| Title:
|
Continuous dependence on parameters and boundedness of solutions to a hysteresis system (English) |
| Author:
|
Kamachkin, Alexander M. |
| Author:
|
Potapov, Dmitriy K. |
| Author:
|
Yevstafyeva, Victoria V. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
67 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
65-80 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We analyze an ordinary differential system with a hysteresis-relay nonlinearity in two cases when the system is autonomous or nonautonomous. Sufficient conditions for both the continuous dependence on the system parameters and the boundedness of the solutions to the system are obtained. We give a supporting example for the autonomous system. (English) |
| Keyword:
|
ODE system |
| Keyword:
|
hysteresis relay |
| Keyword:
|
external disturbance |
| Keyword:
|
bounded solution |
| Keyword:
|
periodic solution |
| MSC:
|
34C11 |
| MSC:
|
34C25 |
| MSC:
|
34C55 |
| MSC:
|
93C15 |
| MSC:
|
93C73 |
| idZBL:
|
Zbl 07478517 |
| idMR:
|
MR4392405 |
| DOI:
|
10.21136/AM.2021.0085-20 |
| . |
| Date available:
|
2022-02-08T10:48:50Z |
| Last updated:
|
2024-03-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149359 |
| . |
| Reference:
|
[1] Andronov, A. A., Vitt, A. A., Khaikin, S. E.: Theory of Oscillators.International Series of Monographs in Physics 4. Pergamon Press, Oxford (1966). Zbl 0188.56304, MR 0198734, 10.1016/C2013-0-06631-5 |
| Reference:
|
[2] Arnold, M., Begun, N., Gurevich, P., Kwame, E., Lamba, H., Rachinskii, D.: Dynamics of discrete time systems with a hysteresis stop operator.SIAM J. Appl. Dyn. Syst. 16 (2017), 91-119. Zbl 1361.37076, MR 3592068, 10.1137/16M1073522 |
| Reference:
|
[3] ström, K. J. Å: Oscillations in systems with relay feedback.Adaptive Control, Filtering, and Signal Processing The IMA Volumes in Mathematics and Its Applications 74. Springer, New York (1995), 1-25. Zbl 0829.93032, MR 1351012, 10.1007/978-1-4419-8568-2_1 |
| Reference:
|
[4] Balanov, Z., Kravetc, P., Krawcewicz, W., Rachinskii, D.: Equivariant degree method for analysis of Hopf bifurcation of relative periodic solutions: Case study of a ring of oscillators.J. Differ. Equations 265 (2018), 4530-4574. Zbl 1397.34121, MR 3843308, 10.1016/j.jde.2018.06.014 |
| Reference:
|
[5] Bertotti, G., (eds.), I. D. Mayergoyz: The Science of Hysteresis. Vol. I. Mathematical Modeling and Applications.Elsevier/Academic Press, Amsterdam (2006). Zbl 1117.34045, MR 2307929, 10.1016/B978-012480874-4/50000-2 |
| Reference:
|
[6] Botkin, N. D., Brokate, M., Behi-Gornostaeva, E. G. El: One-phase flow in porous media with hysteresis.Phys. B 486 (2016), 183-186. MR 3797613, 10.1016/j.physb.2015.08.039 |
| Reference:
|
[7] Brokate, M., Krejčí, P.: Weak differentiability of scalar hysteresis operators.Discrete Contin. Dyn. Syst. 35 (2015), 2405-2421. Zbl 1338.47118, MR 3299005, 10.3934/dcds.2015.35.2405 |
| Reference:
|
[8] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions.Applied Mathematical Sciences 121. Springer, New York (1996). Zbl 0951.74002, MR 1411908, 10.1007/978-1-4612-4048-8 |
| Reference:
|
[9] Burns, R. S.: Advanced Control Engineering.Butterworth-Heinemann, Oxford (2001). 10.1016/B978-0-7506-5100-4.X5000-1 |
| Reference:
|
[10] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Lasiecka, I., Webler, C. M.: Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density.Adv. Nonlinear Anal. 6 (2017), 121-145. Zbl 1373.35045, MR 3641629, 10.1515/anona-2016-0027 |
| Reference:
|
[11] Fang, L., Wang, J., Zhang, Q.: Identification of extended Hammerstein systems with hysteresis-type input nonlinearities described by Preisach model.Nonlinear Dyn. 79 (2015), 1257-1273. Zbl 1345.93046, MR 3302768, 10.1007/s11071-014-1740-3 |
| Reference:
|
[12] Fonda, A., Garrione, M., Gidoni, P.: Periodic perturbations of Hamiltonian systems.Adv. Nonlinear Anal. 5 (2016), 367-382. Zbl 1353.37124, MR 3567850, 10.1515/anona-2015-0122 |
| Reference:
|
[13] Johansson, K. H., Rantzer, A., Åström, K. J.: Fast switches in relay feedback systems.Automatica 35 (1999), 539-552. Zbl 0934.93033, 10.1016/S0005-1098(98)00160-5 |
| Reference:
|
[14] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence.Int. J. Robust Nonlinear Control 27 (2017), 204-211. Zbl 1353.93055, MR 3594931, 10.1002/rnc.3567 |
| Reference:
|
[15] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence.Electron. J. Differ. Equ. 2017 (2017), Article ID 140, 10 pages. Zbl 1370.34066, MR 3665602 |
| Reference:
|
[16] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity.J. Dyn. Control Syst. 23 (2017), 825-837. Zbl 1381.34083, MR 3688896, 10.1007/s10883-017-9368-5 |
| Reference:
|
[17] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Existence of periodic modes in automatic control system with a three-position relay.Int. J. Control 93 (2020), 763-770. Zbl 1435.34048, MR 4077763, 10.1080/00207179.2018.1562221 |
| Reference:
|
[18] Krasnosel'skii, M. A., Pokrovskii, A. V.: Systems with Hysteresis.Springer, Berlin (1989). Zbl 0665.47038, MR 0987431, 10.1007/978-3-642-61302-9 |
| Reference:
|
[19] Leonov, G. A., Shumafov, M. M., Teshev, V. A., Aleksandrov, K. D.: Differential equations with hysteresis operators. Existence of solutions, stability, and oscillations.Differ. Equ. 53 (2017), 1764-1816. Zbl 1394.34004, MR 3804280, 10.1134/S0012266117130055 |
| Reference:
|
[20] Macki, J. W., Nistri, P., Zecca, P.: Mathematical models for hysteresis.SIAM Rev. 35 (1993), 94-123. Zbl 0771.34018, MR 1207799, 10.1137/1035005 |
| Reference:
|
[21] Mayergoyz, I. D.: Mathematical Models of Hysteresis and Their Applications.Elsevier, Amsterdam (2003). MR 1083150, 10.1016/B978-0-12-480873-7.X5000-2 |
| Reference:
|
[22] McCarthy, S., Rachinskii, D.: Dynamics of systems with Preisach memory near equilibria.Math. Bohem. 139 (2014), 39-73. Zbl 1340.34163, MR 3231429, 10.21136/MB.2014.143636 |
| Reference:
|
[23] Paraskevopoulos, P. N.: Modern Control Engineering.Control Engineering (Boca Raton) 10. Marcel Dekker, New York (2001). Zbl 0986.93001, 10.1201/9781315214573 |
| Reference:
|
[24] Pimenov, A., Rachinskii, D.: Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator.Math. Bohem. 139 (2014), 285-298. Zbl 1349.47141, MR 3238840, 10.21136/MB.2014.143855 |
| Reference:
|
[25] Pokrovskii, A. V.: Existence and computation of stable modes in relay systems.Autom. Remote Control 47 (1986), 451-458 translation from Avtom. Telemekh. 1986 1986 16-23. Zbl 0604.93050, MR 0848397 |
| Reference:
|
[26] Potapov, D. K., Yevstafyeva, V. V.: Lavrent'ev problem for separated flows with an external perturbation.Electron. J. Differ. Equ. 2013 (2013), Article ID 255, 6 pages. Zbl 1290.35134, MR 3138830 |
| Reference:
|
[27] Rachinskii, D.: Realization of arbitrary hysteresis by a low-dimensional gradient flow.Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 227-243. Zbl 1330.34074, MR 3426841, 10.3934/dcdsb.2016.21.227 |
| Reference:
|
[28] Solovyov, A. M., Semenov, M. E., Meleshenko, P. A., Reshetova, O. O., Popov, M. A., Kabulova, E. G.: Hysteretic nonlinearity and unbounded solutions in oscillating systems.Procedia Engineering 201 (2017), 578-583. 10.1016/j.proeng.2017.09.634 |
| Reference:
|
[29] Tsypkin, Y. Z.: Relay Control Systems.Cambridge University Press, Cambridge (1984). Zbl 0571.93001, MR 0789077 |
| Reference:
|
[30] Varigonda, S., Georgiou, T. T.: Dynamics of relay relaxation oscillators.IEEE Trans. Autom. Control 46 (2001), 65-77. Zbl 1004.34034, MR 1809466, 10.1109/9.898696 |
| Reference:
|
[31] Visintin, A.: Differential Models of Hysteresis.Applied Mathematical Sciences 111. Springer, Berlin (1994). Zbl 0820.35004, MR 1329094, 10.1007/978-3-662-11557-2 |
| Reference:
|
[32] Visintin, A.: Ten issues about hysteresis.Acta Appl. Math. 132 (2014), 635-647. Zbl 1305.74072, MR 3255072, 10.1007/s10440-014-9936-6 |
| Reference:
|
[33] Visintin, A.: P.D.E.s with hysteresis 30 years later.Discrete Contin. Dyn. Syst., Ser. S 8 (2015), 793-816. Zbl 1304.35357, MR 3356462, 10.3934/dcdss.2015.8.793 |
| Reference:
|
[34] Yevstafyeva, V. V.: On existence conditions for a two-point oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix.Autom. Remote Control 76 (2015), 977-988 translation from Avtom. Telemekh. 2015 2015 42-56. Zbl 1327.93225, MR 3374789, 10.1134/S000511791506003X |
| Reference:
|
[35] Yevstafyeva, V. V.: Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero.Ukr. Math. J. 70 (2019), 1252-1263 translation from Ukr. Mat. Zh. 70 2018 1085-1096. Zbl 1417.34098, MR 3863943, 10.1007/s11253-018-1566-0 |
| Reference:
|
[36] Yu, C.-C.: Autotuning of PID Controllers: Relay Feedback Approach.Advances in Industrial Control. Springer, Berlin (1999). Zbl 0962.93004, 10.1007/b137042 |
| . |
