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Modeling, identification and compensation of complex hysteretic nonlinearities: a modified Prandtl-Ishlinskii approach. (English) Zbl 1293.93213
Summary: Undesired complex hysteretic nonlinearities are present to varying degree in virtually all smart material based sensors and actuators provided that they are driven with sufficiently high amplitudes. This necessitates the development of purely phenomenological models which characterize these nonlinearities in a way which is sufficiently accurate, robust, amenable to control design for nonlinearity compensation and efficient enough for use in real-time applications. To fulfill these demanding requirements, the present paper describes a new compensator design method for invertible complex hysteretic nonlinearities which is based on the so-called modified Prandtl-Ishlinskii hysteresis operator. The parameter identification of this model can be formulated as a quadratic optimization problem which produces the best $$L_2^2$$-norm approximation for the measured input-output data of the real hysteretic nonlinearity. Special linear inequality and equality constraints for the parameters guarantee the unique solvability of the identification problem and the invertibility of the identified model. This leads to a robustness of the identification procedure against unknown measurement errors, unknown model errors and unknown model orders. The corresponding compensator can be directly calculated and thus efficiently implemented from the model by analytical transformation laws. Finally, the compensator design method is used to generate an inverse feedforward controller for a magnetostrictive actuator. In comparision to the conventional controlled magnetostrictive actuator the nonlinearity error of the inverse controlled magnetostrictive actuator is lowered from about 50% to about 3%.

##### MSC:
 93B30 System identification 93C10 Nonlinear systems in control theory 34C55 Hysteresis for ordinary differential equations
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##### References:
 [1] Adriaens, H.; de Koning, W.L.; Banning, R., Mixed feedback/inverse control of piezo-actuated systems, (), 403-408, vol 1. pp [2] Banks, H.T.; Smith, R.C., Hysteresis modeling in smart material systems, J appl mech eng, 5, 1, 31-45, (2000) · Zbl 0959.74044 [3] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, (1996), Springer Berlin-Heidelberg-NewYork · Zbl 0951.74002 [4] Dimmler, M.; Holmberg, U.; Longchamp, R., Hysteresis compensation of piezo actuators, (), (CD-ROM) [5] Ge, P.; Jouaneh, M., Generalized preisach model for hysteresis nonlinearity of piezoceramic actuators, J precision eng, 20, 99-111, (1997) [6] Goldfarb, M.; Celanovic, N., Modeling piezoelectric stack actuators for control of micromanipulation, IEEE control systems, 69-79, (1997) · Zbl 0900.93029 [7] Krasnosel’skii, M.A.; Pokrovskii, A.V., Systems with hysteresis, (1989), Springer Berlin [8] Krejci, P., Hysteresis convexity and dissipation in hyperbolic equations, Gakuto int. series math. sci. appl, 8, (1996), Gakkotosho, Tokyo · Zbl 1187.35003 [9] Kuhnen, K.; Janocha, H., Adaptive inverse control of piezoelectric actuators with hysteresis operators, (), (CD-ROM) [10] Mayergoyz, I.D., Mathematical models of hysteresis, (1991), Springer New York · Zbl 0723.73003 [11] Papageorgiou, M., Optimierung, (1991), Oldenbourg-Verlag Miinchen, Wien [12] Schafer, J.; Janocha, H., Compensation of hysteresis in solid-state actuators, Sensors and actuators physical A, 49, 97-102, (1995) [13] Tao, G.; Kokotovic, P.V., Adaptive control of plants with unknown hysteresis, IEEE trans autom control, 40, 2, 200-212, (1995) · Zbl 0821.93047 [14] Visintin, A., Differential models of hysteresis, (1994), Springer Berlin-Heidelberg-New York · Zbl 0820.35004 [15] Webb, G.V.; Lagoudas, D.C.; Kurdila, A.J., Hysteresis modeling of SMA actuators for control applications, J intell mater systems struct, 9, 432-448, (1998)
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