Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
ODE system; relay hysteresis; nonperiodic external disturbance; two-point oscillatory solution
ODE system; relay hysteresis; nonperiodic external disturbance; two-point oscillatory solution
Summary:
We study an $n$-dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system is reduced to the form with the diagonal matrix and the vector being opposite to the unit vector. We establish a necessary and sufficient condition for the existence of two-point oscillatory solutions, i.e., the solutions with two fixed points on the hyperplanes of the relay switching in phase space. Also, we give the sufficient conditions under which such solutions do not exist. We provide a supporting example, which demonstrates how to apply the obtained results.
References:
[1] Andronov, A. A., Vitt, A. A., Khaikin, S. E.: Theory of Oscillators. International Series of Monographs in Physics 4. Pergamon Press, Oxford (1966). DOI 10.1016/C2013-0-06631-5 | MR 0198734 | Zbl 0188.56304
[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. DOI 10.1137/16M10735 | MR 3592068 | Zbl 1361.37076
[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. DOI 10.1007/978-1-4419-8568-2_1 | MR 1351012 | Zbl 0829.93032
[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. DOI 10.1016/j.jde.2018.06.014 | MR 3843308 | Zbl 1397.34121
[5] Bertotti, G., (Eds.), I. D. Mayergoyz: The Science of Hysteresis. Vol. I. Mathematical Modeling and Applications. Elsevier/Academic Press, Amsterdam (2006). MR 2307929 | Zbl 1117.34045
[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. DOI 10.1016/j.physb.2015.08.039 | MR 3797613
[7] Brokate, M., Krejčí, P.: Weak differentiability of scalar hysteresis operators. Discrete Contin. Dyn. Syst. 35 (2015), 2405-2421. DOI 10.3934/dcds.2015.35.2405 | MR 3299005 | Zbl 1338.47118
[8] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121. Springer, New York (1996). DOI 10.1007/978-1-4612-4048-8 | MR 1411908 | Zbl 0951.74002
[9] Burns, R. S.: Advanced Control Engineering. Butterworth-Heinemann, Oxford (2001). DOI 10.1016/B978-0-7506-5100-4.X5000-1
[10] 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. DOI 10.1007/s11071-014-1740-3 | MR 3302768 | Zbl 1345.93046
[11] Fursov, A. S., Mitrev, R. P., Krylov, P. A., Todorov, T. S.: On the existence of a periodic mode in a nonlinear system. Differ. Equ. 57 (2021), 1076-1087. DOI 10.1134/S0012266121080127 | MR 4316860 | Zbl 1471.93133
[12] Fursov, A. S., Todorov, T. S., Krylov, P. A., Mitrev, R. P.: On the existence of oscillatory modes in a nonlinear system with hystereses. Differ. Equ. 56 (2020), 1081-1099. DOI 10.1134/S0012266120080108 | MR 4147119 | Zbl 1451.34057
[13] Johansson, K. H., Rantzer, A., Åström, K. J.: Fast switches in relay feedback systems. Automatica 35 (1999), 539-552. DOI 10.1016/S0005-1098(98)00160-5 | Zbl 0934.93033
[14] Kamachkin, A. M., Chitrov, G. M., Shamberov, V. N.: Normal matrix forms to decomposition and control problems for multidimensional systems. Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr. 13 (2017), 417-430 Russian. DOI 10.21638/11701/spbu10.2017.408 | MR 3750121
[15] 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. DOI 10.1002/rnc.3567 | MR 3594931 | Zbl 1353.93055
[16] 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. MR 3665602 | Zbl 1370.34066
[17] 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. DOI 10.1007/s10883-017-9368-5 | MR 3688896 | Zbl 1381.34083
[18] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Dynamics and synchronization in feedback cyclic structures with hysteresis oscillators. Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr. 16 (2020), 186-199 Russian. DOI 10.21638/11701/spbu10.2020.210 | MR 4160031
[19] 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. DOI 10.1080/00207179.2018.1562221 | MR 4077763 | Zbl 1435.34048
[20] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Method for the transformation of complex automatic control systems to integrable form. Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr. 17 (2021), 196-212 Russian. DOI 10.21638/11701/spbu10.2021.209 | MR 4311880
[21] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Continuous dependence on parameters and boundedness of solutions to a hysteresis system. Appl. Math., Praha 67 (2022), 65-80. DOI 10.21136/AM.2021.0085-20 | MR 4392405 | Zbl 07478517
[22] Kamachkin, A. M., Potapov, D. K., Yevstafyeva, V. V.: Fixed points of a mapping generated by a system of ordinary differential equations with relay hysteresis. Differ. Equ. 58 (2022), 455-467. DOI 10.1134/S0012266122040024 | MR 4464618 | Zbl 1503.34093
[23] Krasnosel'skij, M. A., Pokrovskij, A. V.: Systems with Hysteresis. Springer, Berlin (1989). DOI 10.1007/978-3-642-61302-9 | MR 0987431 | Zbl 0665.47038
[24] 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. DOI 10.1134/S0012266117130055 | MR 3804280 | Zbl 1394.34004
[25] Macki, J. W., Nistri, P., Zecca, P.: Mathematical models for hysteresis. SIAM Rev. 35 (1993), 94-123. DOI 10.1137/10350 | MR 1207799 | Zbl 0771.34018
[26] Mayergoyz, I. D.: Mathematical Models of Hysteresis and Their Applications. Elsevier/Academic Press, Amsterdam (2003). DOI 10.1016/B978-0-12-480873-7.X5000-2 | MR 1083150
[27] McCarthy, S., Rachinskii, D.: Dynamics of systems with Preisach memory near equilibria. Math. Bohem. 139 (2014), 39-73. DOI 10.21136/MB.2014.143636 | MR 3231429 | Zbl 1340.34163
[28] Medvedskii, A. L., Meleshenko, P. A., Nesterov, V. A., Reshetova, O. O., Semenov, M. E., Solovyov, A. M.: Unstable oscillating systems with hysteresis: Problems of stabilization and control. J. Comput. Syst. Sci. Int. 59 (2020), 533-556. DOI 10.1134/S1064230720030090 | MR 4431725 | Zbl 1470.93128
[29] Paraskevopoulos, P. N.: Modern Control Engineering. Control Engineering (Boca Raton) 10. Marcel Dekker, New York (2001). DOI 10.1201/9781315214573 | Zbl 0986.93001
[30] Pimenov, A., Rachinskii, D.: Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator. Math. Bohem. 139 (2014), 285-298. DOI 10.21136/MB.2014.143855 | MR 3238840 | Zbl 1349.47141
[31] Pokrovskij, A. V.: Existence and computation of stable modes in relay systems. Autom. Remote Control 47 (1986), 451-458. MR 0848397 | Zbl 0604.93050
[32] 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. MR 3138830 | Zbl 1290.35134
[33] Rachinskii, D.: Realization of arbitrary hysteresis by a low-dimensional gradient flow. Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 227-243. DOI 10.3934/dcdsb.2016.21.227 | MR 3426841 | Zbl 1330.34074
[34] 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 Eng. 201 (2017), 578-583. DOI 10.1016/j.proeng.2017.09.634
[35] Tsypkin, Ya. Z.: Relay Control Systems. Cambridge University Press, Cambridge (1984). MR 0789077 | Zbl 0571.93001
[36] Varigonda, S., Georgiou, T. T.: Dynamics of relay relaxation oscillators. IEEE Trans. Autom. Control 46 (2001), 65-77. DOI 10.1109/9.898696 | MR 1809466 | Zbl 1004.34034
[37] Vasquez-Beltran, M. A., Jayawardhana, B., Peletier, R.: Recursive algorithm for the control of output remnant of Preisach hysteresis operator. IEEE Control Syst. Lett. 5 (2021), 1061-1066. DOI 10.1109/LCSYS.2020.3009423 | MR 4211636
[38] Visintin, A.: Differential Models of Hysteresis. Applied Mathematical Sciences 111. Springer, Berlin (1994). DOI 10.1007/978-3-662-11557-2 | MR 1329094 | Zbl 0820.35004
[39] Visintin, A.: Ten issues about hysteresis. Acta Appl. Math. 132 (2014), 635-647. DOI 10.1007/s10440-014-9936-6 | MR 3255072 | Zbl 1305.74072
[40] Visintin, A.: P.D.E.s with hysteresis 30 years later. Discrete Contin. Dyn. Syst., Ser. S 8 (2015), 793-816. DOI 10.3934/dcdss.2015.8.793 | MR 3356462 | Zbl 1304.35357
[41] 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. DOI 10.1134/S000511791506003X | MR 3374789 | Zbl 1327.93225
[42] 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. DOI 10.1007/s11253-018-1566-0 | MR 3863943 | Zbl 1417.34098
[43] Yevstafyeva, V. V.: Existence of $T/k$-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvalue. Math. Notes 109 (2021), 551-562. DOI 10.1134/S0001434621030238 | MR 4236227 | Zbl 1472.34081
[44] Yevstafyeva, V. V.: Existence of two-point oscillatory solutions of a relay nonautonomous system with multiple eigenvalue of a real symmetric matrix. Ukr. Math. J. 73 (2021), 746-757. DOI 10.1007/s11253-021-01957-4 | MR 4466489 | Zbl 1483.34061
[45] Yevstafyeva, V. V.: On the existence of two-point oscillatory solutions of a perturbed relay system with hysteresis. Differ. Equ. 57 (2021), 155-164. DOI 10.1134/S001226612102004X | MR 4237004 | Zbl 1464.34063
[46] Yu, C.-C.: Autotuning of PID Controllers: A Relay Feedback Approach. Advances in Industrial Control. Springer, Berlin (1999). DOI 10.1007/b137042 | Zbl 0962.93004
Search
Partner of
