An existence and regularity theorem is proved for integral equations of convolution type which contain hysteresis nonlinearities. On the basis of this result, frequency-domain stability criteria are derived for feedback systems with a linear infinite-dimensional system in the forward path and a hysteresis nonlinearity in the feedback path. These stability criteria are reminiscent of the classical circle criterion which applies to static sector-bounded nonlinearities. The class of hysteresis operators under consideration contains many standard hysteresis nonlinearities which are important in control engineering such as backlash (or play), plastic-elastic (or stop) and Prandtl operators. Whilst the main results are developed in the context of integral equations of convolution type, applications to well-posed state space systems are also considered.
Mots clés : absolute stability, asymptotic behaviour, frequency-domain stability criteria, hysteresis, infinite-dimensional systems, integral equations, regularity of solutions
@article{COCV_2003__9__169_0,
author = {Logemann, Hartmut and Ryan, Eugene P.},
title = {Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {169--196},
publisher = {EDP-Sciences},
volume = {9},
year = {2003},
doi = {10.1051/cocv:2003007},
mrnumber = {1957097},
zbl = {1076.45004},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2003007/}
}
TY - JOUR AU - Logemann, Hartmut AU - Ryan, Eugene P. TI - Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2003 SP - 169 EP - 196 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2003007/ DO - 10.1051/cocv:2003007 LA - en ID - COCV_2003__9__169_0 ER -
%0 Journal Article %A Logemann, Hartmut %A Ryan, Eugene P. %T Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2003 %P 169-196 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2003007/ %R 10.1051/cocv:2003007 %G en %F COCV_2003__9__169_0
Logemann, Hartmut; Ryan, Eugene P. Systems with hysteresis in the feedback loop : existence, regularity and asymptotic behaviour of solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 169-196. doi : 10.1051/cocv:2003007. http://www.numdam.org/articles/10.1051/cocv:2003007/
[1] , Hysteresis operators, in Phase Transitions and Hysteresis, edited by A. Visintin. Springer-Verlag, Berlin (1994) 1-38. | MR | Zbl
[2] and, Hysteresis and Phase Transitions. Springer-Verlag, New York (1996). | MR | Zbl
[3] , Almost Periodic Functions, 2nd Edition. Chelsea Publishing Company, New York (1989). | Zbl
[4] , and, Stability results of Popov-type for infinite-dimensional systems with applications to integral control, Mathematics Preprint 01/09. University of Bath (2001). Proc. London Math. Soc. (to appear). Available at http://www.maths.bath.ac.uk/MATHEMATICS/preprints.html | MR | Zbl
[5] and, Well-posedness of triples of operators in the sense of linear systems theory, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 41-59. | MR | Zbl
[6] , and, Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990). | MR | Zbl
[7] , Stability of Motion. Springer-Verlag, Berlin (1967). | MR | Zbl
[8] and. Systems with Hysteresis. Springer-Verlag, Berlin (1989). | Zbl
[9] and, Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, in Advances in Mathematical Systems Theory, edited by F. Colonius et al. Birkhäuser, Boston (2001) 255-293. | MR
[10] and, Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities. SIAM J. Control Optim. 38 (2000) 1120-1144. | MR | Zbl
[11] , and, Mathematical models for hysteresis. SIAM Rev. 35 (1993) 94-123. | MR | Zbl
[12] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl
[13] , Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | MR | Zbl
[14] , Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. | MR | Zbl
[15] , Well-Posed Linear Systems. Book manuscript (in preparation). Available at http://www.abo.fi/~staffans/
[16] , -energy preserving well-posed linear systems. Int. J. Appl. Math. Comput. Sci. 11 (2001) 1361-1378. | MR | Zbl
[17] , Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. | MR | Zbl
[18] and, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | Zbl
[19] , Nonlinear Systems Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, NJ (1993). | Zbl
[20] , Transfer functions of regular linear systems, Part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR | Zbl
[21] , The representation of regular linear systems on Hilbert spaces, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 401-416. | MR | Zbl
[22] , The conditions for absolute stability of a control system with a hysteresis-type nonlinearity. Soviet Phys. Dokl. 8 (1963) 235-237 (translated from Dokl. Akad. Nauk SSSR 149 (1963) 288-291). | MR | Zbl
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