Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-18T10:30:47.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Systems with hysteresis in the feedback loop: existence,regularity and asymptotic behaviour of solutions

Published online by Cambridge University Press:  15 September 2003

Hartmut Logemann  and
Eugene P. Ryan
Hartmut Logemann
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.; hl@maths.bath.ac.uk. epr@maths.bath.ac.uk.
Eugene P. Ryan
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.; hl@maths.bath.ac.uk. epr@maths.bath.ac.uk.
Get access

Abstract

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.

Keywords

Absolute stabilityasymptotic behaviourfrequency-domain stability criteriahysteresis infinite-dimensional systemsintegral equationsregularity of solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 169 - 196
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Brokate, Hysteresis operators, in Phase Transitions and Hysteresis, edited by A. Visintin. Springer-Verlag, Berlin (1994) 1-38.
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer-Verlag, New York (1996).
C. Corduneanu, Almost Periodic Functions, 2nd Edition. Chelsea Publishing Company, New York (1989).
R.F. Curtain, H. Logemann and O. Staffans, 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
R.F. Curtain and G. Weiss, 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.
G. Gripenberg, S.-O. Londen and O.J. Staffans, Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990).
W. Hahn, Stability of Motion. Springer-Verlag, Berlin (1967).
M.A. Krasnosel'skii and A.V. Pokrovskii. Systems with Hysteresis. Springer-Verlag, Berlin (1989).
H. Logemann and A.D. Mawby, 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.
Logemann, H. and Ryan, E.P., Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities. SIAM J. Control Optim. 38 (2000) 1120-1144. CrossRef
Macki, J.W., Nistri, P. and Zecca, P., Mathematical models for hysteresis. SIAM Rev. 35 (1993) 94-123. CrossRef
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
Salamon, D., Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. CrossRef
Salamon, D., Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431.
O.J. Staffans, Well-Posed Linear Systems. Book manuscript (in preparation). Available at http://www.abo.fi/ staffans/
Staffans, O.J., J-energy preserving well-posed linear systems. Int. J. Appl. Math. Comput. Sci. 11 (2001) 1361-1378.
Staffans, O.J., Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. CrossRef
Staffans, O.J. and Weiss, G., Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. CrossRef
M. Vidyasagar, Nonlinear Systems Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, NJ (1993).
Weiss, G., Transfer functions of regular linear systems, Part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854.
G. Weiss, 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.
Yakubovich, V.A., 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).