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

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PROCEEDING

INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS

AND ITS APPLICATIONS 2012

ICMSA 2012

“MATHEMATICAL

AND STATISTICAL THINKING FOR

TECHNOLOGY DEVELOPMENT”

Published by:

MATHEMATICS DEPARTMENT

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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

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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

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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.

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

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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 participants to the 8th 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

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Message

from the

Chairman

Organizing Committee

On behalf of the organizing committee, it is my great pleasure to welcome all participants of the 8th 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

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IMPLEMENTATION

OF THE ALGORITHM KALMAN FILTER

ON REDUCTION MODEL

Didik Khusnul Arif1, Widodo2 , Salmah3 , and Erna Apriliani4

1

Ph.D Student in Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia and Lecturer of Department of Matematics Institut Technology Sepuluh Nopember (ITS),

didik@matematika.its.ac.id

2

Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia

widodo_math@yahoo.com

3

Department of Mathematics Gadjah Mada University (UGM) Yogyakarta Indonesia

syalmah@yahoo.com

4

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

erna@matematika.its.ac.id

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.

Proceeding International Conference on Mathematics, ISBN 978-979-96152-7-5 Statistics and its Applications 2012 (ICMSA 2012)

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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

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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( ) is the input vector, ( ) , z( ) 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:

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. (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

( ) and ( ) with . Then the

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

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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.

2. Determine matrix reachability gramians P and matrix observability gramians Q. 3. Establish a the realization equilibrium.

Actual equilibrium can be established by performing the following steps: - 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

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measurement data. Phase prediction and phase correction is performed recursively, in a way to minimize the estimation error covariance ( ̂ ), with is the actual state 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

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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 and at the other end were kept constant heat for all . 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.

(20)

(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

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

0 20 40 60 80 100 120

Distribusi Panas SISTEM AWAL Pada Posisi ke-3 Pada semua Iterasi dengan deltaT=0.1 X_asli

Distribusi Panas SISTEM SETIMBANG Pada Posisi ke-3 Pada semua Iterasi dengan deltaT=0.1

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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

While the estimation results in reduced system can be seen in the picture below:

1 2 3 4 5 6 7 8 9 10

Distribusi Panas SISTEM AWAL Pada semua Posisi pada Iterasi ke-50 dengan deltaT=0 X_asli

Distribusi Panas SISTEM SETIMBANG Pada semua Posisi pada Iterasi ke-50 dengan deltaT Xs_asli

kovariansi Kesalahan Estimasi filter Kalman pd SISTEM SETIMBANG

0 20 40 60 80 100 120

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Figure 4: Heat distribution in model reduction

In general, it appears that the estimated heat distribution on all state on the reduced system shows the same pattern of results with the results of the estimates for the initial system and its equilibrium systems. From Figure 4 shows that the estimated state is nearing its actual state value. This is confirmed by the results of the estimation error covariance representation which converges towards zero.

5 Conclusion

From the results of the above description, it can be concluded that the application of Kalman filter algorithm, both the initial system, the system in balance and in its reduced system, showed the same pattern of results estimated for all its state. It should be noted that the results estimated in the third state for the initial system, the system in balance and in its reduced system, all show different results. This shows that the position of state for the initial system and to its equilibrium system is different.

Therefore, for future work, it is necessary to follow the state-state changes its position during the formation of the model reduction.

References

1. Anderson, R, 1979, Optimal Filtering, Prentice-Hall

2. Bierman, G.J., 1977, Factorization Methods for Discrete Sequential Estimation, Academic Press. 3. Greenberg, D.M., 1988, Advanced Engineering Mathematics, Prentice-Hall

4. John C. Doyle and Keith Glover, Robust and Optimal Control, Prentice-Hall. 5. Kemin Zhou., 1998, Essentials of Robust Control, Prentice-Hall.

6. Lewis, F.L., 1992, Applied Optimal Control & Estimation, Prentice-Hall. 7. Nakamura, S., 1991, Applied Numerical Methods with Software, Pentice-Hall.

8. Olsder, G.J., 1994, Mathematical System Theory, Delftse Uitgevers, Maatschappij, Netherlands 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Distribusi Panas Sistem Tereduksi Pada semua Posisi pada Iterasi ke-20 dengan dt=0.01 X_asli

Distribusi Panas Sistem Tereduksi Pada Posisi ke-3 Pada semua Iterasi dengan d X_asli

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”

PROCEEDING

INTERNATIONAL CONFERENCE ON MATHEMATICS, STATISTICS

AND ITS APPLICATIONS 2012

ICMSA 2012

Message

Chairman

Organizing Committee

CONTENTS

IMPLEMENTATION

OF THE ALGORITHM KALMAN FILTER

ON REDUCTION MODEL

1

Introduction

2

Methodology

3

The Result and Discussions

4

Analysis

5

Conclusion

References