1. 472 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
Smart Piezoelectric Film Sensors
for Structural Control
Anthony Faria Vaz, Member, IEEE, and Rafael Bravo
Abstract—This paper describes how structural control informa-
tion can be embedded in a smart piezoelectric film. The control
knowledge is imparted by shaping a piezoelectric film. Shaping in-
volves electrode etching and splicing together sections with varying
poling directions. The shaping can be used to implement a general
class of spatial filters. This technique is successfully demonstrated
in this work using an articulated flexible beam. The controller is
realized in terms of a pair of shaped piezoelectric films. One film
implements a spatial filter of the strain profile of the beam, and
the other implements a spatial filter of the strain rate profile. Elec-
tronic circuits for measuring piezoelectric film signals are analyzed
in terms of their fidelity, and their effect on controller performance.
Although other researchers have explored the concept of imparting
controller information in a shaped piezoelectric film, they have not
explicitly developed a methodology for coping with the noise char-
acteristics of these sensors. In this paper, an controller design
procedure is used to mitigate the effects of sensor noise, torque dis-
turbances, and the limitations of the instrumentation. The major
contribution of this paper is the integrated design methodology
which copes with sensor, actuator, and modeling limitations in a
practical structural control application.
Index Terms—Active vibration damping, flexible structure,
control, piezoelectric, polyvinylidene flouride.
I. INTRODUCTION
THE DESIGN of instrumentation for the control of large
flexible structures is particularly challenging. Flexible
structures have a high number of lightly-damped vibrational
modes.Inaccuratemodelingofthestructuraldynamicscanhavea
drastic effect on controller performance. Typically, the dynamics
of these structures are modeled using either partial differential
equations or finite element methods. The dominant vibrational
modesareidentifiedandafinitedimensionalmodelisdeveloped.
To make the design problem tractable, the model has as low an
order, as possible. Unfortunately, designing controllers based on
reduced order models of the dynamics can result in closed loop
instability. This is termed the “spill-over problem” [1], because
high order neglected modes can be excited and become unstable.
A modal control technique was originally proposed by
Mierovitch [2]. The dynamics of a structure are formulated
in terms of a high order state-space model of the vibrational
models. Mierovitch developed a controller by using a linear
quadratic Gaussian (LQG) design technique. The resulting
Manuscript received December 10, 2002; revised October 29, 2003. This
work was supported in part by the Canadian Space Agency under contracts
9F009-0-4140, 9F011-1-0924, 9F028-6-6162, and 9F028-5-5015, and in part
by the Natural Science and Engineering Research Council of Canada.
A. F. Vaz is with the Applied Computing Enterprises Inc., Mississauga, ON
L5V 1E5, Canada (e-mail: anthonyvaz@idirect.com).
R. Bravo is with the Department of Mechanical Engineering, Universidad del
Zulia, Maracaibo, Venezuela.
Digital Object Identifier 10.1109/TIM.2004.823302
controller has high order dynamics and requires a large number
of strain sensors. Although this approach is theoretically
powerful, it is impractical due to the large number of strain
sensors and the associated real-time processing requirements.
Piezoelectric film sensors can be used to greatly alleviate the
real-time processing requirements required for modal control
techniques. The piezoelectric films have the interesting property
that they can be shaped, by splicing and electrode etching, to
sense and control individual modes in a structure. This enables
a feedback controller to be realized in terms of a suitably shaped
film. In principle, the feedback can be infinite order, as it is only
limited by the fidelity with which the electrode can be etched.
Traditionally, controller design has been arrived at using a tem-
poralparadigmwherecontrollercomplexityismeasuredinterms
of dynamic order. Piezoelectric films allow control design to be
performedusingaspatialparadigm,wherecontrollercomplexity
is measured in terms of film size and electrode geometry.
The implementation of infinite order controllers using piezo-
electric films was first proposed by Miller and van Schoor [3].
They showed that a control law can theoretically be represented
as a convolution kernel of surface strain and strain rate. A
method of observing a single mode of a flexible structure using
piezoelectric films was first investigated by Lee and Moon
[4]. Lee and his coworkers [5] used modal sensors to actively
damp the first mode vibrations in a cantilevered flexible plate.
This work was extended by Miller and his coworkers [6], [7]
to implement spatial filters and travelling wave controllers.
None of these researchers explored the effect of electronic
instrumentation upon controller performance.
In this paper, we investigate the controller that results from a
superposition of modal filters. Furthermore a method of using
the substructure dynamics to analyze the function of piezolec-
tric films is discussed. This method extends the quasistatic
technique of [8] to deal with the case of shaped piezoelectric
films. Futhermore, we show that a design method can
be used to optimize the controller that results from a shaped
piezoelectric film. The relationship between electronic instru-
mentation and the electromechanical models of piezoelectric
behavior are examined in detail. The finite input impedance of
the instrumentation gives rise to a nonideal behavior that can be
mitigated through proper controller design. Piezoelectric films
are used to implement a controller for an articulated flexible
beam. Theoretically, a flexible beam has an infinite number of
vibrational modes that correspond to the eigenfunctions of the
Bernoulli–Euler partial differential equation. Accordingly, this
control problem is representative of the high order dynamics
of more complex structures. The dynamics can be expressed
in terms of a high order state space model with states that
correspond to modal amplitudes and rates.
0018-9456/04$20.00 © 2004 IEEE
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2. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 473
Fig. 1. Articulated flexible beam.
TABLE I
BEAM PROPERTIES
TABLE II
MOTOR PROPERTIES
II. CONFIGURATION OF ARTICULATED FLEXIBLE BEAM
The experimental configuration is illustated in Fig. 1. The
beam is rigidly attached to a bracket that is attached to the shaft
of an electric servomotor. The beam is rotated in the horizontal
plane by the servomotor. Gravity does not contribute to the dy-
namic equations of the beam because the beam is rigid in the
vertical direction and pliant in the horizontal direction. Accord-
ingly, the dynamic model needs to account for the relationship
between shaft torques and flexing of the beam in the horizontal
plane. A polyvinylidene fluoride (PVDF) film is bonded to ei-
ther side of the beam. The PVDF film is a piezoelectric that is
thin and flexible. The PVDF film is contoured in a manner that
implements a spatial filter of the vibrational modes of the beam.
This spatial filtering technique is discussed in more detail in a
later section. A computer based data acquistion system using
the Quanser multi-Q I/O board [9] is used to monitor all the
measureable signals of the experimental apparatus. The beam
properties are listed in Table I. The motor parameters are listed
in Table II. The parameters listed in the above tables are used in
Section III to derive the dynamic equations.
Fig. 2. Coordinate frames.
III. DYNAMICS OF AN ARTICULATED FLEXIBLE BEAM
In this section, the dynamic model of an articulated flexible
beam is derived. The coordinates frames associated with the
beam are illustrated in Fig. 2. Coordinate Frame 1 is fixed in
space; hence, it provides an inertial system of coordinates. It is
composed of the of position vectors , , and .
Coordinate frame 2 rotates with the shaft and is composed of
the of position vectors , , and . The vector is aligned
with . The vector is aligned with the centreline of the
undeflected beam. The angle between vectors and is the
shaft angle .
In Fig. 2, a differential element of the beam is denoted by .
It is located by where vector locates the position of
if the beam were not deflected, and vector gives the beam
deflection of . The deflections of the beam are assumed to be
small so that and are orthogonal. In frame 2 coordinates, , ,
and are represented as , , and
. The supercript denotes the transpose
operator. The variable is the position along the undeflected
beam, is the beam deflection as a function of position and
time , and is the shaft angle as a function of time .
The following coupled partial differential equations and
boundary conditions can be derived through the application of
Hamilton’s principle [10]
(1)
(2)
and (3)
where , is the shaft assembly
inertia, and is the linear beam density. The following equa-
tions must be satisfied due to the geometrical constraints of the
beam
and (4)
The coupled partial differential (1) and (2) can be solved
using eigenfunction analysis [11]. The solution can then
be expressed in terms of the eigenfunction expansion
(5)
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3. 474 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
TABLE III
BEAM EIGENVALUES AND MODAL FREQUENCIES
with mode shapes and mode amplitudes for
where is sufficiently large so that approximation
error is small; that is
The above approximation error approaches zero as . The
mode shapes and the mode amplitudes are solutions
of the following differential equations for
(6)
(7)
where is the corresponding eigenvalue and
Differential (7) with boundary conditions (3) and (4) has a gen-
eral solution of the form
(8)
where is an arbitrary coefficient. The eigenvalues are ob-
tained by solving for the roots of the following characteristic
equation
(9)
The above equation has an infinite set of eigenvalues
. The numerical solution for the first five eigen-
values of (9) and the corresponding modal frequencies are listed
in Table III.
The mode shapes are an orthogonal basis for a Hilbert space
when the inner product is defined. The inner product should
reflect the orthogonal nature of the mode shapes. It can be shown
that for mode indices
An inner product , which is consistent with the
orthogonal properties of the mode shapes, is defined as follows:
Note, for any distinct pair of mode shapes and ,
, . Although mode shapes can be
normalized using the inner product, we use a different pro-
cedure. We select the parameters so that the condition
is satisfied. This allows the amplitudes of
different modes to be compared in a consistent manner. Hence,
.
The modes shapes of the cantilever beam have the important
property that the second derivatives of the mode shapes are or-
thogonal
(10)
This property is essential for the construction of modal sensors
using piezoelectric films, which we discuss in a later section.
Substitute (5) into (1) and integrate over to obtain the fol-
lowing:
(11)
Substitute (5) into (2), integrate over , and then simplify with
(7) to obtain the following:
(12)
Equations (11) and (12) can be expressed in terms of the fol-
lowing matrix differential equation:
(13)
(see the equations at the bottom of the next page). The effects of
viscous damping of the servomotor friction and the beam struc-
tural damping can be included in a modified dynamic model
(14)
The matrix has the structure as seen in the equation at the
bottom of the next page, where of the viscous damping co-
efficient of the servomotor, and is the damping factor of mode
for . The value of is listed in Table II; it is
determined empirically from experiments using the servomotor
without the beam. The damping factors of the beam vibrational
modes are determined from strain gauge measurements when
the beam is cantilevered in a fixed test stand. Empirical results
show that the damping factor of the first mode (cf.
[12]). The higher order modes were assumed to have the same
damping factor.
IV. MODELLING PIEZOELECTRIC FILMS
Consider the rectangular piezoelectric film illustrated in
Fig. 3. Axis 1 and axis 2 define a plane which is parallel to the
film’s surface. The 3 axis is perpendicular to the film’s surface.
Axis 3 points in the opposite direction to the electric field used
to pole the piezoelectric film during its manufacture. Some
piezoelectric films, such as polyvinylidene flouride (PVDF),
are uniaxially stretched during the poling process. In such
a case, the 1 axis is oriented in the direction of the uniaxial
stretching. Piezoelectric film’s which are uniaxially stretched
exhibit properties which differ in the directions of the axes 1
and 2 (e.g., ).
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4. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 475
Fig. 3. Transverse effects for a piezoelectric film.
Films which are not uniaxially stretched, such as piezoce-
ramics, do not exhibit these differences. The transverse effects in
piezoelectric films are illustrated in Fig. 3. The film is assumed
to have a length along axis 1, a width along the axis 2, and
a thickness along the axis 3. A voltage is measured across
the upper and lower electrodes. The upper and lower electrodes
are assumed to have free charges of and , respectively.
A tensile force and an elongation displacement are mea-
sured along axis 1. If the film acts as an actuator, a voltage
is applied across the film; the induced polarization in the piezo-
electric generates a tensile force and an elongation . If
the film acts as a sensor, a tensile force is applied which
stretches the film by ; the induced polarization in the piezo-
electric causes an increase of free charges on the contacts which
generates a voltage through a capacitive effect.
Piezoelectric films can also be modeled in terms of voltage
and displacement [13]
(15)
where is interpreted as the film compliance at constant elec-
tric field intensity; is interpreted as the film capacitance at
constant strain; and , the ratio of film charge to displacement.
The above film parameters are defined as follows:
and
where is the Young’s modulus of the piezoelectric at con-
stant field intensity, is the permittivity of the piezoelectric at
a constant stress, and is the transverse piezoelectric charge
to stress ratio.
Equation (15) can be represented in terms of the electro-
mechanical models based on displacement and voltage. The
charge generated by stretching a piezoelectric film is modeled
by the circuit in Fig. 4(a). The circuit in Fig. 4(b) is obtained
by applying Thévenin’s theorem to the circuit in Fig. 4(a).
In Fig. 4(c), the relation between mechanical reaction force
and mechanical displacement is modeled by a spring of
stiffness . The net force exerted on the spring is the sum
of the mechanical tensile force and the piezolectric force
. It is clear from the inverted voltage source in the circuit
of Fig. 4(b) that the charge generated by the piezoelectric
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5. 476 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
Fig. 4. Electromechanical models based on displacement Y and voltage V .
TABLE IV
POLYVINYLIDENE FLOURIDE FILM PROPERTIES
effect causes a negative piezoelectric force in the mechanical
model of Fig. 4(c). Thus, the piezoelectric effect augments the
mechanical stiffness of the film.
The relevant properties for polyvinylidene flouride (PVDF)
film are summarized in Table IV. The parameters can be substi-
tuted into (15) to establish the relevant piezoelectric models for
a particular implementation using PVDF film. In Section V, a
method of using the geometry of the film to implement signal
processing functions is discussed.
V. SHAPED PIEZOELECTRIC FILMS
A shaped film is idealized as a rectangular piezoelectric film
which has a poling direction that varies throughout the film, and
electrodes that are etched into geometrical shapes. The prop-
erties of a shaped film are determined by the pattern of the
poling direction variation and the electrode geometry. A shaped
film can be constructed from several electrically connected con-
tiguous sections of piezoelectric films. The orientation of the
polling direction varies between sections. The piezoelectric sec-
tions are bonded to an electrical insulator which is bonded to the
beam. The inner electrodes are inaccessible. Copper tape leads
with adhesive epoxy are used to make electrical contact with the
inner electrodes. The outer electrodes can be etched into geo-
metrical patterns. This construction is illustrated in Fig. 5. Note
the composite structure is rectangular.
PVDF is used to construct shaped piezoelectric films due to
its flexibility. The shaping of PVDF films is carried out using a
multistep procedure. First, a computer program is used to gen-
erate a postscript file, the file is then printed on a laser printer to
produce actual size templates of the PVDF electrode geometry
[14]. Kapton adhesive tape templates are then cut using laser
printed templates followed by bonding to PVDF sheets. The
combined PVDF and template then has its exposed electrode re-
moved with acetone. The removal of the electrode eliminates the
short circuiting of the PVDF film. The silver ink on the upper
and lower PVDF electrodes tends to deform and short circuit
Fig. 5. Shaped piezoelectric film.
Fig. 6. Top view of piezoelectric film bonded to a flat beam.
the electrodes during the cutting process. The PVDF sections
are then bonded to the flexible beam using cynoacrylic epoxy.
Kapton tape is bonded to the flexible beam to serve as a sub-
strate for the PVDF sections, which are connected using copper
tape and conductive epoxy. The resulting sensor has the struc-
ture illustrated in Fig. 5.
Shaped films are analyzed in terms of variations of the poling
direction and domain of the outer electrodes. The functions used
to represent these quantities are expressed in terms of the co-
ordinate system of the beam to which they are attached. The
coordinate system of the beam is shown in Fig. 6. The , ,
coordinate frame has its origin on the neutral surface of the
beam. The beam has a length along the axis, a width
along the axis, and a thickness along the axis. The upper
and lower surfaces of the beam have coordinates of and
, respectively.
The shaped film is bonded to the beam as shown in Fig. 6.
The subscript is used to distinquish quantities associated with
the piezoelectric film. The film is situated between
and in the axis, and and in
the axis. The coordinate of the beam surface, to which the
film is bonded, is denoted by . If the beam is bonded to the
top surface, ; otherwise, for the lower
surface.
The bending strain of the flat beam shown in Fig. 6 varies
only in the and directions. Hence beam strain can be
represented as a function . Note the strains have the
following antisymmetric property: .
Accordingly, the strain in a cross section of the piezoelectric
film can represented by a function . The strain vari-
ations of in the direction are negligible due to the thin-
ness of the piezoelectric film. The piezoelectric film is assumed
to be perfectly bonded to the beam [15]; hence,
.
The properties of a shaped film are characterized in terms
a domain function and poling function . The
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6. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 477
domain function defines the region of a shaped film’s
etched electrode (see the first equation at the bottom of the
page). The poling direction is specified with respect to the elec-
tric field intensity associated with a particular electrode polarity.
The electric field intensity points from the positive electrode
to the negative electrode. Accordingly, the poling direction is
aligned to the electric field intensity if it points toward the neg-
ative electrode. The poling function specifies the vari-
ations of the poling direction throughout a shaped film (see the
second equation at the bottom of the page). These two functions
are fundamental to the understanding of interactions between
piezoelectric films and flexible structures.
The shape function of a shaped film is the average of
the poling function over a cross section with a fixed
coordinate. It is defined by
(16)
This function succinctly expresses the key property which de-
termines the operation of piezoelectric film sensors.
The capacitance of the film is given by
where is the permittivity of the piezoelectric, is the area
covered by the film’s electrodes, and is the film thickness.
When the shaped film is strained, an electric flux density is
created due to the polarization in the piezoelectric. The electric
flux varies in the direction, but not in the direction. Inte-
grating electric flux gives charge , given by
(17)
The above equation is derived by applying (15) and the circuit
model in Fig. 4(a) to an infinitesimal element. See [13], [15] for
the details.
For a given strain profile , the corresponding open
circuit voltage and the short circuit current are re-
spectively given by [15]
(18)
and
(19)
VI. SMART PIEZOELECTRIC SENSOR
The effect of vibrational modes upon the open circuit voltage
and short circuit current can be inferred from a modal expansion
of the strain profile [16]
(20)
Substitute (20) into (17), for polarization charge ; into (18),
for open circuit voltage ; and into (19), for short circuit cur-
rent . This yields
and
(21)
where the modal charge is
(22)
The ability to observe particular modes in a beam can be used
to implement a modal feedback control law in terms of a film’s
shape. Suppose a film is shaped so that
where
and
point is covered by both electrodes of the film;
otherwise.
poling direction at points towardthe electrode;
the film does not cover ;
poling direction at points toward the electrode.
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7. 478 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
The open circuit voltage is given by
(23)
which is proportional to the feedback
The short circuit current is given by
(24)
which is proportional to the feedback
This technique can be used to economize on the real-time com-
putations required to implement control algorithms. This was
first done by Lee and Moon [4] for the case . A computer-
aided fabrication procedure for shaping piezoelectric films is
discussed in [14], [17].
The piezoelectric films are inferfaced using the electronic cir-
cuits illustrated in Fig. 7. The circuits are constructed using a
LF351 operational amplifier, which has high impedance JFET
input nodes. In order to function properly, each input must have
a DC current path. The circuit in Fig. 7(a) is used to buffer the
open circuit voltage of the piezolective film. The nonin-
verting terminal has a resistor to provide a DC
current path. The circuit outputs a voltage that is a high pass
filtered approximation of the voltage . The transfer function
can be derived by replacing the piezoelectric film in Fig. 7(a)
with the equivalent circuit in Fig. 4(b) where the voltage source
is . This yields the following Laplace transfer function
(25)
The transfer function has zero at the origin; hence a piezoelectric
film cannot be used to measure static strain. The differential
equation related to (25) is given by the following:
(26)
For completeness, the state space model for control design
should incorporate the above dynamics. As in
circuit Fig. 7(a), in (26). This limit implies
that the input impedance of the buffer amplifier in Fig. 7(a)
should be as high as possible. Furthermore, the piezoelectric
film should also be as large as possible, as this increases the
value of .
The circuit in Fig. 7(b) is used to amplify the short circuit
current of the piezoelectric film . The circuit outputs a voltage
that is proportional to . In particular
(27)
Fig. 7. Electronic circuits for interfacing with piezoelectric film.
Fig. 8. Lower LFT representation.
VII. FEEDBACK CONTROLLER DESIGN
In this section, a controller is designed that enables the artic-
ulated beam to slew from one angle to another in a manner that
rapidly suppresses beam vibration. The controller computes the
weighted sum of four signals: the motor shaft encoder angle and
rate, and the open circuit voltage and the short circuit current
from a pair of shaped piezoelectric films. The controller output
is used to command the motor torque. The controller does not
incorporate any additional dynamics. The controller feedback
complexity is embodied in the shape functions of the piezoelec-
tric films. To ensure that the controller can be realized in terms
of shaped piezoelectric films, our design procedure restricts the
controller structure to state feedback.
The system can be written in lower linear fractional trans-
formation Form (LFT). This form enables the inclusion of the
effects of sensor and control disturbances in a simple manner.
A block representation of the lower LFT form is given in Fig. 8.
The signal is the control input, is the exogenous input, is
the output, and is the exogenous output. The system dynamics
can be written as follows:
(28)
The state space form results when for as defined
in (13). The objective of the methodology is to minimize
the norm of the closed loop transfer function from to .
In this work frequency weighted disturbances and noise signals
are not used as they would introduce loop shaping dynamics in
the controller. For our system, , and .
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8. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 479
TABLE V
DISTURBANCE AND NOISE MAGNITUDES USED FOR LMI OPTIMIZATION
The effects of piezoelectric film dynamics in (26) are treated
as disturbances in the LFT framework. This allows the controller
structure to be state feedback rather than decentralized output
feedback. The design problem for state feedback can be
solved simply using linear matrix inequality (LMI) optimiza-
tion, whereas output feedback requires ad hoc parameter opti-
mization [18].
The optimum state feedback controller that minimizes the
criterion can be solved using a LMI [18]. In particular, the
optimum state feedback is given by
(29)
where and are solutions to the following LMI [18]
[see (30) at the bottom of the page].
The vector is used to account for the uncertainty associ-
ated with the effect of the torque disturbances and sensor noise.
The vector is a 13-dimensional vector composed of the RMS
magnitudes of the torque disturbance, encoder angle error, en-
coder angle rate error, the five modal components of noise,
and the the five modal components of noise. The disturbance
and noise values are listed in Table V. The modal components
of noise are computed by
where
The modal components of noise are computed by
where
The modal noise components are listed in Table VI. The ma-
trices and are given by
and
TABLE VI
MODAL NOISE COMPONENTS USED FOR LMI OPTIMIZATION
The matrix is defined so that represents the torque dis-
turbance. The matrix is defined so that represents
the effect of sensor noise. The sensor noise increases dramati-
cally for higher modes; this has the effect of penalizing the use
of feedback from higher modes.
The control input is penalized slightly with a weight of 0.375.
If the control weight is too small, the resulting controller causes
excessive saturation of the control signal. If the weight is too
high, the resulting controller has a very sluggish response. Ma-
trix is given by
The matrix is defined so that represents the effect of
penalty on control signal .
The states are penalized as follows: a factor of 1 for the angle,
a factor of 1 for mode amplitudes one to five, a factor of 0.1 for
angular rate, and a factor of 0.1 for mode amplitude rates one to
five. This puts emphasis on causing the beam to straighten and
suppress the dominant vibrations. Matrix is given by
The matrix is defined so that represents the effect of
penalty on the output signal .
The LMI procedure yields the following controller
(31)
where
VIII. EXPERIMENTAL RESULTS
The signals and are implemented using an open circuit
and a short circuited PVDF film, respectively. The electrode pat-
terns of the shaped piezoelectric films are shown in Fig. 9. The
(30)
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9. 480 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
Fig. 9. Normalized piezoelectric film shapes for control design.
normalization parameters for the and films are respec-
tively , and . The film width
. The and signals are given by
(32)
and
(33)
The nonideal measurement with and
is obtained by substituting (32) into (26)
(34)
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10. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 481
Fig. 10. Simulated response to angular step.
The simulated closed loop step response is shown in Fig. 10.
The simulations were performed for both the ideal and nonideal
measurement of . The corresponding responses of the vibra-
tional modes are shown in Fig. 11. The responses due to the
nonideal measurement of differ imperceptibly from those
of the ideal measurement. The measurement error is illus-
trated in Fig. 12. The error has a peak value of 0.16 V this cor-
responds to a motor command error of 7.31 mV, which has a
neglible effect. Accordingly, we do not need to iterate the LMI
optimization with a higher value for noise.
The controls and of (31) are respectively implemented
using and PVDF films. The circuits shown in Fig. 7,
with and , were interfaced with
the piezoelectric films. To reduce electrical noise, the outer elec-
trode of each film and the beam is grounded. A computer with
a Quanser multi-Q I/O board [9] is used to measure the signals
from the encoder and the electronic circuits interfaced with the
PVDF films. The encoder signal corresponds to the motor angle.
The angular rate is computed by numerical differentiation. The
angle, angular rate, and the piezolectric film signals are scaled
and summed by the computer to generate the motor command
signal. Fig. 13 shows the response of the system to a step input.
The controller is capable of driving the beam while at the same
time stabilizing the oscillations with small deflections. Due to
the low saturation torque of the actuator employed, the oscilla-
tions of the beam are not suppressed until the gross angular error
is overcome, and the control signal returns to the nonsaturated
range.
The simulations on Fig. 10 closely resemble the experimental
responses on Fig. 13. The simulated shaft angle in Fig. 10(a)
has a similar profile to experimental shaft angle response in
Fig. 13(a). The experimental response has a small steady state
error due to the 4-mV dead band of the motor. The simulated
motor command signal in Fig. 10(b) has a similar profile to
the experimental motor command signal in Fig. 13(b). The ex-
perimental response has high frequency noise spikes that result
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11. 482 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
Fig. 11. Simulated response of vibrational modes to angular step.
Fig. 12. Simulated V measurement error.
from the gain associated the rate derived from numerical differ-
entiation of the shaft angle. The simulation of the piezoelec-
tric film signal in Fig. 10(c) has a similar profile to the
experimental signal in Fig. 13(c). The simulation of the
piezoelectric film signal in Fig. 10(d) has a similar profile
to the experimental signal in Fig. 13(d). Both the experi-
mental and signal are corrupted by high frequency noise
and a small component of the high frequency modes. The high
noise is due to the high impedance circuitry that is susceptible
to ambient noise. It can be reduced with proper shielding. The
high frequency mode component is a result of small errors in
film shape that result from the manual fabrication process. De-
spite the noise and small shaping imperfections, the controller
realized with the piezoelectric films successfully damps vibra-
tions in the flexible beam. Furthermore, the signals of the ex-
perimental response closely match their simulations.
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12. VAZ AND BRAVO: SMART PIEZOELECTRIC FILM SENSORS 483
Fig. 13. Experimental response to angular step.
IX. CONCLUSION
Smart piezoelectric films have been shown to alleviate the
instrumentation and control problems associated with flexible
structures. High order modal feedback controllers for actively
damping vibrations in flexible structures were implemented by
smart piezoelectric films. The signal processing function of the
smart piezoelectric film was characterized in terms of its shape
function. The film implements a physical realization of the spa-
tial integral of strain, or strain rate, weighted by the shape func-
tion. Strain integration corresponds to open circuit voltage ,
and strain rate integration corresponds to short circuit current
of a piezoelectric film. The shape function was designed using a
synthesis procedure which accounts for uncertainty in dy-
namic modeling, disturbances, and sensor noise.
Simple electronic circuits could be used to measure both
and . The finite impedance of a buffer amplifier results
in a high pass filtered measurement of . This makes the
signal unsuitable for static control, but suitable for vibration
suppression. The piezoelectric film output signals and
are highly susceptible to noise. The effects of noise can be sig-
nificantly reduced by insulating the films from the substructure
using Kapton tape.
Smart films have been successfully used in this work to im-
plement a vibration controller for an articulated flexible beam.
The smart films allow the complexity of the controller hardware
to be dramatically simplified. The feedback for a tenth order
model of the vibration dynamics has been implemented exclu-
sively with a pair of smart piezoelectric films. Work is under
way to extend the current results to more complex structures
[19].
ACKNOWLEDGMENT
The authors would like to thank Dr. S. Yeung, Mr. H. Rey-
naud, and Dr. G. Vukovitch at the Canadian Space Agency for
their assistance. We appreciate the assistance of H. Wu in con-
structing the piezoelectric film sensors used for this research.
The authors would also like to thank the anonymous reviewers
for their insightful comments and helpful suggestions.
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13. 484 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 2, APRIL 2004
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Anthony Faria Vaz (S’80–M’81) received the
B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical
engineering from the University of Toronto, Toronto,
ON, Canada.
He is the President of Applied Computing Enter-
prises, Mississauga, ON, Inc., where he specializes
in system theoretic analysis and software develop-
ment. He has done consulting work on smart struc-
tures, signal processing, control systems, and elec-
trooptics for both the space and defense industries.
He is also the Director of Risk Analytics at the Cana-
dian Imperial Bank of Commerce (CIBC). He is responsible for developing
mathematical models for monitoring and controlling the risks of CIBC’s global
trading portfolio. His research interests include the development of signal pro-
cessing and control algorithms for a variety of applications.
Dr. Vaz is a Licenced Professional Engineer in the province of Ontario,
Canada.
Rafael Bravo received the Mechanical Engineering
Diploma from the Universidad del Zulia, Maracaibo,
Venezuela, in 1990 and the M.Eng. and Ph.D. degrees
in mechanical engineering from McMaster Univer-
sity, Hamilton, ON, Canada, in 1996 and 2000, re-
spectively.
He is currently a Postdoctoral Fellow with the
Mechanical Engineering Department, McMaster
University. From 1992 to 2003, he was a faculty
member at the Universidad del Zulia, Maracaibo,
Venezuela. During the summers of 2001 and 2002,
he was a Visiting Professor at McMaster University. His current research
interests are active vibration control, multibody and continuum dynamics, and
applications of the finite element method to nonlinear vibration problems.
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