INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS AND ITS APPLICATIONS 2012 ICMSA 2012 “MATHEMATICAL AND STATISTICAL THINKING FOR TECHNOLOGY DEVELOPMENT”

  

PROCEEDING

  

INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS

AND ITS APPLICATIONS 2012

  

ICMSA 2012

“MATHEMATICAL AND STATISTICAL THINKING FOR

TECHNOLOGY DEVELOPMENT”

  

Published by:

  MATHEMATICS DEPARTMENT

  INSTITUT TEKNOLOGI SEPULUH NOPEMBER SURABAYA, INDONESIA PROCEEDING

  

INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS

AND ITS APPLICATIONS 2012

  ICMSA 2012

  Editor : Prof. Dr. Basuki Widodo, M.Sc. Prof. Dr. M. Isa Irawan, MT. Dr. Subiono Subchan, Ph.D Prof. Nur Iriawan, Ph.D Dr.rer.pol. Heri Kuswanto Dr. Muhammad Mashuri

  ISBN 978-979-96152-7-5 This Conference is held by cooperation with Statistic Department, ITS Mathematics Department, Udayana Secretariat : Mathematics Department Kampus ITS Sukolilo Surabaya 60111, Indonesia

   icmsa2012.org

ORGANIZING COMMITTEE

  PATRONS : Rector of ITS STEERING : Dean Faculty of Mathematics and Natural Sciences ITS

  INTERNATIONAL SCIENTIFIC COMMITTEE Prof. Basuki Widodo (ITS - Indonesia) Prof. Nur Iriawan (ITS - Indonesia) Prof. Nyoman Budiantara (ITS - Indonesia) Prof. M. Isa Irawan (ITS - Indonesia) Dr. Muhammad Mashuri (ITS - Indonesia) Dr. Erna Apriliani (ITS - Indonesia) Dr. Subiono (ITS - Indonesia) Subchan, Ph. D (ITS - Indonesia)

Prof. Dr. Herman Mawengkang (University of Sumatera Utara - Indonesia)

Dr. Hizir Sofyan (Syiah Kuala University - Indonesia) Dr. Saib Suwilo (University of Sumatera Utara - Indonesia) Dr. Tarmizi Usman (Syiah Kuala University - Indonesia) Prof. Dato Dr. Rosihan M. Ali (Universiti Sains Malaysia - Malaysia) Assoc. Prof. Dr. Anton Abdulbasah Kamil (Universiti Sains Malaysia - Malaysia) Assoc. Prof. Adam Baharum (Universiti Sains Malaysia - Malaysia)

Assoc. Prof. Dr. Pachitjanut Siripanitch (National Institute of Development

Administration - Thailand) Assoc. Prof. Dr. Surapong Auwatanamongkol (National Institute of Development Administration - Thailand)

Assoc. Prof. Putipong Bookkamana (Chiang Mai University - Thailand)

COMMITEE

  Chairman : Dr. Suhartono Co-chairman : Dr. Darmaji Secretary : Dr. Santi Wulan Purnami Soleha, M.Si. Finance : Dr. Irhamah Technical Programme :

  Dr. Komang Dharmawan Drs. Daryono Budi Utomo, M.Si. Drs. Lukman Hanafi, M.Sc. Subchan, Ph. D Dr.rar.net. Heri Kuswanto Dr. Sutikno Dr. Agnes Tuti Rumiati

  Web and Publication : Dr. Imam Mukhlas Dr. Brodjol Sutijo S.U. Transportation and Accommodation : Dr. Purhadi

  Drs. I Gusti Ngurah Rai, M.Si. Drs. Bandung Ary, M.IKom.

  

Message

from the

  

Rector

Institut Teknologi Sepuluh Nopember

I would like convey my sincere congratulation to all involved parties for the

successful organization of the IMT-GT International Conference on

Mathematics, Statistics and its Applications. The ICMSA is annually conference

organized by The Indonesia-Malaysia-Thailand Growth Triangle (IMT-GT), and

this year ITS hosts the conference, organized by the Department of Mathematics

and Department of Statistics. I would like also to express my deep appreciation

to Department of Mathematics, Udayana University Bali for the collaboration.

nd This ICMSA 2012 is held as part of our 52 Institute Anniversary.

  

It is great pleasure for me to welcome and thank all keynote speakers for the

worthy time to share your experience and expertise to all conference participants.

I do believe that your participation to this conference is a highlight and give a

significant insight to all of us. I expect that your patronage and support towards

the advancement of knowledge through this event, will contribute to the future

development of Mathematics and Statistics.

  

As we know that the role of Mathematics and Statistics is vital in many aspects

of live. This conference is a means to share and discuss a new knowledge and

inventions among researchers, practitioners and students that may lead to a more

real contribution of Mathematics and Statistics in solving problems arises in

social, business, economic, environment, and many others.

Last but not least, I wish all participants have a very pleasant and valuable

moment during the conference. Moreover, I hope that new collaborations among

participants could be established. To our foreign guests, I wish you a memorable

stay in Bali. We welcome you anytime to visit our university, Institut Teknologi

Sepuluh Nopember (ITS) in Surabaya.

  Prof. Dr. Ir. Triyogi Yuwono, DEA. Rector of the Institut Teknologi Sepuluh Nopember (ITS) Indonesia

iv

  

Message

from the

  

Dean Faculty of Mathematics and Natural

Sciences

Institut Teknologi Sepuluh Nopember

  On behalf of the Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, it is a great honor and sincere to welcome all

  th

  participants to the 8

  IMT-GT International Conference on Mathematics, Statistics, and it’s Applications (ICMSA 2012). This year, Department of Mathematics and Department of Statistics, Institut Teknologi Sepuluh Nopember collaborate with Department of Mathematics, Udayana University, have honor to organize this meaningful international conference. I believe that the purpose of this conference is not only sharing knowledge among mathematician, statisticians, and scholars in related fields but also to hearten new generation of expertise in mathematics and statistics to realize the science and technology advancement. It is undeniable that science and technology are the products of mathematics and statistics applications. Many disciplines like engineering, computer science, information technology, operational research, logistics management, risk management and many others are all the products of mathematics and statistics. Thus, it is essential that we must hold this annual conference as a stage for all scholars in finding new ideas and applications on Mathematics and Statistics. Greatly thank to all supportive session including organizing committee, keynote speakers, invited speakers, paper reviewers, participants and sponsors. This event will not achieve without you all. Finally, I hope that the outcome of ICMSA 2012 will be pleasing and most useful to everybody.

  Sincerely yours, Prof. Dr. R.Y. Perry Burhan

  Dean

v

  

Message

from the

  

Chairman

Organizing Committee

  On behalf of the organizing committee, it is my great pleasure to welcome all

  th

  participants of the 8

  IMT-GT 2012 International Conference on Mathematics, Statistics, and it’s Applications (ICMSA 2012). This conference had been held for seven times in Indonesia, Malaysia and Thailand. It is the fourth time that Indonesia hosts the conference and Department of Mathematics and Department of Statistics, Institut Teknologi Sepuluh Nopember collaborate with Department of Mathematics, Udayana University, are honored to organize this important event.

  The theme of our conference is “Mathematical and Statistical Thinking for Technology Development” highlighting the importance of mathematical and statistical science as the major tools for solving problems and making right decisions. They play vital roles to the development of science and technology in the IMT-GT region and beyond. The regular meeting among researchers in the fields like this conference will promote the progress and advancement of the fields. This conference will surely serve as a venue for researchers in the fields to present their works, exchange ideas and seek collaboration. Participants from the IMT-GT region and many countries around the world will attend the conference. More than 10 distinguished speakers from many countries are invited to give talks in the conference. So, I hope all participants will enjoy attending to the talks and paper presentations as well as have very fruitful discussions.

  I would like to take this opportunity to thank all the keynote and invited speakers for coming and sharing their knowledge with us. I am also very grateful to all international and local scientific committee and many others who have contributed to the accomplishment of the meeting. Without their helps and supports, the preparation for the conference would deem impossible to complete. Finally, I would like to thank all participants for joining the conference. I do hope all participants will have opportunity to explore Bali and enjoy staying in the island of Gods. Sincerely yours, Dr. Suhartono

  Chairman

vi

  

CONTENTS

  (D.K. Syofyan and A.N.M. Salman)

  AM5 Dynamic Stability Model For A Small Submarine

  (Purwanti, B. S. R. , Yusivar, F., and Garniwa, I. M. K.)

  AM4 Determination the Error and Delta Error for Braking Control System of Three Phase Motor

  (Suharmadi Sanjaya )

  AM3 Numerical Solution To Control The Exploitation Of Ground Water

  

(Erna Apriliani, Subchan, Fitri Yunaini, and Santi Hartini)

AM2 Pedestrian flow characteristics in a least developing country. (Khalidur Rahman, Noraida Abdul Ghani, Anton Abdulbasah Kamil, Adli Mustafa)

  AM1 Estimation and Control Design of Mobile Robot Position

  (Intan Muchtadi-Alamsyah)

  PM10 Pollard Rho Algorithm for Elliptic Curves over Composite Fields

  (Irwansyah, Ahmad Muchlis, Djoko Supriyanto, and Intan Muchtadi)

  PM9 On almost weakly self-dual normal bases

  (Gregoria Ariyanti, Ari Suparwanto, and Budi Surodjo)

  PM8 Invertible Matrices over The Symmetrized Max Plus Algebra

  Graphs

  Cover i

  Theoretic Method (Maria Anestasia) PM7 Some Sufficient Conditions for Corona Graphs to be Product Cordial

  (Darmajid, Intan Muchtadi-Alamsyah, and Irawati) PM6 Fractal Dimension: Box Dimension, Hausdorff, Dimension, and Potential

  PM4 Generating functions of polynomial sequences over integral domains and quotient fields (Yusuf Chebao) PM5 Polydule varieties over finite-dimensional algebra

  (I. A. Purwasih, M. Ba_ca, and E. T. Baskoro)

  PM3 The locating chromatic number of strong product of two paths

  PM2 A Study Permutation Theory and Its Application to Enumeration of Latin Square-X (Subiono, Muhammad Syifa'ul Mufid)

  (Supriyadi Wibowo, Muslich)

  ( )

  PM1 Connection Between Parvate-Gangal Mean Value Theorems and Holder Continuous Function of Order

  Mathematics Papers

  Paper of Plenary

  Tentative Schedule xii

  Message from Rector of Institut Teknologi Sepuluh Nopember (ITS) iv Message from Dean Faculty of Mathematics and Natural Sciences v Message from Chairman of Organizing Committee vi Contents vii

  Organizing Committee iii

  (Aries Sulisetyono) AM6 Optimization Model On Quadratic Programming Problem With Fuzzy

  (Sugiyarto)

  AM7 Detecting Fouling In Heat Exchanger By Extended Kalman Filter Method And Ensemble Kalman Filter

  (Lukman Hanafi , Erna Apriliani, and Ana Fadlilah)

  AM8 Review of Asset Return Distribution and Its Application

  (Sandya N. Kumari and A. Tan)

  AM9 Control System Roket Rkx-200 Lapan Using Pid Controller

  (Subchan, Putra Setya Bagus J.N. dan Idris E.P.)

  AM10 Modified Feige-Fiat-Shamir Signature Scheme with Message Recovery

  (Dessi Nursari , Elena Sabarina, and Rizki Yugitama)

  The effect of the use of the MDS matrices in the T-020 block cipher AM11 algorithm

  (Sutoro, Bety Hayat Susanti)

  AM12 RAN Signature Scheme (Novita Loveria, Rizkya Mardyanti, Ayubi Wirara) Studies on Simplified Chaos Hash Algorithm-1 (SCHA-

  AM13 1) using Yuval’s

  Birthday Attack

  (Adrian Admi, Bety Hayat Susanti)

  AM14 The Implement of extracting the partial subkey bits from Linear

  (Bashir Arrohman , Elena Sabarina, and Prilia Trianantia Lestari)

  AM15 UDD Assumption for Life Annuity with m-thly Payments

  (Farah Kristiani)

  AM16 Orchestration of semantic web service using OWL-S for variations of ERP business process

  (Anang Kunaefi, Riyanarto Sarno , Bandung Arry Sanjoyo, Imam Mukhlash, Hanim Maria Astuti)

  AM17 Comparing The Distribution of Non Stationary Processes : A Spatial Dominance Approach

  (Irwan Susanto, Respatiwulan and Supriyadi Wibowo)

  AM18 The Correlation Between Hydrodynamic Of River And Pollutant Dispersion In A River.

  Mixed Geographically Weighted Multivariate Linier Model (Case Study : The Rainfall and Morphometry Effects in the Determination of Water Flow Rate and Sediment in Konto Hulu Watershed) ( Basuki Widodo,

  Bambang Agus S., Setiawan)

  Stochastic Divination Reckoning Enactment on Multi class Queueing AM19 System.

  (K.Sivaselvan and C.Vijayalakshmi )

  AM20 Using SVAR with B-Q Restriction to examine post-tsunami inflation in Aceh

  (Saiful Mahdi)

  AM21 Mathematical Modeling of Circular Cylinder Drag Coefficient with I- Type as a Passive Control

  (Chairul Imron, Suhariningsih, Basuki Widodo, and Triyogi Yuwono)

  AM22 Analyzing portfolio performance of Bangladesh stock market

  (Md. Zobaer Hasan, Anton Abdulbasah Kamil, Adli Mustafa and Md. Azizul Baten)

  AM23 Implementation Of The Algorithm Kalman Filter On Reduction Model

  (Didik Khusnul Arif, Widodo, Salmah, Erna Apriliani) Statistics Papers

  SA1 Dimension reduction with sliced inverse regression as pra-processing in

  (Ni Wayan Dewinta Ayuni and Sutikno)

  SA2 Application of Structural Equation Modelling (SEM) to Analysed the Behaviour of Consumers for the Products of Embroidery and Pariaman Needlework (Lisa Nesti, Irna Ekawati) Comparison of stability long-horizon R-forecasting and V-forecasting

  SA3 SSA

  (Awit M. Sakinah, Toni Toharudin, Gumgum Darmawan)

  SA4 The weibull prior distribution on the information-based approach asset pricing model by brody hughston macrina

  (Mutijah)

  SA5 Hypothesis Testing In Regression Model Bivariate Weibull

  (Andi Quraisy and Purhadi)

  SA6 Identification Correlation Between Economic Variables and Welfare Variables With Canonical Correlation Analysis

  (Asep Rusyana, Nurhasanah,Raudhatul Jannah)

  SA7 Spatial Bayesian Poisson Lognormal Analysis of Dengue Relative Risk Incidence in Surabaya on 2010

  

(Mukhsar, Iriawan, N , Ulama, B. S. S, Sutikno, Kuswanto, H)

  SA8 Survival Analysis With Cox Regression ModeL (Case Study : Dengue Hemorrhagic Fever (DHF) Patients in The Haji Hospital at Surabaya).

  (Ni Putu Lisa Ernawatiningsih and Purhadi)

  SA9 Smoothing Spline Estimators in Semiparametric Multivariable Regression Model

  (Rita Diana, I. Nyoman Budiantara, Purhadi dan Satwiko Darmesto)

  SA10 Multivariate adaptive regression splines (MARS) approach for poverty data in East Java Province (Memi Nor Hayati and Purhadi) SA11 A Copula Approach to Construct Vulnerability Rice Puso Maps in East

  Java With El-Nino Southern Oscillation (ENSO) Indicator

  

(Pratnya Paramitha Oktaviana, Sutikno, and Heri Kuswanto)

  Model Based Clustering Versus Traditional Clustering Methods: A SA12

  Comparison Based On Internal and External Validation Measure

  (I Gede Nyoman Mindra Jaya, Henk Folmer, Budi Nurani Ruchjana)

  SA13 The Identification of the relationship between the area of the rice harvest and rainfall using Copula Approach.

  (Iis Dewi Ratih , Sutikno, dan Setiawan)

  SA14 Simulation of The Stationary Spatio-Temporal Disaggregation using Bayesian State-space with Adjusting Procedure

  (Suci Astutik, Nur Iriawan, Suhartono, and Sutikno)

  SA15 Optim al S moot hing Param et er for Spli ne Parti al Estim ator i n

  Multi response S emiparam et ri c R egression

  (Wahyu Wibowo, Sri Haryatmi, I Nyoman Budiantara)

  SA16 Bayesian Model Comparison : BIC for Nonlinear SEM

  ( Margaretha Ari Anggorowati, Nur Iriawan Suhartono and Hasyim Gautama )

  SA17 Modeling of Gross Regional Domestic Product Manufacturing Industries Sector in East Java : a Spatial Durbin Model Approach

  (Setiawan)

  SA18 Multi Input Intervention Model for Evaluating the Impact of the Asian Crisis and Terrorist Attacks on Tourist Arrivals in Bali

  (Sri Rezeki, Suhartono, Suyadi)

  Param et ers Estim ation of the Additive Outlier of t he Vector SA19

  Autoregression

  (Agus Suharsono, Suryo Guritno, Subanar)

  SA20 Breeder Genetic Algorithm for bi-objective Multiple VRP with Stochastic Demands

  (Irhamah and Zuhaimy Ismail)

  SA21 Forecasting Fruit Sales At Moena Fresh Bali Using Calendar Variation Model

  

(Dwiatmono Agus Widodo , Ni Made Dwi Ermayanthi, and

Suhartono)

  Credit Scoring for Cooperative of Financial Services Using Logistic SB1

  Regression Estimated by Genetic Algorithm

  1

  2

(Sukono , Asep Sholahuddin , Krishna Prafidya Romantica)

  SB2 ANOVA estimation in longitudinal linked data

  (Klairung Samart, Ray Chambers)

  Multivariate Adaptive Regression Splines (Mars) Approach For Analysis SB3

  Of Poverty Data In East Java Province

  

(Erma Oktania Permatasari and Bambang Widjanarko Otok)

  SB4 Exploration of factor analysis using R and SPSS to identify the Potential Factors on Indonesia Community Health Development Index (IPKM).

  (Zurnila Marli Kesuma, Virasakdi Chongsuvivatwong)

  SB5 Optimization In Process Production Of Envelopes By Multirespon Taguchi Method And Fuzzy Logic.

  (Sony Sunaryo, Albertus Laurensius Setyabudhi)

  SB6 Breast Cancer Diagnosis Using Smooth Support Vector Machine and Multivariate Adaptive Regression Splines

  (Shofi Andari, Santi W. Purnami, Bambang W.O)

  SB7 Modeling Long Memory Time Series By Singular Spectrum Analysis ( Case Study : Handymax Price Data)

  (Gumgum Darmawan)

  SB8 Multivarite Poisson Control Chart and its Applications

  (Wibawati, and Muhammad Mashuri)

  SB9 Adaptive Neuro-Fuzzy Inference System for Short Term Load Forecasting In Indonesia.

  (Indah Puspitasari, Suhartono, M. Sjahid Akbar) SB10 Forecasting Inflation in Indonesia using Ensemble Method

  (Mega Silfiani and Suhartono)

  SB11 Weighted Fuzzy Rule Base To Modeling Time Series Data And Its Application In Prediction Of Stock Prices.

  (Nurhayadi, Subanar, Abdurakhman, Agus Maman Abadi)

  SB12 Modeling The Effects of Modern Market Existence of Traders Traditional Market Income by Using Support Vector Regression (SVR)

  (Dwi E. Kusrini, Isnaini P. Dewi, and Irhamah)

  SB13 Nonparametric Classification Method Of Walfare Households In The Province Of East Java

  (Bambang Widjanarko Otok & Suhartono)

  SB14 Optimization in Process Production of Envelopes by Multi-response Taguchi Method and Fuzzy Logic

  (Brodjol S.S. Ulama and Mahmuddin, A.)

  SB15 Air Temperature Modeling By Using Artificial Neural Networks Approach

  (Edy fradinata, ST, MT)

  SB16 Smooth Support Vector Regression for Floating Breakwater Performance

  (Yoyok Setyo Hadiwidodo and Santi Wulan Purnami)

  SB17 Daily

  (Fithriasari, K; Iriawan, N; Ulama, B. S; Sutikno; Kuswanto, H)

  The Classification Of Breast Cancer Malignancy Using Ordinal Logistic SB18 Regression And Support Vector Machine (SVM).

  1 (Farizi Rachman and Santi Wulan Purnami)

  SB19 Optimization of Steel Cutting Process Using Bootstrapping Response Surface

  (Wiwiek Setya Winahju )

  SB20 Estimating costs for oil spills with Simultan Equation Modelling

  (Mukhtasor, Dwi Endah Kusrini, Mauludiyah)

  SB21 Optimization of Design of Experiment for hybrid fuel-system of Premium Benzene and LPG in motorcycling vehicles

  (Hendro Nurhadi)

  TENTATIVE SCHEDULE Monday, 19 November 2012 08.00 - 08.30 : Registration 08.30 : Opening Ceremony

• – 08.50

  09.00 - 09.45 : Keynote session 1 by Prof. Dr. Ir. Arnold W. Heemink (Netherland)

  09.45 - 10.30 : Keynote session 2 by Prof. Virasakdi Chongsuvivatwong (Thailand)

  10.30 - 11.00 : Break 11.00 - 11.45 : Keynote session 3 by Prof. Dato Dr. Rosihan M. Ali (Malaysia) 11.45 - 13.00 : Break 13.00 - 13.30 : Invited Speaker Dr. Darmaji (Class PM)
• Prof. Dr. Md. Azizul Baten (Class AM)
• Dr. Dedi Rosadi (Class SA)
• Dr.rar.net Heri Kuswanto (Class SB)
• 13.30 - 15.30 : Parallel Session

  Tuesday, 20 November 2012 08.00 - 08.45 : Keynote session 4 by Prof. Philipp Sibbertsen (Germany) 08.45 - 09.30 : Keynote session 5 by Prof. Nur Iriawan, Ph.D (Indonesia) 09.30 - 10.00 : Coffee Break 10.00 - 10.45 : Keynote session 6 by Prof. Iwan Pranoto, Ph.D.(Indonesia) 10.45 - 12.30 : Break 12.30 - 14.30 : Parallel Session Wednesday, 21 November 2012 Bali City Tour

  

Proceeding International Conference on Mathematics, ISBN 978-979-96152-7-5

Statistics and its Applications 2012 (ICMSA 2012)

  

IMPLEMENTATION

OF THE ALGORITHM KALMAN FILTER

ON REDUCTION MODEL

  1

  

2

  3

  4 ₁ Didik Khusnul Arif , Widodo , Salmah , and Erna Apriliani

Ph.D Student in Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia

and Lecturer of Department of Matematics Institut Technology Sepuluh Nopember (ITS),

₂

Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia

₃

Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia

₄

Department of Mathematics Institut technology Sepuluh Nopember (ITS) Surabaya Indonesia

  Abstract. In this paper, the first will be the establishment of a model reduction of dynamic stochastic discrete systems. Model reduction is done by cutting method equilibrium. Then we will implement the Kalman filter algorithm on reduced models.

  As a case study for the simulation, this paper selected the distribution of heat conduction problems. The first will be the establishment of a model reduction of heat conduction systems. Further implementation of the Kalman filter algorithm on heat conduction system is reduced. Then we will analyze the comparison of the results of the estimate. Keywords: estimate, filter Kalman, model reduction.

1 Introduction

  Estimation of state variables of a system is necessary because not all state variables of the system can be measured directly. Estimation of the state variable stochastic systems is done by Kalman filter. In its application, Kalman filter has several weaknesses that can be caused due to numerical stability problems or due to a less accurate system modeling.

  To solve the stability problem numerically, we accomplish this by forming the matrix square root of its covariance matrix. This algorithm is known as covariance Square Root Filter (SRCF). To reduce the computational time the square root covariance filter, Verlaan and Heemink (1997) to modify the reduced rank square root of the covariance matrix. The proposed algorithm is known Veerlan Square Root Filter covariance with Rank Reduction (RRSQRT). Reduction of rank in the algorithm is done by singular value decomposition. Algorithm RRSQRT has successfully accelerate computing time on SRCF algorithm.

  Issues of computation time can also be influenced by the order system. therefore, to reduce the computation time, we can do by replacing the first order model with a smaller model orders without significant error. This method is known as the model reduction method. One method for forming reduced models are cutting equilibrium.

  

AM23 Generally seen that from some existing research, the main focus has been discussed by the researchers is how to get the state variable estimation procedure that can get accurate estimation results and require a short computing time.

  Therefore in this paper will attempt to get the outcome variable estimation algorithm accurately estimates and short computing time. First, it will be reduced modeling of dynamic stochastic discrete systems. As an initial study, the method used to establish the model is reduced by cutting method equilibrium. Here will be a search for the model reduction process by the method of cutting equilibrium. The part that needs to be observed is to observe which parts of the state from the initial system will be reduced and the state which is maintained.

  Furthermore, it will formulate a Kalman filter algorithm for reduced models that have been formed. Then to see the results of its estimation and analysis, we can perform the implementation in applied problems, namely the problem of the distribution of heat conduction. First we create a model reduction of heat conduction systems. Next we implement the Kalman filter algorithm on reduced models that have been formed. Then we do a comparative analysis on the estimation of these methods.

2 Methodology

  In this paper, a Kalman filter algorithm will be implemented on the reduction model. As a case study chosen is a matter of estimation of the distribution of heat conduction in the wire conductor. The steps taken are: 1.

  Reviewing the process of establishing a model reduction of discrete systems with cutting methods equilibrium.

  2. Constructing Kalman filter algorithm to model reduction of discrete systems.

  3. Establish a distribution system modeling heat conduction.

  4. Estimate to determine the distribution of heat conduction by using a Kalman filter.

  5. Doing reduction model of heat conduction system.

  6. carry out the implementation of Kalman filter method on the model of heat conduction is reduced.

  7. comparative analysis of the estimation results obtained from the application of

  Kalman filter algorithm at the beginning of the system compared to the estimation results obtained from the application of Kalman filter algorithm on the system reduced its system.

3 The Result and Discussions

3.1 Model Reduction

  Model reduction is a reduction order system problem. The effort to get a simple model with a smaller order is called the reduction model (Karolos M. Grigoriadis, 1995). There are many methods to perform the model reduction, such as singular perturbation method (Yi Liu, 1989), the method of cutting equilibrium (Sigurd, S., 2001), genetic algorithms (Satakshi, 2004), the extended cuts equilibrium (Henrik Sandberg, 2008).

  Equilibrium cutting method is the simplest method. If we give the initial system is stable and observed, then we will also obtain a stable reduction model and observed. As an initial assessment will be used method of cutting is reduced in proportion to get the model.

3.1.1 Model Reduction by Cutting equilibrium

  Given a deterministic dynamic discrete systems with time invariant: (1) . (2) where is a state variable, , u , z

  ( ) ( ) is the input vector, ( ) ( ) is the output vector, , and ( ) respectively is a constant matrix with the corresponding order. For the next system (1) and (2) referred to as a system (A, B, C,

  D).

  Further defined two matrices P and Q, each of which is an achievement gramian matrix and keteramatan gramian matrix of the system (A, B, C, D), ie: (3)

  ) ∑ ( .

  (4) )

  ∑ ( P and Q are each positive definite and is the unique solution of the Lyapunov equation:

  (5) (6) . Then the system (A, B, C, D) is called equilibrium if P and Q are the same and is a diagonal matrix, ie

  ( ) , with is a positive real number which is the Hankel singular value of the i-th system (A, B, C, D). Generally defined √ ( ) . is gramian equilibrium of the system in balance, so it can be written as follows:

  (7) ̃

  (8) ̃ . In general equilibrium system can be written in the form:

  (9) ̃ ̃ ̃ ̃ .

  (10) ̃ ̃ ̃

  Furthermore gramian equilibrium can be partitioned into [ ] where . Then the

  ( ) and ( ) with equilibrium system ( ̃ ̃ ̃ ) be partitioned corresponding to the partition equilibrium gramian , to obtain its system partition as follows:

  ̃ ̃ ̃

  ̃ ̃ ̃

  (11) ( | ) . ̃ ̃

  ̃ ̃

  ̃ where ). ̃ ( ), ̃ ( ̃

  ) , dan ̃ ( ̃ ̃ ̃

  ̃

  2. Determine matrix reachability gramians P and matrix observability gramians Q.

  Actual equilibrium can be established by performing the following steps:

  Finally, to get the model reduction of the system ( ) can be done by cutting the state variable on the equilibrium system

  ( ̃ ̃ ̃ ) associated with Hankel singular values small. Model reduction formed first order r, with , and written in the form:

  ̃ ̃ ̃ ̃ (12)

  ̃ ̃ ̃ .

  (13) Reduction model expressed in equation (12) and (13) is called the reduced system ( ̃ ̃ ̃ )

3.1.2 Model Reduction Algorithm with equilibrium Cutting Methods

  Based on the foregoing, it can be briefly made a simple algorithm to obtain reduced models of discrete systems (A, B, C, D). The steps in the model reduction algorithm is as follows: 1.

  Given input initial system.

  Beginning of a given system is a discrete system, assuming the system to be stable.

  3. Establish a the realization equilibrium.

• Get the factorization matrix R such that .
• Construction of the matrix and doing factorization such that

  , where U is unitary matrix (ie: ) and ( ).

• Determine the transformation matrix T which is a non-singular matrix, ie: .
• Forms of equilibrium systems (

  , , , ) with its matrices are: , .

  4. Furthermore, to get the model reduction, do the cutting state equilibrium systems based on Hankel singular value of the smallest.

  In this case, the reduction process model does not detect the part where the state variables of the initial system is maintained and the state where the reduced variables. So the state variables of the initial system will be different from the state variables of the system is its reduction results.

  Therefore, the author felt the need to assess which parts of the state variable is maintained from the model reduction procedure. It is intended to do a comparative analysis on the estimation of the same state variable between the initial system and the system yield reduction.

3.2 Kalman Filters for Reduction System

  Estimation of state variables in stochastic dynamical systems can be performed using a Kalman filter. In the Kalman filter, an assessment is done by predicting the state variable based on dynamical systems and then made improvements based on measurement data. Phase prediction and phase correction is performed recursively, in a way to minimize the estimation error covariance is the actual state ̂

  ( ), with variables and is the estimate of the state variable. ̂

  Based on the description in the model reduced the formation of a discrete system, it appears that the formation of reduction model begins with the establishment of the equilibrium. Equilibrium system is the result of the transformation of the initial system state but its a different order to the order of the initial system state. In equilibrium systems, the position of the state is already sorted by its Hankel singular value. So the model reduction can be obtained from the system equilibrium by cutting state variables corresponding to the small Hankel singular value.

  Therefore, to apply the Kalman filter to the model reduction, as well as by applying Kalman filter on its equilibrium system. Given initial system as follows: . with and respectively a system noise and measurement noise, and each is a scale stochastic. System noise and measurement noise respectively assumed Normal- Gaussian distributed with mean zero and its covariance, respectively and . further from the initial system can be obtained equilibrium system, which can be written in the following form:

  (14) ̃ ̃ ̃ ̃ .

  (15) ̃ ̃ ̃ with

  ̃ , ̃ , ̃ and T is the transformation matrix obtained from the process of establishing a system of equilibrium.

  So based on the formation of Kalman filter algorithm for stochastic dynamic discrete systems, the Kalman filter algorithm can be developed for the system equilibrium of dynamic stochastic discrete systems as follows:

   System Model and Measurement Model (16) ̃ ̃

  (17) ̃ ̃

  (18) ̃ ( ̅ ) ( ( ). )

   Initialization ; .

  (19) ̃̂ ̅

   Phase prediction (time update) error covariance : (20)

  ) ( estimation : . (21)

  ̃̂ ̃̂  Phase Correction (Measurement Update) error covariance :

  (22) (( ̂ ) ( ) ) estimation :

  (23) ̃̂ ̃̂ ( ̃ ̃̂ ) . ( )

  If used Kalman gain, the Kalman Gain :

  (24) )

  ( ) ( ( ) error covariance : (25)

  ( ) estimation : (26) ̃̂ ̃̂ ( ̃ ̃̂ ) .

3.3 Modeling Heat Conduction System

  Given the problems of heat on a straight wire of length and the heat conduction coefficient

  . Modeling of heat conduction problem can be done as follows. First selected the

• axis to reveal the longitudinal direction of the wire, with and , stating the position of the ends of the wire. Furthermore it is assumed that the sides are perfectly insulated wire, meaning no heat can penetrate the sides of the wire and temperature flowing in the wire is affected by the position and time. The temperature is expressed by u, the position expressed by x, and time is denoted by t. So u is a function of x and t. The temperature variation in the wire can be expressed in an equation called the equation of heat conduction, which is: dengan

  (27) , with is the thermal diffusivity of the wire material. Furthermore it is assumed also that at the end of the left side of the trunk completely isolated, which means no heat change at position for all and at the other end were kept constant heat .

  With the forward Euler method, then from equation (27) can be obtained: (28)

  ( ( ) ( ) ( )) . with i and k respectively determine the position and the time step. Furthermore, the central difference method, then from equation (28) can be obtained for a discrete system, which can be written as follows:

  . (29) ( )

  With .

  To maintain a stable pendiskritan explicitly, it must be selected or . While the boundary conditions and initial conditions are: .

  ( ) , ( ) ̅, ( ) ( ) In the above modeling system used for isolated systems perfectly on the sides of the wire. Whereas in fact they are not, it means there is heat transfer between the wire with air. Such influence is referred to as noise system. The systems include the noise system can be modeled as follows:

  . (30) ( ) with

  ( ) is assumed normally distributed with mean 0 and variance Q. Kalman filter to improve the prediction results based on measurement data. Therefore, it needs to be defined a measurement equation to relate the measurement data to the system. Measurement equation contains measurement noise , and is defined as:

  .

  (31) Measurement noise is also assumed to be Gaussian distributed Normal with mean 0 and variance R.

  System in the form of finite difference equations (30) and measurement (31) can be written in the form of state space systems are invariant with respect to time:

  (32) . (33) with is the state vector, is a vector output

  ] [ [ ] or measurement, is a system disorders and

  ] [ is the measurement disturbance. While matrices A, B, G, and C

  [ ] each is the coefficient matrix corresponding to the dimensions.

4 Analysis

  Given distribution of heat conduction problems in a wire conductor. In this simulation, we assumed the wire length and a length of wire is divided in 10 positions. The given boundary conditions are and with its coefficient of thermal conductivity is . First formed discrete systems by

  ⁄ taking , with the initial estimate ̂ . Estimated initial error variance is and I is the identity matrix. Then from the measurement results, the data obtained by the heat at a certain position ie u

  3 = 1.4 and u 8 = 48.5.

  further by taking Δt = 0.1, formed a discrete system with time. System noise generated from random values with mean 0 and covariance , while the measurement noise generated from random variable with mean 0 and covariance

  .

  By using a matlab program, to do the reduction process model and implementation of Kalman filter algorithms. Analysis of heat conduction distributions estimated by the Kalman filter algorithm can be done as follows. Based on Figure 1 shows the estimated distribution of heat conduction in the initial system and the system of its equilibrium. It appears that even though the estimation is taken to position the same state, that state is taken in position-3 both for the initial system and the system is in balance, it is different estimation. This shows that the position of state-state equilibrium in the system is different from its initial system

  Distribusi Panas SISTEM AWAL Pada Posisi ke-3 Pada semua Iterasi dengan deltaT=0.1 Distribusi Panas SISTEM SETIMBANG Pada Posisi ke-3 Pada semua Iterasi dengan deltaT=0.1 _{30 20 25 X_Estimasi X_asli 10 15 5 Xs_Estimasi Xs_asli pe tu ra te r m 10 15 5 ra te tu pe m} r _{-10 -5 -5 20 40 iterasi ke-k 60 80 100 120} -15 _{20 40 iterasi ke-k 60 80 100 120}

  Figure 1: Heat distribution in position 3 on all iterations

  Figure 2 shows the estimated distribution of heat conduction in all positions for its 50th iteration, for both the initial system and its equilibrium systems. From both figures it appears that the heat distribution in each state, when seen in the same position, seen showing different results. But overall, both the initial system and its equilibrium systems, both show a similar pattern to estimate its overall state.

  Figure 2: Heat distribution at all positions for the 50th iteration

  This is also confirmed by the results of Figure 3 which shows the estimation error covariance. From Figure 3 shows that the estimation error covariance, for both the initial system and to its equilibrium systems, they showed me the pattern of results, the estimation error covariance him longer to get to zero. This means that the estimate converges to zero. In other words, state-state estimation is nearing its true value.

  Figure 3: Estimation error covariance