and Finance

Two-Dimensional Warranty Policies Using Copula ...…………...................………………………………. 671 Adhitya Ronnie Effendie

Consistency of the Bootstrap Estimator for Mean Under Kolmogorov Metric and Its

Implementation on Delta Method ……................................………………….................................. 679 Bambang Suprihatin, Suryo Guritno, Sri Haryatmi

Multivariate Time Series Analysis Using RcmdrPlugin.Econometrics and Its Application for

Finance ..................................................................................................................................... 689 Dedi Rosadi

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.


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


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


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


( ) ( ) 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


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


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


( ) ( ) 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


)(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:


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


-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.

References

[1] BOYD, S., GHAOUI, FERON, LE., E. AND BALAKRISHNAN, V., LMI in System and Control Theory,

SIAM, Philadelphia, 1994.

[2] CHILALI, M., AND GAHINET, P., H Design with Pole Placement Constraints: An LMI Approach,

IEEE Trans. Automatic Control, Vol 41, No 3 pp 358-367,1996.

[3] HONG, S.K. AND NAM, Y., Stable Fuzzy Control System Design with Pole Placement Constraint: An LMI Approach, Computer in Industry, Elsevier Science, 2003.

[4] LAM, H.K., LEUNG, F.H.F. AND TAM, P.K.S., A LMI Approach for Control of Uncertain Fuzzy

System, IEEE Control Systems Magazine, August 2002. [5] MASTORAKIS, N.E., Modeling Dynamical Systems via the Takagi-Sugeno Fuzzy Model, Department of

Electrical Engineering and Computer Science, Hellenic Naval Academy Piraeus, Greece, 2004. [6] MESSOUSI, W.E., PAGES, O. AND HAJJAJI, A.E., Robust Pole Placement for Fuzzy Models with

Parametric Uncertainties: An LMI Approach, University of Picardie Jules Verne, France, 2005.

[7] NARIMANI, M and LAM, HK., Relaxed LMI-Based Stability Conditions for Takagi-Sugeno Fuzzy Control Systems Using Regional-Membership-Function-Shape-Dependent Analysis Approach, IEEE

Transaction on Fuzzy Systems, Vol 17, No 5, 2009.

[8] OGATA, K, Modern Control Engineering, 2nd ed. Englewood Cliffs, N.J,: Prentice Hall, Inc, 1990. [9] OLSDER,J., Mathematical System Theory, Faculty of Technical Mathematics and Informatics, Delf


University, Netherland, 1994.

[10] POLDERMAN J.W., Applied Mathematical Sciences (AMS) series, Enscheda, Groningen, 2006.

[11] SOLIKHATUN, Fuzzy State feedback Control with Multiobjectives, Proceeding IndoMS International Conference Mathematics Applied, Gadjah Mada University, Yogyakarta, 2009.

[12] TANAKA, K., AND SUGENO, M., Stability Analysis and Design of Fuzzy Control System, Fuzzy Set

System, Vol 45, No 2, pp135-156, 1992. [13] TANAKA, K. AND WANG, H., Fuzzy Control System Design and Analysis: A Linear Matrix Inequality

Approach, John Wiley & Sons, Inc, 2001.

[14] WANG, H. O., TANAKA, K., AND GRIFFIN, M. F., An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues, IEEE Trans. On Fuzzy System, Vol 4. No 1 pp 14-23, 1996.

Solikhatun



Gadjah Mada University, Yogyakarta

e-mail : [email protected]

Salmah



Gadjah Mada University, Yogyakarta

e-mail : [email protected]

mailto:[email protected]

mailto:[email protected]

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