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One dynamical input reconstruction problem: tuning of solving algorithm via numerical experiments

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

Topical Section: Mathematical modeling

The input reconstruction problem for a stochastic differential equation is investigated by means of the approach of the theory of dynamic inversion. The statement when the simultaneous reconstruction of disturbances in both the deterministic and stochastic terms of the equation is performed from the discrete information on several realizations of the stochastic process is considered. 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 empirical procedure for the automatic tuning of algorithm’s parameters in order to get best approximation results for a specific dynamical system is proposed. To optimize this time-taking process, the parallelization of calculations is applied. A model example illustrating the method proposed is given.
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Keywords stochastic differential equation; dynamical input reconstruction; controlled model; tuning procedure; parallelization of calculations

Citation: Lidiya Melnikova, Valeriy Rozenberg. One dynamical input reconstruction problem: tuning of solving algorithm via numerical experiments. AIMS Mathematics, 2019, 4(3): 699-713. doi: 10.3934/math.2019.3.699

References

  • 1.R. W. Brockett and M. P. Mesarovich, The reproducivility of multivariable control systems, J. Math. Anal. Appl., 11 (1965), 548-563.    
  • 2.M. K. Sain and J. L. Massey, Invertibility of linear time-invariant dynamical systems, IEEE T. Automat. Contr., 14 (1969), 141-149.    
  • 3.L. M. Silverman, Inversion of multivariable linear systems, IEEE T. Automat. Contr. 14 (1969), 270-276.    
  • 4.L. Ljung and T. Söderström, 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 and X. R. Li, Estimation and Tracking: Principles, Techniques, and Software, Boston: Artech House, 1993.
  • 7.A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems, Moscow: Nauka, 1979; New York: Wiley, 1981.
  • 8.S. I. Kabanikhin, Inverse and Ill-Posed Problems, Berlin: De Gruyter, 2011.
  • 9.A. V. Kryazhimskii and Yu. S. Osipov, Modelling of a control in a dynamic system, Engrg. Cybernetics, 21 (1984), 38-47.
  • 10.Y. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, London: Gordon & Breach, 1995.
  • 11.Y. S. Osipov, A. V. Kryazhimskii and V. I. Maksimov, Dynamic Recovery Methods for Inputs of Control Systems, Yekaterinburg: Izd. UrO RAN, 2011 [in Russian].
  • 12.V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, Utrecht-Boston: VSP, 2002.
  • 13.N. N. Krasovskii, Control of a Dynamical System, Moscow: Nauka, 1985 [in Russian].
  • 14.N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, New York: Springer, 1988.
  • 15.V. I. Maksimov and L. Pandolfi, On a dynamical identification of controls in nonlinear time-lag systems, IMA J. Math. Control I., 19 (2002), 173-184.    
  • 16.M. S. Blizorukova and V. I. Maksimov, On a reconstruction algorithm for the trajectory and control in a delay system, P. Steklov I. Math., 280 (2013), 66-79.    
  • 17.V. I. Maksimov, Game control problem for a phase field equation, J. Optimiz. Theory App., 170 (2016), 294-307.    
  • 18.A. V. Kryazhimskii and Y. 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].
  • 19.Y. S. Osipov and 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.
  • 20.V. L. Rozenberg, Dynamic restoration of the unknown function in the linear stochastic differential equation, Automat. Rem. Contr., 68 (2007), 1959-1969.    
  • 21.V. L. Rozenberg, Reconstruction of random-disturbance amplitude in linear stochastic equations from measurements of some of the coordinates, Comp. Math. Math. Phys., 56 (2016), 367-375.    
  • 22.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. Available from: https://www.sciencedirect.com/journal/ifac-papersonline/vol/51/issue/32?page=2.
  • 23.B. ∅ksendal, Stochastic Differential Equations: An Introduction with Applications, Berlin: Springer, 1985; Moscow: Mir, 2003.
  • 24.V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, et al. Handbook on Probability Theory and Mathematical Statistics, Moscow: Nauka, 1985 [in Russian].
  • 25.F. L. Chernous'ko and V. B. Kolmanovskii, Optimal Control under Random Perturbation, Moscow: Nauka, 1978 [in Russian].
  • 26.G. N. Milshtein, Numerical Integration of Stochastic Differential Equations, Sverdlovsk: Izd. Ural. Gos. Univ., 1988 [in Russian].
  • 27.I. Foster, Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering, Massachusetts: Addison-Wesley Reading, 1995.
  • 28.V. P. Gergel', Theory and Practice of Parallel Computing, Moscow: Binom, 2007 [in Russian].

 

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