**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

doi: 10.17586/2226-1494-2017-17-5-938-946

doi: 10.17586/2226-1494-2017-17-5-938-946

# GENERATION OF SPECTRAL DENSITY MATRICES FOR MULTICHANNEL CONTINUOUS SYSTEMS UNDER WHITE NOISE ACTION

**Read the full article**';

**Article in**Russian

**For citation:**Vunder N.A., Dudarenko N.A., Zaharova P.I., Ushakov A.V. Generation of spectral density matrices for multichannel continuous systems under white noise action.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2017, vol. 17, no. 5, pp. 938–946 (in Russian). doi: 10.17586/2226-1494-2017-17-5-938-946

**Abstract**

**Subject of Research. **Wepropose an algorithm of spectral density matrix generation for an output of multichannel continuous systems under vector white noise action. The spectral density matrix is used for the following cases. In the first case, a separate channel of the system is considered. The channel is excited by a scalar white noise. In this case, the spectral density matrix is scalar. In the second case, the system is excited by a vector white noise with components of different intensities. Stochastic process of the selected output is of interest. In this case, the spectral density matrix of the output is scalar too and becomes the spectral density function. In the third case, the system is excited by a vector white noise with components of different intensities, like in the second case. But, spectral density matrix of vector output is considered. **Method. **The algorithm development is based on the use of Lyapunov matrix equation and Wiener-Kolmogorov-Khinchin integral. Scalarization of frequency representation of stochastic output vector is based on two methods. The first method is a per-channel generation of spectral density functions. The second method is a singular decomposition of the spectral density matrix of the output to form a majorant and a minorant of spectral densities in the output space of the system. **Main Results. **The constructive algorithm is obtained for studies of the system spectral properties both for the case of separate channels and for the case of vector "input-output" ratio. Thus, the results are invariant to the dimension of the input-output ratios. **Practical Relevance. **The results serve a useful purpose for the cases when multichannel systems operate under external actions undefinable by a finite-dimensional representation. The most illustrative application examples of the paper findings are the systems of stabilizing a plant spatial position under stochastic actions with white noise representation.

**Keywords:**stochastic action, white noise, multichannel continuous system, Lyapunov equation, Wiener-Kolmogorov-Khinchin integral, spectral density matrix, correlation matrix, singular value decomposition

**Acknowledgements.**This work was supported by the Government of the Russian Federation, Grant 074-U01, the Ministry of Education and Science of the Russian Federation, Project 14.Z50.31.0031, Russian Federation President Grant No.14.Y31.16.9281-НШ. This work was funded by the RFBR according to the research project No.16-08-00997.

**References**

*Statistics of Random Processes*. Moscow, Nauka Publ., 1974, 696 p. (In Russian)

*Mathematical Bases of Automatic Control Theory. Textbook*. Ed. B.K. Chemodanov. 3

^{rd}ed. Moscow, MGTU named by N.E. Bauman Publ., 2009, vol. 3, 352 p. (In Russian)

*Linear Optimal Control Systems*. Wiley-Interscience, 1972, 608 p.

*Linear Estimation and Stochastic Control*. London, Chapman and Hall Ltd., 1977, 224 p.

*Digital Signal Processing*. New Jersey, Prentice Hall, 1975, 585 p.

*Introduction to the Statistical Theory of Turbulence*. Moscow, MPEI Publ., 2007, 100 p. (In Russian)

*Stochastic Differential Equations: An Introduction with Applications*. 6

^{th}ed. Berlin, Springer, 2003, 379 p.

*Journal of Automation and Information Sciences*, 2013, vol. 45, no. 6, pp. 36–47. doi: 10.1615/JAutomatInfScien.v45.i6.40

*Stochastic Continuous and Discrete Dynamic Systems*

*under*

*Uncertainty*. St. Petersburg, ITMO University Publ., 2016, 297 p.

*Teoriya Sistem Avtomaticheskogo Regulirovaniya*[The Theory of Automatic Control Systems]. St. Petersburg, Professiya Publ., 2003,

752 p. (In Russian)

*Control of Finite Linear Objects*. Moscow, Nauka Publ., 1976, 424 p. (In Russian)

*Desoer Linear System Theory: The State Space Approach*. 4

^{th}ed. NY, Dover Publications, 2008, 656 p.

*The Modern Theory of Multivariable Control: The Unit of the State Space*. Saarbrucken, LAP LAMBERT Academic Publishing, 2011, 428 p.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2015, vol. 15, no. 6, pp. 1036–1044. doi:10.17586/2226-1494-2015-15-6-1036-1044

*Digital and Analog Communication Systems*.

6

^{th}ed. Printing Prentice Hall, 2001.

*Matrix Computations*. 4

^{th}ed. Johns Hopkins University Press, 2012, 790 p.