Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 3, pp. 533-541, Warsaw 2013

ROBUST ADAPTIVE VIBRATION CONTROL FOR A GENERAL CLASS

OF STRUCTURES IN THE PRESENCE OF TIME-VARYING

UNCERTAINTIES AND DISTURBANCES

Hamid Reza Koofigar

University of Isfahan, Faculty of Engineering, Isfahan, Iran

e-mail: koofigar@eng.ui.ac.ir

Shahab Amelian

Department of Mechanical Engineering, Naein Branch, Islamic Azad University, Naein, Iran

e-mail: amelian@naeiniau.ac.ir

The problemof active vibration suppression in awide class of smart structures is addressed.
The dynamical model of a structure may be perturbed by uncertain time-varying parame-
ters and external disturbances. A novel adaptive-based control algorithm is presented here
to satisfy robustness properties with respect to model uncertainties and environmental di-
sturbances. Reflecting practical situations, the upper bound of perturbations is not required
for controller design. The analytical stability of a closed-loop system is presented based on
the Lyapunov stability theorem. Furthermore, numerical analysis is also provided to show
the effectiveness of the proposedmethod.

Key words: vibration control, robust adaptive control, piezoelectric materials

1. Introduction

Vibration control of flexible structures is one of the main topics in the field of engineering. In
general, vibration suppression algorithms developed by researchers may be passive or active.
Passive control, achieved by incorporating mechanical elements into a structure, Skup (2010),
is applied when the effects of external disturbances are known in advance. On the contrary,
the so-called active control schemes present some self-adaptive mechanisms to reduce vibration
of the structure even in the presence of model uncertainties, time-varying loads and unknown
disturbances. From a practical viewpoint, implementing active control of flexible structures by
piezoelectricmaterials hasbeen extensively studiedduring the last decade,Pietrzakowski (2001),
Huang andTseng (2008). This arises from the fact that piezoelectricity is a natural phenomenon
which facilitates transformingmechanical energy to electrical energy and vice versa. In addition,
piezoelectric materials with low weight and low residual effect exhibit considerable flexibility
and can be used in a wide range of temperature. Meanwhile, these materials can be utilized as
distributed sensors and actuators incorporated into the structure.
Active vibration suppression has become the focus of attention generally in mechanical and

civil structures, see Song et al. (2006), Longa et al. (2011), and particularly in beams –Trindade
et al. (2001), Vasques and Rodrigues (2006), Chang (2012), and aircrafts Song and Agrawal
(2001),Wachłaczenko (2010). Establishing the controllability concept for dynamical systems by
Klamka (1991), and particularly for mechanical systems by Klamka (2005), various reported
control schemes for structural systems can be put in the main categories, including: (i) impro-
ved conventional control techniques, e.g., proportional and derivative control, Belouettar et al.
(2008), Fey et al. (2010), (ii) optimal control algorithms based on either classical strategies,
Stavroulakis et al. (2005), Vasques andRodrigues (2006), or stochastic based optimization tech-
niques, Marinaki et al. (2011). Such optimization-based methodologies may ensure the optimal



534 H.R. Koofigar, S. Amelian

performance in the absence of system uncertainties. Moreover, inverse matrix calculation is ne-
eded in thedesignprocedure,whereas the large dimensionof suchmatrices in complex structures
takes a considerable time in the implementation process, (iii) intelligent control algorithmsbased
on fuzzy logic, Sharma et al. (2005), or neural networks, (iv) robust control schemes, e.g. sliding
mode control applied to a simplified dynamics of the structure, Gu et al. (2008).
In practice, the systemparametersmay varywith time due to various circumstances. Among

the reported methods, adaptive-based control techniques are powerful tools especially when
the variations are slow enough, see Astrom and Wittenmark (1994), Krstic et al. (1996). In
fact, conventional adaptive methods including adaptive control together with some parameter
adjustingmechanismmay fail for the case of time-varying perturbations. Investigating into this
field, several results havebeen reportedwhen thevariations areperiodic,Xu (2004),Ding (2007),
or the upper boundof the parameter vector is known in advance,Ge andWang (2003), Cai et al.
(2006).On the other hand, twomain types of disturbances including time-varying ones and those
associatedwithfixeddeformationsmayalso affect theperformanceof a flexible structure, Irschik
(2002). Dealing with the purpose of attenuating or rejecting the influence of disturbances, some
robust control methods and also several adaptive approaches have been introduced, especially
for the case of periodic disturbances, Bodson and Douglas (1997), Ding (2007).
In this paper, a robust adaptive algorithm is developed to achieve active vibration control

of structures. More precisely, an adaptive algorithm is designed to tackle time-varying model
uncertainties and incorporated a robust mechanism to deal with external disturbances. In fact,
a combination of tools from both robust and adaptive approaches is adopted to achieve the
desiredperformance. Some specificproperties of thedeveloped active vibration control algorithm
are: (i) it can be applied to a wide class of flexible structures, (ii) there are no conservative
assumptions, e.g., on the upper bound, the speed of variations and the periodicity of model
uncertainties and external disturbances, (iii) robust stability is ensured by theoretical analysis
and verified by various numerical simulations.
The organization of the paper is as follows. Presenting the mathematical model of flexible

structures, the vibration control problem is formulated inSection 2. Section 3presents the robust
adaptive control algorithm and its stability analysis based on the Lyapunov stability theorem.
In Section 4, various simulation results are given to illustrate the performance of the proposed
vibration suppressionmethod. Finally, the concluding remarks are given in Section 5.
Throughout the paper, ‖ · ‖ denotes the Euclidean vector norm and for a n×1 vector V,

the weighted norm is defined as ‖V‖2Q := V
TQV with a weighting matrix Q. Furthermore,

V ∈L2[0,T ] if
∫T
0
‖V(t)‖2dt<∞, T ∈ [0,∞), and V ∈L

∞
if ‖V(t)‖<∞ for all t∈ [0,∞).

2. Mathematical model and problem statement

Mathematical modeling of beams and structures is performedmainly based on linear piezoelec-
tricity, sensor dynamics and equations ofmotion. Depending on the analysis or control synthesis
purposes, each of the aforementioned factors may lead to adopt a suitable method for math
formulation of themodel.Dealingwith the vibration control problem, the finite elementmethod
can be used to describe the dynamical equation of motion for a smart mechanical structure as,
Trindade et al. (2001)

MẌ+DẊ+CX=Fm+Fe (2.1)

where X represents the state vector of the system, including travnsersal deflection and rotation
variables, M is the mass matrix, D denotes the viscous dampingmatrix and C stands for the
stiffness matrix. The force vector Fe acts as a control input, produced by electromechanical
coupling effects, and Fm is a mechanical point force vector acting as external disturbance.



Robust adaptive vibration control ... 535

From a practical viewpoint, the viscous dampingmatrix D and the stiffness matrix Cmay
not be determined accurately, especially for complex structures, and the existence of uncertainty
in the aforementioned matrices is inevitable. The variation of parameters included in such ma-
trices motivates taking a time-varying uncertain dynamical model for controller design. Hence,
dynamical equation (2.1) takes the form

MẌ+h0(X,Ẋ)+
p∑

i=1

ϕi(X,Ẋ)θi(t)=u(t)+d(t) (2.2)

where

h0(X,Ẋ)=D0Ẋ+C0X

denotes the nominal part produced by known matrices C0 and D0, ϕi(X,Ẋ) is a dimen-
sionally compatible matrix associated with an unknown time-varying parameter vector θi(t),
i=1,2, . . . ,p, u(t) stands for the applied control input, and d(t) denotes the disturbance input.
The following assumptions are made regarding the system.

Assumption 1. The variations of parameters included in C and D can be time-varying with
unknown bounds, i.e., θi(t) belongs to the compact set Ωi = {θi(t) : ‖θi(t)‖ ¬ βi},
i=1,2, . . . ,p, in which βi > 0 is an unknown constant.

Assumption 2. The time-varying external disturbance d(t) is normboundedwith anunknown
value, i.e., ‖d(t)‖¬ δ, where δ is an uncertain parameter.

Assumption 3. Controllability, defined by Klamka (2005), as the possibility to control a dy-
namical system from an arbitrary initial state to an arbitrary final state using a set of
admissible controls, is satisfied here for the presented mechanical systems.

The objective is to design an active control algorithm that ensures vibration suppression in
the presence of model uncertainties and external disturbances. As a preliminary step to design
such a controller, define the tracking error vector as e = Xd −X, where Xd represents the
desired state vector, usually set to zero, for vibration suppression purposes.

3. Robust adaptive controller design

In order to develop the control algorithm and for notational consistency, two error metric func-
tions are defined as S(t) = ė(t)+e(t) and Sr(t) = Ẍd(t)+ ė(t). The general structure for the
control input is proposed as

u=h0+MSr +KS+ua+ur (3.1)

where K is a positive definitematrix, ua presents an adaptive subcontroller, and ur is a robust
subcontroller to be designed. In fact, ua deals with the system parameter uncertainties and
ur ensures robustness with respect to the environmental disturbances.
In the following, using the Lyapunov stability theorem, the subcontrollers ua and ur are

derived. To this end, choose the Lyapunov function

V (e, ė)= eTKe+
1

2
S
T
MS+

1

2γ
β̃2 (3.2)

where β̃=β− β̂ denotes the parameter estimation error and γ > 0 is the adaptation gain.



536 H.R. Koofigar, S. Amelian

The time derivative of V is

V̇ =2eTKė+ST(Më+Mė)+
1

γ
β̃
˙̃
β (3.3)

Substituting ë by Ẍd− Ẍ in (3.3) and replacing MẌ from (2.2), one can obtain

V̇ =2eTKė+ST
(
MẌd+h0+

p∑

i=1

ϕiθi(t)−u(t)−d(t)+Mė

)
+
1

γ
β̃
˙̃
β (3.4)

Incorporating control law (3.1) into (3.4) yields

V̇ =2eTKė−STKS+ST
( p∑

i=1

ϕiθi(t)−ua−d(t)−ur

)
−
1

γ
β̃
˙̂
β (3.5)

Taking assumption 1 into account, V̇ can be written as

V̇ ¬−eTKe− ėTKė+β
p∑

i=1

‖STϕi(X,Ẋ)‖−S
T
ua−S

T
d−STur−

1

γ
β̃
˙̂
β (3.6)

where β=max{β1,β2, . . . ,βp}.
Now, the adaptive and robust terms ua and ur are respectively proposed as

ua = β̂
2

p∑

i=1

ϕi(X,Ẋ)ϕ
T
i (X,Ẋ)S

‖STϕi(X,Ẋ)‖β̂+σe
−rt

ur =
1

2ρ
S (3.7)

where β̂, the estimate of β, is calculated by the adaptation mechanism

˙̂
β= γ

p∑

i=1

‖STϕi(X,Ẋ)‖ (3.8)

Substituting ua, ur, and update law (3.8) into inequality (3.6), gives

V̇ ¬−eTKe− ėTKė+ β̃
p∑

i=1

‖STϕi(X,Ẋ)‖+σe
−rt−

1

γ
β̃
˙̂
β

¬−eTKe− ėTKė−STd−
1

2ρ
S
T
S+σe−rt

(3.9)

Using the equivalence

−STd−
1

2ρ
S
T
S=−

1

2ρ
(S+ρd)T(S+ρd)+

1

2
ρ‖d‖2 (3.10)

inequality (3.9) can be rewritten as

V̇ ¬−eTKe− ėTKė−
1

2ρ
(S+ρd)T(S+ρd)+

1

2
ρ‖d‖2+σe−rt (3.11)

By omitting some strictly negative terms from the right hand side of inequality (3.11), one can
obtain

V̇ ¬−eTKe+
1

2
ρ‖d‖2+σe−rt (3.12)

and

V̇ ¬−ėTKė+
1

2
ρ‖d‖2+σe−rt (3.13)



Robust adaptive vibration control ... 537

The following results are then concluded.
(i) By assumption 2, inequality (3.12) implies that V̇ is boundedas V̇ ¬−λK‖e‖

2+1
2
ρδ2+σ,

where λK is theminimum eigenvalue of K. Choosing λK > (ρδ
2+2σ)(2ε2) for any small

ε > 0, there exists a κ> 0 such that V̇ (e, ė)¬−κ‖e‖2 < 0 for all ‖e‖> ε. Thus, there
is a T > 0 such that ‖e‖ ¬ ε for all t­ T . This implies that the error vectors e(t) are
uniformly ultimately bounded (UUB), Krstic et al. (1995).

(ii) Taking inequality (3.13) into account and following aprocedure similar to that given in (i),
the boundedness of ė(t) is concluded.

(iii) In many practical situations, the disturbance inputs, e.g. a constant load for a specific
time duration, Stavroulakis et al. (2005), a sinusoidal periodic wind-type pressure, Banio-
topoulos andPlalis (2002), and zeromeanwhite Gaussian noise, are energy bounded, i.e.,
d∈L2[0,T ]. Hence, integrating inequality (3.10) from t=0 to t=T yields

T∫

0

‖e(t)‖2K dt+V
(
e(T), ė(T)

)
¬V
(
e(0), ė(0)

)
+
σ

r
(1− δe−rT)+

1

2
ρ

T∫

0

‖d(t)‖2 dt

for all 0 ¬ T < ∞. This implies that e(t) is square-integrable, i.e. e(t) ∈ L2[0,T ],
which together with the boundedness property of e(t) and ė(t), Barbalat’s lemma (see
the appendix) Krstic et al. (1995), ensures the convergence of e(t) and the closed-loop
stability, despite the system uncertainties and external disturbances.

Remark 1. Choosing a smaller ρ> 0 provides the systemwith a faster response. This may be
obtained at the expense of larger control effort. In fact, there exists a trade-off between
the value of subcontroller gain ρ and themagnitude of control input u.

Remark 2. Fromapractical viewpoint, the exponential term in ua formedby σ> 0 and r > 0
provides the smoothness of the control law without violation the convergence property of
the tracking error.

Remark 3. Unlike some previous works, Trindade et al. (2001), Stavroulakis et al. (2005),
Marinaki et al. (2011), the inverse calculation of M, whose large dimension in complex
structures takes a considerable time in the implementation process, is not required here.
Moreover, the effects ofmodel uncertainties and disturbances even with unknownbounds,
are well suppressed.

Remark 4. In order to alleviate the increase in the estimation value β̂ without bound occur-
ring in the case of imperfect implementation of adaptation mechanism (3.8), an effective
modification is adopted here. To this end, substitute update law (3.8) with

˙̂
β=µ

p∑

i=1

‖STϕi(X,Ẋ)‖ (3.14)

where

µ=

{
γ if ‖e‖>ε

0 otherwise

which ensures that all the signals and states of the closed loop systemareboundedand ‖e‖
is robustly converged to a (small) prescribed bound ε> 0. In fact, this modification acts
as a projection algorithm and, therefore, the stability analysis can be followed similar to
that of conventional projectionmethods in the literature, Astrom andWittenmark (1994),
Khalil (1996). Briefly discussing, adaptation mechanism (3.14) is activated whenever the
norm of the tracking error e exceeds the prescribed bound ε > 0 and, consequently, the
instability due to the increase in β̂ is alleviated.



538 H.R. Koofigar, S. Amelian

The aforementioned analysis shows the capability of the proposed control algorithm for
vibration suppression in awide class of flexible structures in the presence ofmodel uncertainties
and environmental disturbances.

4. Results and discussion

In order to verify the effectiveness of the proposed robust adaptive vibration control algorithm,
the controller is applied to aflexible beam instrumentedwith apiezoelectric sensor and actuator,
as schematically shown in Fig. 1.

Fig. 1. Configuration of the beamwith piezoelectric patches

In simulation studies, the viscous damping and the stiffness matrices are imposed to a si-
ne variation with an amplitude of 30% of the nominal values and period of 0.5. The beam
specifications are listed in Table 1.

Table 1.Beam specifications

Parameter Value

Length [mm] 200

Height [mm] 10

Width [mm] 1

Young’s modulus [GPa] 200

Density [kg/m2] 7800

Three cases are here considered to evaluate the performance of the designed vibration sup-
pression algorithm. At the first step, no disturbances are imposed to the model perturbed by
parameter uncertainties.
The control signal is activated at t = 50ms as demonstrated in Fig. 2. The capability of

the method in vibration control is illustrated in Fig. 3 and is focused in the steady state for
exact analysis. In the second case, a zero mean white noise, as shown in Fig. 4, is imposed to
the systemwhich is well suppressed by the proposed robust control algorithm, see Fig. 5. Since
the sinusoidal signals can effectively model the effects of wind on structures, Baniotopoulos
and Plalis (2002), Stavroulakis et al. (2005), a sine disturbance with a period of 0.2s and an
amplitude of 1 is considered in the third case. Figure 6 illustrates a comparison between the
time response of vibrations in the absence of control effort and by activating the control input
at t=50ms, showing the achievement of the vibration suppression task.

5. Conclusions

The problem of active vibration control is addressed for smart structures. Removing drawbacks
of some previous investigations, a novel robust adaptive vibration suppression algorithm is pre-



Robust adaptive vibration control ... 539

Fig. 2. Control signal for robust adaptive vibration suppression activated at t=50ms

Fig. 3. Time response of vibration with the designed controller (—-), and without control (– –),
(a) transient response, (b) steady state response

Fig. 4. Zeromean white Gaussian noise

Fig. 5. Time response of vibration in the presence of white noise using the designed controller (—-) and
without control (– –), (a) transient response, (b) steady state response



540 H.R. Koofigar, S. Amelian

Fig. 6. Time response of vibration in the presence of a sinusoidal disturbance applying the designed
robust adaptive controller (—-), and without control (– –)

sented, and the effectiveness of themethod is shown by both analytical and simulation analysis.
The model uncertainties and disturbances may have unknown bounds. Some kinds of external
disturbances, mostly imposed in practical situations, are considered in the analysis. The nume-
rical studies show that the goal of vibration control is achieved by the designedmethod, despite
the model uncertainties and environmental disturbances.

Appendix

One of the results of Barbalat’s lemma, used in the stability proof in this paper, is stated by
Krstic et al. (1995), Ioannou and Sun (1996), as

Lemma: If e, ė∈L
∞
and ė∈L2, then e(t)→0 as t→∞.

Acknowledgment

Thisworkhasbeen supportedby IranNationalScienceFoundation (INSF)undergrantNo.89001149.

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Manuscript received June 26, 2012; accepted for print August 29, 2012