On a problem of dynamical input reconstruction for a system of special type under conditions of uncertainty

Valeriy Rozenberg; Valeriy Rozenberg

doi:10.3934/math.2020263

AIMS Mathematics

2020, Volume 5, Issue 5: 4108-4120. doi: 10.3934/math.2020263

Previous Article Next Article

Research article

On a problem of dynamical input reconstruction for a system of special type under conditions of uncertainty

Valeriy Rozenberg ^{1,2
,
,}

1.
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskoi str. 16, Yekaterinburg, 620990 Russia
2.
Ural Federal University, Mira str. 19, Yekaterinburg, 620002 Russia

Received: 30 December 2019 Accepted: 17 April 2020 Published: 28 April 2020
MSC : 49K15, 60H10, 93E12

From the viewpoint of the approach of the theory of dynamic inversion, an input reconstruction problem in a differential system of special type is under investigation. The first equation of the system is a linear stochastic Ito equation, whereas the second is a linear ordinary equation containing an unknown disturbance. The statement when the reconstruction is performed from the discrete information on several realizations of the stochastic process being a solution of the first equation is considered. The problem is reduced to an inverse problem for the system of ordinary differential equations, which includes, instead of the stochastic equation, the equation describing the dynamics of the mathematical expectation of the desired process. A finite-step software-oriented solving algorithm based on the method of auxiliary feedback controlled models is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. An illustrating example is given, for which the calibration of algorithm's parameters is discussed.
- stochastic differential equation,
- ordinary differential equation,
- dynamical input reconstruction,
- lack of information,
- controlled model
Citation: Valeriy Rozenberg. On a problem of dynamical input reconstruction for a system of special type under conditions of uncertainty[J]. AIMS Mathematics, 2020, 5(5): 4108-4120. doi: 10.3934/math.2020263

Related Papers:

Abstract

From the viewpoint of the approach of the theory of dynamic inversion, an input reconstruction problem in a differential system of special type is under investigation. The first equation of the system is a linear stochastic Ito equation, whereas the second is a linear ordinary equation containing an unknown disturbance. The statement when the reconstruction is performed from the discrete information on several realizations of the stochastic process being a solution of the first equation is considered. The problem is reduced to an inverse problem for the system of ordinary differential equations, which includes, instead of the stochastic equation, the equation describing the dynamics of the mathematical expectation of the desired process. A finite-step software-oriented solving algorithm based on the method of auxiliary feedback controlled models is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. An illustrating example is given, for which the calibration of algorithm's parameters is discussed.

References

[1]	R. W. Brockett, M. P. Mesarovich, The reproducivility of multivariable control systems, J. Math. Anal. Appl., 11 (1965), 548-563. doi: 10.1016/0022-247X(65)90104-6
[2]	M. K. Sain, J. L. Massey, Invertibility of linear time-invariant dynamical systems, IEEE Trans. Automat. Contr., AC-14 (1969), 141-149.
[3]	L. M. Silverman, Inversion of multivariable linear systems, IEEE Trans. Automat. Contr., AC-14 (1969), 270-276.
[4]	L. Ljung, T. SöderstrSöm, Theory and Practice of Recursive Identification, Massachusetts: M.I.T. Press, 1983.
[5]	J. P. Norton, An Introduction to Identification, London: Academic Press, 1986.
[6]	Y. Bar-Shalom, X. R. Li, Estimation and Tracking: Principles, Techniques, and Software, Boston: Artech House, 1993.
[7]	V. K. Ivanov, V. V. Vasin, V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications, Moscow: Nauka, 1978 [in Russian].
[8]	A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems, Moscow: Nauka, 1979; New York: Wiley, 1981.
[9]	S. I. Kabanikhin, Inverse and Ill-Posed Problems, Berlin: De Gruyter, 2011.
[10]	A. V. Kryazhimskii, Yu. S. Osipov, Modelling of a control in a dynamic system, Engrg. Cybernetics, 21 (1984), 38-47.
[11]	Yu. S. Osipov, A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, London: Gordon & Breach, 1995.
[12]	Yu. S. Osipov, A. V. Kryazhimskii, V. I. Maksimov, Dynamic Recovery Methods for Inputs of Control Systems, Yekaterinburg: Izd. UrO RAN, 2011 [in Russian].
[13]	V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, Utrecht-Boston: VSP, 2002.
[14]	N. N. Krasovskii, Control of a Dynamical System, Moscow: Nauka, 1985 [in Russian].
[15]	N. N. Krasovskii, A. I. Subbotin, Game-Theoretical Control Problems, New York: Springer, 1988.
[16]	V. I. Maksimov, L. Pandolfi, On a dynamical identification of controls in nonlinear time-lag systems, IMA J. Math. Control Inform., 19 (2002), 173-184. doi: 10.1093/imamci/19.1_and_2.173
[17]	M. S. Blizorukova, V. I. Maksimov, On a reconstruction algorithm for the trajectory and control in a delay system, Proc. Steklov Inst. Math., 280 (2013), S66-S79.
[18]	K. J. Keesman, V. I. Maksimov, On feedback identification of unknown characteristics: A bioreactor case study, Int. J. Contr., 81 (2008), 134-145. doi: 10.1080/00207170701385843
[19]	V. I. Maksimov, Game control problem for a phase field equation, J. Optim. Theory and Appl., 170 (2016), 294-307. doi: 10.1007/s10957-015-0721-0
[20]	A. V. Kryazhimskiy, V. I. Maksimov, Resource-saving infinite-horizon tracking under uncertain input, Appl. Math. Comp., 217 (2010), 1135-1140. doi: 10.1016/j.amc.2010.01.014
[21]	Yu. S. Osipov, A. V. Kryazhimskii, Positional modeling of a stochastic control in dynamical systems, in Stochastic Optimization: Proceedings of the International Conference, Kiev, Ukraine, 1984, Ser. Lecture Notes in Control and Information Sciences, 81, 696-704, Berlin: Springer, 1986.
[22]	V. L. Rozenberg, Dynamic restoration of the unknown function in the linear stochastic differential equation, Autom. Remote Control, 68 (2007), 1959-1969. doi: 10.1134/S0005117907110069
[23]	V. L. Rozenberg, Dynamical input reconstruction problem for a quasi-linear stochastic system, Proc. of the 17th IFAC Workshop on Control Applications of Optimization, CAO 2018, October 15-19, 2018, Yekaterinburg, Russia, 727-732.
[24]	A. V. Kryazhimskii, Yu. S. Osipov, On a stable positional recovery of control from measurements of a part of coordinates, in Some Problems of Control and Stability: Collection of Papers, Sverdlovsk: Izd. AN SSSR, 1989, 33-47 [in Russian].
[25]	M. S. Blizorukova, V. I. Maksimov, A control problem under incomplete information, Automat. Remote Control, 67 (2006), 461-471. doi: 10.1134/S0005117906030106
[26]	V. L. Rozenberg, Reconstruction of random-disturbance amplitude in linear stochastic equations from measurements of some of the coordinates, Comput. Math. Math. Phys., 56 (2016), 367-375. doi: 10.1134/S0965542516030143
[27]	V. I. Maksimov, On dynamical reconstruction of an input in a linear system under measuring a part of coordinates, J. Inverse Ill-Pose P., 26 (2018), 395-410. doi: 10.1515/jiip-2017-0118
[28]	V. L. Rozenberg, On a problem of dynamic reconstruction under incomplete information, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 25 (2019), 207-218. doi: 10.21538/0134-4889-2019-25-1-207-218
[29]	L. Melnikova, V. Rozenberg, One dynamical input reconstruction problem: Tuning of solving algorithm via numerical experiments, AIMS Math., 4 (2019), 699-713. doi: 10.3934/math.2019.3.699
[30]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Berlin: Springer, 1985; Moscow: Mir, 2003.
[31]	F. L. Chernous'ko, V. B. Kolmanovskii, Optimal Control under Random Perturbation, Moscow: Nauka, 1978 [in Russian].
[32]	V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, et al. Handbook on Probability Theory and Mathematical Statistics, Moscow: Nauka, 1985 [in Russian].
[33]	G. N. Milshtein, Numerical Integration of Stochastic Differential Equations, Sverdlovsk: Izd. Ural. Gos. Univ., 1988 [in Russian].

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)