An adaptive algorithm for a stable online identification of a disturbance in a fractional-order system on an infinite time horizon

P. G. Surkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 172-188

Résumé

The problem of online identification of an uncontrolled external disturbance (noise) in a system of differential equations with a fractional Caputo derivative is considered on an infinite time horizon. Information on the position of the system is available for observations only during its functioning, and only a part of the coordinates of the system's phase vector can be measured. The case of measuring all phase coordinates is also considered. The measurements are carried out at discrete, sufficiently frequent times with a certain error. Therefore, the problem of finding the unknown disturbance is ill-posed. To solve it, an adaptive online identification algorithm is constructed using the dynamic inversion approach, which is based on a combination of regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method with a smoothing functional of special form and the Krasovskii extremal aiming method. The algorithm is based on the choice of an appropriate auxiliary control system and a feedback control law in this system. The proposed algorithm approximates the external disturbance and is stable under information noises and computational errors. A model example demonstrating the application of the developed technique is considered.

Détail

Keywords: online identification, external disturbance, Caputo fractional derivative, infinite time interval.

Bibliographie

[1] Bar-Shalom Y., Li X.R., Kirubarajan T., Estimation with applications to tracking and navigation: Theory algorithms and software, John Wiley Sons, NY, 2004, 592 pp.

[2] Keesman K.J., System identification. An introduction, Springer-Verlag, London, 2011, 323 pp. | MR | Zbl

[3] Norton J.P., An Introduction to identification, Dover Publ. Inc., NY, 2009, 310 pp. | MR | Zbl

[4] Pandolfi L., Systems with persistent memory: Controllability, stability, identification, Springer, Cham, 2021, 356 pp. | MR | Zbl

[5] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 285 pp. | MR

[6] Tikhonov A.N., Arsenin V.Ya., Metody resheniya nekorrektnykh zadach, URSS, M., 2022, 288 pp.

[7] Butera S., Di Paola M., “A physically based connection between fractional calculus and fractal geometry”, Annals. of Physics, 350 (2014), 146–158 | DOI | MR | Zbl

[8] Podlubny I., “Geometrical and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus and Applied Analysis, 5:4 (2002), 367–386 | MR | Zbl

[9] Stanislavskii A.A., “Veroyatnostnaya interpretatsiya integrala drobnogo poryadka”, Teoret. i mat. fizika, 138:3 (2004), 491–507 | DOI | MR | Zbl

[10] Tarasov V.E., “Geometric interpretation of fractional-order derivative”, Fractional Calculus and Applied Analysis, 19:5 (2016), 1200–1221 | DOI | MR | Zbl

[11] Gomoyunov M.I., “Differential games for fractional-order systems: Hamilton–Jacobi–Bellman–Isaacs equation and optimal feedback strategies”, Mathematics, 9:14 (2021), 1667 | DOI

[12] Matychyn I., Onyshchenko V., “Time-optimal control of linear fractional systems with variable coefficients”, Internat. J. Appl. Math. Comp. Sci., 31:3 (2021), 375–386 | DOI | MR | Zbl

[13] Boumenir A., Kim Tuan V., Al-Khulaifi W., “Reconstructing a fractional integro-differential equation”, Math. Methods Appl. Sci., 44:4 (2021), 3159–3166 | DOI | MR | Zbl

[14] Samko S.G. Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp.

[15] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier Science, NY, 2006, 540 pp. | MR | Zbl

[16] Vasin V.V., Osnovy teorii nekorrektnykh zadach, Izd-vo SO RAN, Novosibirsk, 2020, 313 pp.

[17] Kabanikhin S.I., Inverse and Ill-posed problems: Theory and application, De Gruyter, Berlin, 2012, 459 pp. | MR

[18] Kryazhimskii A.V., Osipov Yu.S., “O modelirovanii upravleniya v dinamicheskoi sisteme”, Izv. AN SSSR. Tekhnicheskaya kibernetika, 1983, no. 2, 51–60 | Zbl

[19] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[20] Osipov Yu.S., Kryazhimskii A.V., Maksimov V.I., Metody dinamicheskogo vosstanovleniya vkhodov upravlyaemykh sistem, Izd-vo IMM UrO RAN, Ekaterinburg, 2011, 292 pp.

[21] Maksimov V.I., Osipov Yu.S., “O granichnom upravlenii raspredelennoi sistemoi na beskonechnom promezhutke vremeni”, Zhurn. vychisl. matematiki i mat. fiziki, 56:1 (2016), 16–28 | DOI | Zbl

[22] Blizorukova M.S., Maksimov V.I., “O rekonstruktsii vkhodnogo vozdeistviya parabolicheskogo uravneniya na beskonechnom promezhutke vremeni”, Izv. vuzov. Matematika, 2014, no. 8, 30–41 | Zbl

[23] Blizorukova M.S., Maksimov V.I., “Ob odnom algoritme dinamicheskogo vosstanovleniya vkhodnogo vozdeistviya”, Differents. uravneniya, 49:1 (2013), 88–100 | DOI | MR | Zbl

[24] Rozenberg V.L., “Dinamicheskaya rekonstruktsiya vozmuschenii v kvazilineinom stokhasticheskom differentsialnom uravnenii”, Zhurn. vychisl. matematiki i mat. fiziki, 58:7 (2018), 1121–1131 | DOI

[25] Maksimov V.I., “O rekonstruktsii upravlenii v eksponentsialno ustoichivykh lineinykh sistemakh, podverzhennykh malym vozmuscheniyam”, Prikl. matematika i mekhanika, 71:6 (2007), 945–955 | MR | Zbl

[26] Maksimov V.I., “The methods of dynamical reconstruction of an input in a system of ordinary differential equations”, J. Inverse and Ill-posed Problems, 29:1 (2021), 125–156 | DOI | MR | Zbl

[27] Surkov P.G., “Zadacha dinamicheskogo vosstanovleniya pravoi chasti sistemy differentsialnykh uravnenii netselogo poryadka”, Differents. uravneniya, 55:6 (2019), 865–874 | DOI | Zbl

[28] Surkov P.G., “Real-time reconstruction of external impact on fractional order system under measuring a part of coordinates”, J. Comp. Appl. Math., 381:3 (2021), 113039 | DOI | MR | Zbl

[29] Surkov P.G., “Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach”, Fractional Calculus and Applied Analysis, 24:3 (2021), 895–922 | DOI | MR | Zbl

[30] Fagnani F., Maksimov V., Pandolfi L., “A recursive deconvolution approach to disturbance reduction”, IEEE Transactions on Automatic Control, 49:6 (2004), 907–921 | DOI | MR | Zbl

[31] Subbotina N.N., Krupennikov E.A., “Slaboe so zvezdoi reshenie zadachi dinamicheskoi rekonstruktsii”, Tr. MIAN, 315 (2021), 247–260 | DOI | Zbl

[32] Barbashin E.A., Vvedenie v teoriyu ustoichivosti, URSS, M., 2022, 230 pp.

[33] Matignon D., “Stability results for fractional differential equations with applications to control processing”, Computational engineering in systems applications, 2:1 (1996), 963–968

[34] Gomoyunov M.I., “Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems”, Fractional Calculus and Applied Analysis, 21:5 (2018), 1238–1261 | DOI | MR | Zbl