Abstract. Stochastic control systems are considered under pulsed input entrance influence generated Poisson streams of events; examples are systems such as devices using ionizing radiation, the systems of mass service etc. A given nonlinear functional of the Poisson flow intensity with time limitation is synthesized. The proposed method employs a probabilistic form of the mathematical description of Poisson process. Calculations of nonlinear functionals use an estimate of stream intensity as an argument, the estimate is based on calculation of random number of events for the fixed quantization interval. An analytical solution of the problem of setting calculating device with the account for Poisson distribution of input signal is presented. Calculation formulas are given that make it possible to find a concrete solution of the problem for several typical nonlinearities. The results are confirmed by modeling the process of formation of a nonlinear functional described by a third order polynomial using the applied mathematical programs package Scilab. It is anticipated that the proposed algorithm for calculating the tuning function of a computing device for reproducing a given analytically nonlinear functional can be used to solve practical problems in stochastic systems with Poisson input signal.

Keywords: problem of synthesis, Poisson process, intensity, nonlinear functionality, stochastic system, control function

References:

1. Rashid R., Hoseini S.F., Gholamian M.R., Feizabadi M. Journal of Industrial Engineering International, 2015, no. 4(11), pp. 485–494. DOI:10.1007/s40092-015-0115-9.
2. Miller B.M., Avrachenkov K.E., Stepanyan K.V., Miller G.B. Problems of Information Transmission, 2005, no. 2(41), pp. 150–170. DOI:10.1007/s11122-005-0020-8.
3. Vodovozov A.M., Polyanskiy A.V.Journal of Instrument Engineering, 1978, no. 8, pp. 17–20. (in Russ.)
4. Bukhalev V.A., Skrynnikov A.A., Fedotov A.Y. Journal of Computer and Systems Sciences International, 2013, no. 4(52), pp. 599–607. DOI:10.1134/S1064230713030040/.
5. Suchkova L.I., Yurchenko A.V. Kontrol’. Diagnostika (Testing. Diagnostics), 2012, no. 13, pp. 136–140. (in Russ.)
6. Poloskov I., Malanin V. International Journal of Dynamics and Control, 2016, no. 2(4), pp. 195–203. DOI:10.1007/s40435-015-0172-3.
7. Zol'nikov V.K., Safonov A.V. Programmnye produkty i sistemy (Software & Systems), 2011, no. 3, pp. 47. (in Russ.)
8. Arsenyev V.N., Silantev S.B., Yadrenkin А.А.Journal of Instrument Engineering, 2017, no. 5(60), pp. 391–397. DOI: 10.17586/0021-3454-2017-60-5-391-397. (in Russ.)
9. Levkina S.V. Journal of Instrument Engineering, 2017, no. 2(60), pp. 132–135. DOI: 10.17586/0021-3454-2017-60-2-132-135. (in Russ.)
10. Grigor'ev Yu.A., Pluzhnikov V.L. Herald of the Bauman Moscow State Technical University. Series Instrument Engineering, 2010, no. 4, pp. 78–90. (in Russ.)
11. Rybakov K.A. Civil Aviation High Technologies, 2013, no. 194, pp. 55–62. (in Russ.)
12. Sobolev V.I. Measurement Techniques, 2014, no. 3(57), pp. 247–254. DOI:10.1007/s11018-014-0440-1
13. Kingman J.F.C. Poisson Processes, Oxford, Clarendon Press, 1993.
14. Korn G.A., Korn T.M. Mathematical handbook: For scientists and engineers, NY, McGraw-Hill, 1968.
15. Schuster M., Unbehauen R. Electrical Engineering, 2006, no. 3(88), pp. 229–239. DOI:10.1007/s00202-004-0278-7.