Construction and Experimental Implementation of a
Model-Based Inverse Filter to Attenuate Hysteresis
in Ferroelectric Transducers
Andrew G. Hatch and Ralph C. Smith Tathagata De and Murti V. Salapaka
Department of Mathematics Electrical Engineering Department
Center for Research in Scientific Computation Iowa State University
North Carolina State University Ames, IA 50011
Raleigh, NC 27695 tatha@iastate.edu, murti@iastate.edu
rsmith@eos.ncsu.edu, aghatch@eos.ncsu.edu
Abstract
Hysteresis and constitutive nonlinearities are inherent properties of ferroelectric transducer ma-
terials due to the noncentrosymmetric nature of the compounds. In certain regimes, these effects
can b e mitigated through restricted input fields, charge- or current-controlled amplifiers, or feed-
back designs. For general operating conditions, however, these properties must be accommodated
in mo dels, transducer designs, and model-based control algorithms to achieve the novel capabilities
provided by the compounds. In this paper, we illustrate the construction of inverse filters, based on
homogenized energy models, which can be used to approximately linearize the piezoceramic trans-
ducer behavior for linear design and control implementation. Attributes of the inverse filters are
illustrated through numerical examples and experimental open loop control implementation for an
atomic force microscope stage.
i
1 Introduction
Ferroelectric materials, including the compound lead zirconate titanate (PZT), exhibit novel actuator
and sensor capabilities due to the unique electromechanical coupling which they exhibit. This pro-
vides them with the capability for providing broadband transduction and nanometer-level set point
accuracy. Furthermore, piezoelectric transducers are moderately inexp ensive and can be designed to
minimally affect the passive dynamics of underlying structures. However, the noncentrosymmetric
ion structure that imbues the materials with unique actuator and sensor properties also produces
hysteresis and constitutive nonlinearities at all drive levels.
To illustrate, consider the prototypical atomic force microscope (AFM) stage depicted in Fig-
ure 1(a) which employs stacked piezoceramic actuators to position the sample in the x and y di-
rections. An additional PZT mechanism provides transverse p ositioning capabilities. Nested minor
loops collected at 0.1 Hz are plotted in Figure 1(b) and data collected at frequencies ranging from
0.279 Hz to 27.9 Hz is plotted in Figure 2 to illustrate the frequency-dependent nature of the hys-
teresis inherent to field-displacement data.
At low frequencies, the inherent hysteresis can be accommodated through proportional-integral-
derivative (PID) or robust control designs [5, 6, 15, 19]. However, at the higher frequencies required
for applications ranging from real-time monitoring of biological processes — e.g., protein unfolding
— to comprehensive product diagnostics, increasing noise-to-data ratios and diminishing high-pass
characteristics of control filters preclude a sole reliance on feedback laws to eliminate hysteresis.
Alternatively, it is illustrated in [12, 13], that use of charge- or current-controlled amplifiers can
essentially eliminate hysteresis. However, this mode of operation can be prohibitively expensive when
compared with the more commonly employed voltage-controlled amplifiers, and current control is
ineffective if maintaining DC offsets as is the case when the x-stage of an AFM is held in a fixed
position while a sweep is performed with the y-stage.
This motivates the development of models and model-based control designs which incorporate
and accommodate the hysteresis and constitutive nonlinearities. Numerous approaches have been
employed to characterize these nonlinear effects including Preisach models [7, 18], domain wall mod-
els [25, 26], micromechanical models [4, 10, 11], mesoscopic energy relations [3, 9] and homogenized
energy models [23, 30]. We employ the homogenized energy framework due to its energy basis, its
capability to q uantify a wide range of physical phenomena and operating regimes, its unified nature
0 2000 4000 6000 8000
−6
−4
−2
0
2
4
x 10
−5
Electric Field (V/m)
Displacement (m)
(a) (b)
Figure 1: (a) PZT-based AFM stage, and (b) nested minor loops in data collected at 0.1 Hz.
1
0 2000 4000 6000 8000
−5
0
5
x 10
−5
Electric Field (V/m)
Displacement (m)
0 2000 4000 6000 8000
−5
0
5
x 10
−5
Electric Field (V/m)
Displacement (m)
0 2000 4000 6000 8000
−5
0
5
x 10
−5
Electric Field (V/m)
Displacement (m)
(a) (b) (c)
Figure 2: Frequency-dependent field-displacement behavior of a stacked PZT stage: sample rates of
(a) 0.279 Hz, (b) 5.58 Hz, and (c) 27.9 Hz.
for characterizing hysteresis in ferroelectric, ferromagnetic and ferroelastic compounds [28, 29], and
the potential it provides for real-time implementation. Details regarding the development of this
modeling framework and its relation to other characterization techniques can be found in [21, 30].
Model-based control design for piezoceramic transducers operating in highly hysteretic and non-
linear regimes can be roughly segregated into two categories: (i) nonlinear control designs, and (ii)
linear control designs employing nonlinear inverse filters. Examples of the first technique in the
context of optimal control design for smart material transducers are provided in [20,32]. The second
technique is based on the concept of employing either an exact or approximate inverse model to lin-
earize the transducer behavior in the manner depicted in Figure 3. This approach has been employed
with a variety of models and control designs — e.g., see [31] for details regarding the development
of adaptive control designs utilizing piecewise linear Preisach models and their inverses — and is the
technique which we focus on in this paper.
In Section 2 we summarize constitutive relations developed in [21, 30] for ferroelectric materi-
als and provide a highly efficient algorithm for implementing the model when thermal relaxation
is negligible. A corresponding inverse polarization-field algorithm is summarized in Section 3 and
illustrated through a numerical example. The constitutive model is subsequently employed in Sec-
tion 4 to develop a lumped model for the stacked actuator employed in the AFM stage shown in
Figure 1(a) to illustrate the construction of a macroscopic transducer model. The accuracy of the
transducer model is illustrated through a comparison with the frequency-dependent data plotted
in Figure 2. In Section 5, an algorithm for the inverse displacement-field relation to linearize the
transducer response is developed and, in Section 6, the algorithm is experimentally implemented
as an inverse filter for the open loop tracking of a triangular input signal. It is demonstrated that
this model-based filter design effectively linearizes the nonlinear and hysteretic transducer dynamics
and provides an approximately tenfold increase in accuracy at higher frequencies as compared with
uvu
d
Filter
Inverse
Actuator
Physical
Figure 3: Use of an inverse filter to linearize the response u of a hysteretic actuator to achieve a
desired output u
d
.
2
the unfiltered case. This significantly improves the accuracy of the transducer and diminishes the
sole reliance on feedback laws whose authority decrease as tracking speeds and noise-to-data ratios
increase.
2 Constitutive Relations
Constitutive relations quantifying the electromechanical behavior of piezoceramic materials are con-
structed in two steps. In the first, Helmholtz and Gibbs energy relations at the lattice level are used
to characterize the local field-polarization and field-strain behavior of ferro electric compounds for
thermally inactive and active operating regimes. In the second step of the development, material
nonhomogeneities and variable effective field effects are incorporated through the assumption that
certain material properties are manifestations of underlying distributions rather than constants. This
yields low-order macroscopic constitutive relations which are efficient to implement.
2.1 Local Constitutive Relations
Let E,P,ε and σ respectively denote the electric field, polarization, strain and stress. It is illustrated
in [30] that an appropriate formulation of the Helmholtz energy for fixed temperatures in the absence
of stresses is
ψ
P
(P )=
1
2
η(P + P
R
)
2
,P≤−P
I
1
2
η(P − P
R
)
2
,P≥ P
I
1
2
η(P
I
− P
R
)
³
P
2
P
I
− P
R
´
, |P | <P
I
.
As shown in Figure 4, P
I
is the positive inflection point which delineates the transition between
stable and unstable regions, P
0
denotes the unstable equilibrium, and P
R
is the value of P at which
the positive local minimum of ψ occurs. The parameter η is the reciprocal of the slope of the E-P
relation after switching occurs. This fact can b e used to establish an initial parameter value for η
when modeling a specific data set.
G
(E
,P)
2
G (E
1
,P)
ψ
(P)=G(0,P)
R
(a)
0
R
I
R
R
c
I
(b)
P
E
P
−
P
P
−
P
P
P
P
P
P
P
P
P
E
E
E
Figure 4: (a) Helmholtz energy ψ and Gibbs energy G for σ = 0 and increasing fields E. (b) Switch
in the local p olarization
P that occurs as E is increased beyond the local coercive field E
c
given by
(5) in the absence of thermal activation.
3
The corresponding Gibbs energy relation
G
P
(E,P)=ψ
P
(P ) − EP (1)
incorporates the electrostatic energy due to the applied field E when σ =0.
Elastic effects and electromechanical coupling are incorporated in the Helmholtz energy relation
ψ(P, ε)=ψ
P
(P )+
1
2
Yε
2
− a
1
εP − a
2
εP
2
.
The Gibbs energy is then given by
G(E,σ,P,ε)=ψ
P
(P )+
1
2
Yε
2
− a
1
εP − a
2
εP
2
− EP − σε (2)
where σε incorporates the elastic energy. Note that Y denotes the Young’s modulus and a
1
,a
2
are
ferroelastic coupling coefficients associated with linear piezoelectric and quadratic electrostrictive
effects.
Polarization Kernel — Negligible Thermal Activation
In the case of negligible thermal activation, the local average polarization kernel
P is determined
from the necessary conditions
∂G
∂P
=0,
∂
2
G
∂P
2
> 0.
Applying these conditions to (1) yields the piecewise linear E-P relation
P (E)=
1
η
E + P
R
δ (3)
where δ = −1 for negatively oriented dipoles and δ = 1 for those with positive orientation. To specify
δ, and hence
P , more specifically in terms of the initial dipole orientations and previous switches,
we employ Preisach notation and take
[
P (E; E
c
,ξ)](t)=
[
P (E; E
c
,ξ)](0) ,τ(t)=∅
E
η
− P
R
,τ(t) 6= ∅ and E(max τ (t)) = −E
c
E
η
+ P
R
,τ(t) 6= ∅ and E(max τ (t)) = E
c
.
(4)
Here
[
P (E; E
c
,ξ)](0) =
E
η
− P
R
,E(0) ≤−E
c
ξ,−E
c
<E(0) <E
c
E
η
+ P
R
,E(0) ≥ E
c
defines initial kernel values in terms of the parameter ξ =
E
0
η
± P
R
, ∅ designates the empty set, and
the set of switching times is given by
τ(t)={t
s
∈ (0,t] | E(t
s
)=−E
c
or E(t
s
)=E
c
}.
The local coercive field
E
c
= η(P
R
− P
I
)(5)
quantifies the field at which the negative well ceases to exist and hence a dipole switch occurs. To
illustrate, the condition τ 6= ∅ and E(max τ (t)) = E
c
designates that switching has occurred and the
last switch was at E
c
; hence the local polarization is [P (E; E
c
,ξ)](t)=
E(t)
η
+ P
R
4
