doi: 10.17586/2226-1494-2015-15-6-1036-1044

# СREATION OF CORRELATION FUNCTIONS OF LINEAR CONTINUOUS SYSTEMS BASED ON THEIR FUNDAMENTAL MATRICES

N. A. Vunder, E. A. Nad’kina, A. V. Ushakov, J. V. Chugina

Article in Russian

For citation: Vunder N.A., Nad’kina E.A., Ushakov A.V., Chugina J.V. Сreation of correlation functions of linear continuous systems based on their fundamental matrices. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 6, pp. 1036–1044.

Abstract
The paper presents a method of creating correlation matrices and functions of the state vectors and outputs of the linear continuous systems functioning under the conditions of stochastic stationary, in a broad sense, effects. Fundamental matrices form the basis of the method. We have shown that for the linear continuous systems with single dimensional input and single dimensional output the correlation output function of such system can be found as the free movement of this system generated by its initial state. This state is constructed from the variance matrix of the state vector and the transposed output matrix. We have elucidated that when a continuous system belongs to a class of multi-dimensional input – multi-dimensional output systems, the following options are available for solving the problem of creation of the correlation function of a linear system. The first option is to partition the system into separate channels. Then the approach developed for systems with onedimensional input and one-dimensional output is applied to each of the separate channels. The second option is used to preserve the vector nature of the stochastic external influence. It consists in partition of output vector to scalar components by separating output matrix into separate rows with subsequent formation of the correlation function according to the scheme for single dimensional input and single dimensional output type systems. The third option is based on the scalarization of matrix correlation output function by applying the singular value decomposition to it. That gives the possibility to form scalar majorant and minorant of correlation output functions. We have established that a key component of a computational procedure of creating the correlation function of continuous linear system is a variance matrix of the system state vector. In the case of functioning under an exogenous stochastic effect like "white noise" the variance matrix is calculated by solving the matrix Lyapunov equation. We have found out that in the case of an exogenous stochastic effect like "colored noise"
capability of using the Lyapunov equation to find the variance matrix of system state arises after aggregated system design
composed of the system itself and the shaping filter with "colored noise" at the output. Examples illustrate obtained
procedures of creating correlation functions.

Keywords: stochastic effects, continuous system, Lyapunov equation, fundamental matrix, correlation function.

Acknowledgements. This work was supported by the Government of the Russian Federation, Grant 074-U01 and the Ministry of Education and Science (Project 14. Z50.31.0031).

References

1. Liptser R.Sh., Shiryaev A.N. Statistika Sluchainykh Protsessov [Statistics of Random Processes]. Moscow, Nauka Publ., 1974, 696 p.
2. Ivanov V.A., Medvedev V.S., Chemodanov B.K., Yushchenko A.S. Matematicheskie Osnovy Teorii Avtomaticheskogo Upravleniya: Uchebnoe Posobie [Mathematical Bases of Automatic Control Theory. Textbook]. Ed. B.K. Chemodanov. 3rd ed. Vol. 3. Moscow, MGTU named by N.E. Bauman Publ., 2009, 352 p.
3. Kwakernaak H., Sivan R. Linear Optimal Control Systems. Wiley-Interscience, 1972, 608 p.
4. Davis M.H.A. Linear Estimation and Stochastic Control. London, Chapman and Hall Ltd., 1977, 224 p.
5. Oppenheim A.V., Schafer R.W. Digital Signal Processing. New Jersey, Prentice Hall, 1975, 585 p.
6. Golub G.H., Van Loan C.F. Matrix Computations. 4th ed. Johns Hopkins University Press, 2012, 790 p.
7. Genin L.G., Sviridov V.G. Vvedenie v Statisticheskuyu Teoriyu Turbulentnosti [Introduction to the Statistical Theory of Turbulence]. Moscow, MPEI Publ., 2007, 100 p.
8. Besekerskii V.A., Popov E.P. Teoriya Sistem Avtomaticheskogo Regulirovaniya [The Theory of Automatic Control Systems]. St. Petersburg, Professiya Publ., 2003, 752 p.
9. Oksendal B.K. Stochastic Differential Equations: An Introduction with Applications. 6th ed. Berlin, Springer, 2003, 379 p.
10. Ushakov A., Dudarenko N., Slita O. Sovremennaya Teoriya Mnogomernogo Upravleniya: Apparat Prostranstva Sostoyanii [The Modern Theory of Multivariable Control: The Unit of the State Space]. Saarbrucken, LAP LAMBERT Academic Publishing, 2011, 428 p.
11. Dudarenko N.A., Ushakov A.V. Matrix formalism of the degeneration control problem of multichannel dynamical systems under vector stochastic exogenous impact of the colored noise type. Journal of Automation and Information Sciences, 2013, vol. 45, no. 6, pp. 36–47. doi: 10.1615/JAutomatInfScien.v45.i6.40
12. Andreev Yu.N. Upravlenie Konechnomernymi Lineinymi Ob"ektami [Control of Finite Linear Objects]. Moscow, Nauka Publ., 1976, 424 p.
13. Dudarenko N.A., Poliniva N.A., Ushakov A.V. Fundamental matrix of linear continuous system in the problem of estimating its transport delay. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2014, no. 5 (93), pp. 32–37. (In Russian)
14. Typysev V.A., Stepanov O.A., Loparev A.V., Litvinenko Y.A. Guaranteed estimation in the problems of navigation information processing. Proc. IEEE Int. Conf. on Control Applications, CCA'09. St. Petersburg, 2009, art. 5281081, pp. 1672–1677. doi: 10.1109/CCA.2009.5281081
15. Loparev A.V., Stepanov O.A., Tupysev V.A., Tosikova T.P. Sintez algoritmov obrabotki navigatsionnoi informatsii s garantirovannym kachestvom otsenivaniya [Synthesis of algorithms of navigation information processing with assured estimation quality]. Trudy XVI Mezhdunarodnoi Konferentsii po Integrirovannym Navigatsionnym Sistemam [Proc. XVI Int. Conf. on Integrated Navigation Systems]. St. Petersburg, 2009, pp. 207–210.