Abstract
We solve the problem of output process analysis for nonlinear stochastic control systems with continuous time. The proposed method is based on a spectral form of mathematical description which provides a unified approach to solving linear operator equations. The techniques we have developed allow for reducing the problem of finding the state vector distribution density to solving a system of linear algebraic equations on the decomposition coefficients for the desired density into a series by the functions of a certain complete orthonormal system for both the case when no constraints are imposed on state vector coordinates and the case of problems with constraints.
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References
Pugachev, V.S. and Sinitsyn, I.N., Teoriya stokhasticheskikh sistem (Theory of Stochastic Systems), Moscow: Logos, 2004.
Panteleev, A.V. and Bortakovskii, A.S., Teoriya upravleniya v primerakh i zadachakh (Control Theory in Examples and Problems), Moscow: Vysshaya Shkola, 2003.
Tikhonov, V.I. and Mironov, M.A., Markovskie protsessy (Markov Processes), Moscow: Sovetskoe Radio, 1977.
Risken, H., The Fokker-Planck Equation: Methods of Solution and Applications, New York: Springer, 1996.
Kazakov, I.E. and Mal'chikov, S.V., Analiz stokhasticheskikh sistem v prostranstve sostoyanii (Stochastic Systems Analysis in the State Space), Moscow: Nauka, 1983.
Krasovskii, A.A., Fazovoe prostranstvo i statisticheskaya teoriya dinamicheskikh sistem (The Phase Space and Statistical Theory of Dynamical Systems), Moscow: Nauka, 1974.
Sinitsyn, V.I., Finding Multidimensional Distributions of a Process Defined by a Stochastic Differential Equation: Ellipsoidal Approximation Method, Dokl. Akad. Nauk SSSR, 1991, vol. 320, no. 2, pp. 289–294.
Di Paola, M. and Sofi, A., Approximate Solution of the Fokker-Planck-Kolmogorov Equation, Probab. Eng. Mech., 2002, vol. 17, no. 4, pp. 369–384.
Wei, G.W., A Unified Approach for the Solution of the Fokker-Planck Equation, J. Phys. A: Math. Gen., 2000, vol. 33, pp. 4935–4953.
Kuznetsov, D.F., Stokhasticheskie differentsial'nye uravneniya: teoriya i praktika chislennogo resheniya (Stochastic Differential Equations: Theory and Practice of a Numerical Solution), St. Petersburg: Politekhn. Univ., 2009.
Semenov, V.V., Equation of the Generalized Characteristic Function for the State Vector in Automated Control Systems, in Analytic Methods of Regulator Synthesis, Saratov: SPI, 1977, no. 2, pp. 3–36.
Pugachev, V.S., Lektsii po funktsional'nomu analizu (Lectures on Functional Analysis), Moscow: Mosk. Aviats. Inst., 1996.
Solodovnikov, V.V. and Semenov, V.V., Spektral'naya teoriya nestatsionarnykh sistem upravleniya (Spectral Theory of Nonstationary Control Systems), Moscow: Nauka, 1974.
Solodovnikov, V.V., Dmitriev, A.N., and Egupov, N.D., Spektral'nye metody rascheta i proektirovaniya sistem upravleniya (Spectral Methods of Computing and Designing Control Systems), Moscow: Mashinostroenie, 1986.
Lapin, S.V. and Egupov, N.D., Teoriya matrichnykh operatorov i ee prilozhenie k zadacham avtomaticheskogo upravleniya (Matrix Operator Theory and Its Application to Automated Control Problems), Moscow: Mosk. Gos. Tekhn. Univ., 1997.
Panteleev, A.V., Rybakov, K.A., and Sotskova, I.L., Spektral'nyi metod analiza nelineinykh stokhasticheskikh sistem upravleniya (The Spectral Method for Analyzing Nonlinear Stochastic Control Systems), Moscow: Vuz. Kniga, 2006.
Rybin, V.V., Opisanie signalov i lineinykh nestatsionarnykh sistem upravleniya v bazisakh veivletov i ikh analiz v vychislitel'nykh sredakh (Representing Signals and Linear Nonstationary Control Systems in Wavelet Bases and Their Analysis in Computational Environments), Moscow: Mosk. Aviats. Inst., 2003.
Romanov, V.A. and Rybakov, K.A., Spectral Characteristics of the Multiplication, Differentiation, and Integration Operators in the Basis of Generalized Hermite Functions, in Tr. Mosk. Aviats. Inst., 2010, no. 39 (http://www.mai.ru).
Kazakov, I.E., Artem'ev, V.M., and Bukhalev, V.A., Analiz sistem sluchainoi struktury (Analyzing Systems with Random Structure), Moscow: Fizmatlit, 1993.
Rybakov, K.A. and Sotskova, I.L., Spectral Method for Analysis of Switching Diffusions, IEEE TAC, 2007, vol. 52, no. 7, pp. 1320–1325.
Anulova, S.V., Veretennikov, A.Yu., Krylov, N.V., Liptser, R.Sh., et al, Stochastic Calculus, Itogi Nauki Tekhn., Sovr. Probl. Mat., Fundam. Napravl., Moscow: VINITI, 1989, vol. 45.
Sotskova, I.L., Applying the Method of Generalized Characteristic Functions to Analyzing Stochastic Control Systems for Flying Vehicles, in Problems of Stochastic Control, Moscow: Mosk. Aviats. Inst., 1986, pp. 71–78.
Semenov, V.V. and Sotskova, I.L., The Spectral Method for Solving Fokker-Planck-Kolmogorov Equation for Stochastic Control System Analysis, Second IFAC Sympos. Stochast. Control. Preprints, Part 1, 1986, pp. 131–136.
Rybakov, K.A., Software for the Spectrum Spectral Method, Tr.. Mosk. Aviats. Inst., 2003, no. 14 (http://www.mai.ru).
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Original Russian Text © A.V. Panteleev, K.A. Rybakov, 2011, published in Avtomatika i Telemekhanika, 2011, No. 2, pp. 183–194.
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Panteleev, A.V., Rybakov, K.A. Analyzing nonlinear stochastic control systems in the class of generalized characteristic functions. Autom Remote Control 72, 393–404 (2011). https://doi.org/10.1134/S0005117911020159
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DOI: https://doi.org/10.1134/S0005117911020159