Passivity of Magnetostrictive Materials

Abstract

Magnetostrictive materials display large force and displacement in response to an applied field, as well as short response time. However, their nonlinear and hysteretic behavior has hindered their use. We prove, using the physics of the material, that these materials are passive. The corresponding energy storage function is shown to be the Helmholtz energy. This result is independent of the model used. The effect of varying load is included. Passivity is important because it can be used to obtain control systems that maintain stability despite uncertainties and disturbances. The minima of the storage function are also obtained. The storage function is written explicitly in the case of a common model for these materials, the Preisach model.

Stability and robust position control of hysteretic systems

“…Similarly, the assumptions that u(0) = 0 and |y(0)| r ∞ are simply requirements on the initial settings in the system to ensure that large initial conditions do not affect the inputoutput gain. For the Preisach models, the assumption u(0) = 0 implies that the hysteretic system is in a state of minimum energy [8,9]. The condition |y(0)| r ∞ can be easily satisfied by y(0) = 0.…”

“…This result is used to establish L 2 -stability of a velocity controller. In [9], a physics-based argument is used to prove the passivity of a magnetostrictive actuator. In this proof, no specific model is used and the results apply to any hysteresis model for magnetostrictive actuators.…”

On control of voltage-actuated piezoelectric beam: A Krasovskii passivity-based approach

“…The expressions (11) and (12) are used in writing the energies (6) in scalar form applicable to the considered voltage actuated piezoelectric actuator. The explicit forms of the energies (6) of a voltage-actuated piezoelectric actuator are as follows,…”

“…In [8], a method which compensates for the ferroelectric hysteretic behaviour by calculating an inverse operator based on the Prandtl-Ishlinski theory is proposed. Another approach, based on passivity, can be found in [9]- [11] and [12] for Preisach hysteresis models. In [13], the inverse operator based on the Prandtl-Ishlinski theory is adjusted to compensate for the non-linear creep phenomena.…”

Load-dependent hysteresis of magnetostrictive materials

“…The parameter γ 4 is a constant, Y is the Young's modulus at a constant magnetization, ε is strain, σ is stress, and γ 1 (σ) and γ 2 (σ) are functions of stress depending on the material. As detailed in [14,15], the Gibbs energy is minimized when the dipole is in equilibrium. Since the dipole dynamics are very fast, it can be assumed that the dipoles are always in equilibrium and hence, the Gibbs energy is minimized.…”