4TH INTERNATIONAL EURASIAN CONFERENCE ON

MATHEMATICAL SCIENCES AND APPLICATIONS

BOOKS OF ABSTRACTS

31 AUGUST - 03 SEPTEMBER 2015

ATHENS-GREECE

HONORARY COMMITTEE Prof. Dr. Muzaffer ELMAS (Sakarya University Rector) Prof. Dr. Azmi ÖZCAN (Bilecik Seyh Edebali University Rector) Prof. Dr. Sadettin HÜLAGÜ (Kocaeli University Rector) Prof. Dr. Idriss MANSOURI (Hassan University Rector) Prof. Dr. Saaid AMZAZI (Mohammed University Rector) Prof. Dr. Ismail KOCAYUSUFOĞLU (Intenational Balkan Univeristy Rector)

SCIENTIFIC COMMITTEE

Prof. Dr. Abdullah Aziz ERGİN (Akdeniz University) Prof. Dr. Adnan TERCAN (Hacettepe University) Prof. Dr. Alexander ZLOTNIK (National Research University Higher School of Economics) Prof. Dr. Arif SALIMOV (Ataturk University) Prof. Dr. Arkadii KIM (Institute of Mathematics and Mechanics of Russian Academy of Sciences) Prof. Dr. Bayram ŞAHİN (Inonu University) Prof. Dr. Bo Wun HUANG (Cheng Shiu University) Prof. Dr. Claudio Rodrigo Cuevas HENRIQUEZ (Federal University of Pernambuco) Prof. Dr. Danal O’REGAN (National University of Ireland) Prof. Dr. Ekrem SAVAŞ (Istanbul Commerce University) Prof. Dr. Emin ÖZÇAĞ (Hacettepe University) Prof. Dr. Emine MISIRLI (Ege University) Prof. Dr. Fethi ÇALLIALP (Dogus University) Prof. Dr. Fikret ALİYEV (Baku State University) Prof. Dr. George BOGOSLOVSKY (Moscow State University) Prof. Dr. Gennady A. LEONOV (Saint-Petersburg State University) Prof. Dr. H. Hilmi HACISALİHOĞLU (Bilecik Seyh Edebali University) Prof. Dr. Halis AYGÜN (Kocaeli University) Prof. Dr. Hellmuth STACHEL (Vienna Technical University) Prof. Dr. Jan Van MILL (VU Amsterdam University)

Prof. Dr. Kadri ARSLAN (Uludag University) Prof. Dr. Kailash Chander MADAN (Ahlia University) Prof. Dr. Kil Hyun KWON (Korea Advanced Institute of Science and Technology) Prof. Dr. Khalil EZZENBI (Cadi Ayyad University) Prof. Dr. Laszlo LEMPERT (Purdue University) Prof. Dr. Mahir RESULOV (Beykent University) Prof. Dr. Mahmut ERGÜT (Firat University) Prof. Dr. Maksymilian DRYJA (Waraw University) Prof. Dr. Muhammad Rashid Kamal ANSARI (Sir Syed University of Engineering and Technology) Prof. Dr. Mukut Mani Tripathi (Banaras Hindu University) Prof. Dr. Murat ALTUN (Uludag University) Prof. Dr. Mustafa ÇALIŞKAN (Gazi University) Prof. Dr. Muvarskhan JENALIYEV (Institute of Mathematics and Mathematical Modeling) Prof. Dr. Müjgan TEZ (Marmara University) Prof. Dr. Naime EKİCİ (Cukurova University) Prof. Dr. Nuri KURUOĞLU (Bahcesehir University) Prof. Dr. Ömer AKIN (TOBB University of Economy and Technology) Prof. Dr. Pedro MACEDO (University of Aveiro) Prof. Dr. Prasanta SAHOO (Jadavpur University) Prof. Dr. Ram N. MOHAPATRA (University of Central Florida ) Prof. Dr. Reza LANGARI (Texas A&M University) Prof. Dr. Roberto BARRIO (University of Zaragoza) Prof. Dr. Sadık KELEŞ (Inonu University)

Prof. Dr. Saeid ABBASBANDY (Imam Khomeini International University) Prof. Dr. Sergeev Armen GLEBOVIC (Steklov Mathematical Institute) Prof. Dr. Snezhana HRISTOVA (Plovdiv University) Prof. Dr. Toka DIAGANA (Howard University) Prof. Dr. Walter RACUGNO (University of Cagliari) Prof. Dr. Wolfgang SPROESSING (Freiberg University of Mining and Technology) Prof. Dr. Varga KALANTAROV (Koc University)

ORGANIZING COMMITTEE

Prof. Dr. Murat TOSUN (Sakarya University) Prof. Dr. Cihan ÖZGÜR (Balikesir University) Prof. Dr. Kazım İLARSLAN (Kirikkale University) Prof. Dr. Ahmet KÜÇÜK (Kocaeli Univeristy) Assoc. Prof. Dr. Erdal ÖZÜSAĞLAM (Aksaray University) Assoc. Prof. Dr. Mehmet Ali GÜNGÖR (Sakarya University) Assoc. Prof. Dr. Melek MASAL (Sakarya University) Assoc. Prof. Dr. Nesip AKTAN (Necmettin Erbakan University) Assoc. Prof. Dr. Soley ERSOY (Sakarya University) Assoc. Prof. Dr. Şevket GÜR (Sakarya University) Assist. Prof. Dr. Ayşe Zeynep AZAK (Sakarya University) Assist. Prof. Dr. Mahmut AKYİĞİT (Sakarya University) Assist. Prof. Dr. Mehmet GÜNER (Sakarya University) Assist. Prof. Dr. Murat SARDUVAN (Sakarya University) Dr. Canay AYKOL YÜCE (Ankara University)

SECRETARIAT Prof. Dr. Murat TOSUN (Sakarya University) Research Assist. Hidayet Hüda KÖSAL (Sakarya University) Research Assist. Tülay ERİŞİR (Sakarya University) Researcher Selman HIZAL (Sakarya University)

CONTENTS INVITED TALKS

Authenticated Encryption Based on Prime ModuliD. Giri.............................................................................................................................. 22

An Algebraic Description of Gradient Descent DecodingE. Martínez-Moro........................................................................................................... 23

Elliptic Diophantine Equations and the Elliptic Logarithm MethodN. Tzanakis..................................................................................................................... 24

Möbius Transformations and the Circle-Preserving PropertyN. Yılmaz Özgür.............................................................................................................. 26

Advances in Frames, Riesz Bases and Frames on Hilbert *C ModulesR. N. Mohapatra............................................................................................................. 27

- 1 -

Analytical Problem Solution about Initial Step of Pressing Powder Material TubeM. Ya. Flax, A. V. Bochkov, V. A. Goloveshkin, A. V. Ponomarev............................. 301

X-ray and Computational Studies of (2Z, 3E)-3-(((E)-3-Ethoxy-2-Hydroxybenzylidene) Hydrazono)Butan-2-One OximeN. Çalışkan, Ç. Yüksektepe Ataol, H. Batı, N. Kurban, P. Kurnaz, G. Ekmekçi......... 303

Fixed Points of Certain Automorphisms of Free Solvable Lie AlgebrasN. Ekici............................................................................................................................ 304

Generalizations of Sherman's inequality by Lidstone's interpolating polynomialR. P. Agarwal, S. I. Bradanović, J. Pečarić................................................................... 305

A simple State Observer Design for Linear Dynamic Systems by Using Taylor Series Approximation S. Aksoy, H. Kızmaz........................................................................................................ 306

On the Classical Zariski Topology over Prime Spectrum of a ModuleS. Çeken, M. Alkan......................................................................................................... 308

Strongly k -SpacesS. Ersoy, İ. İnce, M. Bilgin............................................................................................. 309

On the Study of the Matrix in a Model of Economic DynamicsS. I. Hamidov.................................................................................................................. 311

Numerical Solutions of Steady Incompressible Dilatant Flow in an Enclosed Cavity Region S. Şahin, H. Demir.......................................................................................................... 312

Smarandache Curves of Mannheim Curve Couple According to Frenet FrameS. Şenyurt, A. Çalışkan................................................................................................... 313

Smarandache Curves of Involute-Evolute Curve Couple According to Frenet FrameS. Şenyurt, S. Sivas, A. Çalışkan.................................................................................... 315

The Analysis of the Mathematical Modelling Activities in the Ninth Grade Mathematics Coursebook S. Urhan, Ş. Dost............................................................................................................ 317

On Geodesic Paracontact CR-Lightlike SubmanifoldsS. Yüksel Perktaş, B. Eftal Acet..................................................................................... 319

- 20 -

A simple State Observer Design for Linear Dynamic Systems by Using Taylor Series

Approximation

Saadettin Aksoy 1 and Hakan Kizmaz 2

Abstract. State variables that determine a system’s dynamics should be known for

analysis and control of dynamical systems [1-2]. Specifically, dynamics feedback for pole

placement is required. Furthermore, estimation of state variables in real time is a very

important problem in adaptive control applications [3]. Unfortunately, all of the state variables

cannot be measured in practice. As a result, use of a suitable state observer or estimator is

unavoidable in order to obtain immeasurable state variables. There exit a variety of state

observers in the literature [4-5]. Implementation of state observers that use only input and

output measurements of the systems are carried out via solution of the observer state integral

equations pertinent to the observer. There are several numerical solution algorithms for a

solution of the observer state integral equations in the literature [6]. Even though the Runge-

Kutta numerical integration algorithm is frequently used for this purpose, it has several

drawbacks that depend on the step-size h. First, accuracy gets poorer as h increases. Second,

computation time becomes an issue if h is too small. Third, round-off errors may become

important for small values of h because the number of cycles required to cover the desired

time interval [0, t] increases. Note that equations are evaluated for each t in the interval [0, t]

in all of the above mentioned algorithms.

In this study, a simple general algorithm is proposed for state variables estimation of

linear, time-invariant multi-input multi output systems. The proposed algorithm is based on

Taylor series approximation and has an analog solution. The solution that results from the

proposed algorithm gets closer to the true solution when more and more terms are kept in the

Taylor series. Finally, the proposed method gives the approximate solution of the estimation

vector ˆ (t)x as a function of time in the interval [0,t]. Consequently, computation of the state

integral equations for each t is eliminated. The Taylor series are defined on the interval

t [0,1] and have the orthogonality property like the Walsh, Chebyshev and Legendre series

[7-8]. The proposed algorithm uses some important properties such as the operational matrix

of integration for Taylor vector [9-10]. The algorithm consists of four steps. In the first step,

the feedback gain matrix G, which will force the estimation error to go to zero in a short time,

is determined by using a suitable method [4]. In the second step, the observer state equation is

1 Siirt University, Siirt, Turkey, aksoy@siirt.edu.tr 2 Sakarya University, Sakarya, Turkey, hkizmaz@sakarya.edu.tr

- 306 -

converted into integral equation by integrating the terms on either side of the equation. After

some algebraic manipulations, the time dependent terms on either side of the integral equation

are removed. Hence, the problem is reduced to a set of nonlinear equations with constant

coefficients. System outputs are used by the observer equations. Therefore, we have to

calculate it’s as the function of time. They can obtained from plant output measurement by

using curve fitting methods such as Linear Least Squares, Lebenberg-Marquardt and Gauss-

Newton [11]. Finally, in the last step, nonlinear equations for unknown state vector are

converted into a recursive form whose solution can be obtained easily by a computer program.

The proposed estimation algorithm was implemented in MATLABTM and it was applied to

different cases. Results obtained by the proposed algorithm are in harmony with the real

results.

Keywords. State Estimation, Taylor Series, State Observers, Curve Fitting

References

[1] Daughlas B. Miron, Design of Feedback Control Systems, Harcourt Brace Jovanovic Inc.,

USA, 1989.

[2] Brasch, F.M., and Pearson, J. B., Pole Placement Using Dynamic Compensators, IEEE

Trans. Automatic Control, AC -15, pp. 34-43, 1970.

[3] Aström, K.J., and Wittermark, B., Adaptive Control, Addison Wesley Pub. Inc., USA,

1989.

[4] Kailath, T., Linear Systems, Prentice-Hall Inc., Tokyo, 1980.

[5] O. Reilly, J., Observers for Linear Systems, London Academic Press, London,1983.

[6] Hildebrand,F.B., Introduction to Numerical Analysis, 2d edition, McGraw-Hill, New

York, 1937.

[7] G. Sansone, Orthogonal Functions, Interscience Publishers, Inc., New York, 1991.

[8] Cheng-Chilan Liu and Y. P. Shih, Analysis and optimal control of time-varying systems

via Chebyshev polynomials, Int. J. Control, Vol.38, No.5, pp. 1003-012, 1983.

[9] Spares P.D., and Moutroutsas S. G., Analysis and Optimal Control of Time-Varying

Linear Systems via Taylor Series, Int. Journal Cont., Vol.41, pp. 831-842, 1985.

[10] Mouroutsos S. G. and Sparis P. D., Taylor Series Approach to System Identification

Analysis and Optimal Control, Journal of Frank. Inst., vol. 319, no. 3, pp. 359-371, 1985.

[11] Steven C. Chapra, Raymond P. Canale, Numerical Methods for Engineers, McGraw-Hill,

USA, 2009.

- 307 -