JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

43, 3, pp. 487-500, Warsaw 2005

CONTROLLER DESIGN AND IMPLEMENTATION FOR

ACTIVE VIBRATION SUPPRESSION OF A PIEZOELECTRIC

SMART SHELL STRUCTURE

Ulrich Gabbert

Tamara Nestorović-Trajkov

Heinz Köppe

Institut für Mechanik, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

e-mail: ulrich.gabbert@mb.uni-magdeburg.de; tamara.nestorovic@mb.uni-magdeburg.de;

heinz.koeppe@mb.uni-magdeburg.de

The paper is dealing with the model based controller design for a shell
structure attached with piezoelectric patches as actuators and sensors.
The state-spacemodel used for the controller designwas obtained using
the finite element (FE) approach, modal analysis and modal reduction
resulting in a form convenient for the controller development. The opti-
mal LQ controller was designed for the vibration suppression purposes
of a funnel shaped shell structure. The design model for the controller
development was augmented with additional dynamics which takes into
account excitations/disturbances, contributing thus to a better vibra-
tion suppression. The controller was implemented on a funnel-shaped
piezoelectric structure. The structure was intensively investigated expe-
rimentally, and the achieved results of the controlled behaviour with
respect to vibration suppression are presented and discussed.

Key words: active vibration suppression, optimal controller design,
piezoelectric smart structure

1. Introduction

The vibration suppression task represents an important issue, especially in
smart structures, which demand active adaptation of the structural behaviour
in accordance with environmental conditions. The controller design and its
implementation play an important role in achieving this task. In this paper,
the controller design for a piezoelectric shell structure is presented as well as



488 U.Gabbert et al.

its implementation for the vibration suppressionof the structureunder investi-
gation. For the model based controller design approach, a state-space model
of a piezoelectric structure can be developed using the FE analysis, which
represents a convenient technique, especially in early phases of the structure
development, controller design and simulation verification. This approachwas
employed in this paper for obtaining a model of the piezoelectric structure.

Piezoelectric sensing and control have been addressed in different papers
(e.g. Gabbert and Tzou, 2001; Rao and Sunar, 1993; Tzou and Tseng, 1990;
etc.). This paper concerns vibration control of a shell structure using distri-
buted piezoelectric actuator/sensor patches and optimal LQ tracking with
additional dynamics implemented as a part of the overall design procedure of
smart structures.

In the first part of the paper, themodel development using the FE appro-
ach is explained, resulting in a state space representation convenient for the
controller design.Themodel based controller design is presented subsequently,
combining the optimal LQ controller with an augmented design model which
takes into account the influence of periodic excitations in the frequency range
of interest, contributing thus to a better vibration suppression. The control-
ler was implemented and tested on a funnel-shaped shell structure attached
with piezoelectric actuators and sensors. Experimental results were obtained
applying Hardware-in-the-Loop simulation. Finally, the influence of the actu-
ator/sensor placement was also considered and illustrated with an example.

1.1. Finite element based development of a state space model for piezo-

electric structures

The approach is based on the development of equations of motion for a
piezoelectric structure approximated by a given number of finite elements. A
semi-discrete form of the equations of motion of the finite element can be
derived using an approximation method of displacements and electric poten-
tial (Berger et al., 2000; Gabbert, 2002; Görnandt and Gabbert, 2002). That
approach was used to develop a comprehensive library of multi-field finite
elements (1D, 2D, 3D elements, thick and thin layered composite shell ele-
ments etc.), which was implemented in our finite element package COSAR
(see: http://www.femcos.de). For simulation of thin lightweight structures,
curvedmulti-layer shell elements developed on the basis of the classical Semi-
Loof element family (Irons, 1976) have proved to give good results. Following
the Kirchhoff-Love hypothesis, three different approaches were developed to
include fully coupled electro-mechanical behaviour into the SemiLoof elements
(Gabbert et al., 2002; Seeger et al., 2002). Thematerial properties of an acti-



Controller design and implementation... 489

ve fibre composite can be calculated by applying a homogenisation procedure
(Berger et al., 2003).
As a result of theFEanalysis, behaviour of a structure approximatedbyan

arbitrary number of finite elements can be describedwith assembled equations
of motion in a semi-discrete form (Gabbert et al., 2002; Nestorović-Trajkov et
al., 2003a)

Mq̈+Ddq̇+Kq=F (1.1)

where M, Dd and K are the mass matrix, the damping matrix and the stif-
fness matrix, respectively, and vector q represents the vector of generalized
displacements (includingmechanical displacements andelectric potential), and
contains all degrees of freedom. The total load vector F is divided into the
vector of external forces FE and the vector of control forces FC

F =FE +FC =Ew(t)+Bu(t) (1.2)

where the forces are generalized quantities including also electric charges. The
vector w(t) represents the vector of external disturbances and u(t) is the
vector of the controller influence on the structure. The matrices E and B
describe the positions of forces and control parameters in the finite element
structure, respectively.As a convenient procedure for obtaining the state space
model, modal truncation is adopted since the high order of the FE model
represented by equation (1.1) is not suitable for the controller design and the
reduction of the model order is required. A decoupled system of equations
(1.3) in modal coordinates z is obtained by performing ortho-normalization
with Φ>mMΦm = I and Φ

>

mKΦm = Ω, where the modal matrix Φm and
the spectral matrix Ω are obtained from the solution to the linear eigenvalue
problem for (1.1). In the decoupled system of equations

z̈+∆ż+Ωz=Φ>mF (1.3)

thematrix ∆=Φ>mDdΦm represents themodal dampingmatrix.Generalised
displacements q are related to modal coordinates z by

q(t)=Φmz(t) (1.4)

In the modal truncation procedure, the modal displacement vector z is par-
titioned and only a part zr corresponding to selected eigenmodes of interest
for the control is retained. Introducing the modal reduced state vector

x(t)=

[
zr(t)
żr(t)

]
(1.5)



490 U.Gabbert et al.

the modal reducedmodel is obtained in the state space form

ẋ=

[
0 I

−Ωr −∆r

]
x+

[
0

Φ>r B

]
u(t)+

[
0

Φ>r E

]
w(t) (1.6)

where thematrices Ωr,∆r and Φr are obtained fromanappropriatepartition
of thematrices Ω,∆ and Φm, respectively.Written in a standard state space
form, the state equation of the model becomes

ẋ(t)=Ax(t)+Bu(t)+Ew(t) (1.7)

Using a similar procedure, a state space formulation of the output equation is
obtained

y=Cx(t)+Du(t)+Fw(t) (1.8)

which assumes, in a general case, the influence of the control and external
inputs on the outputs.

2. Controller design

A state-space model of the controlled plant, obtained from the finite ele-
ment analysis andmodal reduction, can be used for the controller design. The
model in the form of equations (1.7), (1.8) represents a starting point for the
controller designwhere, in a general case, the presence of the disturbance vec-
tor w in the state and output equations is assumed. The optimal LQ control
lawbasedon the tracking systemdesignwithadditional dynamics (Nestorović-
Trajkov et al., 2003b; Vacarro, 1995) can be viewed as a successful means for
the vibration suppression in smart structures.

The controller design assumes rejection of disturbances/excitations, which
cause vibrations, providing thus the controlled system outputs with asympto-
tically reduced oscillation magnitudes.

Thediscrete-time state space equivalent of the state spacemodeldeveloped
through the FEM procedure and modal reduction is used for the controller
design

x[k+1]=Φx[k]+Γu[k]+εw[k]
(2.1)

y[k] =Cx[k]+Du[k]+Fw[k]



Controller design and implementation... 491

where

Φ=eAT Γ=

T∫

0

eAτB dτ ε=

T∫

0

eAτE dτ (2.2)

and T is the sampling time.

The controller design includes available a priori knowledge about the oc-
curring disturbance type contained in the additional dynamics, which repre-
sents an important part of the controller design procedure. The additional
dynamics is introduced in order to compensate for the presence of the distur-
bance providing at the same time the tracking of the reference trajectories
described by models with the same poles as those of disturbances. Such a
controller with the additional dynamics serves for controlling purposes if the
reference input to be tracked and the disturbance acting upon the structure
can be described by a rational discrete function. This condition is fulfilled by
a sine function used as a disturbancemodel. A special interest in investigation
of this type of disturbances has arisen from the fact that periodic disturban-
ces with frequencies corresponding to eigenfrequencies of a smart structure
can cause resonance. Taking it into account, this type of disturbance can be
considered the worst study case.

The additional dynamics is formed from the coefficients of the polynomial

δ(z) =
∏

i

(z− eλiT)mi = zs+ δ1z
s−1+ . . .+ δs (2.3)

where λi are the poles of the reference input and/or excitation/disturbance.
A state space realization of the additional dynamics is expressed in the form
of matrices

Φa =




−δ1 1 0 · · · 0
−δ2 0 1 · · · 0
...

...
...
...
...

−δs−1 0 0 · · · 1
−δs 0 0 · · · 0



Γa =




−δ1
−δ2
...

−δs−1
−δs




(2.4)

In the case ofmultiple-inputmultiple-output (MIMO) systems, the additional
dynamics must be replicated in q parallel systems (once per each output),
where q is the number of outputs.

The replicated additional dynamics is described by

Φ
def
= diag(Φa, . . . ,Φa︸ ︷︷ ︸

q times

) Γ
def
= diag(Γa, . . . ,Γa︸ ︷︷ ︸

q times

) (2.5)



492 U.Gabbert et al.

The discrete-time design model (Φd,Γd) is formed as a cascade combination
of the additional dynamics (Φa,Γa) or (Φ,Γ) and the discrete-time plant
model (Φ,Γ)

xd[k+1]=Φdxd[k]+Γdu[k] (2.6)

where

Φd =

[
Φ 0

Γ∗C Φ∗

]
Γd =

[
Γ

0

]
xd =

[
x[k]
xa[k]

]
(2.7)

and Φ∗ denotes Φa or Φ, and Γ
∗ represents Γa or Γ, depending on whe-

ther the controlled structure is modelled as a single-input single-output or
a multiple-input multiple-output system, respectively. The gain matrix L of
the optimal LQ regulator is calculated on the basis of the design model (2.6)
in such a way that the feedback law u[k] = −Lxd[k] minimizes performance
index (2.8) subject to constraint (20), where Q and R are symmetric, positive-
definite matrices

J =
1

2

∞∑

k=0

(
x
>

d [k]Qxd[k]+u
>[k]Ru[k]

)
(2.8)

The control system is designed for the realization (Φd,Γd). The feedback gain
matrix L in the control law is partitioned into submatrices L1 and L2 formed
from the first n and last q×s columns of the matrix L, respectively

L=
[
L1 L2

]
(2.9)

Thus the feedback gain matrix L1 corresponds to the state variables of the
controlled structure, while the feedback gainmatrix L2 pertains to the rest of
the state variables in the design state vector xd introduced by the additional
dynamics.

Thedesignof the controller involves the estimationof the state variables. In
the state spacemodel obtained using the FE approach, the state variables are
modal variables which are not measurable and their estimation is therefore
necessary. For the estimation of the state variables, a Kalman filter can be
used.Given the covariances Qw and Rv of the process andmeasurement noise
respectively, the Kalman estimator is defined with the following equations

x̂[k] =x[k]+Lest[k](y[k]−Cx[k])
(2.10)

x[k] =Φx̂[k−1]+Γu[k−1]



Controller design and implementation... 493

where the Kalman gain matrix is

Lest[k] =P[k]C
>
R
−1
v (2.11)

and

P[k] =Mk[k]−Mk[k]C
>(CMk[k]C

>+Rv)
−1
CMk[k]

(2.12)

Mk[k+1]=ΦP[k]Φ
>+εQwε

>

Thematrices P and Mk are determined by solving equations (2.12).

The optimal LQ tracking control systemwith the additional dynamics and
Kalman’s estimator is implemented as shown in Fig.1.

Fig. 1. The optimal LQ tracking systemwith additional dynamics and a state
estimator

3. Implementation in a funnel-shaped structure

Active vibration control using the described controller design and model
development procedure was implemented for the vibration suppression of a
funnel-shaped shell structure shown in Fig.2. The funnel is the inlet part
of a magnetic resonance image (MRI) tomograph (Fig.3) used in medical
diagnostics.



494 U.Gabbert et al.

Fig. 2. (a) The funnel of anMRI tomographwith actuator/sensor placement,
(b) the finite element mesh of the funnel

Six actuator-groups and six sensors attached to the surface of the funnel
are used for experimental modal analysis and vibration control of the structu-
re. They are denoted as 1L, 2L, 3L for the left-hand side actuators and sensors
and 1R, 2R, 3R for the right-hand side ones. Each of six actuators represents a
group consisting of four piezoelectric patches (function modules). Each of six
sensors is a single piezoelectric patch. The function modules are made of pie-
zoceramic films (PZTfilmSonoxP53) with standard dimensions 50×25×0.2
[mm]. Applying the FEM-based approach for the analysis of the funnel be-
haviour and numeric model development (using the finite element software
COSAR), the eigenfrequencies were determined and, on the basis of the com-
parison with experimental results, a reduced order state space model of the
funnel was adopted for the control of the funnel eigenmodes in the frequency



Controller design and implementation... 495

Fig. 3. AnMRI tomograph

range up to 35Hz, where the eigenfrequencies of interest are: f1 = 9.573Hz,
f2 =23.333Hz, f3 =31.439Hz. As the worst study case of the controller de-
sign (with respect to resonance), the excitations brought to the funnel using
the shaker are selected as sine signals with frequencies corresponding to the
eigenfrequencies of the funnel. The experimental rig for the implementation
of the designed optimal LQ controller with the additional dynamics, including
the funnel as a part of the Hardware-in-the-Loop (HiL) system is represented
in Fig.4.

Fig. 4. The funnel of theMRI tomographwithin the HiL systemwith dSPACE



496 U.Gabbert et al.

Assuming the sine excitation with the frequency f1, the control system
was designed with the controller parameters Q= I8×8 and R=10. The time
response of the sensor S1Rand the control effort of the actuatorA2Robtained
experimentally with this control system in the presence of the sine excitation
with the frequency f1 is shown in Fig.5. The periodwithout control is clearly
indicatedby the zero control inputandobviously greater vibrationmagnitudes
of the sensor response.

Fig. 5. The response of sensor S1R and control (actuator A2R) in the presence of
the sine excitation with the frequency f1

In the presence of the excitation obtained as a sum of the sinusoids with
frequencies f1, f2, f3, the designed controller exhibits behaviour as shown in
Fig.6.

Fig. 6. The sensor response and control with an excitation containing three
eigenfrequencies (the control system designed to control the first eigenfrequency)

With the excitation obtained as a sum of the sinusoids with frequencies
f1, f2, f3, the control system is designed with the order of the design model
12 and the weighting matrices Q = I12×12 and R = 10. This control system
performs better vibration suppression (Fig.7) in comparison with Fig.6 using
a slightly greater control effort.

The influence of the acutator/sensor placement is shown through the com-
parison of the results for acutaror/sensor pairs A2R–S1R and A2R–S2L
(Fig.8 and Fig.9). According to the criterion of the overlapped deforma-
tion, distributions for the first five eigenmodes of the funnel, regions of ac-



Controller design and implementation... 497

Fig. 7. The sensor signal and control with an excitation containing three
eigenfrequencies (the controller designed for simultaneous control of the first three

eigenfrequencies)

tuator/sensor position 2L, 2R are determined as the best regions for the actu-
ator/sensor placement. The results shown inFig.8 andFig.9 confirmthis opti-
mal actuator/sensor placement. Actuator/sensor pair A2R–S2L gives better
vibration suppression results: in the controlled case the vibration amplitudes
are reduced to approximately 7%of the uncontrolled amplitudes (Fig.9), whi-
le with actuator/sensor pair A2R–S1R this reduction reaches approximately
60% of the uncontrolled amplitudes (Fig.8).

Fig. 8. Actuator/sensor pair A2R–S1R: the excitation obtained as a sum of three
sinusoidal signals and response of the sensor in the presence of this excitation

Fig. 9. Actuator/sensor pair A2R–S2L: sensor response and control with an
excitation containing three eigenfrequencies

The presented results show efficiency of the control system applied for the
reduction of the vibration amplitude. According to expectations, the control
systemdesigned taking into account three different frequencies of the periodic



498 U.Gabbert et al.

excitation exhibits better behaviour in the presence of excitations obtained as
a sum of three sinusoidal signals than the controller which takes into account
only one eigenfrequency in the presence of the same excitation.

4. Conclusion

In this paper, the model based controller design as a part of the overall
design procedure for the development of piezoelectric smart structures is pre-
sented. The state space model used as the starting point for the controller
design is obtained using the FE approach and modal analysis resulting after
appropriatemodal reduction in a form convenient for the optimal LQ control-
ler design.

The innovation in the optimal LQ controller design is achieved by intro-
ducing additional dynamics to the formulation of the design model for the
controller design and connecting the two approaches in order to achieve bet-
ter controlled performance of the closed-loop systemwith respect to vibration
suppression in the presence of considered excitations. The influence of periodic
excitations/disturbances taken into account through the additional dynamics
is considered the worst study case due to the possibility of the resonance. The
suppression of vibrations caused by such excitations and disturbances repre-
sents therefore an important task which was, in this case, demonstrated to
be successfully accomplished by applying the control system to the considered
funnel shaped piezoelectric shell structure.

Acknowledgement

This work has been partially supported by a postgraduate program of the Ger-

manFederal State of Saxony-Anhalt. It wasmotivated and supported by cooperation

with the Siemens company in the frame of the German industrial Research project

’Adaptronik’ supported by the GermanMinistry for Education and Research. These

supports are gratefully acknowledged.

References

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500 U.Gabbert et al.

Projektowanie i implementacja sterownika do aktywnej redukcji drgań

powłokowej struktury ”inteligentnej” z elementami piezoelektrycznymi

Streszczenie

Praca dotyczy problemu modelowego projektowania struktury powłokowej z na-
klejonymi do niej elementami piezoelektrycznymi. Zastosowany w procesie projek-
towania model, który opisano w przestrzeni stanu, został sformułowany za pomocą
metody elementów skończonych, analizy modalnej i modalnej redukcji, co dało wy-
godny do implementacji rezultat. Optymalny sterownik LQ zaprojektowano w celu
ograniczenia amplitudy drgań struktury powłokowej o lejkowatym kształcie. Użyty
model został uzupełniony o zjawiska dynamiczne uwzględniające zewnętrzne wymu-
szenia i zakłócenia oddziaływujące na powłokę, co poprawiło docelową efektywność
aktywnego tłumienia drgań.Ostatecznie, dokonanopraktycznegowdrożenia sterowni-
ka, którywspółpracowałz lejkowato-kształtnąpowłokąwyposażnąwpiezoelektryczne
czujniki i elementywykonawcze.Powłokowastruktura została następnie poddana ba-
daniom eksperymentalnym, a rezultaty pomiaróww kontekście skuteczności redukcji
drgań uzyskanej wskutek implementacji analizowanego sterownika zaprezentowano
i omówionow pracy.

Manuscript received December 6, 2004; accepted for print February 8, 2005