V.I. Maksimov. On an algorithm for the reconstruction of a perturbation in a nonlinear system ... P. 156-166

A problem of reconstruction of an unknown perturbation in a system of nonlinear ordinary differential equations is considered. The methods of solution of such problems are well known. In this paper we study a problem with two peculiarities. First, it is assumed that the phase coordinates of the dynamical system are measured (with error) at discrete sufficiently frequent times. Second, the only information known about the perturbation acting on the system is that its Euclidean norm is square integrable; i.e., the perturbation can be unbounded. Since the exact reconstruction is impossible under these assumptions, we design a solution algorithm that is stable under information noise and computation errors. The algorithm is based on the combination of elements of the theory of ill-posed problems with the extremal shift method known in the theory of positional differential games.

Keywords: linear control systems, dynamic reconstruction

Received November 5, 2019

Revised January 13, 2020

Accepted January 20, 2020

Vyacheslav Ivanovich Maksimov, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: maksimov@imm.uran.ru

REFERENCES

1.   Kurzhanski A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under the conditions of uncertainty]. Moscow: Nauka Publ., 1977, 392 p.

2.   Kurzhanski A.B., Valyi I. Ellipsoidal calculus for estimation and control. Basel: Birkhauser, 1997, 321 p. ISBN: 978-0-8176-3699-9 .

3.   Ananyev B.I., Gusev M.I., Filippova T.F. Upravlenie i otsenivanie sostoyanii dinamicheskikh sistem s neopredelennost’yu [Control and estimation of dynamical systems states with uncertainty]. Novosibirsk: Siberian Branch of RAS Publ., 2018, 193 p.

4.   Osipov Yu.S., Kryazhimskii A.V., Maksimov V.I. Some algorithms for the dynamic reconstruction of inputs. Proc. Steklov Inst. Math., 2011, vol. 275, suppl. 1, pp. 86–120. doi: 10.1134/S0081543811090082 

5.   Osipov Yu.S., Kryazhimskii A.V. Inverse problems for ordinary differential equations: dynamical solutions. Basel: Gordon and Breach, 1995, 625 p. ISBN: 2881249442 .

6.   Osipov Yu.S., Kryazhimskii A.V., and Maksimov V.I. Metody dinamicheskogo vosstanovleniya vkhodov upravlyaemykh sistem [Methods for dynamic reconstruction of inputs of control systems]. Ekaterinburg: Ural Branch of RAS Publ., 2011, 291 p.

7.   Maksimov V.I., Pandolfi L. The reconstruction of unbounded controls in nonlinear dynamical systems. J. Appl. Math. Mech., 2001, vol. 65, no. 3, pp. 371–376.

8.   Blizorukova M.S., Maksimov V.I. On an algorithm for dynamic reconstruction of the input. Diff. Eq., 2013, vol. 49, no. 1, pp. 88–100. doi: 10.1134/S0012266113010096 

9.   Maksimov V.I. On dynamical reconstruction of an input in a linear system under measuring a part of coordinates. J. Inverse and Ill-Posed Problems, 2018, vol. 26, no. 3, pp. 395–410. doi: 10.1515/jiip-2017-0118 

10.   Blizorukova M.S., Maksimov V.I. On a reconstruction algorithm for the trajectory and control in a delay system. Proc. Steklov Inst. Math.., 2013, vol. 280, no. 1, pp. 66–79. doi: 10.1134/S0081543813020065 

11.   Kappel F., Kryazhimskii A.V., Maksimov V.I. Dynamic reconstruction of states, and the guaranteeing control of a reaction-diffusion system. Dokl. Math., 2000, vol. 61, no. 1, pp. 143–145.

12.   Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y: Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow: Nauka Publ., 1974, 456 p.

13.   Tikhonov A.N., Arsenin V.Ya. Methods for Solutions of Ill-Posed Problems. N Y: Wiley, 1977, 258 p. ISBN: 0470991240 . Original Russian text (2nd ed.) published in Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach. Moscow: Nauka Publ., 1978, 285 p.

14.   Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Inverse and Ill-Posed Problems Series. Utrecht: VSP, 2002, 281 p. ISBN: 90-6764-367-X/hbk . Original Russian text published in Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya. Moscow: Nauka Publ., 1978, 206 p.

15.   Maksimov V.I. Calculation of the derivative of an inaccurately defined function by means of feedback laws. Proc. Steklov Inst. Math., 2015, vol. 291, pp. 219–231. doi: 10.1134/S0081543815080179 

16.   Fang H., Shi Y., Yu J. On stable simultaneous input and state estimation for discrete-time linear systems. Internat. J. Adaptiv. Contr. Signal Proc. 2011, vol.  25, no. 8, no. 671–686.

17.   Keller J.Y., Chabir K., Sauter D. Input reconstruction for networked control systems subject to deception attacks and data losses on control signals. Int. J. Syst. Sci., 2016, vol. 47, no. 4, pp. 814–820.

18.   Keller J.Y., Sauter D. Kalman filter for discrete-time stochastic linear systems subject to intermittent unknown inputs. IEEE Trans. Autom. Contr., 2013, vol. 58, no. 7, pp.1882–1887.

19.   Chabir K., Sid M.A., and Sauter D. Fault diagnosis in a networked control system under communication constraints: A quadrotor applications. Int. J. Apll. Math. Comput. Sci., 2014, vol.  24, no. 4, pp. 809–820.

Cite this article as: V.I. Maksimov. On an algorithm for the reconstruction of a perturbation in a nonlinear system, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 1, pp. 156–166.