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Firsova Alisa

Computer science and informatics faculty

Applied Mathematics department

Speciality: Economical cybernetics

 Theme of master's work:

Dynamic System Simulation in Economics

Scientific supervisor: Dmitrieva Olga Anatolievna

Master's portal of DonNTU| Autobiography |

Abstract

Urgency of the theme

During the last years the governments have realigned the policy of socio-economic truck in Ukraine with macroeconomic stabilization. The significance of effective management in microlevel was underestimated. The crises and standard of well-being decrease stemmed from it.  There has arisen necessity to reconsider the ways and methods of transformations at microlevel ­− the level of separate enterprises making up the basis for present-day economy. [1]

In the course of market truck development the demand for new economical and mathematical tool formation has appeared for both analysis of economical dynamics and a strategy generation of economical process management. [2]

Many economic systems are characterized by long-term memory. That is the behavior of the system when t>t₀ is defined not only by the set of parameters at the exact moment but also by time history at previous moments. [3] Thus, in this connection, there should be some quantitative methods enabling to disclose the dynamics of the processes at the market, factors that influence on market rates formation, taking into account the specific character of Ukraine economy. [4] Considering the above mentioned we can come to a conclusion that the master’s research issue is timely and practically.

The purpose and objectives

The present paper aims developing a dynamic model of economical system that meets the requirements of rapidly changing conditions in the country.

Project tasks:

• to reveal the factors influencing state of national economy;

• to study the economical situation;

• to analyse the methods applied to the given system and reveal the most practically valuable;

• to develop a dynamic system correlating to the current economical situation.

The review of existing researches

    The question of Dynamic System Simulation in Economics is being studied all round the world by universities such as National science academy of Ukraine , Institute of cybernetics named after V.Glushkov, etc.

Scientific novelty

     Scientific novelty:

• the robust ellipsoid estimation of nonlinear dynamic system algorithms have been applied to the national economy;

•  the mathematical model which reacts to changes in economy has been worked out.

The theoretical data of the master’s project can find an application to any sphere of economy not only for the sake of estimation of state dynamic systems and for optimal control under them and also can act as strategy for economy management.

 The basic idea of the research issue

Let the evolution of discrete dynamic system and its measuring line are described by equations

x_k+1 = A_kx_k + f_k( u_k ),      (1)

y_k = h_k^Tx_k + ξ_k.               (2)

Here k = 0, 1, …— discrete time; x_k — n-dimensional state vector inaccessible to immediate change, x_k Î Rⁿ; Rⁿ — n-dimensional substantial Euclidean space; u_k — known vector of management, u_kÎ R^m, m ≤ n; y_k — vector of observed signals, which are the results of measuring of outcome of object h_k^Tx_k, deadened by action of additive disturbance ξ_k, y_k^T = (y_1k,…,y_mk), ξ_k^T =(ξ_1k,…, ξ_mk);

T — transposition operation symbol; A_k and h_k^T= — known n´ n and m´ n order matrixes accordingly, det A_k ¹ 0 "k = 0, 1,…, h_ik Î Rⁿ; f_k(u_k) — given n-dimensional 

function. It is assumed that disturbance ξ_k satisfies component or standard constraint

ξ_ik² ≤ c_ip², i = 1,…, m, k = 0, 1, …,     (3)

(ξ_ik – ξ_i,k-1)² ≤ c_ir², i = 1,…, m, k = 1, 2, …,     (4)

ξ_k^Tξ_k ≤ c_p², k = 0, 1, …,     (5)

(ξ_k – ξ_k-1)^T(ξ_k – ξ_k-1) ≤ c_r², k = 1, 2, …,     (6)

where c_ip ,c_i, c_ir, c_r — given constants.

Assumed that the results of current measuring y_k, k = 0, 1, …, are known, there is a problem of ellipsoid estimation of unknown state vector x_k

x_k Î E[ , H_k] = E_k ,

where E[ , H_k] ={x: σ(x, , H_k) ≤ 1}, σ(x, , H_k) = (x - )^TH_k^-1(x - ), — ellipsoid centre, H_k = H_k^T > 0. The characteristic of  priori ellipsoid , H_k when k = 0 is considered to be given.

            State estimation of discrete dynamic system. From equalizations (1), (2) we have

ξ_k = y_k - h_k^Tx_k  ,                (10)

ξ_k-1 = y_k-1 - h_k-1^Tx_k-1 .         (11)

            Having used equalization (1), from (10) and (11) we found

ξ_k - ξ_k-1 = z_k- r_k^Tx_k,           (12)

where

z_k=(z_1k,…,z_mk)^T= y_k - y_k-1 – h_k-1^TA_k-1^-1f_k-1(u_k-1), r_k^T= =h_k^T-h_k-1^TA_k-1^-1.

Let ζ_k = ξ_k - ξ_k-1, relation (12) can be represented as additional measuring line of system (1)

z_k=r_k^Tx_k+ ζ_k,             (13)

where disturbance ζ_k = (ζ_1k, …, ζ_mk)^T satisfies  component or standard constraint

ζ_ik² ≤ c_ir², i = 1,…, m, k = 1, 2, …,     (14)

ζ_k^Tζ_k ≤ c_r², k = 0, 1, …,     (15)

Robust algorithm of construction of ellipsoid family  is of the form

,                                 (16)

,     (17)

.                             (18)

Here e_k=h_k^TH_kh_k, , , I_m – unity m´m order matrix, b_kÎ(0,1) and a_k >0 – algorithm parameters. The centre  and the matrix  of ellipsoid É ÉS_r(x_k) is calculating according to formulas (16) – (18) by replacement in their right parts variablesn, H_k, c_p and  by variables τ, , , c_r and  accordingly.[5, 6]

Work results

At this stage of the present master’s project the analysis of the current state of the national economy has been carried out. The results of the present investigation in question have testified to the necessity to apply the methods of mathematical modeling analysed. Under the condition of the constantly changing economical situation and durative crises it was appropriate to select the robust ellipsoid estimation of dynamic system method with disturbance constraint.

Note: at the time of writing this abstract the master's project has not been finished yet. The Final date is December 2009.The full text and materials on a theme can be received from the author or the tutor of the project.

 References

1. Doodko V.A. The simulation of situational management of industrial enterprise http://www.nauka-shop.com/mod/shop/productID/15509/

2. Tuition program of «Mathematic models of dynamic processes in economy». http://www.nauka-shop.com/mod/shop/productID/15777/

3. Booltova L. V. Mathematical modeling of multicriterion economy tastks with hysteresis nonlinearityhttp://www.mil.ru/files/autoreferatbutova.doc

4. Zelenskiy V.M. Mathematical modeling of informational processes under the conditional of transformed economy http://bankrabot.com/work/work_30390.html

5. Volosov V.V. Robust algorithms of ellipsoidal state estimation of one type of nonlinear dynamic systems. // The problems of management and informatics, 2008, ¹1

6. Volosov V.V. Shevchenko V.N. Robust methods of ellipsoidal estimation of dynamical systems state in the case of restrictions on the noise of measured output and the speed of its changing. // The problems of management and informatics, 2008, ¹5