vaz-ESM-2002

ENCYCLOPEDIA OF

SMART MATERIALSVOLUME 1 and VOLUME 2

Mel Schwartz

The Encyclopedia of Smart Materials is available Online at

www.interscience.wiley.com/reference/esm

A Wiley-Interscience Publication

John Wiley & Sons, Inc.

1066 TRUSS STRUCTURES WITH PIEZOELECTRIC ACTUATORS AND SENSORS

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TRUSS STRUCTURES WITH PIEZOELECTRICACTUATORS AND SENSORS

RAFAEL BRAVO

Universidad del Zulia

Maracaibo, Venezuela

ANTHONY FARIA VAZ

Applied Computing Enterprises Inc.

Mississauga, Ontario, Canada

&

University of Waterloo

Waterloo, Ontario, Canada

MOHAMED DOKAINISH

McMaster University

Hamilton, Ontario, Canada

INTRODUCTION

Structural systems that have a built-in capability to sense

external stimulus and respond to it depending on a prede-

termined criteria are commonly referred to as smart struc-

tures. An example of a smart structure is a flexible satellite

antenna that can sense its own vibration and apply correc-

tive action to dampen it out. A smart structure consists of

four major components: the structure itself, sensors, actua-

tors, and the control system. Smart structures can achieve

performance objective that cannot be achieved by a tra-

ditional passive structure. Through the use of an active

control system, a lightweight structural member can repli-

cate the vibration suppression characteristics of a heavy

structural member. To be effective, the design of the struc-

ture and the achievable control performance must be con-

sidered in an integrated manner. This approach enables

the trade-offs among structural weight, geometry, and con-

trol performance to be optimized to achieve a specified op-

erational characteristic.

Large flexible space structures (LFSS) have lightly

damped vibrational modes that range from very low

to high frequencies. The degradation to mission perfor-

mance caused by structural flexibility must be elimi-

nated. Conflicting requirements of structural stiffness and

lightweight must be balanced by the structural engineer.

There are two possible ways of dealing with flexibility in a

structure: use stiffer structural members, which increase

weight, or augment the inherent damping in the structure.

Due to weight restrictions, only the second approach is fea-

sible. Viscoelastic materials can be used to passively aug-

ment the damping of the high-frequency modes. Unfortu-

nately, these materials do not provide significant damping

for low-frequency modes. As it is the low-frequency modes

that most severely impact structural performance, active

damping is required.

Active damping techniques make use of sensors and ac-

tuators to measure structural deflection and perform cor-

rective action. This minimizes the maximum deflection am-

plitude and the settling time of the undesired oscillations.

These techniques require a knowledge of the dynamic be-

havior of a structure. Complex structures are modeled

mathematically using analytical techniques based on fi-

nite element analysis, or experimental model identification

based on modal testing. In our research, we use analytical

techniques to develop a dynamic model. The model is then

validated by empirical modal tests.

One issue that arises, when modeling a dynamic system

for control purposes, is the accuracy of the model behavior,

compared to that of the actual structure. The dynamics

of a LFSS are theoretically characterized by infinite-order

eigenfunction expansions. In practice, some form of model

reduction is used to develop a finite-order state-space

model. The state space model is used for both controller de-

sign and simulation. The modeling error of dynamics takes

two forms: neglected high-order modes and inaccurately

modeled low-order modes. The errors associated with the

low-order modes are due to uncertainty in the knowledge of

mechanical parameters, and simplifications taken during

the modeling process. The analytical modeling approaches

assume an a priori damping factor of zero. Empirical tests

must be used to determine the correct damping factors.

Empirical testing of the LFSS in a terrestrial laboratory

is problematic, due to the effects of gravity and the atmo-

sphere, as the structures are intended for use in the zero

gravity and vacuum of space. Accordingly, the controller

design methodology must be able to cope with the inher-

ent uncertainty in the model of the structural dynamics.

Active Vibration Control of Flexible Structures

The linear–quadratic–Gaussian control method, or LQG

control, was developed in the 1960s and is capable of ob-

taining controllers for multiple input–multiple output sys-

tems such as the ones in consideration in this article.

LQG controllers give good response when model dynam-

ics are known exactly. However, LQG controllers do not

necessarily cope well with model uncertainty (1). Further-

more, the infinite bandwidth of LQG controllers can excite

unmodeled higher-order dynamics. This effect is termed

the “spillover problem” (2). Allen et al. (3) applied the

LQG method directly to the problem of vibration sup-

pression in a truss structure. They employed a frequency-

weighted LQG method in which the system dynamics are

augmented with bandpass filters for the input and output

signals. This compensated for the traditional deficiencies

of LQG controllers: the bandpass filters restricted the con-

trol bandwidth to frequencies over which the structural

TRUSS STRUCTURES WITH PIEZOELECTRIC ACTUATORS AND SENSORS 1067

dynamics were well modeled. They demonstrated the lim-

ited bandwidth and robustness characteristics of their con-

troller design on an experimental truss structure.

The H∞ control design method (4–6) has recently been

applied to the problem of control of large flexible space

structures. This method enables the design of a controller

that robustly stabilizes a system in the presence of bounded

structured and unstructured uncertainties. Furthermore,

the control design can also incorporate performance re-

quirements that must be met in the presence of uncertain-

ties. Current research is focused on the manner in which

uncertainties and performance can be specified to obtain a

useful controller design.

Buddie et al. (7) used a “weighted gap optimization”

approach for the synthesis of the H∞ controller, employ-

ing input and output weighting functions to “shape” the

open loop response of the system. Experiments performed

on both a flower-shaped structure and a truss structure

successfully achieved an increased damping for partic-

ular modes while maintaining a specified level of ro-

bustness. Their experiments made use of momentum ex-

change actuators that were operated by linear dc motors.

The accelerometer sensors were collocated at the actua-

tor locations, and their signals are integrated to provide

velocity.

A good survey on the subject of vibration problems asso-

ciated with flexible spacecraft is provided by van Woerkom

(8). In particular, he discusses the OGO series of satellites

and the Hubble space telescope. He gives an overview of

the various control techniques that were applied to stabi-

lize these satellites.

Piezoelectric Materials in Smart Structures

The use of piezoelectric materials as sensors and actuators

in flexible systems has been a topic of research since the

mid-1980s. Hubbard and his collaborators (9,10) proposed

the application of piezoelectric polymer (PVDF) as actuator

for the active damping of beams with different boundary

conditions. The sensors employed were tip-mounted lin-

ear and angular accelerometers for the case of cantilever

beams and a base-mounted angular accelerometer for the

case of the simply supported beam. They used several lin-

ear and nonlinear velocity–feedback control techniques to

artificially increase the modal damping ratios of the flexi-

ble beams. These control laws were applied to rectangular

and spatially varying distributed PVDF films, with volt-

age levels of up to ±250 V. A significant level of damping

increase was attained, particularly when using nonlinear

on–off velocity–feedback control.

Crawley and de Luis (11) developed a quasi-static

analysis procedure for modeling the interaction between

rectangular piezoelectric materials and the structure to

which they are attached. They considered the cases of

Table 1. Dimensions and Mechanical Properties of Polycarbonate Tubing

Outside Diameter Thickness Young’s Modulus Density

Material (mm) (mm) (N/m2) (kg/m3)

Polycarbonate 12.7 1.5875 2.35 × 109 1.2 × 103

surface-bonded and embedded actuators, in bending and

extension. Their predictions were compared to experi-

ments performed on cantilevered beams of various mate-

rials and actuator placement. Lee and Moon (12) analyzed

the effect of etching and splicing to shape the electrode

of a piezoelectric film. Lee and Moon constructed a modal

sensor that was able to observe a single mode of a flexible

plate. Miller and his co-workers (13,14) showed that spa-

tial filters could also be realized in terms of etched piezo-

electric films. A state feedback control law can be imple-

mented by etching piezoelectric films; this has the poten-

tial of simplifying the hardware required for control. Vaz

(15–18) used the quasi-static analysis procedure of Craw-

ley and de Luis to develop interaction equations for model-

ing the behavior of shaped piezoelectric sensors and actu-

ators. The interaction equations were validated through

experimental testing; (19–21). The interaction equa-

tions were generalized into a finite element framework

in (7,22,23).

TRUSS STRUCTURE CONFIGURATION

A truss structure was designed to be a testbed for evalu-

ating potential control techniques for LFSS using piezo-

electric actuators and sensors. The design criteria for the

selection of the final configuration were the following:

1. Have similarity to structures used in flexible struc-

ture literature.

2. Include at least 10 modes under 100 Hz.

3. Not include local bending of truss members in first 5

modes of vibration.

4. Meet cost, weight, and actuation power require-

ments.

The structure was constructed from polycarbonate tub-

ing with 12.7 mm outer diameter, connected at the nodes by

joining blocks. The polycarbonate material was selected be-

cause of its advantages over metals such as aluminum and

steel in this application. The main selection criteria were

cost, machinability, weight, strength, electrical properties,

ease of assembly, and bonding of piezoelectric materials

to the structure. Table 1 shows the mechanical properties

and the dimensions of the polycarbonate tubes used in the

structure, including the parameters needed for the finite

element simulation. The configuration of the structure is

shown in Fig. 1. The truss members represented by the

dark lines undergo the most strain during vibration of the

truss structure. These truss members are ideally suited to

sensor and actuator placement.

The truss structure was vertically cantilevered at the

bottom, and it was composed of six vertical bays, each a

300-mm cube. The thick lines represent the active truss


00.1

0.20.20.10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 1. Configuration of the truss structure.

members, which are fitted with sensors and actuators. Not

all the active truss members are to be used in the control

design simultaneously. The design was iterated until the

desired design criteria were met. The structure was ap-

proximately 1800 mm high. Figure 2 gives a detailed view

of two bays of the structure, showing more clearly the topo-

logy of the truss. At each junction there is a joining block,

also made of polycarbonate material.

Figure 2. Detail of two bays.

Figure 3. Experimental truss structure.

The experimental truss structure is shown in Fig. 3. The

truss members are bonded to the junction blocks using a

liquid plastic solvent containing methylene chloride. The

associated electronics are shown in the background. The

five gray boxes contain the custom designed high-voltage

amplifiers. The amplifiers are powered by an HP 6521A dc

power supply. Accelerometers are placed at the top of the

structure. The lower portion of the truss structure is shown

in closeup in Fig. 4. The active truss members are the lower

truss members that have bonded piezoelectric films.

The active truss members are fitted with two identi-

cal polyvinyldene fluoride (PVDF) rectangular piezoelec-

tric sensors. One sensor measures the open circuit voltage,

which is proportional to the strain. The second measures

the short circuit current, which is proportional to the strain

rate. In the midspan of the active truss member, there is

a piezoceramic actuator, which receives the control signal

and transforms it into actuation forces.

FINITE ELEMENT AND MODAL ANALYSISOF THE TRUSS STRUCTURE

In the finite element method, a continuous structure is dis-

cretized into a finite number of elements (24). This con-

verts the dynamics of the continuous structure, with infi-

nite number of degrees of freedom, to a finite-dimensional

representation. The discretized structure has dynamics

that are characterized by a system of ordinary differential


Figure 4. Active truss members.

equations, whereas the continuous structure dynamics

are characterized by a system of partial differential

equations.

Assembly of Finite Element Model

The truss members are modeled by an interconnected set of

rod elements. A rod element has two nodes with one degree

of freedom per node that corresponds to the displacement

in the axial direction. Figure 5 shows a diagram of the

rod element. The displacement is varies linearly along the

rod element. The nodes are labeled with a local numbering

scheme LNODE = {1, 2}. The axial displacement of at local

node i is denoted by ξi for i = 1, 2.

The mesh used for the finite element model with rod el-

ements is shown in Fig. 6. The nodes are placed at the junc-

tions, and are indicated by star symbols. The cross section

of the rod elements is indicated by the ovals at midspan.

Using rod elements is equivalent to assuming that the

structure is pin-jointed, and thus the strain profile along

each element is constant. For this reason, increasing mesh

density, by partitioning the truss members with additional

rod elements, would not alter the strain profile. Concen-

trated mass elements are added at each junction to sim-

ulate the effect of the extra weight of the junction blocks,

and are denoted in by small squares coincident with the

nodes. The finite element model has 83 rod elements and

28 nodes. Each node has 3 degrees of freedom; hence the

structure has a total of 84 degrees of freedom. The struc-

ture is fixed at the base; this restricts 3 degrees of freedom

at each of the four base nodes, for a total of 12 constrained

degrees of freedom.

Local Node 1 Local Node 2

k

1ξ

k

2ξ

Figure 5. Rod element k.

Figure 6. Finite element mesh for the rod element model.

The nodes are labeled with a global numbering scheme

GNODE = {1, 2, . . . , n}, where n = 28. Each truss member

is associated with a unique rod element. The rod ele-

ments are labeled with a global numbering schemeGELEM =

{1, 2, . . . , m}, where m = 83. The function GNODE maps a

global element number and a local node number to global

node number; that is, GNODE: GELEM × LNODE → GNODE.

Each node has three coordinates that define its unde-

formed position with respect to a global coordinate frame.

We let pα1 , pα

2 , and pα3 respectively denote the x, y, and z po-

sition coordinates of node α, α ∈ GNODE. The 123 coordinate

frame is an alternate representation of the x-y-z coordinate

frame: 1 corresponds to x, 2 to y, and 3 to z. We consider

a rod element k with nodes α and β, where k ∈ GELEM, and

α, β ∈ GNODE. The projection of the length of rod element k

on axis i of the global the coordinate coordinate frame is

given by

1pki =

∣

∣

∣pα

i − pβ

i

∣

∣

∣, i = 1, 2, 3. (1)

The length of rod element k, denoted by Lk, is given by

Lk =

√

(

1pk1

)2+

(

1pk2

)2+

(

1pk3

)2. (2)

The direction cosines for rod element k are computed as

cos θki =

1pki

Lk, i = 1, 2, 3. (3)

When the truss structure is subjected to forces, the rod

elements deform and the nodes move from their rest posi-

tion. To illustrate this, we let qα1 , qα

2 , and qα3 , respectively,

denote the x, y, and z coordinates of the displacement of

the deformed position of node α, α ∈ GNODE, from its unde-

formed position. We consider a rod element k with nodes α

and β, where k ∈ GELEM, and α, β ∈ GNODE. The components

of the deformation of rod element k, in the global coordinate

frame, can be found by subtracting the node displacements

as follows:

1qki =

(

qβ

i − qαi

)

sgn(

pβ

i − pαi

)

, i = 1, 2, 3. (4)

The component of deformation 1qki corresponds to an elon-

gation if 1qki > 0, and a compression if 1qk

i < 0. The total


deformation 1qkTOTAL of the rod element k is given by

1qkTOTAL = 1qk

1 cos θk1 + 1qk

2 cos θk2 + 1qk

3 cos θk3 , (5)

where cos θki is the direction cosine of element k and axis i.

The strain of element k, denoted by εk, is given by

εk =1qk

TOTAL

Lk. (6)

Now, let qα denote the displacement vector at node

α, α ∈ GNODE. The composite displacement vector q com-

prises the displacement vectors at each node of the struc-

ture. These vectors are defined as

q =

q1

...

qn

, qα =

qα1

qα2

qα3

. (7)

Let 1qk denote the deformation vector of element

k, k ∈ GELEM. The composite deformation vector 1q com-

prises the deformation vectors at each element of the struc-

ture. These vectors are defined as

1q =

1q1

...

1qm

, 1qk =

1qk1

1qk2

1qk3

. (8)

The composite total deformation vector 1qTOTAL comprises

the total deformations of each element of the structure. In

particular, 1qTOTAL is defined as

1qTOTAL =

1q1TOTAL

...

1qmTOTAL

. (9)

The composite strain vector ε comprises the strains of each

element of the structure. In particular, ε is defined as

ε =

ε1

...

εm

. (10)

The 3m× 3n matrix S is constructed so that

1q = Sq. (11)

In particular, the ab element of S, denoted by (S)ab, is given

by

where 1 ≤ a ≤ 3m and 1 ≤ b ≤ 3n.

The m× 3m block diagonal matrix R is constructed so

that

1qTOTAL = R1q. (12)

In particular, R is given by

R= diag[

(

cos θ11 ,cos θ1

2 ,cos θ13

)

, . . . ,(

cosθm1 ,cos θm

2 ,cosθm3

)

]

.

(13)

The m× m matrix N is constructed so that

ε = N1qTOTAL. (14)

In particular, matrix N is given by

N = diag

[

1

L1,

1

L2, · · · ,

1

Lm

]

. (15)

Consider further an element k with local displacements

ξk1 and ξk

2 as shown on Fig. 5. The local displacements

can be determined from the displacement vector q. In

particular,

ξk1 = cos θk

1qGNODE(k ,1)1 + cos θk

2 qGNODE(k,1)2

+ cos θk3qGNODE(k ,1)

3 , (16)

ξk2 = cos θk

1qGNODE(k ,2)1 + cos θk

2 qGNODE(k ,2)2

+ cos θk3qGNODE(k ,2)

3 .

In accordance with the above expressions, a 2 × 3n matrix

Gke is constructed so the following relationship holds.

[

ξk1

ξk2

]

= G ke q. (17)

In particular, the ab element of Gke , denoted by (Gk

e)ab, is

given by

(

Gke

)

ab=

cos θki

0

if some i ∈{1, 2, 3} and k ∈ GELEM satisfies

a = 1 and b = 3 [GNODE (k, 1) − 1] + i,

or

a = 2 and b = 3 [GNODE (k, 2) − 1] + i,

otherwise,

The following mass and stiffness matrices are obtained

from finite element modeling (24): the rod element k has

a consistent mass matrix Mke , and a consistent stiffness

(S)ab =

sgn(

pGNODE(k,1)i − pGNODE(k,2)

i

)

if some i ∈ {1, 2, 3} and k ∈ GELEM satisfies

a = 3(k − 1) + i and b = 3[GNODE (k, 1) − 1] + i,

sgn(

pGNODE(k,2)i − pGNODE(k,1)

i

)

if some i ∈ {1, 2, 3} and k ∈ GELEM satisfies

a = 3(k − 1) + i and b = 3[GNODE (k, 2) − 1] + i,

0 otherwise,


matrix K ke given by

Mke =

ρk AkCSLk

6

[

2 1

1 2

]

and Kke =

Ek AkCS

Lk

[

1 −1

−1 1

]

.

(18)

Element k has the parameters density ρk, cross-sectional

area AkCS, Young’s modulus of elasticity Ek, and length Lk.

The mass matrix of the complete structure, M, is given by

M =

m∑

k=1

(

G ke

)TMk

e G ke . (19)

The matrix M is positive definite. To incorporate the effect

of the junction block masses, concentrated mass elements

are added at the nodes. The stiffness matrix of the complete

structure, K, is given by

K =

m∑

k=1

(

G ke

)TK k

e G ke . (20)

The matrix K is positive semidefinite.

The result of a finite element discretization of a struc-

ture is a system of ordinary differential equations, which

are called the free system model:

Mq + Kq = f, (21)

where q is the displacement vector and f is a vector of actu-

ation forces applied to the structure. The vector f is defined

in a later section where the operation of piezoelectric film

actuators is explained. Equation (21) does not incorporate

the boundary conditions associated with the structure. Ac-

cordingly, Eq. (21) has n = 84 degrees of freedom.

In the structure shown in Fig. 1, the bottom four nodes

are clamped to the support base. Hence the displacements

at nodes 1, 2, 3, and 4 are zero; that is, qαi = 0 for i =1,

2, 3 and α =1, 2, 3, 4. Hence there are 12 restricted de-

grees of freedom. There are n = 72 constrained degrees of

freedom. The constrained displacement vector q only in-

cludes the components qαi , i =1, 2, 3, and α ∈ GNODE, that

are not forced to be zero. The n × n matrix P maps q to q

by q = Pq. For our truss structure, the corresponding P

matrix is given by

P = [0n×12, In×n]. (22)

Note that the following relationship also holds: q = PTq.

The constrained mass matrix M and the constrained stiff-

ness matrix K are given by

M = PMPT and K = PK PT. (23)

The matrices M and K can be from obtained from M and K,

respectively, by deleting the rows and columns associated

with the clamped nodes. The constrained system model is

M q + K q = f , (24)

where f = Pf .

Modal Analysis

The constrained system model, Eq. (24), is used to model

the dynamics of the structure. The vibrational mode shapes

and their frequencies are obtained from an eigenvalue

analysis of Eq. (24). Since M is positive definite, the fol-

lowing eigenvalue problem has n solutions:

ω2j Mv j + K v j = 0 for j = {1, 2, . . . ,n}. (25)

Order the eigenvalues ω2j and eigenvectors v j so that

ω2j ≤ ω2

j+1. Construct the modal matrix 8 so that 8 =

[v1, . . . , vn]. Construct the modal frequency matrix � so

that � = diag[ω21, . . . , ω

2n]. The vectors v j, j = {1, 2, . . . ,n},

are normalized according to the following relationships:

8TM8 = I,(26)

8TK8 = �.

The modal matrix 8 defines a coordinate transformation

from the modal coordinate vector η to the physical coordi-

nate vector q, that is,

q = 8η. (27)

Substituting this in the system model and premultiplying

by 8−1, the system model yields the modal model of the

system:

8TM8η + 8TK 8η = 8T f ,

which is equivalent to

η + �η = 8T f . (28)

To account for structural damping, a diagonal modal damp-

ing matrix H can be added.

η + Hη + �η = 8T f . (29)

The matrix H has the following form

H = diag[2ζ1ω1, 2ζ2ω2, . . . , 2ζnωn], (30)

where ζ j is the damping factor for mode j, 0 ≤ ζ j ≤ 1, j =

{1, 2, . . . ,n}. The form of the model in equation (30) has

the advantage that it is uncoupled, since the matrices

� and H are both diagonal. The damping factors ζ j, j =

{1, 2, . . . ,n}, can be determined from either material prop-

erties or experimentation.

The finite element modeling program I-DEAS (25) is

used to analyze the truss structure. The program allows a

structure to be defined in a graphical manner. The program

constructs a representation of Eq. (24) once the material

properties and boundary conditions are defined. The modes

of the structure are obtained by solving Eq. (25) to obtain

the mode shapes v j and frequencies ω j for j = {1, 2, . . . ,n}.

The graphical representations of the first four mode shapes

are given in Fig. 7.


0

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

Mode 1 Freq: 11.1035 Hz

0.20

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4


00.2

0.4

0

0.5

1

1.5

0

0.2

0.4


00.2

0.4

0

0.5

1

1.5

0

0.2

0.4


Figure 7. Mode shapes and frequencies of the structure.

ACTUATOR AND SENSOR CONSTRUCTION

Table 2 shows some of the relevant mechanical and piezo-

electric properties of the piezoelectric materials used in

this research. For sensor construction, polyvinyldene flu-

oride (PVDF) piezoelectric film was selected. This mate-

rial was chosen because it produces relatively large volt-

ages for a given strain (high g31 coefficient). For actuator

construction, the piezoceramic material lead zirconate ti-

tanate (PZT) was selected. Piezoceramics produce a rela-

tively high strain for a given electric field (high d31 coeffi-

cient).

The sensors and actuators piezoelectric films are at-

tached to the tubular truss member as illustrated on Fig. 8.

The actuators are cylindrical shells of ceramic PZT mate-

rial, split longitudinally to facilitate the installation in the

active truss members. The sensors are rectangular pieces

of PVDF. The sensors are installed in pairs: one PVDF film

measures strain, whereas the other measure strain rate.

The dimensions of the sensors and actuators are given

in Table 3. The length is measured along the axis of the

truss member; the width refers to the dimension along the

circumference of the member, and the thickness is mea-

sured across the radius, as shown on Fig. 8. A sensor film

has length Lks , width Wk

s , and thickness tks , whereas an ac-

tuator has length Lka, width Wk

a , and thickness tka .

For later convenience, we introduce the following nota-

tion: Let SSEN = {1, 2, . . . , r} be the index set for the piezo-

electric film sensor pairs on a truss member. The function

SEN maps a sensor number to a global element number;

that is, SEN: SSEN → GELEM. Let SACT = {1, 2, . . . , r} be the

index set for the piezoelectric film actuators on a truss

member. The function ACT maps an actuator number to

a global element number; that is, ACT: SACT → GELEM.

FORMULATION OF THE STATE SPACE DYNAMIC MODEL

In this section the dynamic interaction equations for the

piezoelectric sensors and actuators with the finite element

model are derived. The interaction equations are used to

formulate the state space model of the dynamic equations.

The state space model is the canonical form for designing

control systems and simulation.


Table 2. Properties of Piezoelectric Materials

Young’s

d31 g31 Modulus Permitivitty

Material Application (C/N) (V m/N) E (N/m2) ∈ (F/m)

PVDF Sensor 23 × 10−12 216 × 10−3 2 × 109 106 × 10−12

BM527 (PZT) Actuator 215 × 10−12 7 × 10−3 6.9 × 1010 3.54 × 10−8

Observation Matrix forthe Piezoelectric Sensor

The open circuit voltage voc(t) of a piezoelectric film sensor

is proportional to the strain of the film. The short circuit

current isc(t) of a piezoelectric film sensor is proportional

to the strain rate of the film. Let vkoc(t) and ik

sc(t) denote the

open circuit voltage and short circuit current associated

with the piezoelectric film attached to the truss member

k, k ∈ GELEM. Lee and Moon (12) developed equations relat-

ing the deformation of a piezoelectric film and the resulting

open circuit voltages and short circuit currents. Adapting

these equations yields the following equations for the open

circuit voltage vkoc(t) and short circuit current ik

sc(t):

vkoc(t) = −

dk31s Ek

s Wks

Cks

Lks

∫

0

εks (x, t)dx, (31)

iksc(t) = −dk

31s Eks Wk

s

Lks

∫

0

∂εks (x, t)

∂tdx, (32)

where dk31s is the transverse piezoelectric charge to stress

ratio, Eks is the Young’s modulus of the film, Wk

s is the width

of the piezoelectric film, Cks is the capacitance of the film,

and εks (x, t) is the strain profile of the piezoelectric film.

Note that the subscript s denotes parameters of the sensor

film, while superscript k denotes the rod element associ-

ated with the truss member to which the sensor is attached.

Modeling the truss member with rod elements yields a

strain profile that is constant along the tubular truss mem-

ber. Hence the strain for element k can be represented as

εk(t), and determined from Eq. (6). Due to the negligible

thickness of the PVDF sensor films, and the hardness of

kr

k

sL

Piezoelectric Element

Tubular Truss

Member k

k

sW

k

st

Figure 8. Piezoelectric film attached to tubular truss member.

the epoxy, we assume εks (x, t) = εk(t). Accordingly, the ex-

pressions for the open circuit voltage and closed circuit

current simplify as

vkoc(t) = γ k

Vocεk

s (t), γ kVoc

= −dk

31s Eks Wk

s Lks

Cks

, (33)

iksc(t) = γ k

Iscεk

s (t), γ kIsc

= −dk31s Ek

s Wks Lk

s . (34)

>From the preceding equations, it is easy to see that mea-

surements of open circuit voltage and short circuit current

can be employed to obtain the strain and strain rate in a

truss member. By using Eqs. (6), (10), and (11), the above-

mentioned voltage and currents can be related to the dis-

placement of vector q in Eq. (7).

Let the composite vector of open circuit voltages Voc(t),

and the composite vector of short circuit currents Isc(t) be

given by

Voc(t) =

vSEN(l)oc

...

vSEN(r)oc

and Isc(t) =

iSEN(l)sc

...

iSEN(r)sc

. (35)

Matrices MVoc and MIsc of size r × m can be constructed so

that

Voc = MVocε and Isc = MIsc ε. (36)

In particular, the ab elements of matrices MVoc and MIsc are

given by

(

MVoc

)

ab=

{

γSEN(a)Voc

if b = SEN(a),

0 otherwise,(37)

(

MIsc

)

ab=

{

γSEN(a)Isc

if b = SEN(a),

0 otherwise,(38)

where 1 ≤ a ≤ r and 1 ≤ b ≤ m.

Control Matrix for the Piezoelectric Actuator

In the following derivation, perfect bonding between a

piezoelectric film and the truss member is assumed. This

is a valid assumption due to the hardness of the epoxy used

to bond the piezoelectric films. The manner in which the

Table 3. Dimensions of Sensors and Actuators

Length Width Thickness

Application Material (mm) (mm) (mm)

Sensor PVDF 42 17.0 28×10−3

Actuator BM527 (PZT) 25.4 39.9 1.0


strain in a substructure varies when it is bonded to a piezo-

electric film is given by a coupled set of partial differential

equations analyzed in (11,16). For the case of perfect bond-

ing, with a substructure undergoing axial deformation, the

following simplification is possible. The piezoelectric strain

εa

(

ξ)

at the normalized coordinate ξ , where ξ = −1 de-

notes one end of the piezoelectric film, and ξ = +1 denotes

the other end, is given by the formula

εa

(

ξ)

=9

9 + 2

[

ε+

SUB + ε−

SUB

2−

(

ε+

SUB − ε−

SUB

2

)

ξ +2

93

]

(39)

The strain of the substructure at ξ = −1 and ξ = +1 are

denoted by ε−

SUB and ε+

SUB, respectively. The stiffness ratio

between the piezoelectric film and the substructure is de-

noted by 9. The effective strain of due to the piezoelectric

effect is denoted by 3. Although the strain of the piezo-

electric film is the same as the strain of the substructure

underneath the film, the strain of the substructure is dif-

ferent in locations not covered by the film.

We will now adapt Eq. (39) to the case where the sub-

structure is a truss member modeled by a rod element.

In such a case, the substructure strains at either end of

the piezoelectric film are identical; this corresponds to

ε−

SUB = ε+

SUB. Hence the strain over the piezoelectric film

is uniform. The strain of a piezoelectric film bonded to the

truss member associated with rod element k is denoted by

εka, and given by

εka =

(

9k

9k + 2

)

εk +

(

2

9k + 2

)

3ka, (40)

where εk is the strain of the truss member k on the por-

tion not underneath the piezoelectric film, and 3ka is the

effective strain due to the piezoelectric effect. The effective

strain 3ka is given by

3ka =

(

dk31a

tka

)

vka, (41)

where vka is the voltage applied to the piezoelectric film

associated with element k. The stiffness ratio for element

k is given by

9k =Ek tk

Eka tk

a

, (42)

where Ek and Eka are, respectively, Young’s modulus of elas-

ticity for the truss member k and the piezoelectric film ac-

tuator attached to it; tk and tka are, respectively, the thick-

ness of truss member k and the piezoelectric film actuator

attached to it.

When there is perfect bonding between a piezoelectric

film and its substructure, the force exerted by the piezo-

electric on the structure is concentrated at the ends of the

piezoelectric film (11,16). Let gke denote the force exerted

by the ends of piezoelectric film on the truss member as-

sociated with element k. Summing the stress at the end of

the piezoelectric yields

gke = Ek

a AkaCS

(

εka − 3k

a

)

, (43)

where AkaCS is the cross-sectional area of the piezoelectric

actuator film bonded to truss member k. The transmitted

forced can be rearranged as

gke = gk

eSTIFF + gkeACTIVE, (44)

where

gkeSTIFF = µk

STIFF εk, (45)

gkeACTIVE = µk

ACTIVE vka, (46)

and

µkSTIFF = Ek

a AkaCS

(

9k

9k + 2

)

, (47)

µkACTIVE = −Ek

a AkaCS

(

9k

9k + 2

) (

dk31a

tka

)

. (48)

The force gke has two components. The component gk

eSTIFF

is the passive force due to the intrinsic elasticity of the

piezoelectric film, which augments the stiffness of the truss

member. The component gkeACTIVE is the active force con-

tributed by the piezoelectric effect that varies with the ap-

plied voltage vka.

Equations (45) and (46) only define the forces gkeSTIFF and

gkeACTIVE for truss members k that have a piezoelectric film

actuator; that is, k = ACT( j) for some j ∈ SACT. We extend

the definition to all truss members in the following manner.

For all k ∈ GELEM,

gkeSTIFF =

{

µkSTIFF εk, if k = ACT ( j) for some j ∈ SACT,

0 otherwise;

and

gkeACTIVE =

{

µkACTIVE vk

a, if k = ACT ( j) for some j ∈ SACT,

0 otherwise.

Let the composite vector of passive element forces gkeSTIFF,

and the active element forces geACTIVE be given by

geSTIFF =

g1eSTIFF

...

gmeSTIFF

and geACTIVE =

g1eACTIVE

...

gmeACTIVE

.

(49)

Let the composite vector of applied voltages Va be given

by

Va =

vACT(l)a ...

vACT(r)a

. (50)


The m× m matrix YSTIFF is constructed so that

geSTIFF = YSTIFF ε, (51)

where strain vector ε is given by Eq. (10). In particular, the

ab element of matrix YSTIFF is given by

(YSTIFF)ab =

µACT(a)STIFF if a = b and a = ACT ( j) for some

j ∈ SACT,

0 otherwise,

for 1 ≤ a ≤ m and 1 ≤ b ≤ m.

The m× r matrix YACTIVE is costructed so that

geACTIVE = YACTIVEVa. (52)

In particular, the ab element of matrix YACTIVE is given

by

(YACTIVE)ab =

{

µACT(b)ACTIVE if a = ACT (b) and b ∈ SACT,

0 otherwise,

for 1 ≤ a ≤ m and 1 ≤ b ≤ r.

In the context of finite element model, Eq. (21), the ap-

plied force f has two components:

f = fSTIFF + fACTIVE, (53)

where

fSTIFF = ST R TYSTIFF NRS q (54)

and

fACTIVE = ST R TYACTIVEVa. (55)

Matrices S, R, and N are defined in Eqs. (11), (13), and (15),

respectively.

State Space Model Construction

In this section, we construct a state space model for the

modal model of Eq. (29). The canonical form of a state space

is shown below.

x = Ax + Bu,

(56)y = Cx + Du.

The state x of the modal model is the composite vector

of mode amplitudes η and mode amplitude rates η. The

input vector u is the vector of voltages applied to piezo-

electric actuators Va. The output vector y is the composite

vector of open circuit voltages and short circuit currents

for the piezoelectric film sensors. In particular, x, u, and y

are given by

x =

[

η

η

]

, u = Va, and y =

[

Voc

Isc

]

.

The dimension of the modal amplitude vector η is the num-

ber of modes in the model. Although the constrained sys-

tem model in Eq. (29) has n =72 modes, we can construct

a model with a reduced set of modes. This is typically done

to simply the computational complexity of control design.

For a n mode model, 1 ≤ n ≤ n, state space matrices are

A, B, C, and D are given by

A =

[

0 In×n

−�n −Hn

]

+ 8Tn ST RTYSTIFF NRSPT8n, (57)

B = 8Tn ST RTYACTIVE, (58)

C =

[

Mνoc NRSPT8n 0

0 MIsc NRSPT8n

]

, (59)

D = 02r×r, (60)

where

8n = [v1, v2, . . . , vn], (61)

�n = diag[

ω21, ω

22, . . . , ω

2n

]

, (62)

Hn = diag [2ζ1ω1, 2ζ

2ω2, . . . 2ζnωn]. (63)

The modal matrix 8n contains the first n eigenvectors of

Eq. (25).

Two state space models are used for modeling the dy-

namics of the truss structure. A low-order design model

is used for controller design, and a high-order evalua-

tion model is used to simulate the controller performance.

The design model retains the first five modes (n = 5) and

assumes no damping (ζ j = 0, j = 1, 2, . . . , 5). This is con-

sistent with the fact that the true damping of structure

is not known until it is tested. The evaluation model

retains the first 30 modes (n = 30) and uses the esti-

mated modal damping factors (ζ j = 0.004, j = 1, 2, . . . , 30)

obtained from empirical tests on the structure.

CONTROL DESIGN AND SIMULATIONS

LFT Framework for H∞ Feedback Control Design

A robust H∞ feedback controller has been designed for the

truss structure. To obtain the controller using this tech-

nique, the dynamic model of the system has to be written

in lower linear fractional transformation (LFT) form. This

formulation framework is convenient because it allows for

the inclusion of the effects of the sensor and control dis-

turbances, as well as loop-shaping weights on the inputs

and outputs. Figure 9 shows the block representation of the

lower LFT form of the control problem. The signal z is the

exogenous output, y is the controlled output, w is the exo-

genous input, and u is the control input. The relationship


Figure 9. Lower LFT representation of control problem.

between these signals can be represented in the following

manner:

x = Ax + B1 w + B2 u,

z = C1x + D11 w + D12 u, (64)

y = C2x + D21 w + D22 u.

In the LFT framework, it is desired to minimize the H∞

norm from w to z. The nature of these signals and their

relationship to controller design is explained later.

Consider the state space model of the truss dynamics in

Eq. (56). The model has the matrix transfer function P(s)

that is given by

P(s) = C(sI − A)−1 B + D (65)

Figure 10 shows the conventional block diagram of the

closed loop control system, including the sensor noise nand

disturbances d to the control. In order to put the problem

in the LFT framework, we aggregate the unwanted inputs

n and d into an exogeneous input signal w. We also aggre-

gate the signals that we wish to penalize, the state x and

the control input u, into an exogenous output signal z. The

resulting exogenous input and output vectors are

w =

[

d

n

]

, z =

[

x

u

]

. (66)

Different frequency ranges in the input signals and outputs

can be penalized through the use of loop-shaping filters.

This is illustrated in Fig. 11 for a system in lower LFT

form with an input filter W1 and an output filter W2.

)(sP )(sK

n+

+

+

+

d

y u

Figure 10. Block diagram of the closed loop system.

Figure 11. Shaping of the open loop.

The system model can be uncertain over different fre-

quency ranges. For example, uncertainty in the high-

frequency region can result from modal truncation. Uncer-

tainty in the low frequency can result from low-frequency

disturbances being applied to the system due to ground

movements. The uncertainty can be handled by specify-

ing appropriate filters in W1, such as a bandpass filter,

to introduce robustness to the perturbations in the model

dynamics. The output filter W2 can be used to introduce

performance requirements by placing weights on different

elements of the control signal and state space. These speci-

fication techniques are explained by Zhou et al. (6)

The dynamic controller that minimizes the H∞ norm of

the closed loop is given by (26)

K = Ck(sI − Ak)−1 Bk + Dk (67)

The state space matrices that form the controller are found

from the solution of the following matrix inequality

HXL+ Q∗ J∗ PXL

+ PXLJQ < 0, (68)

where the unknowns are XL and J. The matrices in equa-

tion (65) are defined as follows.

J =

[

Ak Bk

Ck Dk

]

, HXL=

A∗XL + XL A XLB C

∗

B∗XL −I D∗

11

C D11 −I

,

A =

[

A 0

0 0nk×nk

]

, B =

[

B1

0

]

, B =

[

0 B2

I 0

]

,

C =

[

0 I

C2 0

]

, C = [C1 0],

D21 =

[

0

D21

]

, D12 = [0 D12],

PXL=

⌊

B∗ XL 0 D∗12

⌋

, Q =⌊

C D21 0⌋

.

Gahinet (26) shows that if a controller with the struc-

ture in Eq. (67) exists, then XL is given by

XL =

[

X X2

X∗2 I

]

, X − Y−1 = X2 X∗2, (69)


where X and Y are the solutions to the following linear

matrix inequalities.

[

Ny 0

0 I

]∗

A∗ X + XA XB1 C1

B∗1 X −I D∗

11

C1 D11 −I

[

Ny 0

0 I

]

< 0,

(70)

[

Nx 0

0 I

]∗

AY + YA∗ B1 YC∗1

B∗1 −I D∗

11

C1Y D11 −I

[

Nx 0

0 I

]

< 0,

(71)

The matrices Nx and Ny are full rank and satisfy the fol-

lowing.

Im(Ny) = Ker[B∗2 D∗

12],

Im(Nx) = Ker[C2 D21].

Once XL has been found, the matrix inequality given in

Eq. (68) becomes linear. Accordingly, Eq. (65) can be solved

using LMI solution techniques to obtain the matrix J. The

controller matrices (Ak, Bk, Ck, Dk) are obtained as parti-

tions of J. The numerical solutions to the above LMI prob-

lems can be obtained using the LMI Lab toolbox (27) in

MATLAB (28).

Controller Design for Vibration Suppression of Truss

In the simulations, an impact test was performed. Fig-

ure 12 shows the location of the impact. Sensor-actuator

pairs were placed on the four truss members indicated by

thick lines. This selection is made due to the high axial

strains that these truss members undergo compared to

other members in the first five vibrational modes. All the

considered modes are controllable and observable with this

sensor and actuator configuration.

The controller for the truss is designed using a five mode

reduced order model of the dynamics. The truss members

with both sensors and actuators are shown in Fig. 12. The

state space matrices A, B, C, and D are, respectively, given

by Eqs. (57) to (60) with n = 5. The corresponding LFT rep-

resentation is obtained as follows:

B1 = [0·1B 010×4], B2 = B,

C1

[

Q

04×10

]

, C2 = C,

D11 = 014×8, D12 =

[

010×4

R

]

,

D21 = 0.01 × [04×4 I4×4], D22 = D.

The Q and Rare weights on the state and control are given

by

Q = diag [20 20 30 40 50 0 0 0 0 0] and

R = 1 × 104 I4×4.

This choice corresponds to a 10% actuation disturbance

and 1% sensor noise. The control input is heavily weighted

to avoid saturation of the control input. The modal ampli-

tudes are penalized, whereas modal amplitude rates are

00.1

0.20.20.10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 12. Impact test: Location of disturbance and active truss

members.

not. This LFT formulation provides reasonable trade-off

between performance and robustness to sensor noise and

amplifier nonlinearities. Figure 13 shows a maximum sin-

gular value plot. The decreased peaks of the closed loop

indicate the increased damping of the first five modes of

the closed loop system relative to the open loop system. It

also shows that outside this range, the frequency response

0 10 20 30 40 50 60 70 80 90 100-90

-85

-80

-75

-70

-65

-60

-55

-50

-45

-40

Freq. (Hz)

Magnitude

Solid: Open LoopDashed: Closed Loop

Figure 13. Maximum singular value plot for H∞ control.


Figure 14. Displacement of node 24

during H∞ control (simulation).

0 1 2 3 4 5 6 7 8-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

x-D

isp. (m

)

0 1 2 3 4 5 6 7 8-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

Time (s)

y-D

isp. (m

)

of the system is almost unchanged. This shows that the

controller does not destabilize high frequency unmodeled

modes.

The high-order evaluation model is used for the dynam-

ics of the truss structure. The model includes the first 30

modes of vibration. The simulated actuation voltage was

limited to a range of ±265 volts to match the voltage swing

available from the high voltage amplifiers. Simulink (29)

was used to simulate the closed loop response of the truss

with a dynamic controller. The displacement and control

signal response are shown on Figs. 14 and 15.

EXPERIMENTAL RESULTS

The architecture of the truss structure testbed is shown

on Fig. 16. The feedback controller is performed by a high-

performance DSP card in a Pentium-based computer. The

control logic is specified using the Simulink (29) graphi-

cal interface, which runs in the Matlab (28) environment.

The control logic is compiled into a real-time executable

by and Wincom (30) software, which also runs in the Mat-

lab environment. The executable code is downloaded by

the computer to a Quanser MultiQ-3 I/O card. The Multi-

Q card has a DSP and supports 8 input ADC and 8 out-

put DAC channels with a 12 bit resolution at a sampling

frequency of 15 kHz. Piezoelectric film actuators and sen-

sors are bonded to the truss structure as shown in Fig. 4.

Each piezoceramic actuator is driven by a custom-designed

high-voltage amplifier, with a maximum voltage output of

560 V. A HP 6521A dc high-voltage power supply is used

to power the high-voltage amplifiers. For calibration pur-

poses, strain gauges are fitted to some of the truss mem-

bers. Two accelerometers are installed at the top corner

of the structure to measure accelerations in the X and Y

directions as shown in Fig. 12. The accelerations are inte-

grated to obtain displacement measurements.

Two types of experiments are performed: impact tests

and shaker tests. The impact test uses a impact hammer to

strike the truss structure, and thereby impart an initial de-

flection. The shaker test uses one of the active truss mem-

ber actuators as a disturbance source. A disturbance signal

is applied to the piezoelectric actuator to cause resonant vi-

brations in the truss structure. The open loop and closed

loop responses of the structure are compared for both tests.

Impact Test

The impact test is performed by using an impact ham-

mer to strike the location shown on Fig. 12. The impact

hammer is equipped with an internal accelerometer. By

integrating the accelerometer signal, the initial deflection

resulting from the impact can be determined. Since the

dominant time constant of the truss dynamics is much

lower than the impact duration, the impact behaves like

it imparts an initial deflection to the truss structure. The

open loop and closed loop responses of the structure can

then be compared.

The open loop tests are carried out to have a reference

against which to compare the performance of the closed

loop controlled system. The results were also used to cali-

brate the modal damping ratios in the test model used


0 1 2 3 4 5 6 7 8-5

0

5A

ctu

ato

r 1 (

V)

0 1 2 3 4 5 6 7 8-5

0

5

Actu

ato

r 2 (

V)

0 1 2 3 4 5 6 7 8-5

0

5

Actu

ato

r 3 (

V)

0 1 2 3 4 5 6 7 8-5

0

5

Time (s)

Actu

ato

r 4 (

V)

Figure 15. Control signals during H∞ control

(simulation).

in the simulations. Figure 17 shows the displacement re-

sponse of the system to the impact, obtained from dou-

ble integration of the accelerometer signals. The small in-

herent structural damping of the system is visible. The

structure oscillates without control for 8 seconds. Figure 18

shows the frequency response of the open loop system. The

dominant resonant peak corresponds to the second mode

of vibration, at 13.3 Hz.

The H∞ controller design explained earlier was imple-

mented on the testbed. The displacement and control sig-

nal are shown in Figs. 19 and 20, respectively. The settling

time of the vibrations is dramatically shorter for the closed

loop case. The oscillations damp out in less than 2.5 sec-

onds. The improvement in performance is easier to notice

Computer

Storage

Basic Signal

Conditioning

8 Channel A/D

ConversionController

u

Piezoelectric

Actuators

Piezoelectric

Sensors

8 Channel D/A

Conversion

High Voltage

Amplifiers

ocV

scI

Structure

Computer

and Multi-Q

Accelerometers

Accelerometer

amplifiersFigure 16. Schematic of testbed system

architecture.

in the frequency response plot, shown in Fig. 21. The maxi-

mum resonant peak, at 13.3 Hz, is reduced by 25 dB when

compared to the open loop case.

Shaker Test

In the shaker experiment, a continuous sinusoidal distur-

bance signal is applied to a piezoceramic actuator on the

structure. The active truss member, whose actuator is used

as a disturbance source, is shown on Fig. 22. The sinu-

soidal disturbance signal is adjusted to the dominant res-

onant frequency of 13.3 Hz with the maximum amplitude

of 560 V. The acceleration measurements were taken at

node 24 in both the X and Y directions. Double integration


Figure 17. Displacement of node 24 of open

loop (experimental).

0 1 2 3 4 5 6 7 81.5

1

0.5

0

0.5

1

1.5x 10

3

xD

isp

.(m

)

0 1 2 3 4 5 6 7 81.5

1

0.5

0

0.5

1

1.5x 10

3

yD

isp

.(m

)

Time (s)

of the acceleration signals yields displacement measure-

ments at node 24. The resonant vibrations are allowed

to build up to a steady state before the H∞ controller is

applied.

The displacement at node 24 in the X and Y directions is

shown in Fig. 23. The control signals applied to actuators

1 to 4 are shown in Fig. 24. The first 4.6 seconds of both

responses shows the open loop resonant behavior of the

structure when no control is applied. At 4.6 seconds, the H∞

controller is applied. The oscillation amplitude is decreased

to its steady state value about 1.4 seconds later. The steady

state oscillation amplitude is about a factor 10 smaller than

the open loop amplitude.

0 10 20 30 40 50 60 70 80 90 100110

100

90

80

70

60

50

40

30

20

10

0

Frequency

Tra

nfe

r F

un

ctio

n E

stim

ate

(d

B)

Figure 18. Frequency response of open loop (experimental).

The experiments match well with the numerical simula-

tion, confirming the findings obtained earlier the optimal

H∞ controller is effective in providing a considerable in-

crease in the amount of vibration suppression, due to its

ability to target the control energy on a few specific modes

of vibrations, as stipulated by the designer. The attenua-

tion provided by the control law for the first five modes of

the nominal structure, compared to the open loop case is

shown in Table 4.

CONCLUSION

The use of piezoelectric film actuators and sensors for ac-

tive vibration suppression in truss structures has been in-

vestigated. Techniques for modeling truss dynamics and

the design robust controllers for suppressing vibrations

have been developed. The techniques have been success-

fully used on an experimental truss structure. The truss

structure was designed to be representative of the com-

plex dynamics of the large flexible space structures under

consideration by the Canadian Space Agency.

The dynamics of truss structures with piezoelectric ac-

tuators and sensors are modeled using a two-step proce-

dure. First, a finite element model of the truss dynamics,

without piezoelectric actuators and sensors, is developed.

Second, the interaction equations for modeling the cou-

pling dynamics of piezoelectric film actuators and sensors

to the finite element model of a truss are developed. This

two-step analysis procedure allows the sensor and actuator

placement to be optimized for a given finite element model.

The dynamic modeling methodology has been validated by

tests on an experimental truss structure.


0 1 2 3 4 5 6 7 81.5

1

0.5

0

0.5

1

1.5x 10

3

xD

isp.(

m)

0 1 2 3 4 5 6 7 81.5

1

0.5

0

0.5

1

1.5x 10

3

yD

isp.(

m)

Time (s)

Figure 19. Displacement of node 24 during H∞ control (experimental).

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 1 (

V)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 2 (

V)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 3 (

V)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 4 (

V)

Time (s)

Figure 20. Control signals during H∞ control (experimental).


0 10 20 30 40 50 60 70 80 90 100110

100

90

80

70

60

50

40

30

20

10

0

Frequency

Tra

nfe

r F

un

ctio

n E

stim

ate

(d

B)

Figure 21. Frequency response during H∞ control (experi-

mental).

Figure 22. Application point

for the continuous disturbance. 00.1

0.20.20.10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Flexible structures are difficult to control because their

dynamics are characterized by a large number of vibra-

tional modes. To reduce computational complexity, con-

troller design is typically performed using a reduced order

model. The H∞ controller design procedure yields a con-

troller that concentrates the control energy on the modes

included in the design model. The design procedure ac-

counts for sensor noise and disturbances resulting from

nonlinearities in the amplifier and piezoelectric actuators.

Control input saturation can be avoided by using a high

penalty on the control energy during the controller design.

The H∞ controller performance is analyzed using a high-

order evaluation model. The simulations showed that the

H∞ controller provides significant increase in damping to

the modes included in the design model, but does affect

the higher-order excluded modes. This behavior is ideal,

as it ensures that the structure will not become unstable

through the excitation of higher-order modes.


0 1 2 3 4 5 6 7 82

1

0

1

2x 10

4xD

isp

.(m

)

0 1 2 3 4 5 6 7 82

1

0

1

2x 10

4

yD

isp

.(m

)

Time (s)

Figure 23. Displacement of node 24 during continuous distur-

bance test with H∞ control (experimental).

Experimental results with the truss structure have con-

firmed the validity of the simulations. Two tests have been

performed, an impact test and shaker test. Comparisons

between the open loop and the closed loop responses show

that the H∞ controller significantly decreases the vibra-

tional mode amplitudes. The controller targets its efforts

on the modes retained in the design model.

Piezoelectric materials are ideally suited for construct-

ing actuators and sensors for vibration suppression in

flexible structures. Polyvinylidene fluoride (PVDF) is ide-

ally suited for sensor construction. It is lightweight, flexi-

ble, and provides a high voltage for a given strain. Piezo-

ceramic materials are suited to actuator construction.

Piezoceramics are stiff, rugged, and provide relatively

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 1

(V

)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 2

(V

)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 3

(V

)

0 1 2 3 4 5 6 7 85

0

5

Actu

ato

r 4

(V

)

Time (s)

Figure 24. Control signals during continuous dis-

turbance test with H∞ control (experimental).

Table 4. Mode Attenuation

Mode Attenuation (dB)

1 13

2 25

3 12

4 29

5 15

large strains when subjected to an electric field. The pre-

dominately linear behavior of the piezoelectric materials,

and the simple manner in which they can be integrated

into a structure, make them a good choice for actuators

and sensors in a vibration suppression control system.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Steven Yeung, Mr.

Howard Reynaud, Dr. George Vukovitch, and Dr.Yan-Ru

Hu at the Canadian Space Agency for their technical and

financial assistance.

This work has been supported by the Canadian Space Agency

under contracts 9F009-0-4140, 9F011-0-0924, 9F028-5-5106, and

9F028-6-6162, and the Natural Sciences and Engineering Re-

search Council of Canada.

BIBLIOGRAPHY

1. J.C. Doyle. IEEE Trans. Autom. Contr. AC-23(4): 756–757

(1978).

2. M.J. Balas. IEEE Trans. Autom. Contr. 27(3): 522–535 (1982).

3. J.J. Allen and J.P. Lauffer. J. Dyn. Syst. Meas. Contr. 119:

(September 1997).

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