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


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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).

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