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Integral control of infinite-dimensional systems in thepresence of hysteresis: an input-output approach

Published online by Cambridge University Press:  05 June 2007

Hartmut Logemann ,
Eugene P. Ryan  and
Ilya Shvartsman
Hartmut Logemann
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK; [email protected]; [email protected]; [email protected]
Eugene P. Ryan
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK; [email protected]; [email protected]; [email protected]
Ilya Shvartsman
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK; [email protected]; [email protected]; [email protected]
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Abstract

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection ofeither (a) a hysteretic input nonlinearity, an L 2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2-stable, time-invariantlinear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certainconstant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

Keywords

Actuator nonlinearitieshysteresisinfinite-dimensionalsystemsinput-output analysisintegral controlsensornonlinearities
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 458 - 483
Copyright
© EDP Sciences, SMAI, 2007

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References

H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation, Control. Masson, Paris (1996).
M. Brokate, Hysteresis operators, in Phase Transitions and Hysteresis, A. Visintin Ed., Springer, Berlin (1994) 1–38.
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York (1996).
Byrnes, C.I., Gilliam, D.S., Shubov, V.I. and Weiss, G., Regular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Syst. 8 (2002) 341–370. CrossRef
Curtain, R.F., Logemann, H. and Staffans, O., Stability results of Popov-type for infinite-dimensional systems with applications to integral control. Proc. London Math. Soc. 86 (2003) 779–816. CrossRef
Fliegner, T., Logemann, H. and Ryan, E.P., Low-gain integral control of well-posed infinite-dimensional linear systems with input and output nonlinearities. J. Math. Anal. Appl. 261 (2001) 307–336.
Gorbet, R.B., Morris, K.A. and Wang, W.L., Passivity-based stability and control of hysteresis in smart actuators. IEEE Trans. Control Systems Technology 9 (2001) 5–16. CrossRef
B.Z. Guo and Z.C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Syst. Control Lett. 54 (2005) 1135–1142.
B.Z. Guo and Z.C. Shao, Regularity of an Euler-Bernoulli plate with Neumann control and colocated observation, J. Dynam. Control Syst. 12 (2006) 405–418.
B.Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim. 44 (2005) 1598–1613.
Ikhouane, F. and Rodellar, J., A linear controller for hysteretic systems. IEEE Trans. Auto. Control 51 (2006) 340–344. CrossRef
Ikhouane, F., Mañosa, V. and Rodellar, J., Adaptive control of a hysteretic structural system. Automatica 41 (2005) 225–231. CrossRef
Jönson, U., Stability of uncertain systems with hysteresis nonlinearities. Int. J. Robust Nonlinear Control 8 (1998) 279–293. 3.0.CO;2-N>CrossRef
M.A. Krasnosel'skii and A.V. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989).
Lasiecka, I. and Triggiani, R., The operator $B^*L$ for the wave equation with Dirichlet control. Abstract Appl. Anal. 2004 (2004) 625–634. CrossRef
H. Logemann and A.D. Mawby, Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, F. Colonius et al. Eds., Birkhäuser, Boston, Advances in Mathematical Systems Theory (2001) 255–293.
Logemann, H. and Mawby, A., Discrete-time and sampled-data low-gain integral control of infinite-dimensional linear systems in the presence of input hysteresis. SIAM J. Control Optim. 41 (2002) 113–140. CrossRef
H. Logemann and E.P. Ryan, Time-varying and adaptive integral control of infinite-dimensional regular systems with input nonlinearities, SIAM J. Control Optim. 38 (2000) 1120–1144.
Logemann, H. and Ryan, E.P., Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behaviour of solutions. ESAIM: COCV 9 (2003) 169–196. CrossRef
Logemann, H., Ryan, E.P. and Townley, S., Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain. IEEE Trans. Auto. Control 44 (1999) 1315–1319. CrossRef
R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica 39 (2003) 1555–1569.
D. Salamon, Control and Observation of Neutral Systems. Pitman, London (1984).
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. Cambridge University Press, Cambridge (2005).
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.
Tan, X. and Baras, J.S., Modeling and control of hysteresis in magnetostrictive actuators. Automatica 40 (2004) 1469–1480. CrossRef
G. Tao and P.V. Kokotović, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley, (1996)
Tucsnak, M. and Weiss, G., How to get a conservative well-posed system out of thin air, part II: Controllability and stability. SIAM J. Control Optim. 42 (2003) 907–935. CrossRef
Weiss, G., Transfer functions of regular linear systems, part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827–854.
Weiss, G. and Rebarber, R., Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2000) 1204–1232. CrossRef