Nonlinear Model of a Stochastic Control System

Abstract

In this work, a model of a stochastic control system is obtained based on the method of canonical expansions of random processes. The algorithm for calculating the parameters of the system allows one to take into account an arbitrary order of nonlinear links and the amount of a posteriori information about the studied sequence of changing the coordinates of the control object. The mathematical model also does not impose any restrictions on the behavior properties of the controlled object: linearity, stationarity, monotonicity, scalarity, Markov property, etc. The paper presents a block diagram of an algorithm for calculating the parameters of a stochastic control system. The formula for the mean square of the extrapolation error of the future coordinates of the object under study allows us to estimate the accuracy of the solution to the control problem.

Keywords

• 1 Stochastic control system
• random sequence
• canonical decomposition

1. Introduction

Functioning of the objects of different nature is in most cases carried out in the conditions of influence and interaction of multiple random factors as a result of which the coordinates of a control object are changed randomly [1-4]. Methods of analysis of random processes has a wide range of applications in modern industrial technologies [5,6], local and global energy systems [7,8], communication technologies [9,10], cybersecurity [11,12], applied problems of the economy [13, 14], etc. The task of controlling and predicting the state of the system under study is strictly mathematical and is dual [15]. The solution to such a problem is based on the methods of extrapolation of random processes used to construct stochastic control systems. The basic method of stochastic control theory is the Wiener-Hopf method [16,17], which involves solving an integral equation. However, such equations for real problems, most often, do not have an analytical solution. Recursive methods have found wide application [18, 19], which are fairly easily extended to nonstationary processes and allow the use of computer technology. However, the optimal solution using these methods (for example, the Kalman extrapolator filter [20]) can be obtained only for Markov random processes. The most universal mathematical model for solving the problem of nonlinear forecasting is the KolmogorovGabor polynomial [21], but finding its parameters for a large number of coordinates of the control object and the order of nonlinear connections is a very difficult and time-consuming task. In practice, when implementing extrapolation algorithms, various simplifications and restrictions are used that are imposed on the properties of random processes. For example, it is assumed that the random sequence of changes in the coordinates of the control object is stationary, scalar, or Markov, which leads to a significant limitation of accuracy. In this regard, the problem of synthesizing a control system for an arbitrary number of object state parameters and an arbitrary order of nonlinear connections is urgent.

2. Formulation of the problem

Let us assume that a stochastic system has nonlinear relations and its properties are fully set at a discrete number of points ti , i  1, I by moment functions M P   P i  , M C   P i , M C   C i  ,  , =1, N;  , j  1, I ( P i  – a random sequence of changes in the coordinates of the object under study, C i  – sequence of control parameters). It is necessary to obtain a mathematical model of a stochastic control system for an arbitrary number of known coordinates p i  and c i  control actions, taking into account nonlinear connections.

3. Mathematical model of a stochastic control system

The most common tool for analyzing random sequences is the canonical decomposition tool. To obtain a canonical model of a vector random sequence P i  , C i , i  1, I taking nonlinear relations into account, we consider an array of random values

The correlation moments of array elements ( 1 ) fully describe the probabilistic relations of sequence P i  ;C i  , i  1,I at an investigated number of points ti , i  1, I . Therefore, the application of vector linear canonical expansion to the first line P i  ,i  1,I allows one to obtain a canonical expansion with a full taking into account of a priori information for each component [21-23]:  P 1 P 2 ... P  I 1 P  I    P2 1 P2 2 ... P2  I 1 P2  I    ... ... ... ... ...   PN 1 PN 2 ... PN  I 1 PN  I    C 1 C 2 ... C  I 1 C  I    C 2 1 C 2 2 ... C 2  I 1 C 2  I    ... ... ... ... ...   C N 1 C N 2 ... C N  I 1 C N  I  

i1 2 N P i   M P i   1 l1 1 G(l )l(1,1)  , i   Gi(11) , i  1, I ;

 1 2 N G(1)  P    M P    1 m1 j1 G(mj) m(1j, )  ,   1 G(1j) (1, )  ,  ,   1, N ,  1, I; j1 1 j

 1 2 N G(2)  C    M C       G(mj)( 2j, )  ,  

 1 m1 j1  G(1j) (2, )  ,   1 G( 2j) (2, )  ,  ,   1, N ,  1, I;

N j1 1 j j1 2 j ( 1 ) ( 2 ) ( 3 ) ( 4 ) 2  D1 j   (1, )  ,  ,   1, N ,  1, I ;

1 j D2,    M G( ) 2   M C 2    M 2 C      2   

N j1 mj

2 2   (l, )  ,    D1 j   (2, )  ,  

mj 1 j 2 j

2   (2, )  ,  ,   1, N ,  1, I ;

2 j  1 H N    D coefficients contains information about the corresponding values P( )   , C ( )   and coordinate functions  (h, )  , i  describe probabilistic relations of   s order between components P t  and l C t  in the sections t and t .

 i

The block diagram

of the algorithm to calculate parameters D   , l  1, 2,   1, N,   1, I and l,  (h, )  , i  , l, h  1, 2,  ,  1, N ,  , i  1, I of canonical expansion ( 2 ) is presented in Fig.1. l

Let us assume that as a result of a measurement the first value p 1 of the sequence at point t is 1 known. Consequently, the values of coefficients G1(1 ) ,  1, N are known: g1(1 )  p 1  M P 1  1 g1(1j) (1, ) 1,1 ,   1, N

1 j j1

Substituting w1(11) into ( 2 ) allows one to obtain a posteriori canonical expansion of the first component P( 1,1 )  P i / p1 1 of a random sequence P i , C i, i  1, I : forecasting. P( 1,2 )  P i / p1 1 , p1 12  : where m1( ,11,1 )  , i  is optimal estimation of a future value p i provided that value p 1 is used for Specification of the second value g1(12) in ( 10 ) gives canonical expansion of a posteriori sequence The application of the operation of mathematical expectation to ( 10 ) gives an optimal (by the criterion of minimum of mean-square error of extrapolation) estimation of future values of sequence P provided that one value p 1 is used to determine the given estimation: m1,1( 1,1 ) 1, i   M P i / p 1  M P i    p 1 - M P 1 1( 11,1 ) 1, i , i  1, I.

Taking the fact that coordinate functions  l(h, )  , i  , l, h  1, 2,  ,  1, N ,  , i  1, I are determined from the condition of minimum of a mean-square error of approximation in the intervals between arbitrary values P   and C h i  into account, expression (11) can be generalized in case of the forecasting p i  ,   1, N , i  1, I : m1( ,11,1 )  , i   M P i / p 1  M P i    p 1  M P 1 1(11, ) 1, i .

P( 1,1 ) i   P i / p 1  M P i    p 1 - M P 1 1( 11,1 ) 1,i  

N N i1 2 N  G1(1 )1(1,1) 1, i   G1(2 )1(1,1) 1, i     G(l )l(1,1)  , i   Gi(11) , i  1, I.

 2  1  2 l1  1 ( 9 ) ( 10 ) (11) (12) (13) (14) P( 1,2 ) i   P i / p 1 , p 12   M P i    p 1 - M P 1 1( 11,1 ) 1,i  

N   p2 1   p 1  M P( 1 )1( 11,2 ) 1,11( 21,1 ) i    G1(1 ) ( 1,1 ) 1, i     3 1

N i1 2 N  G1(2 )1(1,1) 1, i     G(l ) l(1,1)  , i   Gi(11) , i  1, I.

 1  2 l1  1

Application of an operation of mathematical expectation to (13) allows to obtain the algorithm of extrapolation by two values p 1 , p 12 using expression (12): m1( ,11,2 )  , i   M P i / p 1 , p 12  

   m1( ,11,1 )  , i    p2 1  m1( ,11,1 ) 2,11(21 ) 1, i  , i  1, I.

After N iterations, the value of the random coefficient G1(21)  g1(21) can be calculated based on the information about control action С 1  с( 1 ) :

N g1(12)  c 1   g1(1 ) ( 2,1 ) 1,1

1 2 N and forecasting algorithm with the use of values p 1 , p 1 ,..., p 1 , c 1 takes form m( 1,1 )  , i   m(1,N )  , i   с 1  m(1,N ) 1,1 (1, ) 1, i .

2,1 1,1 1,2 2,1 (15) (17) (18) M P i  , if   0,  m(j,h,q1)  , i   cq    m(j, ,jq1) q,  (j,hq, )  , i  ,  if q  1, j  2  m(j,1N,h)  , i   c    m(j,1N,j) 1,  j(,hq, )  , i  ,  m(j,h,q)  , i   if q  1, j  2  m(j,h,q1)  , i    pq    m(j, ,jq1) q,  j(,hq, )  , i  , if q  1, j  1 m2(,h1,N )  , i    p    m2(,11,N ) 1,  (j,hq, )  , i  , for  q  1, j  1,  E( ,N ) i   M P2 i   M 2 P i  

   2 N Dmj   m(1j, )  k, 2 , i  1, I.

k 1 m1 j1 where m2(,1,N ) 1, i   M P i  / pn   , un   , n  1, N ,  1,  is an optimal in a mean-square sense estimation of future values of an investigated random sequence provided that posteriori information pn   , cn  , n  1, N ,  1, is applied for the forecasting.

The expression for the mean-square error of the extrapolation using algorithm (17) by known values pn   , cn   , n  1, N ,  1, is of the form

The mean-square error of extrapolation E( ,N ) i is equal to the dispersion of the posteriori random

i1 2 N     G(l ) l(1,1)  , i   Gi(11) , i  1, I.   1 l1  1 (19)

The synthesis and application of mathematical model (17) of a stochastic control system presuppose the realization of the following stages:

Stage 1. The gathering of data on an investigated random sequence P i  , C i , i  1, I ;

Stage

The estimation of moment functions

M P  ; P i  ,

M C   C i  , M C   C i  ,  , =1, N;  , j  1, I based on the accumulated realizations of a random sequence P i  , C i , i  1, I ;

Stage 3. The forming of canonical expansion (2);

Stage 4. The calculation of the parameters of extrapolation algorithm (17); Stage 5. The estimation of the future values of extrapolated realization based on expression (17); Stage 6. The estimation of the quality of solving the forecasting problem for an investigated sequence using expression (18).

4. Сonclusions

In this paper, a mathematical model of a stochastic control system is obtained, which has a nonlinear structure. The apparatus of canonical expansions, which is the basis of the model, makes it possible to use the entire prehistory of the functioning of the system and makes it possible not to apply hypotheses (linearity, stationarity, monotonicity, scalarity, Markov property, etc.) that limit the properties of a random sequence of dynamics of system parameters. The proposed algorithm provides the most accurate characteristics, which leads to a significant improvement in the quality of solving the problem of stochastic systems control in various applied areas: reliability of technical systems [2426], robotics [27, 28], economics [29-32], medicine [33], etc.

5. References

[11]Tomer Fishman, Heinz Schandl and Hiroki Tanikawa, Stochastic Analysis and Forecasts of the Patterns of Speed, Acceleration, and Levels of Material Stock Accumulation in Society Environ. Sci. Technol., 50, 7, pp. 3729-3737, 2016. [12]Cheng-Zong Bai, Fabio Pasqualetti, Vijay Gupta, Data-injection attacks in stochastic control systems /

Detectability and performance tradeoffs Automatica, V 82, pp. 251-260, 2017. [13]Tomas Björk, Agatha Murgoci A theory of Markovian time-inconsistent stochastic control in discrete time / Finance and Stochastics, V. 18, pp. 545-592, 2014 [14]Luz R. Sotomayor, Abel Cadenillas, Classical and singular stochastic control for the optimal dividend policy when there is regime switching / Insurance: Mathematics and Economics, 2011, V. 48, Is. 3, pp. 344-354. [15] A.N. Tkachenko, N.M. Brovinskaya, Y.P. Kondratenko, Evolutionary adaptation of control processes in robots operating in non-stationary environments, Mechanism and Machine Theory, Printed in Great Britain, Vol. 18, No. 4. , pp. 275-278. 1983. DOI: 10.1016/0094-114X(83)90118-0 [16] N. Wiener,Extrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering

Applications. MIT Press, New-York, 1949.

[17] Kuntsevich, V.M. et al. (Eds): Control Systems, Theory and Applications. Book Series in Automation,

Control and Robotics, River Publishers, Gistrup, Delft, 2018.

[18] R.E. Kalman, A New Approach to Linear Filtering and Prediction Problems. Trans. ASME Series D, J.

Basic Engg., v. 82 (Series D), pp. 35-45, 1960. [19] D. Simon, Training fuzzy systems with the extended Kalman filter, Fuzzy Sets and Systems, 132, pp. 189199, 2002. [20] G.E.P. Box, G.M. Jenkins, Time–series Analysis, Forecasting and Control, Holden–Day, San Francisco, 1970. [21] V.D. Kudritsky, Filtering, Extrapolation and Recognition Realizations of Random Functions,FADA Ltd.,

Kyiv, 2001. [22] I.P. Atamanyuk, Algorithm of extrapolation of a nonlinear random process on the basis of its canonical decomposition. J. Kibernetika i Sistemnyj Analiz, №2, pp. 131-138, 2005. [23] I.P. Atamanyuk, Polynomial algorithm of optimal extrapolation of stochastic system parameters. J.

Upravlyayushchie Sistemy i Mashiny, №1, pp. 16-19, 2002.

[24] O. Lozynsky, S. Shcherbovskykh, Consideration effect of repair strategy on availability for non-redundant system using extended Markov model, Proceeding International Conference CADSM 2011, Polyana,

Svalyava, Ukraine, 2011.

[25] Igor Atamanyuk, Yuriy Kondratenko, Vyacheslav Shebanin, Vladimir Mirgorod, Method of Polynomial

Predictive Control of Fail-Safe Operation of Technical Systems. Proc. XIIIth International Conference

The Experience of Designing and Application of CAD Systems in Microelectronics, CADSM 2015, 1923 Febr., 2015, Svalyava, Ukraine, pp. 248-251, 2015. DOI: 10.1109/CADSM.2015.7230848 [26] O. Lozynsky, S. Shcherbovskykh, Failure intensity determination for renewal doubling system with parallel redundancy. Proceeding International Conference TCSET' 2010, Lviv-Slavske; Ukraine, 2010. [27]Y. Kondratenko, D. Simon, I. Atamanyuk, University Curricula Modification Based on Advancements in Information and Communication Technologies. Proceedings volume of ICTERI 2016, Kyiv on the 21st-24 th of June, 2016, pp. 184-199, 2016. [28]Y. Kondratenko, G. Khademi, V. Azimi et al., Robotics and prosthetics at cleveland state university:

Modern information, communication, and modeling technologies, 12th International Conference on

Information and Communication Technologies in Education, Research, and Industrial Applications,

ICTERI 2016, Kiev, Ukraine, Vol. 783, pp. 133-155, 2016. DOI: https://doi.org/10.1007/978-3-31969965-3_8 [29] I. Atamanyuk, Y. Kondratenko, N. Sirenko, Forecasting Economic Indices of Agricultural Enterprises

Based on Vector Polynomial Canonical Expansion of Random Sequences, 5th International Workshop

on Information Technologies in Economic Research (ITER) in ICTERI 2016, Kyiv on the 21st-24th of June, pp. 458-468, 2016. [30] A. Kalinichenko, V. Havrysh, V. Perebyynis, Sensitivity analysis in investment project of biogas plant, J.

Applied ecology and environmental researches, 15(4), pp. 969-985, 2017.

[31] I. Atamanyuk, Y. Kondratenko, N. Sirenko, Management System for Agricultural Enterprise on the Basis of Its Economic State Forecasting, Complex Systems: Solutions and Challenges in Economics, Management and Engineering, Vol. 125, pp. 453-470, 2018. DOI: https://doi.org/10.1007/978-3-31969989-9_27 [32] O.V. Dovgal and other, Methods of competitiveness assessment of agricultural enterprise in Eastern

Europe, J. Regional Science Inquiry, Volume 9, Issue 2, December 2017, рр. 231-242.

[33] Zeifman A. et al. (Eds), Stochastic processes: theory and applications, MDPI AG, 2019.

References

[1] R. Dorf , Modern control systems , Laboratory of Basic Knowledge , Moscow, 832 p. ( 2004 )

[2] Chen Guanrong , Chen Goong, Hsu Shih-Hsun, Linear stochastic control systems . CRC Press, New-York, 2000 .

[3] Aviation Turbulence , Processes, Detection, Prediction. By Robert Sharman (Editor), Todd Lane (Editor) Edition, Kindle Edition , 2016 .

[4] T.A. Filippova , Optimization of modes of power plants and power systems , Publishing house of NSTU , 2007 .

[5] Khader M. Hamdia , Mohammad Silani, Xiaoying Zhuang, Pengfei He & Timon Rabczuk Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions / International Journal of Fracture , V. 206 , pp. 215 - 227 , 2017 .

[6] Mangey Ram , Suraj Bhan Singh, Vijay Vir Singh , Stochastic Analysis of a standby system with waiting repair strategy , IEEE Transactions on Systems and Cybernetics: Systems , V. 43 , 2013 .

[7] Barai Pallab , P. Partha , Mukherjee Stochastic analysis of diffusion induced damage in lithium-ion battery Electrodes , The Electrochemical Society Journal of The Electrochemical Society , V. 160 , № 6 . 2013 .

[8] Qiao Zhang , Weiwen Deng, Gang Li , Stochastic control of predictive power management for battery Supercapacitor Hybrid Energy Storage Systems of Electric Vehicles , IEEE Transactions on Industrial Informatics, V. 14 , Is . 7, pp. 3023 - 3030 , 2018 .

[9] Huaming Wu , Katinka Wolter, Stochastic Analysis of delayed mobile offloading in heterogeneous networks / IEEE Transactions on Mobile Computing, V. 17 , Is . 2 , № 1, pp. 461 - 474 , 2018 .

[10] Jinlin Peng , Peilin Hong, Kaiping Xue, Stochastic analysis of optimal base station energy saving in cellular networks with sleep mode , IEEE Communications Letters, V. 18 , Issue 4 , April 2014 , pp. 612 - 615 , 2014 .