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Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects

K. A. Morris, A. O. Ozer

arXiv:1312.0260v1math.AP

TL;DR

Piezoelectric beam models commonly omit magnetic effects, although magnetic effects alter the system’s stabilization behavior. This paper derives a well-posed model that includes them and finds strong but generally non-exponential stabilization under electrical feedback, with exponential stability confined to a zero-measure parameter set.

  • Problem

    Most piezoelectric beam models omit magnetic effects, and the paper investigates how including them affects observability and stabilizability.

    Source: §1 · p. 2, §5 · p. 23

  • Method

    A variational approach derives a piezoelectric beam model with magnetic effects, using a voltage-driven Lagrangian formulation.

    Source: §1 · p. 2, Lagrangian · p. 5

  • Results

    For almost all system parameters, current feedback yields strong stability but not exponential stabilization; exponential stability occurs only for a zero-measure parameter set.

    Source: §1 · pp. 2–3, §5 · p. 23

  • Takeaways & Limitations

    The electrical feedback corresponds to total electrode current, which is typically easier to measure than beam-end velocity.

    Source: §5 · p. 24

    The paper considers no damping.

    Source: §5 · p. 24

Abstract

from arXiv · show

Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable, and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proven that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These results are compared to those of a beam without magnetic effects.

1. Introduction.

Common piezoelectric beam models omit magnetic effects, while the paper studies a fully dynamic model that couples stretching and magnetic waves under voltage control. It finds that stabilization depends strongly on system parameters.

1. Introduction.

  • Electrostatic and quasi-static models omit magnetic effects and their coupling with electrical and mechanical effects.The introduction notes that magnetic effects are experimentally minor in the overall dynamics for polarized ceramics.Source: §1 · pp. 1–2
  • Without magnetic effects, the wave model is exactly observable and exponentially stable under mechanical boundary damping.The cited baseline feedback is V(t) = v_t(L,t).Source: §1 · p. 2
  • The fully dynamic model couples stretching and magnetic wave equations, with voltage control entering through one boundary condition.The control problem is one of simultaneous stabilization of two coupled wave equations.Source: §1 · p. 2
  • For almost all system parameters, electrical current feedback gives strong stability, but the system is not exponentially stabilizable in the energy space.Exponential stabilization is possible only for a set of parameters of zero Lebesgue measure.Source: §1 · pp. 2–3

2. Piezoelectric beam model with magnetic effects.

The beam model starts from a linear piezoelectric constitutive law and reduces it using transverse-isotropy, thin-beam, and Euler-Bernoulli assumptions. The resulting formulation distinguishes mechanical, electrical, and magnetic energy.

2. Piezoelectric beam model with magnetic effects.

  • The constitutive relationship is presented as a widely used linear model for piezoelectric beams.Source: §2 · p. 3
  • Under transverse isotropy and polarization in the x3 direction, the constitutive matrices reduce to a smaller set of beam variables.The beam is assumed thin, forces in the x2 direction are zero, and T33 is also set to zero.Source: §2 · p. 4
  • With the Euler-Bernoulli assumption S13 = 0, the linear constitutive equations relate stress to strain and electric field, and electric displacement to field and strain.The equations include the elastic, piezoelectric, and dielectric coefficients.Source: §2 · p. 4
  • The formulation identifies kinetic, potential, electrical, and magnetic energies, with P − E + B termed electrical enthalpy.Source: Lagrangian · p. 4

Lagrangian.

The Lagrangian formulation is selected according to the electrical quantity prescribed at the electrodes. For voltage actuation, a Legendre transformation yields a formulation in which the applied voltage enters the external work term.

Lagrangian.

  • The choice between L and its transformed form depends on whether charge density/current density or voltage is prescribed at the electrodes.This paper controls voltage at the electrodes.Source: Lagrangian · pp. 4–5
  • For voltage-driven electrodes, a Legendre transformation of L yields a new Lagrangian in independent variables (S, D), with P + E as total stored energy.Source: Lagrangian · p. 5
  • The transformed constitutive relationship expresses stress and electric field in terms of strain and electric displacement.The parameters are redefined using elastic stiffness, piezoelectric coefficients, and dielectric constants.Source: Lagrangian · p. 5
  • The formulation incorporates magnetic energy through Maxwell’s equations and specifies the displacement field and external-force work terms.The listed external quantities include lateral and transverse forces and prescribed surface charge at the electrodes.Source: Lagrangian · p. 4, Magnetic effects · p. 6

Magnetic effects.

The paper uses a fully dynamic electromagnetic model rather than electrostatic or quasi-static approximations. It adds magnetic energy to the beam formulation while imposing simplifying assumptions on the field and charge distributions.

Magnetic effects.

  • The model imposes voltage-related electrical boundary conditions at the electrodes along with a chosen mechanical boundary condition at the beam edges.The text identifies charge and current as alternative electrical boundary conditions and selects the voltage-driven case.Source: Magnetic effects · p. 6
  • Electrostatic modeling sets B and displacement current to zero, whereas quasi-static modeling permits them but omits some magnetic effects.Source: Magnetic effects · p. 6
  • The fully dynamic approach retains the magnetic potential and its time derivative, unlike the quasi-static approach, which sets them to zero.The paper assumes no external body charges or body currents and reduces the magnetic field to its x1-dependent component B2.Source: Magnetic effects · p. 7
  • The magnetic energy is included in the stored-energy formulation alongside potential, electrical, and kinetic energy.Source: Magnetic effects · p. 7, Hamilton’s Principle · p. 8
  • The model assumes D3 is constant through the beam thickness, consistent with an electric potential linear in that direction.The total charge at x is identified as p, with p_x = D3.Source: Magnetic effects · p. 7

Hamilton’s Principle.

Hamilton’s principle yields equations for stretching and bending with associated boundary conditions. Because voltage affects only stretching, the analysis that follows considers the stretching equations.

Hamilton’s Principle.

  • The variation of the Lagrangian under admissible displacements yields separate stretching and bending equations with associated boundary conditions.The displacements considered are v, w, and p.Source: Hamilton’s Principle · p. 8
  • The stretching problem specifies boundary conditions at both beam ends and initial data for displacement and velocity.The boundary conditions include the applied voltage V(t); the initial data are given for v and p.Source: Hamilton’s Principle · p. 8
  • The bending equation is the Rayleigh beam equation; neglecting its moment-of-inertia term gives the Euler-Bernoulli equation.Mindlin-Timoshenko assumptions change the bending and rotation equations but yield the same stretching equation.Source: Hamilton’s Principle · p. 9
  • Bending and rotation decouple from stretching, and voltage affects only stretching, so the paper subsequently studies the stretching equations.The same stretching equation is obtained under Euler-Bernoulli and Mindlin-Timoshenko assumptions.Source: Hamilton’s Principle · p. 9
  • With static magnetic effects, eliminating the electrical variable recovers the classical single-beam stretching equation; the paper assumes positive coupling and magnetic parameters.The classical model is described as exactly observable and stabilizable.Source: Hamilton’s Principle · p. 9

3. Well-posedness. Define

The paper formulates the beam as an operator control system and establishes well-posed input-to-state and input-to-output maps. It also gives operator properties and an alternative port-Hamiltonian representation.

3. Well-posedness.

  • The operator A is positive and self-adjoint, and the state-space construction defines fractional-domain spaces and their duals.For θ ≥ 0, Xθ is defined as Dom(Aθ); X−θ is its dual with respect to X.Source: §3 · p. 10
  • The system operator is skew-adjoint, and its compact resolvent implies a spectrum consisting of isolated eigenvalues.The proof establishes skew-adjointness and uses compactness of the operator domain embedding.Source: §3 · p. 12
  • The control and observation operators are treated as admissible, linking bounded observations to admissible control.The paper states that admissibility of the observation operator is equivalent to admissibility of the control operator.Source: §3 · p. 12
  • For any finite time horizon, the maps from voltage input in L2 to the state in H and to the output are bounded.The theorem states existence of positive bounds for these maps.Source: §3 · p. 13
  • With an alternative state definition, the control system is well-posed on [L2(0, L)]4 and is a port-Hamiltonian system.The passage presents this as an alternative formulation.Source: §3 · p. 13

Damped system.

The paper analyzes a damped system driven by current feedback and shows that strong stability depends on the system parameters. Imaginary-axis modes occur at simultaneous odd half-wave resonances; otherwise the controlled system is strongly stable.

Damped system.

  • The damped formulation is a well-posed conservative linear system with a contraction semigroup and admissible control and observation operators.Its transfer function is bounded by 1 in the open right half-plane.Source: Damped system · p. 14
  • The analysis reduces strong stability to ruling out nontrivial eigenmodes on the imaginary axis.The proof applies the Arendt-Batty stability theorem once such eigenvalues are excluded.Source: Damped system · p. 15
  • Strong stability fails when both modal frequencies equal odd half-wave values; if either differs from those values, the controlled system is strongly stable.The resonant values are a1 = (2n + 1)π/(2L) and a2 = (2m + 1)π/(2L).Source: Damped system · p. 17
  • The current-feedback system is strongly stable unless its parameters fall in an exceptional set of Lebesgue measure zero.The feedback is V(t) = k ṗ(L, t), with k > 0.Source: Damped system · p. 17

4. Exact observability and exponential stabilizability.

This section links exact observability to exponential stability and characterizes parameter regimes where the magnetic beam does or does not meet that condition. It contrasts those regimes with the beam model without magnetic effects.

4. Exact observability and exponential stabilizability.

  • Exact observability of (A, B∗) in finite time is equivalent to exponential stability of the semigroup, and optimizability of (A, B) is equivalent to exact observability.Source: §4 · p. 18
  • For irrational ζ1, (A, B∗) is not exactly observable, so (A, B) cannot be exponentially stabilized on H.Source: §4 · pp. 20–21
  • When ζ1 is a ratio of odd integers, the system is not even approximately observable.Source: §4 · p. 21
  • When ζ1 is a reduced ratio with numerator and denominator of opposite parity, the system is exponentially stabilizable on H.The pair (A, B∗) is exactly observable in this parameter case.Source: §4 · p. 23
  • Without magnetic effects, the beam is exactly observable and exponentially stabilizable by −B∗.Source: §5 · p. 23

5. Conclusions.

The −B* electrical feedback stabilizes the beam strongly for irrational ζ1, but not exponentially; the feedback measures total electrode current rather than beam-end velocity. The conclusions also note that damping and piezoelectric patch placement and system parameters affect the broader modeling and control context.

5. Conclusions.

  • For a coprime odd/even parameter ratio, −B* stabilizes the beam; for irrational ζ1, it yields strong but not exponential stabilization.Explicit polynomial estimates for the irrational case are reported in [23].Source: §5 · p. 23
  • With magnetic effects, B* measures total electrode current, typically easier to measure than beam-end velocity in models without magnetic effects.Source: §5 · pp. 23–24
  • At high frequencies, voltage actuation exhibits hysteresis, while current and charge actuation lead to much less hysteresis [7, e.g.; 10, 11, 22].The current-control model in is quite different, with a bounded control operator.Source: §5 · p. 24
  • No damping is considered; adding damping would make the system stable, while B* would remain electrical and control effectiveness restricted.Source: §5 · p. 24
  • For piezoelectric patches, prior work suggests controllability and stabilizability depend on both patch location and system parameters.This dependence is noted as an ongoing research topic.Source: §5 · p. 24
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