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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
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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
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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, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
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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.
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