«Automation and Informatization of the fuel and energy complex»

Algorithm for determining the income from a complex technical system with multiple eigenvalues of probability matrices

UDC: 004.052
DOI: -

Authors:

¹ National University of Oil and Gas "Gubkin University", Moscow, Russia

Keywords: complex technical system, pipeline transport, homogeneous Markov process with "income", vector of marginal probabilities, matrix small deformation, eigenvalues

Annotation:

The authors of the article consider the income from the operation of a complex technical system over a finite time interval. This technical system is a pipeline transport serviced by an automated process control system. The behavior of such a system in time is described by a homogeneous Markov process with an income that has a finite set of communicating states that coincide with the possible states of the system. It is indicated that under these conditions, the said income significantly depends on the stationary characteristics of the Markov process and some small value. This small value tends to zero with an exponential rate, which is largely determined by the simple eigenvalues of some matrices, which is the difference between the matrices of transition and stationary probabilities of the Markov process. However, a situation often arises when the eigenvalues of the mentioned matrices are multiple. In the given situation, an algorithm and a block diagram of a computer program are proposed that allow such a small deformation of matrices that makes their multiple eigenvalues simple. An example of such a deformation is given.

Bibliography:

1. Sukharev M.G., Karasevich A.M. Tekhnologicheskiy raschet i obespechenie nadezhnosti gazo- i nefteprovodov. – M.: Neft' i gaz RGU nefti i gaza im. I.M. Gubkina, 2000. – 271 s.
2. Nadezhnost' sistem energetiki i ikh oborudovaniya: v 4 t. T. 1, 3 / pod obshchey red. Yu.N. Rudenko. – M.: Energoatomizdat, 1994. – T. 1. – 473 s.; M.: Nedra, 1994. – T. 3: v 2 kn. – Kn. 2. – 287 s.
3. Khovard R.A. Dinamicheskoe programmirovanie i markovskie protsessy. – M.: Sovetskoe radio, 1964. – 189 s.
4. Orlova K.P. Otsenka skorosti skhodimosti dokhoda odnorodnoy Markovskoy tsepi // Sovremennaya nauka: aktual'nye problemy teorii i praktiki. Seriya: Estestvennye i tekhnicheskie nauki. – 2023. – № 10. – S. 87–90. – DOI: 10.37882/2223-2982.2023.10.26
5. Karmanov A.V., Orlova K.P. Ustoychivost' statsionarnogo dokhoda na odnorodnom markovskom protsesse s pereotsenkoy i nepolnymi iskhodnymi dannymi // Avtomatizatsiya i informatizatsiya TEK. – 2024. – № 8(613). – S. 54–58.
6. Kemeni Dzh.Dzh., Snell Dzh.L. Konechnye tsepi Markova / per. s angl. S.A. Molchanova, N.B. Levinoy, Ya.A. Kogana; pod red. A.A. Yushkevicha. – M.: Nauka, 1970. – 271 s.
7. Faddeev D.K., Faddeeva V.N. Vychislitel'nye metody lineynoy algebry. – Izd. 3-e, ster. – SPb.: Lan', 2002. – 736 s.
8. Prasolov V.V. Zadachi i teoremy lineynoy algebry: ucheb. posobie. – M.: MTsNMO, 2016. – 576 s.
9. Braman K., Byers R., Mathias R. The Multishift QR Algorithm. Part I: Maintaining Well Focused Shifts, and Level 3 Performance // SIAM J. of Matrix Analysis and Applications. – 2002. – Vol. 23, Issue 4. – P. 929–947. – DOI: 10.1137/S0895479801384573
10. LAPACK Users' Guide / E. Anderson, Bai Zhuoxuan, C. Bischof [et al.]. – Third Edition. – Philadelphia, PA, US: Society for Industrial and Applied Mathematics, 1999. – 430 p.