# Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

Hartmut Logemann; Eugene P. Ryan; Ilya Shvartsman

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 3, page 458-483
- ISSN: 1292-8119

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topLogemann, Hartmut, Ryan, Eugene P., and Shvartsman, Ilya. "Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 458-483. <http://eudml.org/doc/249991>.

@article{Logemann2007,

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 of
either (a) a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L2-stable, time-invariant
linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain
constant 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.
},

author = {Logemann, Hartmut, Ryan, Eugene P., Shvartsman, Ilya},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Actuator nonlinearities; hysteresis; infinite-dimensional
systems; input-output analysis; integral control; sensor
nonlinearities; actuator nonlinearities; infinite-dimensional systems; sensor nonlinearities},

language = {eng},

month = {6},

number = {3},

pages = {458-483},

publisher = {EDP Sciences},

title = {Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach},

url = {http://eudml.org/doc/249991},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Logemann, Hartmut

AU - Ryan, Eugene P.

AU - Shvartsman, Ilya

TI - Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/6//

PB - EDP Sciences

VL - 13

IS - 3

SP - 458

EP - 483

AB -
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 of
either (a) a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L2-stable, time-invariant
linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain
constant 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.

LA - eng

KW - Actuator nonlinearities; hysteresis; infinite-dimensional
systems; input-output analysis; integral control; sensor
nonlinearities; actuator nonlinearities; infinite-dimensional systems; sensor nonlinearities

UR - http://eudml.org/doc/249991

ER -

## References

top- 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).
- C.I. Byrnes, D.S. Gilliam, V.I. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Syst.8 (2002) 341–370.
- R.F. Curtain, H. Logemann and O. Staffans, Stability results of Popov-type for infinite-dimensional systems with applications to integral control. Proc. London Math. Soc.86 (2003) 779–816.
- T. Fliegner, H. Logemann and E.P. Ryan, Low-gain integral control of well-posed infinite-dimensional linear systems with input and output nonlinearities. J. Math. Anal. Appl.261 (2001) 307–336.
- R.B. Gorbet, K.A. Morris and W.L. Wang, Passivity-based stability and control of hysteresis in smart actuators. IEEE Trans. Control Systems Technology9 (2001) 5–16.
- 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.
- F. Ikhouane and J. Rodellar, A linear controller for hysteretic systems. IEEE Trans. Auto. Control51 (2006) 340–344.
- F. Ikhouane, V. Mañosa and J. Rodellar, Adaptive control of a hysteretic structural system. Automatica41 (2005) 225–231.
- U. Jönson, Stability of uncertain systems with hysteresis nonlinearities. Int. J. Robust Nonlinear Control8 (1998) 279–293.
- M.A. Krasnosel'skii and A.V. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989).
- I. Lasiecka and R. Triggiani, The operator ${B}^{*}L$ for the wave equation with Dirichlet control. Abstract Appl. Anal.2004 (2004) 625–634.
- 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.
- H. Logemann and A. Mawby, 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.
- 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.
- H. Logemann and E.P. Ryan, Systems with hysteresis in the feedback loop: existence, regularity and asymptotic behaviour of solutions. ESAIM: COCV9 (2003) 169–196.
- H. Logemann, E.P. Ryan and S. Townley, Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain. IEEE Trans. Auto. Control44 (1999) 1315–1319.
- R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica39 (2003) 1555–1569.
- D. Salamon, Control and Observation of Neutral Systems. Pitman, London (1984).
- D. Salamon, 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).
- O.J. Staffans and G. Weiss, Transfer functions of regular linear systems, part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc.354 (2002) 3229–3262.
- X. Tan and J.S. Baras, Modeling and control of hysteresis in magnetostrictive actuators. Automatica40 (2004) 1469–1480.
- G. Tao and P.V. Kokotović, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley, (1996)
- M. Tucsnak and G. Weiss, 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.
- G. Weiss, Transfer functions of regular linear systems, part I: Characterization of regularity. Trans. Amer. Math. Soc.342 (1994) 827–854.
- G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim.39 (2000) 1204–1232.

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