Robust adaptive vibration control of a flexible structure

Research Article

Robust adaptive vibration control of a flexible structure

A.M. Khoshnood n, H.M. MoradiAerospace Engineering Department, P.O. Box 16765-3381, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 17 December 2013Received in revised form18 February 2014Accepted 17 March 2014This paper was recommended forpublication by Jeff Pieper

Keywords:Robust adaptive controlVibration controlFlexible structureModel reference adaptive systemL1 small gain theorem

a b s t r a c t

Different types of L1 adaptive control systems show that using robust theories with adaptive controlapproaches has produced high performance controllers. In this study, a model reference adaptive controlscheme considering robust theories is used to propose a practical control system for vibrationsuppression of a flexible launch vehicle (FLV). In this method, control input of the system is shapedfrom the dynamic model of the vehicle and components of the control input are adaptively constructedby estimating the undesirable vibration frequencies. Robust stability of the adaptive vibration controlsystem is guaranteed by using the L1 small gain theorem. Simulation results of the robust adaptivevibration control strategy confirm that the effects of vibration on the vehicle performance considerablydecrease without the loss of the phase margin of the system.

& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Vibration and flexibility in large scale industrial products areinevitable problems and many research have been devoted todesigning active vibration control systems [1,2]. Structural flex-ibility is more important for aerospace systems because of the sizeof structures and accuracy of missions. In this regard, launchingvehicles from aerospace systems significantly encounter withvibrational challenges because of large size structures. The maintypes of vibration are bending vibration, fuel sloshing, and servo-oscillation [3]. Vibration control, including adaptive, robust, andoptimal approaches, has been employed to remove the destructiveeffects of flexibility and undesired oscillations. Literature reviewson attitude control of flexible launch vehicles (FLVs) have beenpresented in several investigations [4–6].

Adaptive control is one of the most practical approaches forvibration control of aerospace structures [7]. Direct methods ofadaptive control, indirect methods and adaptive filtering have beenused for active vibration control and vibration suppression [8]. In newstrategies, adaptive filtering as a practical and useful method has beenapplied to closed loop control systems of aerospace devices. Fromthese approaches, Englehart and Krause have proposed an analognotch filter with a least square algorithm to reduce the bendingvibrational effects on an aerospace launch vehicle [9]. Choi et al. [7,10]have also designed an adaptive control approach for the attitudecontrol of a flexible aerospace vehicle. They used the root mean

square method to estimate the bending frequency and one digitalnotch filter to reduce the flexible behaviors, considering the first andthe second bending vibration modes. Oh et al. have proposed anattitude control of a flexible launch vehicle using an adaptive notchfilter [4]. In another work, Khoshnood et al. have studied a modelreference adaptive control for reducing undesired effects of bendingvibration for an aerospace launch vehicle [5]. Elmelhi has designed amodified adaptive notch filter based on neural network for a flexibledynamic system [11]. In spite of simplicity and accuracy of thesefiltering strategies, linear and nonlinear changes in the phase marginof the closed loop control system as a result of the adaptive filterbandwidths produce considerable limitations. On the other hand,application of adaptive filters may lead to violation of the stability andperformance of the preliminary control system. In this regard, Zhi-jian et al. have presented an interpolated Fourier transform based onan adaptive notch filter to solve the problem of phase lag in aconventional notch filter [12].

In the present study, a new framework of adaptive filteringstrategy is proposed to improve the performance of this strategyfor vibration control. The main idea of this method is theconstruction of vibration control input using undesirable vibrationfrequencies. All of the recent works related to the vibration controlof FLVs have used the adaptive filtering strategy in the case ofinserting the filters in the feedback of the main control system. Inthis paper, for vibration control the adaptive filtering approach isimplemented based on modification of the control input of thesystem. Remarkably, this idea has been taken from the medicalapproaches in which anti-viruses like a noise rejection con-troller are made from the same viruses. Therefore, two basicimprovements are applied to the adaptive filtering strategy. The

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ISA Transactions

http://dx.doi.org/10.1016/j.isatra.2014.03.0040019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: [email protected] (A.M. Khoshnood).

Please cite this article as: Khoshnood AM, Moradi HM. Robust adaptive vibration control of a flexible structure. ISA Transactions (2014),http://dx.doi.org/10.1016/j.isatra.2014.03.004i

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first improvement is construction of the control input of thesystem using the results of filtering (estimated frequencies)instead of inserting the filter into the feedback of the closed loopcontrol system. This change leads to set the undesired phase delayeffects of the filter out of the closed loop control system. Thesecond change is to analyze the robustness of the closed loopcontrol system and the estimation algorithm. These two revisionsproduce significant results for vibration suppression, as shown in anumerical simulation.

The robust adaptive control design in which the adaptivecontrol approaches are implemented based on robust theoremshas been developed in the framework of L1 adaptive controlsystems [13]. Regarding the robust adaptive approaches, in thisstudy, a model reference adaptive filtering scheme consideringrobust theories is investigated to extract a practical control systemfor vibration control of an FLV. The components of the controlinput are adaptively tuned with estimating undesirable vibrationfrequencies. The bending vibration parameters such as vibrationfrequencies are changed according to the mass consumption rateof the vehicle and accurate estimation of undesirable vibrationfrequencies is essential. As a result of this, sub-band adaptivefiltering is considered instead of fullband analysis. On the otherhand, sub-band analysis is employed to expand the estimationfrom one vibration frequency to two or more frequencies. There-fore, a sub-band adaptive filter is designed using the discreteFourier transform (DFT) technique. This technique can be used formulti-frequency estimation with satisfactory persistence of excita-tion properties [14].

The vibration control system is employed for attitude control ofthe FLV. The FLV considered in this study is a multi-vibration modeflexible structure. The inertial navigation system (INS) of thevehicle measures the bending vibration as well as the rigid bodymotion. Simultaneous measurement of the rigid and the flexiblebody rotations may violate the closed loop control system, whichin the worst cases may lead to resonance. The primary aim of therobust adaptive control system is to decrease this resonance.

In this study, a linear model of the FLV in the pitch channel isconsidered for vibration and attitude control. The control system isalso applied to the yaw channel of the vehicle at the samestructure as a result of dynamic symmetric properties of thevehicle. Moreover, according to flight conditions, three vibrationmodes are analyzed for modeling the bending vibration using theassumed modes method.

This paper is organized as follows: Section 2 provides linearequations of motion of the FLV. Section 3 describes the designprocedure of the robust adaptive control system. Robust estima-tion of the control components and construction of the controlinput are given in this section. Section 4 discusses the robuststability of the new vibration control system and Section 5provides the simulation results of the proposed control approach.The conclusions of the work are presented in Section 6.

2. Equations of motion

Equations of motion of an FLV (rigid and flexible body) aregiven in this section. In one of the studies, nonlinear equations ofmotion of a launch vehicle and the method of linearization havebeen addressed by Roshanian et al. [15]. Linear equations ofmotion of the rigid body dynamic are

_α¼ 12ms

ρU0SðCzα�Cx0Þ�T

msU0

� �αþ 1

4msρSDCzqþ1

� �q� CT

msU0ðδ2þδ4Þ

_β¼ 12ms

ρU0SðCyβ�Cx0Þ�T

msU0

� �βþ 1

4msρSDCyr�1

� �rþ CT

msU0ðδ1þδ3Þ

_p¼ 14Ix

ρU0SD2Clp

� �pþCT dy

Ixðδ1�δ2�δ3þδ4Þ

_q¼ 12Iy

ρU20SxacCzα

� �αþ 1

4IyρU0SD

2Cmq

� �q�CT dx

Iyδ2þδ4ð Þ

_r¼ � 12Iy

ρU20SxacCyβ

� �βþ 1

4IyρU0SD

2Cnr

� �r�CT dx

Iyðδ1þδ3Þ ð1Þ

where α and β represent the angle of attack and the side slipangles, respectively; p; q and r are the angular velocities of thevehicle; U0 is the magnitude of the velocity in the x direction; ms

and Ix;y are the mass and the mass moment of inertia in the x and ydirections, respectively; and ρ is the air density. In addition, S; Dand T are the reference surface, the reference length, and thethrust, respectively; Cij is the aerodynamic coefficient; δi is the ithactuator deflection in the pitch and yaw directions; dx, dy and CT

are the thrust coefficients; and xac is the position of the aero-dynamic center of the vehicle.

Using an arbitrary state space representation, the transferfunction between the pitch (yaw) angular velocity and the pitch(yaw) actuator deflections are extracted as

qδ2

¼ a1sþa2s2þa3sþa4

ð2Þ

where ai is derived from Eq. (1). Because the vehicle is dynamicallysymmetric, only the pitch channel of the vehicle is considered. Inaddition, the actuator deflections of the pitch channel are assumedto operate uniformly (δ2 ¼ δ4).

The attitude determination system of the vehicle in the pitchchannel as shown in Fig. 1 measures the rigid body motion and theelastic deflection. In this figure, θ is the pitch angle associated withthe rigid body motion and θb is the pitch angle associated with theflexible body motion. The rigid and the elastic motions can beintegrated as

qT ¼ qr�qb ) _qT ¼ _qr� _qb ð3Þwhere qb is the angular velocity arisen from the bending vibration;qr is the pure rigid body angular velocity of the vehicle in the pitchchannel (q from Eq. (2)); and qT is the total angular velocity. Themathematical model of the vehicle structure is assumed to be afree–free Euler–Bernoulli beam as follows:

ρA∂2y∂t2

þ ∂2

∂x2EI∂2y∂x2

� �¼ f ðx; tÞ ð4Þ

where ρΑ is the mass of the vehicle per unit length; y is thedeflection of the vehicle structure from the bending moment; E isthe Young modulus; I is the area moment of inertia about theneutral axis; and f(x,t) is the external force per unit length. Usingthe assumed mode method, Eq. (4) can be separated into a timedependent part as

€gbjþ2ςjωj _gbjþω2j gbj ¼

Qj

Mj¼ Kj f jδ2 ð5Þ

where gbj is the generalized coordinate for the jth mode of thebending vibration; Mj and Qj are the generalized mass and theforce of the jth bending vibration mode, ωj is the natural frequencyof the jth bending vibration mode; ςj is the damping ratio of thejth bending vibration mode; Kj is a proportional constant; and f j is

b

Z

x

Fig. 1. Rigid and flexible coordinates defined on the vehicle body.

A.M. Khoshnood, H.M. Moradi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





the external force per actuator deflection. According toEqs. (2)–(4), the transfer function of the pitch channel can beexpressed as

qTδ2

¼ a1sþa2s2þa3sþa4

� ∑n

j ¼ 1

ðKjf jψ jÞss2þ2ςjωjþω2

j

ð6Þ

where ψ j is the differentiation of the jth vibrational mode shape inthe x direction; qT is the total angular velocity of the vehicle; andθT is the total pitch angle defined as θr�θb.

3. Robust adaptive vibration control

Construction of the control input of the system is based on theequations of motion of the vehicle. This type of control input ischosen to reduce the vibrational effects on the pitch channel of thevehicle. The FLV must track the desired dynamic model of the rigidlaunch vehicle that can be as a model reference. The robustadaptive controller can be constructed step-by-step as follows:

_qT ¼ _qr� _qb

¼ 12Iy

ρU20SxacCzα

� �αþ 1

4IyρU0SD

2Cmq

� �q�CT dx

Iyðδ2þδ4Þ

� ∑n

j ¼ 1Kjf jψ jδ2�2ςjωjqbj�ω2

j θbj ð7Þ

) _qT ¼ �CT dxIy

ðδ2þδ4|fflfflffl{zfflfflffl}2δ2

Þ� ∑n

j ¼ 1Kjf jψ jδ2

264

375

þ 12Iy

ρU20SxacCzα

� �αþ 1

4IyρU0SD

2Cmq

� �qþ ∑

n

j ¼ 12ςjωjqbjþω2

j θbj

ð8Þ

) _qT ¼ ½Kδ2�þ12Iy

ρU20SxacCzα

� �αþ 1

4IyρU0SD

2Cmq

� �qþ ∑

n

j ¼ 12ςjωjqbjþω2

j θbj

ð9Þwhere K is a constant parameter and δ2 is a control input signalapplied to the pitch channel of the vehicle. In this case, δ2 is

δ2 ¼1

K

�∑n

j ¼ 1�2ςjωjqbj�ω2

j θbj

�ð10Þ

By substituting Eq. (10) into Eq. (9), the effect of the bendingvibration is excluded. Eq. (10) can be modified using an additiveproportional-derivative action (PD-action) controller as

δ2 ¼1

K

�∑n

j ¼ 1�2ςjωjqbj�ω2

j θbjþKd _qd� _qT� �þKpðqd�qT Þ

�ð11Þ

where qd is the desired angular velocity.At the next step, the controller components illustrated in

Eq. (11) must be obtained. These components are as follows:

(a) parameters of the PD-action controller;(b) vibration characteristics, including frequencies and damping

ratios;(c) generalized coordinates associated with vibration (qbj).

The parameters of the PD-action controller are simply tunedbased on the rigid body dynamic model. The components of parts(b) and (c) must be robustly estimated to construct a robustadaptive control input. The first part of Eq. (11) can be removedbecause of the negligible value of the damping ratio in FLVstructures. This considerably reduces the calculated value becausethe control input only uses the pitch angle (θb), instead of both qband θb. Thus, it is not necessary to feedback the rate of thegeneralized coordinates to the control system.

3.1. Robust estimation of vibration frequencies

One of the practical methods of frequency estimation is therecursive least square algorithm commonly used for online para-meter estimation. In this algorithm, when the vibrational signalincludes unknown variance noise, the accuracy of estimation is notadequate for the control system. In this case, the vibrational ornoise signal can be defined as the output of the INS (angularvelocities) as follows:

xðnÞ ¼ sðnÞþvðnÞ ð12Þwhere vðnÞ is a white noise with unknown variance (R) and sðnÞis the nominal vibrational signal (angular velocity). Theexpected value of the square of x(n) according to Eq. (12) can beexpressed as

EfxðnÞ2g ¼ EfðsðnÞþvðnÞÞ2g ð13ÞBecause there is virtually no correlation between the noise and thenominal signal, Eq. (13) can be extended as

EfxðnÞ2g ¼ EfðsðnÞÞ2gþEfðvðnÞÞ2g ð14ÞThe filtered output signal can be used to minimize the vibrationaleffects as

EfxðnÞ2g ¼ EfðHðzÞxðnÞÞ2g ¼ EfðHðzÞsðnÞÞ2þðHðzÞvðnÞÞ2g ð15Þwhere H(z) is a finite impulse response (FIR) filter represented asfollows:

HðzÞ ¼ 1�2K0z�1þz�2 ð16Þand

EfxðnÞ2g ¼ fðsðnÞ�2K0sðn�1Þþsðn�2ÞÞ2gþfðvðnÞ�2K0vðn�1Þþvðn�2ÞÞ2gð17Þ

It can be assumed that the nominal signal (s(n)) is a sinusoidfunction, hence, using the linear prediction method [16], thesinusoid signal can be predicted and Eq. (15) extended as

sðnÞ ¼ 2 cos ω sðn�1Þþsðn�2Þ ð18Þ

) EfxðnÞ2g ¼ gðn;ω;K0Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}if ðω ¼ ωÞffi0

þ2ð2þ cos ð2ωÞÞR ð19Þ

where ω is the estimated frequency, gðn;ω;K0Þ is derived fromEq. (17) and the center frequency of filter H(z) is defined as

K0 ¼ � cos ðωÞ ð20ÞIn Eq. (19), the expected value is equal to zero without noise(R¼0); however, in the presence of noise, the second term of theright side is not zero. Consequently, considering the nonzero partof Eq. (19), a new cost function can be defined as

J ¼ 12∑

eðnÞ22ð2�K0ðnÞÞ

!ð21Þ

The estimation algorithm for the new cost function is derived andthe center frequency of the filter is extracted in the recursivealgorithm as

K0ðnþ1Þ ¼ K0ðnÞ�γeðnÞxðn�1Þð2�K0ðnÞÞ�eðnÞ2

2ð2�K0ðnÞÞ2

" #ð22Þ

where γ is a constant parameter. This estimation algorithm isimplemented in block (1) of Fig. 3. This algorithm is approximatelysimilar to a first order discrete time system where to ensure thestability of this algorithm, roots of its specific polynomial shouldbe held in the unit circle. The stability analysis of this algorithmis known as “thought of experiment” in system identificationapproaches [17].

A.M. Khoshnood, H.M. Moradi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





To simultaneously estimate three vibration frequencies, theprocedure of estimation is implemented in sub-band unit. Thesub-band unit is produced using a multi-rate filter bank. Hence,a 10-channel sub-band unit is designed based on a real valued DFTfilter bank, as shown in Fig. 2. In this filter bank, the bandwidth ofeach sub-band is set equal to 2.5 Hz with regard to the sampletime of the closed loop control system (0.02 s) and the domain ofeach vibration frequency. In the current vehicle, three dominantbending vibration modes of the system are respectively distrib-uted as follows: ω1ffi½1�2:5� Hz; ω2ffi½2:5�5� Hz andω2ffi½5�7:5� Hz. Therefore, the channels number of the designedfilter bank is selected regarding the bending frequencies. In Fig. 2,the number of channels is assigned considering three dominantvibration frequencies. Consequently, each vibration frequency isinserted in one independent channel and the estimation perfor-mance and accuracy are considerably enhanced. Noticeably, allcalculations of the estimation procedure are implemented in thesub-band frequency.

3.2. Estimation of generalized coordinates

The second part of the control input is the generalizedcoordinates associated with the bending vibration. To estimatethese parameters, the error between the reference model (ym) andthe actual one (y) is assumed as follows:

e¼ y�ym ð23Þ

Using Eq. (6) and choosing the desired rigid body dynamic modelas a reference model, Eq. (23) for one vibration mode can berewritten as

eðsÞ ¼ yðsÞ�ymðsÞ ¼ uðsÞðGrðsÞþGbðsÞÞ�uðsÞðGðsÞÞ

¼ uðsÞ ðGrðsÞþGbðsÞÞ�ðGðsÞÞj k

ð24Þ

where GrðsÞ is the rigid body dynamic model of the pitch channel;GbðsÞ is the bending vibration model added to the rigid bodydynamic; u(s) is the overall input of the system; and GðsÞ is themodel reference. In addition, ⌊ðGrðsÞþGbðsÞÞ�ðGðsÞÞc is approxi-mately equal to GbðsÞ. Moreover, it can be asserted that GffiGr;hence, using Eq. (6) for one vibration mode, the error can be

expressed as

eðsÞ ¼ uðsÞGb ¼ CKjf jψ j

s2þ2ςjωjsþω2j

¼ θbjðsÞ ð25Þ

where C is a constant parameter.The parameters of the PD-action controller for the model

without flexibility are tuned using the gain scheduling method.These equations are implemented in block (2) of Fig. 3. Therefore,the control input applied to the pitch channel of the vehicle isfinally proposed as

δ2 ¼1

K

�∑n

j ¼ 1�ω2

j θbjþKdð_qd� _qT ÞþKpðqd�qT Þ�

ð26Þ

4. Robust stability of vibration control system

The L1 small gain theorem is used to guarantee robust stabilityof the proposed vibration control system. This theorem has beenproposed as follows [9]:

Theorem 1. (L1 Small Gain Theorem) The interconnected systemshown in Fig. 4 is stable if jjH1ðsÞjjL1jjH2ðsÞjjL1o1 where jjH1ðsÞjjL1denotes the L1 norm of H1ðsÞ. The L1 gain of a stable proper single-input–single-output system H(s) is defined as

R10 jhðtÞjdt and h(t) is

the inverse Laplace transform of H(s). The proof of this theorem canbe found in [18].

Considering Theorem 1 and Fig. 3, which is a block diagram ofthe closed loop vibration control system, the L1 small gaintheorem for the represented structure cab be written as

jjGðsÞ�CðsÞjjL1jjPðsÞjjL1o1 ð27Þwhere G(s) is the transfer function of the overall vibration controlsystem as shown in Fig. 3 and C(s) is the nominal or preliminarycontrol system of the vehicle. In addition, P(s) is the main dynamicsystem of the vehicle. Eq. (27) can be rewritten regarding the norminequality properties and stuffy conditions as

jjGðsÞjjL1o1�jjPðsÞjjL1jjCðsÞjjL1

jjPðsÞjjL1ð28Þ

If the preliminary closed loop control system is designed based onthe L1 robust stability as the following:

jjCðsÞjjL1jjPðsÞjjL1o1; ) jjGðsÞjjL1o1

jjPðsÞjjL1ð29Þ

then, for the input/output of the G(s), the following inequality isextracted

yðsÞGðsÞ ¼ vðsÞ ) jjvðsÞjjL1 ¼ jjyðsÞGðsÞjjL1) jjvðsÞjjL1o jjyðsÞjjL1jjGðsÞjjL1 ) jjvðsÞjjL1

jjyðsÞjjL1o jjGðsÞ L1jj ð30Þ

where vðsÞ ¼ η∑nj ¼ 1ω

2j θbjðsÞ and η is a tunable gain for better

performance. Moreover, y(s) is the output of the closed loopcontrol system. Eq. (29) can be rewritten as

jjvðsÞjjL1jjyðsÞjjL1

o 1jjPðsÞjjL1

) jjvðsÞjjL1ojjyðsÞjjL1jjPðsÞjjL1

ð31Þ

.

.

.

Fig. 2. Block diagram of the real valued DFT filter bank.

v(s)

u(s)uc(s)

G(s)

C(s)

PD-Controller

c

Model Reference

e

ymy

Robust Adaptive

Algorithm

++

Flexible body

dynamic

(P(s))

Control Input

Control input Construction

1

2

Fig. 3. Block diagram of the closed loop control system (pitch channel).

H1(s)

H (s)

-

+

2

Fig. 4. Block diagram of the small gain theorem.






For the main dynamic system of the vehicle (P(s)), Eq. (30) is againapplicable as

uðsÞPðsÞ ¼ yðsÞ ) jjyðsÞjjL1 ¼ jjuðsÞPðsÞjjL1

) jjyðsÞjjL1o jjuðsÞjjL1jjPðsÞjjL1 ) jjyðsÞjjL1jjPðsÞjjL1

o j uðsÞ jL1 ð32Þ

Then, the final and important inequality is defined as

jjvðsÞjjL1o jjuðsÞjjL1 ) jjvðsÞjjL1o jjucðsÞþvðsÞjjL1 ð33Þwhere ucðsÞ is the preliminary input control of the system or theinput without a vibration controller. It is interesting that the finalinequality is always corrected. However, the best conclusion of therobust analysis can be stated as: if the preliminary closed loopcontrol system of the vehicle is robust stable regarding the L1small gain theorem, the closed loop control system of the vibrationcontrol strategy is also robust stable.

5. Simulation results

The governing equations of motion presented in Section 2 aresimulated in a Matlab/Simulink environment. The FLV is a two-stage aerospace vehicle with a length-to-diameter ratio ofapproximately 15 and bending vibration effects that are notnegligible. Only the first stage of the flight (lift-off phase) in whichthe effects of flexibility are significant is simulated. The criticaloperating point of the vehicle is set on the maximum dynamicpressure point.

It is important to know that the effects of flexibility only appearin the first part of the lift-off phase as a result of increasing thebending vibration frequencies. The pitch channel of the FLV isinfluenced by the significant effects of bending vibration accordingto the frequency range of the elements of the closed loop controlsystem. The new robust adaptive control strategy is implemented

5 10 15 20 25

50

52

54

56

58

60

62

64

66

time [sec]

Pitc

h A

ngle

[deg

]

5 10 15 20 25

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time [sec]

Pitc

h ra

te [r

ad/s

]

Fig. 5. The effects of vibration on the pitch angle (left) and the pitch rate (right) without the vibration controller (for one vibration mode with Fre¼3 Hz).

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

time [sec]

Gen

eral

ized

Coo

rdin

ate

Fig. 6. Generalized coordinate associated with the first bending vibration of thevehicle without the vibration controller (Fre¼3 Hz).

5 10 15 20 25 30

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

time [sec]

Act

uato

r def

lect

ion

[rad

]

Frequency=2.5

Frequency=3

Fig. 7. Actuator deflection for the robust adaptive approach in two range ofbending vibration frequencies.

5 10 15 20 25 30

1

1.5

2

2.5

3

3.5

4

time [sec]

Vib

ratio

n fr

eque

ncie

s [H

z]

Fig. 8. Performance of the sub-band vibration frequency estimation for threevarious frequencies.






19 20 21 22 23 24 25 26 27 28 29-3

-2

-1

0

1

2

3x 10-4

time [sec]

Gen

eral

ized

Coo

rdin

ate

Estimated Generalized Co.

Actual generalized Co.

Fig. 9. Estimation performance of the generalized coordinate associated with thebending vibration in contrast with the actual one (Fre¼3 Hz).

5 10 15 20 25 30

50

52

54

56

58

60

62

64

66

time [sec]

Pitc

h an

gle

[deg

]

Fig. 10. Reduction of the effects of vibration on the pitch angle of the vehicle byusing the robust adaptive control system in comparison with the system withoutvibration controller (Fre¼3 Hz).

0 5 10 15 20 25 30-12

-10

-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [sec]

Mod

el re

fere

nce

erro

r [ra

d]

Fig. 11. Model reference tracking error of the robust adaptive controller(Fre¼3 Hz).

5 10 15 20 25 30 35 40 45

50

55

60

65

70

time [sec]

Pitc

h A

ngle

[deg

]

24 26 28 30 32

56

57

58

59

Without control systemWith robust adaptive controller

Fig. 12. Performance of the robust adaptive control system in application to thevehicle with three vibration modes.

5 10 15 20 25

35

40

45

50

55

60

65

70

75

time [sec]

Pitc

h A

ngle

[deg

]

Robust adaptive controllerAdaptive filtering method

Fig. 13. Comparison of the robust adaptive vibration control system and theadaptive filtering approach in application to the FLV with phase delay.

2 3 4 5 6 7 8 9 1050

51

52

53

54

55

56

57

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Fig. 14. Sensitivity analysis of the adaptive notch filter strategy in the presence offrequency estimation error (accurate frequency estimation¼2.5 Hz).






to assess the properties of vibration suppression and systemperformance. From the vibrational characteristics of the FLVstructure, the first bending vibration frequency set is determinedto be WA ½0; 4� Hz.

The undesirable effects of the first bending vibration on thepitch angle of the vehicle are illustrated in Fig. 5. The bendingvibration, like sinusoidal disturbance, significantly affects theperformance of the angular velocity and pitch angle of the system.Although the nominal or preliminary control system is designed toguarantee the robust stability, the vibrational effects lead toundesirable performance. The generalized coordinates associatedwith the first bending vibration of the FLV in the lateral positionare shown in Fig. 6. This response is extracted without the use ofthe new vibration control system.

The characteristics of the control system based on the robustadaptive approach are shown in Figs. 7–9. Actuator deflection isshown in Fig. 7 for two vibration frequencies. This figure alsoshows the robustness of the method against the frequency change.The components of the control input (estimated vibration fre-quencies and the first generalized coordinate) are illustrated inFigs. 8 and 9, respectively. Note that the transient part of theestimation and its accuracy are adequate for generating a controlinput signal at different frequencies. In addition, the accuracy ofthe estimated generalized coordinate is adequate for constructingthe control input of the system.

The pitch angle of the vehicle is shown in Fig. 10 with andwithout the robust adaptive control system. This figure clearlydemonstrates the valuable performance of the vibration controlstrategy based on the robust adaptive theories. The error definedby Eq. (23) is displayed in Fig. 11. This parameter shows theperformance of the model reference tracking error. Fig. 12 demon-strates the performance of the robust adaptive controller for threeundesirable vibration modes. The existence of at least two domi-nant undesirable vibration modes and the satisfactory decrease of

their effects using the proposed control system are shown in thisfigure.

The performance of the robust adaptive control system incontrast with the adaptive filtering method is demonstrated inFig. 13 for 15% change in the phase delay of the closed loop controlsystem. As shown in this figure, the output of the closed loopcontrol system based on the adaptive filtering method is violatedby the phase delay. The new vibration control system significantlydecreases the undesirable effects of bending vibration withoutconsiderable change in the phase of the system. In addition, forconfirming the preference of the proposed vibration control,a sensitivity analysis is shown in Figs. 14 and 15. For this purpose,in Fig. 14, sensitivity analysis of the adaptive notch filter strategy inthe presence of frequency estimation error is demonstrated. Thesimulation results show that the adaptive filtering system is verysensitive to inaccurate frequency estimation. In another analysis,the same investigation for the robust adaptive control strategy isillustrated in Fig. 15. From this figure, robustness of the newvibration control system against the frequency estimation error isconsiderably guaranteed.

6. Conclusions

Undesirable vibration effects are a major challenge for mechan-ical systems. In aerospace vehicles, this phenomenon is moreimportant because of their complicated sub-systems. On the otherhand, vibrations can affect the navigation system of the vehicle.The present study proposes a control system for decreasing theeffects of bending vibration on an FLV using robust adaptiveapproaches. The controller is designed based on online estima-tions of vibration frequencies. The robust adaptive controlleremploys a model reference adaptive scheme to construct thecontrol input of the vibration control strategy. In this way,

0 5 10 15 20 25 30-10

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Fig. 15. Sensitivity analysis of the robust adaptive control strategy in the presence of frequency estimation error (accurate frequency estimation¼2.5 Hz).






algorithms for frequency and generalized coordinate estimationare developed to shape the control input. One advantage of theproposed vibration control system is that it guarantees the robuststability of the system using the L1 small gain theorem. Therobustness and direct application of the control system are twopractical advantages over previous adaptive control strategies.

Numerical simulation of a linear model of the vehicle demon-strates the performance of the new vibration control strategy. Theresults of the simulation show satisfactory performance of theproposed robust adaptive vibration control system. The controlsystem can significantly decrease the destructive effects of thebending vibration without considerable change in the phasemargin of the closed loop system, which is usually a challenge inadaptive filtering approaches.

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Robust adaptive vibration control of a flexible structure