Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 12, N

o
 3, 2014, pp. 235 - 249 

ROBUST MIXED H2/H ACTIVE VIBRATION CONTROLLER 

IN ATTENUATION OF SMART BEAM  

UDC 62-135:534.1, 624.9 +519.6 

Atta Oveisi, Tamara Nestorović


Mechanics of Adaptive Systems, Ruhr-University Bochum, Germany 

Abstract. The lack of robustness of the mechanical systems due to the unmodeled 

dynamics and the external disturbances withholds the performance and optimality of the 

structures. In this paper, this deficiency is obviated in order to reach the desired robust 

stability and performance on smart structures. For this purpose a multi-objective robust 

control strategy is proposed for vibration suppression of a clamped-free smart beam with 

piezoelectric actuator and vibrometer sensor in an LMI framework which is capable of 

handling weighted exogenous input signals and provides desired pole placement and 

robust performance at the same time. An accurate model of a homogeneous beam is 

derived by means of the finite element modal analysis. Then a low order modal system is 

considered as the nominal model and remaining modes are left as the multiplicative 

unstructured uncertainty. Next, a robust controller with a regional pole placement 

constraint is designed based on the augmented plant composed of the nominal model and 

its accompanied uncertainty by solving a convex optimization problem. Finally, the 

robustness of the uncertain closed-loop model and the effect of performance index weights 

on the system output are investigated both in simulation and practice.  

Key Words: Piezoelectric, Vibration Suppression, Robust Control, Smart Beam, Finite 

Element Method 

1. INTRODUCTION

The concept of adaptive materials has changed the possibilities for structure design, 

particularly self-diagnosis and self-controlled arrangements, namely smart structures. 

This perception is achieved in practice by introduction of multifunctional material based 

transducers which allows the structure to be sensitive towards the environmental stimuli. 

Adaptive structures play a crucial role in challenging areas of applied science where high 

quality performance in extreme environments is an urgent requirement. An active structure 

Received September 24, 2014 / Accepted November 2, 2014 
Corresponding author: Tamara Nestorović  

Universitätsstr. 150, D-44801 Bochum, Germany 
E-mail: tamara.nestorovic@rub.de 

Original scientific paper



236 A. OVEISI, T. NESTOROVIĆ 

 

contains elements such as sensors and actuators, which delivers data in forms of the states 

of the system and will affect the passive response of the structure. 
The evolution of mechanical and aeronautical structures requires them to be lighter 

and at the same time controllable. Overcoming the defect of these systems, specifically 
their sensitivity to unwanted disturbances, has attracted many researchers over the past 
couple of decades in the fields of structural vibration analysis, damage detection, vibration 
control and noise control [1, 2]. 

Of various suggested methods of dynamics control, the use of active control techniques 
in vibration suppression of the light structures is proven to be more effective, where the 
additional masses of stiffeners or dampers should be avoided. Active techniques are also 
more suitable in the cases where the disturbance to be cancelled or the properties of the 
controlled system vary with time [3]. 

Piezoelectric actuators are broadly employed in many practical applications due to 

their capability of coupling strain and electric field. In order to control structural 

vibrations, piezoelectric actuators can be easily bonded on the vibrating structures [4].  
In terms of the dynamic performance, the high-efficient dynamic modeling and 

appropriate control law design are the two key points. For the purposes of dynamical 
modelling, the finite element method has recognized to be one of the most popular methods. 
Reviews such as the one presented by Benjeddou [5] provide a condensed overview of the 
development in the field of the Finite Element Modelling (FEM) modelling of active 
structures. The development of the FEM tools has proceeded at the same rapid pace in the 
next decade, followed by the development of active structural control techniques, as 
reported in the overview by Le Gao et al. [6]. Various types of controller design methods 
such as velocity feedback control [7], high gain feedback regulator [8], linear quadratic 

regulator (LQR) approach [9], H2 control [10], H control [11] have been studied by former 
scientists. In addition, some others evaluate the performance of control algorithms in 
vibration suppression of flexible structure experimentally [12]. The authors of this paper 
have made a contribution to the research field of piezoelectric adaptive structures by 
dedicating their work to the development of necessary finite elements for piezoelectric 
coupled-field problems [13], demonstrating advantages of the FEM approach over other 
methods [14], investigating different aspects of modelling active structures [15], 
implementing developed tools into commercially available software packages [16], dealing 
with control techniques for adaptive structures [17], etc. 

In this work, an accurate model of a piezolaminated cantilever beam is derived by 
means of the finite element modal analysis. The derived formulation provides the state 
space model relating the actuator voltage to sensor voltage. The obtained model is 
capable of offering a finite order model that shall be considered as nominal system while 
the remaining high order states are left as multiplicative unstructured uncertainty of 
modeling. Then, a multi-objective robust controller is designed based on the augmented 
plant composed of the nominal model and its accompanied uncertainty. In addition, a 
regional pole placement constraint is included within the Linear Matrix Inequality (LMI) 
framework to improve closed-loop transient performance. The rest of the paper has the 
following order. In section 2 the configuration of the experimental setup is described. 
This will be used to verify the performance of the regulated controller in real time 
implementation. In section 3 the finite element based modal analysis is performed in 
order to calculate the eigen frequencies and mode shapes of the coupled electro-elastic 
system. Then in section 4 the aforementioned robust controller will be introduced and 
finally the performance of the closed loop system will be evaluated in the next section. 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 237 

 

2. EXPERIMENTAL SETUP 

The structure of experimental smart beam is presented in Fig. 1. The piezo-laminated 

beam consists of a cantilever aluminum beam with Young’s modulus 70 GPa and density 

2.7 g/cm³. In addition since the ultimate goal is to suppress the vibration two piezoelectric 

actuators (DuraActTM P-876.A15) are attached to the beam at the same side. (see Fig. 1) 

 

Fig. 1 Geometry of the smart beam 

 The feedback channel entails the measurement signal namely, the signal measured by a 

scanning digital laser Doppler vibrometer VH-1000-D. This will provide the measurement 

of the velocity of the lateral vibration at a point, near the free end of the beam. Schematic 

configuration of closed-loop vibration control system is presented in Fig. 2. 

 

Fig. 2 Sketch of experimental setup 

It is worthwhile mentioning that the plant has two inputs: the control input which acts 

on the actuator piezo-patch and the disturbance signal which excites the system through 

the disturbance channel. Moreover, the only output of the system is recorded using the 

previously mentioned vibro-meter. 



238 A. OVEISI, T. NESTOROVIĆ 

 

For implementing the controller in real time, dSPACE digital data acquisition and 

real-time control system with DS1005 digital signal process board are used. Connection 

of the digital data acquisition system with the actuators and the computer is provided by 

an ADC Board DS2004 (Analog to Digital converter) and a DAC Board DS2102 (Digital 

to Analog converter). To increase the working range of the DAC boards the control input 

is amplified (PI E-500). The control law, for the active vibration control of the smart 

beam, is then implemented on MATLAB platform. Finally, the control system is 

downloaded to the dSPACE digital data acquisition and real-time control system. 

3. SYSTEM MODELING 

One should notice that the torsional modes are not considered in controller design 

because they are not relevant for the bending vibration. It should be mentioned that due to 

the previous research the dominant mode shape of the flexible beam is the first mode 

shape [18]. 

The dynamics of the actuation is addressed by means of the FEM analysis in coupled 

electro-mechanical domain. This leads to an ordinary differential equation which then 

will be converted to a linear time invariant (LTI) system since it is a convenient model 

for the work in the computer aided control system design. It is assumed that the 

displacements are small enough so that the dynamics of the system remains in linear 

piezo-elasticity. The finite element method presents the dynamic equation of motion in 

matrix representation as: 

 ,Mq Cq Kq F    (1) 

with M, C and K being the mass, damping and stiffness matrices. Also, q represents the 

nodal states of displacement ( , 1, 2,...)
T

i
u i   and electric potential ( , 1, 2,...)

T

j
j   : 

 
1 1

[ ]
T T T

n n
q u u    (2) 

F shows the applied excitation which contains the external forces that is assumed to be 

zero because the external input disturbance is expected to affect the system from the same 

channel as the control input. The vector of control forces is therefore: 

 ( ),F Bu t  (3) 

B matrix describes the position of the generalized control effort in the finite element 

structure with u consisting of all modal inputs. For the control design purposes the 

measurement signal is represented in terms of system states and plant inputs as: 

 
0 0

,
q v

y C q C q   (4) 

In which C0q and C0v are the output displacement and output velocity matrices, 

respectively; they are calculated using the FE procedure and choosing the appropriate 

sensor location. By applying the conventional harmonic solution of q = e
it

 one can 

easily find natural frequencies j and mode shapes j (j = 1,2,...,n) solving the determinant 
of homogenous system of algebraic equations. The solution can be represented in matrix 

form as: 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 239 

 

1

2

0 0

0 0
,

0 0
n

 
 


  
 
 

 

 

11 21 1

12 21 2

1 2 3 4

1 2

,

n

n

n n nn

   
 
  
          
 
   

 

(5) 

The nodal model representation (1) can be transformed to modal coordinates by 

applying the following conversion: 

 ,
m

q q   (6) 

where qm is the vector of generalized modal displacement. Using the symmetricity of the 

mass and stiffness matrices one can easily obtain the transformed matrices as [19] : 

( ),
T

m j
M M diag m     

2
( ),

T

m j j
K K diag m      

(7) 

Similarly, by using the same transformation and the orthogonality of mode shape one 

can find the modal damping matrices under the assumption of the proportional damping 

to be: 

 ,C M K    (8) 

By selecting the state vector to be  
T

m m
x q q   the state space model will be: 

,

,

x Ax Bu

y Cx Du

 

 
 (9) 

where: 

 
00

, , , 0,
2

mq mv

m

A B C C C D
BZ

   
            

 (10) 

while 
2 1

m m
M K


  and ( )

j
Z diag   with j being the damping ratio of jth mode, 

Bm = 
T
B, Cmq = C0q, Cmv = C0v. 

4. CONTROLLER DESIGN 

The closed-loop system by considering multiplicative uncertainty will be as shown in 

Fig. 3. 



240 A. OVEISI, T. NESTOROVIĆ 

 

P(s)

K(s)

W(s) 

+

+

u

y

u

yw

 

Fig. 3 Closed-loop system with multiplicative uncertainty 

where P(s) is the nominal plant, K(s) is desired controller, ∆ is a stable transfer function, 

where |||| < 1 and W(s) is the weighting function for multiplicative uncertainty, that 

satisfies following equation: 

 
( )

1 ( )
( )

realP s W s
P s

   (11) 

where Preal(s) is the transfer function of the real system, by considering all or some of the 

higher modes. Note that, reduction of the order of the nominal plant will hold the 

designed controller’s order in a lower value, but the price will be reduced performance. 

For robust stability, one should have ||Tyu|| < 1, where Tyu is the transfer function 

from u to y when  is removed [20]. However, to handle the stochastic aspects such as 

measurement noise and random disturbance, despite robust H, only H2 performance is 

functional. And finally, for appropriate disturbance rejection and control effort the 

conventional optimization problem is to minimize ||y
T
Qy + u

T
Ru||, where Q and R are 

two weighting functions that indicate the relative importance of disturbance rejection and 

control effort, respectively. For minimizing performance index ||y
T
Qy + u

T
Ru||, we should 

minimize ||T[y u]Tw||2 instead, where w is a bounded H2 norm exogenous disturbance. This 

will be addressed later. The transient response of a linear system is well known to be related 

to the locations of its closed-loop poles. This is the next issue that has to be addressed. 

Since Tyu is equivalent to TuwW(s) [21], the above system can be represented in Fig. 

4 with all of the constraints that have to be satisfied in order to reach the predefined  

H2 / H performance and optimal control effort. 

P(s)

K(s)

W(s)

+

+ 2z

u

z

y





w
Q

R

 

Fig. 4 Desired input and outputs of augmented plant 

Now assume that a state space representation of the open-loop system in Fig. 4 (by 

ignoring K(s)) is: 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 241 

 

 

1 2

1 2

2 2 21 22

1y y

x Ax B w B u

z C x D w D u

z C x D w D u

y C x D w

   

  


  


  
  

 (12) 

where u and w are control input and disturbance, respectively. Our objective is to design 

a dynamic output-feedback controller with the state space realization: 

 
K K

K K

A B y

u C D y

 



 


 
 (13) 

where   is the state variable of the controller. Therefore, the corresponding closed-loop 
system containing the performance and robustness channels will be:  

 
1 1

2 2 1

cl cl cl cl

cl cl cl

cl cl cl

x A x B w

z C x D w

z C x D w



 


 
  

 (14) 

Our three design objectives can be expressed as follows: 

H Performance: the closed-loop RMS gain from w to z does not exceed  if and 

only if there exists a symmetric matrix X such that [22] : 

 

1

1

2

1 1

0

0

T T

cl cl cl cl

T T

cl cl

cl cl

A X X A B X C

B I D

C X D I

X



  





 
 

  
  



 (15) 

This LMI constraint is used to minimize ||Tzw|| (closed-loop H gain from disturbance 

to z output channel). 

H2 Performance: the H2 norm of the closed-loop transfer function from w to z2 does not 

exceed v if and only if Dcl2 = 0 and there exist two symmetric matrices X2 and Q such that 

[23]: 

 

2 2

2 2

2 2 2

2

0

0

( )

T

cl cl cl

T

cl

cl

T

cl

A X X A B

B I

Q C X

X C X

Trace Q 

 
 

 

 
 

 



 (16) 

Pole placement: the closed-loop poles lie in the LMI region: 

 { : 0}
T

D z C L Mz M z      (17) 

with L = L
T
 = {ij}1i, jm and M = [ij]1i, jm if and only if there exists a symmetric matrix 

Xpol satisfying: 



242 A. OVEISI, T. NESTOROVIĆ 

 

1 ,
0

0

T

ij pol ij cl pol ji pol cl
i j m

pol

X A X X A

X

  
 

    


 (18) 

For tractability in the LMI framework, we must seek a single Lyapunov matrix: 

 
2

:
pol

X X X X


    (19) 

that enforces all three sets of constraints. Factorizing X as: 

 
1

1 2 1 2

0
, : , :

0
T T

R I S
X X X X X

M I N


   

     
   

 (20) 

and, introducing the change of controller variables [25]: 

 
2

2 2

:

:

( )

K K K

T

K K K y

T T

K K K y K K y

B NB SB D

C C M D C R

A NA M NB C R SB C M S A B D C R

  


 


    

 (21) 

the inequality constraints on X are readily turned into LMI constraints in the variables R, 

S, Q, AK, BK, CK and DK [22], [24]. This leads to the suboptimal LMI formulation of our 

multi-objective synthesis problem, which is defined as: 

Minimize 
2

. . ( )trace Q   over variables R, S, Q, AK, BK, CK, DK and 
2
 satisfying [26]: 

2 2 2

2 2

1 2 1

1 1

2

1 2 1

2 22 2 22

2 2

0

0

T T T T

K K K K y

T T T

K y y K

K K y

K y

K y

K y

K K y

K K y

ij ij

K

AR RA B C C B A A B D C

H A S SA B C C B

H H

C R D C C D D C

B B D D H

SB B D H

I H

D D D D I

Q C R D C C D D C

H R I

H I S

AR B C A B D CR I

A SAI S



 

   

 

     


  


  

 


  


  

  
 

 
 
 

  
 

 

2

2
1 ,

2

0

2 2

0

21 22 1

0
( )

( )

0

K y

T T T T

K K

ij T T T T

K y y K
i j m

K y

B C

RA C B A

A B D C A S C B

Trace Q

D D D D





 

 

  
  

  

 
      





 

 

(22) 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 243 

 

Given optimal solutions  
*
,Q

*
 of this LMI problem, the closed-loop H and H2 

performances are bounded by: 

 
* *

2 2
, ( )T T trace Q

 
   (23) 

5. CASE STUDY AND DISCUSSION 

This section presents the vibration damping quality of the proposed method both in 

simulation and experiment. For implementation of the controller, a structure consisting of 

an aluminum clamped beam with two piezoelectric patches is used. The patches are 

attached on the same side of the beam (see Fig. 1). The model of the structure for control 

design purposes is obtained based on the method described before. Since the actuator 

placement plays an important role in vibration control performance the optimal placement 

of the actuator is addressed based on the mixed H2 / H method that is described by 

Nestorović and Trajkov [27]. 

Firstly, two shape numbers of the clamped beam are considered as nominal model and 

higher order modes remain as unstructured uncertainty. In addition, a weighting function 

for multiplicative unstructured uncertainty that satisfies Preal(s)/P(s) = Wunc(s) + 1 is 

considered. With Preal(s), P(s) and Wunc(s) being the full order transfer function of the 

system, nominal transfer function and frequency based appropriate weighting function 

representing the unstructured uncertainty, respectively. Fig. 5 shows the weighting 

functions that are considered for modeling unstructured uncertainty, disturbance and 

H2 / H performance. 

10
-1

10
0

10
1

10
2

10
3

-80

-60

-40

-20

0

20

40

60

M
a

g
n

it
u

d
e

 (
d

B
)

 

 
Bode Diagram

Frequency  (rad/s)

Plant

Wunc

Wperf

Wdist

 

Fig. 5 The relation of weighting function to real system 



244 A. OVEISI, T. NESTOROVIĆ 

 

The desired controller design is carried out by solving convex optimization problem 

that is formulated in Eq. (22). For obtaining an appropriate H performance, the 

magnitude of  | | should be under unit and for increasing the performance one should 
minimize H2 norm from exogenous disturbance to performance index. The relative 

magnitudes of Q and R determine the relative importance of disturbance rejection (vibration 

suppression) to control effort (actuator saturation). To improve transient performance, as 

mentioned before, one shall resort to an additional regional pole placement constraint in 

order to achieve a better closed-loop damping across the uncertainty range. This places 

the closed-loop poles into a suitable sub-region of the left-half plane that can be 

expressed as an additional LMI constraint. A typical example of LMI region that is 

commonly treated in multi-objective synthesis that guarantees H2 stability is the conic 

sector centered at the origin and with inner angle 2 = 2cos
1

() [22]. In this work, the 

closed-loop damping coefficient is assumed to be  = 0.1. 
The controller is designed by setting Q  = 10. Comparison of the impulse response of 

the closed-loop system with this controller and the impulse response of the open-loop 

system (Fig. 6) shows the performance of the controller in suppressing the vibration.  

0 1 2 3 4 5
-40

-30

-20

-10

0

10

20

30

40

 

 
Impulse Response

Time (seconds)

A
m

p
li
tu

d
e

Open-loop system

closed-loop system

 

Fig. 6 Impulse response of the open-loop and closed-loop system 

 

Actuator voltage of this controller during the impulse response is plotted in Fig. 7. 

As one can see, the maximum amplitude of the actuator voltage is under about 20 Volts. 

In addition, comparison of frequency responses of closed-loop system and open-loop 

system is shown in Fig. 8 which shows that the amplitude is reduced in the nominal 

model natural frequencies. 

 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 245 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-80

-60

-40

-20

0

20

40

Time (seconds)

A
c
tu

a
to

r 
V

o
lt
a

g
e

 (
V

.)

 

Fig. 7 Input control for impulse response of the closed-loop system 

10
1

10
2

10
3

-60

-50

-40

-30

-20

-10

0

10

20

30

M
a

g
n

it
u

d
e

 (
d

B
)

 

 
Bode Diagram

Frequency  (rad/s)

Open-loop system

closed-loop system

 

Fig. 8 Bode diagram of closed-loop system and open-loop system 

For investigation of the robust performance of the uncertain closed-loop system with 

the designed controller by structured singular value analysis Fig. 9 is obtained. This plot 

shows upper/lower bounds of uncertain closed-loop structured singular values in 

frequency domain.  



246 A. OVEISI, T. NESTOROVIĆ 

 

10
1

10
2

10
3

0

0.5

1

1.5

2

Frequency (rad/sec)

M
u

 u
p

p
e

r 
b

o
u

n
d

s

 

 

 

Fig. 9 µ bounds of uncertain closed-loop system 

The performance margin is the reciprocal of the structured singular value and if the 

magnitude of the structured singular value were under unit, in entire frequency range, the 

system would have robust performance. Therefore, upper bounds from structured singular 

value become lower bounds on the performance margin and critical frequency associated 

with the upper bound of the structured singular value, here is critical = 87rad/sec. In 
addition, the system can tolerate up to 557% of the modeled uncertainty without losing 

desired performance.  

Through the experimental implementation of the control law on the smart structure 

the possibility of the successful vibration control performance is evaluated on full order 

system. The vibration amplitude suppression will be demonstrated under the harmonic 

excitation of the piezo-beam through the control channel and the results obtained using 

hardware in loop system with dSPACE RTI platform. Experimental excitation is 

considered to be harmonic F(t) = Asin(2fjt), with fj being the first bending resonant 

frequency of the clamped piezo-beam. The closed-loop system is implemented on the real 

time data acquisition platform of the dSPACE with sampling frequency of 10 kHz. The 

predefined task of the controller is to guarantee the robust stability and performance in 

conjugation with real time vibration amplitude suppression in frequency ranges close to 

resonance eigenvalues. Therefore, investigations are carried out in time domain by means 

of the experimental setup shown in Fig. 10. 

 

Fig. 10 Experimental rig of the closed-loop system 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 247 

 

For the analysis in time domain the sinusoidal excitation signal is generated in 

Simulink and lead out through the dSPACE DAC. The frequency of the excitation is 

adjusted experimentally to reach the highest vibration amplitude representing the actual 

eigenfrequency. The response of the system for controlled and uncontrolled case is 

shown in Fig. 11 based on the measurement signal generated by Doppler vibro-meter. 

0 1 2 3 4 5
-40

-30

-20

-10

0

10

20

30

40

Time (seconds)

A
m

p
li
tu

d
e

 

 

Open-loop system

Closed-loop system

 

Fig. 11 Experimental comparison of velocity response 

This diagram shows the velocity magnitudes of the beam measured by dSPACE ADC 

board. In addition the corresponding control effort generated for piezo-actuator patches 

by the dSPACE DAC board is shown in Fig. 12. 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-10

-5

0

5

10

Time (seconds)

A
c
tu

a
to

r 
v
o
lt
a
g
e
 (

V
.)

 

Fig. 11 Control effort of the piezo-patch actuator 

The experimental results show the obvious performance of the robust control system 

in attenuating the vibration amplitude. 



248 A. OVEISI, T. NESTOROVIĆ 

 

6. CONCLUSION 

Vibration control of a clamped-free beam with piezoelectric actuator and vibrometer 

has been achieved by using a multi-objective robust output feedback control strategy with 

regional pole placement constraints in an LMI framework, based on H2 / H weighting 

objective functions. The robustness of the closed loop smart beam with respect to 

external input disturbance increased to 557% of the modeled uncertainty. The regional 

pole placement constraints guaranteed the improvement of the transient response of the 

closed-loop system and the optimality of the control effort is achieved by satisfying the 

appropriate H2 LMI based performance index. All these constraints are presented in a 

LMI formulation, which is solvable in the MATLAB environment. Finally, the 

performance of the approach is proven to be effective and robust on the experimental set 

up where the higher order modes take effect in the dynamics of the smart beam.  

 

REFERENCES 

1. Milovančević, M., Veg, A., Makedonski, A., Marinović, J. S., 2014, Embedded systems for vibration 
monitoring, Facta Univesitatis Series Mechanical Engineering, 12(2), pp. 171-181. 

2. Oveisi, A., Gudarzi, M., 2013, Adaptive sliding mode vibration control of a nonlinear smart beam: A 
comparison with self-tuning Ziegler-Nichols PID controller, Journal of Low Frequency Noise Vibration 

and Active Control, 31(1-2), pp. 41 – 62. 
3. Carra, S., Amabili, M., Ohayon, R., Hutin, P. M., 2008, Active vibration control of a thin rectangular 

plate in air or in contact with water in presence of tonal primary disturbance, Aerospace Science and 

Technology, 12, pp. 54-61. 
4. Caruso, G., Galeani, S., Menini, L., 2003, Active vibration control of an elastic plate using multiple 

piezoelectric sensors and actuators, Simulation Modeling Practice and Theory, 11, pp. 403-419. 

5. Benjeddou, A., 2000, Advances in piezoelectric finite element modeling of adaptive structural elements: 
a survey, Computers & Structures, 76(1-3), pp. 347-363. 

6. Gao, L., Lu, Q., Fei, F., Liu, L., Liu, Y., Leng, J., 2013, Active vibration control based on piezoelectric 
smart composite, Smart Materials and Structures, 22(12) 125032. 

7. Panda, S., Ray, M. C., 2009, Active control of geometrically nonlinear vibrations of functionally graded 
laminated composite plates using piezoelectric fiber reinforced composites, Journal of Sound and 
Vibration 325, pp. 186-205. 

8. Tavakolpour, A. R., Mailah, M., Mat Darus, I. Z., 2009, Active vibration control of a rectangular flexible 
plate structure using high gain feedback regulator, International Review of Mechanical Engineering, 
3(5), pp. 579-587. 

9. Narayanan S., Balamurugan V., 2003, Finite element modeling of piezolaminated smart structures for active 
vibration control with distributed sensors and actuators, Journal of Sound and Vibration, 262 pp. 529-562. 

10. Caruso G., Galeani S., Menini L., 2003, Active vibration control of an elastic plate using multiple 
piezoelectric sensors and actuators, Simulation Modelling Practice and Theory, 11, pp. 403-419. 

11. Oveisi, A., Gudarzi, M., Mohammadi, M.M., Doosthoseini, A., 2013, Modeling, identification and active 
vibration control of a funnel-shaped structure used in MRI throat, Journal of Vibroengineering, 15(1), 

pp. 438-450. 

12. Qiua, Z., Wub, H., Zhanga, D., 2009, Experimental researches on sliding mode active vibration control 
of flexible piezoelectric cantilever plate integrated gyroscope, Thin-Walled Structures, 47(8-9) pp. 836-846. 

13. Marinkovic, D., 2007, A new finite composite shell clement for piezoelectric active structures, Ph.D. 
thesis, Otto-von-Guericke-Universität Magdeburg Fakultät für Maschinenbau. 

14. Marinkovic, D., Marinkovic, Z., 2011, FEM and Ritz Method-A Piezoelectric Active Shell Case Study, 
Transactions of FAMENA, 35(3), pp. 39-48. 

15. Marinkovic, D., Koppe. H., Gabbert. U., 2009, Aspects of modeling piezoelectric active thin-walled 
structures, Journal of Intelligent Material Systems and Structures, 20(15), pp.1835-1844. 



 Robust Mixed H2/H ACTIVE Vibration Controller in Attenuation of Smart Beam 249 

 

16. Nestorovic, T., Marinkovic, D., Chandrashekar, G., Marinkovic, Z., Trajkov, M., 2012, Implementation 
of a user defined piezoelectric shell clement for analysis of active structures, Finite Elements in Analysis 
and Design, 52, pp.11-22. 

17. Oveisi, A., Gudarzi, M., 2013, Nonlinear robust vibration control of a plate integrated with piezoelectric 
actuator, International Journal of Mathematical Models and Methods in Applied Sciences, 7(6), pp. 638-646. 

18. Nestorović, T., Durrani, N., Trajkov, M., 2012, Experimental model identification and vibration control of a 
smart cantilever beam using piezoelectric actuators and sensors, Journal of Electroceramics, 29(1), pp. 42-55. 

19. Géradin, M., 1997, Mechanical vibrations theory and application to structural dynamics, 2
nd

 Edition, 
Wiley, Chichester. 

20. Zhou, K., Doyle, J. C., 1997, Essentials of robust control, Prentice Hall. 
21. Sivrioglu, S., Tanaka, N., 2002, Acoustic power suppression of a panel structure using H∞ output 

feedback control, Journal of Sound and Vibration, 249(5), pp. 885-897. 

22. Chilali, M., Gahinet, P., 1995, H∞ design with pole placement constraints: H∞ an LMI approach, IEEE 
Transactions on Automatic Control, 41(3), pp. 358-367. 

23. Banjerdpongchai, D., How J. P., 1998, Parametric robust H2 control design with generalized multipliers 
via LMI synthesis, International Journal of Control, 70(3), pp. 481-503. 

24. Scherer, C., 1995, Mixed H2/H∞ Control, Trends in Control: A European Perspective, volume of the 
special contributions to the ECC. 

25. Gahinet, P., 1996, Explicit controller formulas for LMI-based H∞ synthesis, Automatica, 32(7), pp. 1007-1014. 
26. Gudarzi, M., Oveisi, A., Mohammadi, M. M., 2012, Robust active vibration control of a rectangular 

piezoelectric laminate flexible thin plate: an LMI-based approach, International Review of Mechanical 

Engineering, 6(6), pp. 1217-1227. 

27. Nestorović, T., Trajkov, M., 2013, Optimal actuator and sensor placement based on balanced reduced 
models, Mechanical Systems and Signal Processing, 36(2), pp. 271-289. 

ROBUSTNO H2/H UPRAVLJANJE VIBRACIJAMA 

U CILJU PRIGUŠENJA AKTIVNE KONZOLE 

Ograničena robustnost u odnosu na neprecizno modeliranje dinamike i spoljašnjih poremećaja 

mehaničkih sistema značajno utiče na njihove performanse i optimalna svojstva aktivnih struktura. U 

ovom radu prikazujemo na koji način se ovaj nedostatak može prevazići  u cilju postizanja željenih 

performansi aktivnih struktura. U tom cilju predložen je robustni upravljački zakon za redukciju 

vibracija aktivne konzole sa piezoelektričnim aktuatorom i laserskim senzorom u LMI okruženju, koji 

se uspešno može primenjivati u prisustvu eksternih ulaza sa težinskim funkcijama,a koji obezbeđuje 

željeno podešavanje polova i robustne performanse u isto vreme. Model homogene konzole dobijen je 

primenom modalne analize metodom konačnih elemenata.  Zatim je modalni sistem redukovanog reda 

posmatran kao nominalni model, dok su modeli višeg reda su razmatrani kao mulitiplikativna 

nesigurnost. Potom je na osnovu proširenog modela, koji sačinjava nominalni model sa pratećom 

neizvesnošću, projektovan robustni kontroler sa lokalnim podešavanjem polova, rešavanjem konveksnog 

optimizacionog problema. Na kraju je simulacijom i eksperimantalno analizirana robustnost nizvesnog 

modela zatvorenog kola, kao i uticaj težinskog indeksa performansi na izlaz sistema. 

Ključne reči: piezoelektrični materijali, redukcija vibracija, robustno upravljanje, aktivna konozla, 

metod konačnih elemenata.