active vibration control using optimal lq tracking system with additional dynamics ·...

International Journal of ControlVol. 78, No. 15, 15 October 2005, 1182–1197

Active vibration control using optimal LQ tracking system

with additional dynamics

T. NESTOROVIC-TRAJKOV*, H. KOPPE and U. GABBERT

Institut fur Mechanik, Otto-von-Guericke Universitat Magdeburg, Germany

(Received 30 September 2004; in final form 10 May 2005)

This paper focuses on the vibration suppression task using an appropriate controller designfor active control of structures with distributed piezoelectric actuators and sensors. The

problem arises from the need to control undesired vibrations caused by disturbances orexcitations acting upon a structure in an efficient and at the same time a simple way. A specialclass of disturbances/excitations (periodical, with frequencies equal or near to the eigen-frequencies of the controlled structure) may cause undesired resonant states. In order to reject

such disturbances and suppress vibrations in the presence of excitations an optimal LQcontroller based on tracking systems with additional dynamics is proposed for the vibrationcontrol problem. The controller was tested in the presence of excitations with different

frequencies. Controller design is model-based, where for the numeric modelling of thestructure the finite element approach was used. Besides, subspace-based model identificationwas used as well. Controller design, testing and implementation were performed on the

funnel-shaped shell structure, the inlet part of the magnetic resonance tomograph.Simulation results as well as the real-time implementation of the controller as a part of theHardware-in-the-Loop system show considerable vibration suppression in the presence ofexcitations and confirm the efficiency of the controller.

1. Introduction

Development of actively controlled structures involvesmany subsequent and complex steps. This paper focuseson the controller design problem for the active controlof structures with distributed piezoelectric actuatorsand sensors with the aim of vibration suppression. Theproblem arises from the need to control undesired vibra-tions caused by disturbances or excitations acting upona structure in an efficient and a simple way. Overalldesign of active (smart) structures represents a broadand complex field with many directions of developmentwhere, besides modelling, controller design for a specificcontrol purpose plays an important role. Proposed con-troller design technique is aimed at vibration suppres-sion but can be extended to other control problems

in smart structures (like acoustic problems) as well. Aspecial class of disturbances/excitations (periodical,with frequencies equal or near to the eigenfrequenciesof the controlled structure) may cause undesired reso-nant states. In order to reject such disturbances andsuppress vibrations in the presence of excitations anoptimal LQ controller based tracking system withadditional dynamics is proposed for the vibrationcontrol problem.

Design of an optimal LQ or a feedback-gain control-ler has been addressed in literature as a means used insolving active control problems (Lim et al. 1999, Raoand Sana 2001, Gabbert et al. 2003, Chandiramaniet al. 2004, Seeger 2004). The novel approach to optimalcontrol presented in this paper combines a trackingsystem with additional dynamics (Vaccarro 1995) withthe optimal LQ controller (Ogata 1995, Vaccaro 1995,Franklin et al. 1998) and Kalman filter. This approachhas been successfully used for control of vibration inactive structures with distributed piezoelectric actuators

*Corresponding author. Email: [email protected]

magdeburg.de

International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online � 2005 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207170500163383

and sensors in the sense of considerable vibrationmagnitudes suppression. The advantage of the pro-posed method can be seen through the better vibrationmagnitudes reduction in the steady state comparedwith the standard feedback and optimal LQ controllerowing to the additional dynamics incorporated in theoverall design model as a part of the compensator incascade combination with the controlled structure. Thecontroller is especially aimed at vibration suppressionin the presence of periodic excitations and disturbanceswith frequencies corresponding to the eigenfrequenciesof the controlled structure, because they are responsiblefor possible resonant states, which should be prevented.An a priori knowledge about such disturbances orexcitations reflecting real operating conditions is avail-able from the modal analysis of the structure underinvestigation, which can be performed numerically orexperimentally.An optimal LQ tracking system with additional

dynamics was tested on a funnel-shaped shell structurewhich represents the inlet part of the magnetic resonanceimage (MRI) tomograph used in medical diagnostics.Since the proposed controller design is model based, asa starting point for the control system development astate space model was used, which was obtained usingtwo approaches: (i) finite element (FE) based modellingusing the FE analysis software COSAR� (COSAR1992) and modal truncation and (ii) subspace based iden-tification (Viberg 1995, Van Overschee and De Moor1996). Controller implementation showed comparableand good results in both cases confirming at the sametime the feasibility of the numeric model obtained usingthe FE approach. Since themodel-based optimal control-ler design assumes state feedback, a Kalman filter wasused for the modal states estimation in the case whenthe controller design is based on the numeric FE model-ling and modal truncation and for the estimation of thestate variables of the identified model.The influence of the actuator/sensor placement was

also considered and optimal placement at specifiedregions was confirmed by simulation and experimentalresults. For the real-time implementation and testingof the controller Hardware-in-the-Loop (HiL) systemwith dSPACE� was used incorporating the funnel inletof the MRI tomograph as a real structure.Besides the vibration control proposed controller

opens the possibility of extended use in solving acousticproblems in a similar manner depending on thefrequency ranges of interest aimed for control.

2. State space model for the controller design

As a starting point for the controller design a state spacerepresentation of the controlled structure is used.

General continuous-time or discrete-time form of thestate space model used for the model based controllerdesign can be derived using the FE approach or thesubspace based identification.

2.1. State space model obtained using the FE approach

As a result of the FE analysis, behaviour of a structureapproximated by an arbitrary number of finite elementscan be described with assembled equations of motion ina semi-discrete form (Gabbert et al. 2002, Nestorovic-Trajkov et al. 2003a)

M€qqþDd _qqþ Kq ¼ �FF, ð1Þ

where M, Dd and K are the mass matrix, the dampingmatrix and the stiffness matrix, respectively. The totalload vector �FF is divided into the vector of the externalforces FE and the vector of the control forces FC

�FF ¼ FE þ FC ¼ �EEwðtÞ þ �BBuðtÞ, ð2Þ

where the forces are generalized quantities also includ-ing electric charges. Vector w(t) represents the vectorof external disturbances and u(t) is the vector of the con-troller influence on the structure. Matrices �EE and �BBdescribe the positions of the forces and the controlparameters in the finite element structure, respectively.As a convenient procedure for the state space modelobtaining modal truncation is adopted, since the highorder of the FE model represented by equation (1) isnot suitable for the controller design and the reductionof the model order is required. A decoupled systemof equations (3) in modal coordinates z is obtainedby performing ortho-normalization with (T

m M(m ¼ I

and (Tm K(m ¼ :, where the modal matrix (m and

the spectral matrix : are obtained from the solutionof the linear eigenvalue problem for (1). In decoupledsystem of equations

€zzþ D_zzþ:z ¼ (Tm

�FF, ð3Þ

D ¼ (Tm Dd(m represents the modal damping matrix.

Generalized displacements q are related to modalcoordinates z by

qðtÞ ¼ (mzðtÞ: ð4Þ

In the modal truncation procedure the modal displace-ment vector z is partitioned and only a part zr cor-responding to selected eigenmodes of interest for the

Active vibration control 1183

control is retained. Introducing the modal reduced statevector

xðtÞ ¼zrðtÞ_zzrðtÞ

� �, ð5Þ

the modal reduced model is obtained in the state spaceform

_xx ¼0 I

�:r �Dr

� �xþ

0

(Tr

�BB

� �uðtÞ þ

0

(Tr

�EE

� �wðtÞ,

ð6Þ

where matrices :r, Dr and (r, and are obtained fromthe appropriate partition of the matrices :, D and (m

respectively. Written in a standard state space formthe state equation of the model becomes

_xxðtÞ ¼ AxðtÞ þ BuðtÞ þ EwðtÞ: ð7Þ

Using a similar procedure a state space formulation ofthe output equation is obtained

y ¼ CxðtÞ þDuðtÞ þ FwðtÞ, ð8Þ

which assumes in a general case the influence of thecontrol and external inputs on the outputs.Discrete-time equivalent of the continuous-time state

space model (7), (8) is used as a starting point for thecontroller design

x½kþ 1� ¼ (x½k� þ !u½k� þ ew ½k�,

y½k� ¼ Cx½k� þDu½k� þ Fw½k�, ð9Þ

where

( ¼ eAT, ! ¼

ðT0

eA�Bd�, e ¼

ðT0

eA�Ed� ð10Þ

and T is the sampling time.

2.2. Subspace identification of the state space model

For the identification of the state space model fromthe measured input-output data the subspace-basedidentification is used (Viberg 1995, Van Overscheeand De Moor 1996, McKelvey et al. 2002). Generaldiscrete-time combined deterministic-stochastic form ofthe model to be identified corresponds to modifiedform of equations (9)

x½kþ 1� ¼ (x½k� þ !u½k� þ w½k�,

y½k� ¼ Cx½k� þDu½k� þ v½k�, ð11Þ

where w[k] and v[k] denote the process and the measure-ment noise, respectively.

Input-output measurement data are organized in theform of appropriate input and output block Hankelmatrices (Van Overschee and De Moor 1996). InputHankel matrix is defined as

U ¼ U0 2i�1j ¼

u0 u1 u2 � � � uj�1u1 u2 u3 � � � uj� � � � � � � � � � � � � � �

ui�1 ui uiþ1 � � � uiþj�2

ui uiþ1 uiþ2 � � � uiþj�1uiþ1 uiþ2 uiþ3 � � � uiþj� � � � � � � � � � � � � � �

u2i�1 u2i u2iþ1 � � � u2iþj�2

2666666666664

3777777777775:

ð12Þ

Output measurement matrix Y is defined in a similarway and the measurement data are organized inthe form of the following input-output relation(Viberg 1995)

Y½k� ¼ !�x½k� þ(�U½k�, ð13Þ

where !� represents the observability matrix for thesystem (11), (� is the Toeplitz matrix (Viberg 1995) ofimpulse responses from u to y

(� ¼

D 0 � � � 0

C D 0

..

. . .. . .

. ...

C(��2! � � � C! D

2664

3775 ð14Þ

and � is a specified number greater than the statedimension but much smaller than the data length. Fora deterministic case (Franklin et al. 1998) the problemis simplified to determining !� and (� by computingthe singular value decomposition (SVD) of U in thefirst step

U ¼ PRQT ¼ ½Pu1 Pu2 � ½Ru 0 �QT

u1

QTu2

" #: ð15Þ

If matrix U has dimension m� n and rank r, then thepartition in (15) is performed as follows:

P ¼ p1 � � � pr j prþ1 � � � pm� �

¼ Pu1 Pu2

� �ð16Þ

Q ¼ q1 � � � qr j qrþ1 � � � qn� �

¼ Qu1 Qu2

� �ð17Þ

1184 T. Nestorovic-Trajkov et al.

where pi are called the left singular vectors ofU. It can beshown that they are eigenvectors of UUT. Vectors qi arethe right singular vectors of U. It can be shown that theyare eigenvectors of UTU (Vaccaro 1995). Multiplying(13) by Qu2, matrix !� can be determined from a SVDof YQu2. Then matrix C is obtained as the first row(in a sense of a block-row) of the observability matrix!�, and matrix( is calculated from: !� ¼ �!!�( applyingpseudo inverse, where �!!� is obtained by dropping thelast row of !�. Matrix !� represents the matrix obtainedby dropping the first row of !�. For the calculation of �and D matrices, equation (13) is multiplied by thepseudo inverse of U on the right and by PT

u2 from (15)on the left. Thus the equation is reduced to

PTu2YU

�1 ¼ PTu2(�: ð18Þ

After rearranging, the equation (18) can be solved for� and D using the least squares. In this way thesystem parameters in the form of state space matricesof the model (11) are identified using the subspacebased identification method.

3. Optimal LQ tracking system with

additional dynamics

Proposed control methodology combines trackingsystem with additional dynamics, optimal LQ controllerand Kalman filter. The idea of introducing additionaldynamics in order to make the system output tracka given reference input or to reject a disturbance/excitation relies on tracking zero-input trajectories(Vaccaro 1995, Franklin et al. 1998). Thus the trackingcontrol problem converts to a regulation problem oftracking zero-input trajectories of a design modelaugmented with additional dynamics. The advantageof a tracking system with additional dynamics can beviewed in the light of the fact that once the knowledgeof the specified reference input and/or the excitation/disturbance is incorporated in additional dynamics, thedesigned control system can handle both types ofinputs. It will be shown through the control systemdesign procedure.Design of the optimal LQ tracking system with

additional dynamics assumes inputs to the system (refer-ence inputs and excitations/disturbances) which can berepresented in the form of a rational transfer function(step, ramp, sinusoidal and exponential signals). Inthe vibration control of smart structures sinusoidalexcitations play a significant role in the controllerdesign procedure. The excitation frequency correspond-ing to an eigenfrequency of the structure under controlor to a combination of eigenfrequencies can be regarded

as a possible worst study case due to the resonance.The rejection of such excitations/disturbances is there-fore significant for the vibration control of smart struc-tures. For defining additional dynamics disturbance/excitation and reference input are assumed to be theoutputs of the models described with the followingstate space equations, respectively.

Disturbance: _zzdðtÞ ¼ AdzdðtÞ, fðtÞ ¼ cdzdðtÞ: ð19Þ

Reference input: _zzrðtÞ ¼ ArzrðtÞ, rðtÞ ¼ crzrðtÞ: ð20Þ

The following interpretation is implied by the theoremin (Vaccaro 1995, p. 332). For a system described by adiscrete-time linear state space realization (�, �, C),see equation (11), and the system input (referenceinput or disturbance/excitation) expressed in the formof a rational transfer function R(z)¼ n(z)/d(z) if atleast one of input poles zi is not an eigenvalue of �,then additional dynamics must be used to have a track-ing system with zero steady-state error. The proof can befound in (Vaccaro 1995, p. 332). Introducing additionaldynamics an augmented design plant is obtained. Theregulator designed for the augmented plant achievesthan exact tracking. In the case of multiple input polesadditional dynamics are defined on the basis of thesets �r and �d, which contain the reference input anddisturbance/excitation poles respectively together withtheir multiplicities

�r ¼ ð�r1 , mr1Þ, ð�r2 , mr2 Þ, . . .� �

,

�d ¼ ð�d1 , md1 Þ, ð�d2 , md2 Þ, . . .� �

, ð21Þ

where �ri is the eigenvalue of the Ar matrix (20) withthe multiplicity mri and �di is the eigenvalue of theAd matrix (19) with the multiplicity mdi . The set �is defined as the union of the sets �d and �r. If acommon eigenvalue appears in both the sets �d and�r, only the one with the higher multiplicity is includedin the set �. Eigenvalues �i in the set � define the polesof additional dynamics.

An important property of the control systems withadditional dynamics is that the system designed totrack a specified reference input rejects at the sametime the disturbance of the same type and vice versa.Even if it is not required from the system to reject dis-turbances, by the nature of its structure it will rejectthe disturbances/excitations which are defined with thesame poles as the specified reference input. This resultsfrom the definition of the set �. If no disturbancepoles are specified, the set �d is an empty set. Thenthe set � coincides with the set �r since it is obtainedas a union of �r and the empty set. The same set � isobtained if �d¼�r, i.e. if it is required that the system


rejects the disturbances/excitations of the same typeas the reference input. Similar reasoning implies thatthe tracking systems designed in this way operatewell in cases when the reference input and disturbance/excitation switch the roles. The sets�r and�dwould thenreverse the elements, but their union remains the same.Mapping the poles �i from the set � into z-domain,

the polynomial �(z) in the denominator of the addi-tional dynamics transfer function is determined in thefollowing way:

�ðzÞ ¼Yi

z� e�iT� �mi

¼def

zs þ �1zs�1 þ �2z

s�2 � � � þ �s,

ð22Þ

where s ¼P

mi is the total number of the poles and mi isthe multiplicity of the pole �i. The additional dynamicssystem is then defined by the transfer function

HaðzÞ ¼zs

�ðzÞ¼

zs

zs þ �1zs�1 þ � � � þ �s, ð23Þ

and implemented as a part of the compensator in a cas-cade combination with the plant. In this way the transferfunction of the design plant Ha(z)H(z) (cascade combi-nation of the plant and additional dynamics) has poleswhich are at the same time eigenvalues of the statematrix for the design plant in z-domain. In order torepresent the state space model of the additionaldynamics in an observable canonical form, the nomina-tor of the transfer function Ha(z) is chosen as zs.Matrices �a and �a of the additional dynamics arethen determined from the coefficients of the polynomial� (z)

�a ¼

��1 1 0 � � � 0��2 0 1 � � � 0

..

. ... ..

. . .. ..

.

��s�1 0 0 � � � 1��s 0 0 � � � 0

266664

377775, �a ¼

��1��2

..

.

��s�1��s

266664

377775:

ð24Þ

In the case of multiple-input multiple-output (MIMO)systems additional dynamics has to be replicated in qparallel systems (once per each output), where q is thenumber of outputs. Replicated additional dynamicsare then formed as a block matrix with �a, �a on thediagonal:

�� ¼def

diag ð�a, . . . ,�a|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}q times

Þ, �� ¼def

diag ð�a, . . . ,�a|fflfflfflfflfflffl{zfflfflfflfflfflffl}q times

Þ: ð25Þ

The discrete-time design model (�d,�d) is formed as

a cascade combination of additional dynamics and thediscrete-time plant model (�, �) and expressed as a

state-space formulation

xd½kþ 1� ¼ (dxd½k� þ !du½k�, ð26Þ

�d ¼� 0

��C ��

� �, �d ¼

�0

� �, xd ¼

x½k�xa½k�

� �: ð27Þ

Matrices (�and !� in the design model realizationð(d, !dÞ denote additional dynamics of a single-input

single-output system ð(a, !aÞ or replicated additionaldynamics ð ��, ��Þ of a multiple-input multiple-output

system. Dimension of design model is nþ qs where n isthe order of the plant model, q the number of outputs

and s the order of the additional dynamics.The control system is designed for the realization

(�d,�d). The feedback gain matrix L of the control law

u½k� ¼ �Lxd½k� ð28Þ

is partitioned into submatrices L1 and L2 formedfrom the first n and last q� s columns of the matrix L,

respectively

L ¼ ½L1 L2 �: ð29Þ

Thus the feedback gain matrix L1 corresponds to

the state variables of the controlled structure, whilethe feedback gain matrix L2 pertains to the rest of the

state variables in the design state vector xd introducedby additional dynamics. Control system with additional

dynamics implemented as a part of the compensator isrepresented in figure 1.

Design of the controller involves the estimation of thestate variables. If the state space model of the structure

is obtained using the FE approach, state variables aremodal variables which are not measurable and their

estimation is therefore necessary. On the other hand inthe identified state space model state variables are not

measurable either, which imposes the need for theirestimation. For this purpose the Kalman filter is emplo-

yed (Preumont 1997, Franklin et al. 1998, Nestorovic-Trajkov et al. 2003a, b). Kalman filter design is based

on the assumed plant model in the form (11). Theprocess and measurement noise w and v are assumed

to be white noise with the zero mean. The covariancesof the process and measurement noise are denoted as

EðwwTÞ ¼ Qw and EðvvTÞ ¼ Rv, respectively. Then the


Kalman estimator is defined by the following equations

xx½k� ¼ �xx½k� þ Lest½k�ðy½k� � C �xx½k�Þ

�xx½k� ¼ (xx½k� 1� þ !u½k� 1�,

)ð30Þ

where the Kalman gain matrix is

Lest½k� ¼ P½k�CTR�1v ð31Þ

and

P½k� ¼Mk½k� �Mk½k�CT

� CMk½k�CTþ Rv

� ��1CMk½k�

Mk½kþ 1� ¼ (P½k�(T þ eQweT:

9>>=>>; ð32Þ

Matrices P and Mk are determined by solvingequations (32). For the Kalman filter implementationthe initial conditions for �xx½0� and Mk½0� are required.These initial values represent the a priori estimate ofthe state and of the accuracy of this a priori estimate.In certain cases some test data can be used as a supportfor the choice of these initial values, but it is not typical.Usually it is assumed that the components of �xx½0�contained in y are equal to the first measurement andthe remaining components are equal to zero. In a similarway the components of Mk corresponding to measuredcomponent can be set to Rv and the remaining compo-nents can be set to a high value.Besides the knowledge of the initial value for Mk,

the a priori knowledge of the process noise magnitudeQw and the measurement noise magnitude Rv is alsorequired. The value for Rv in a given actual design pro-blem can be chosen based on the sensor accuracy. Hereit should also be noted that the assumption about theprocess noise being the white noise is introduced inorder to simplify solving of the optimization problem.Physically Qw is often associated with unknown distur-bances. In a case when a random disturbance is acolored noise (time correlated), it can be accurately

modelled by augmenting � with a coloring filter whichconverts a white noise input into time correlated noise(Franklin et al. 1998). Due to complexity it is oftennot done in practice. Rather than that the disturbancesare assumed to be white and the noise intensity isadjusted to give acceptable results in the presence ofexpected disturbances.

Incorporating the Kalman filter in the controllerdesign, besides the necessary state estimation, a reduc-tion of the observation spillover (the influence of theresidual modes on the sensor output) can be achieved(Preumont 1988, 1997).

Feedback gain matrix L in the control law (28) isdetermined designing the optimal LQ regulator for thedesign model represented by the realization (�d,�d),equation (26). The feedback gain matrix L has to bedetermined in such a way that the control law (28)minimizes the performance index

J ¼1

2

X1k¼0

xd½k�TQxd½k� þ u½k�TRu½k�

� �ð33Þ

respecting the constraint (26) where Q and R aresymmetric, positive-definite matrices. Optimal controllerdesign problem is solved minimizing the cost function(33) under constraint (26) using the method of Lagrangemultipliers (Vaccaro 1995, Franklin et al. 1998). Thechoice of the weighting matrices Q and R in the perfor-mance index is designer dependant and is based on therelative importance of the various states and controls.The trade-off between the control effort and the systemresponse determines the choice of the weighting matrices.In general, the weighting matrices are chosen in sucha way that large input signals are penalized by increas-ing the value of the matrix R and faster response ofappropriate state variables is achieved by increasingthe values of appropriate elements in the weightingmatrix Q.

With all the steps of the optimal LQ controller design,including defining additional dynamics, estimator designand finally optimal control law design, the procedure

Figure 1. Control system with additional dynamics implemented as a part of the compensator.


of the model based optimal control design can beconcluded resulting in a control system which cansuccessfully be used for the vibration suppression insmart structures.Optimal LQ tracking control system with additional

dynamics and Kalman estimator is represented infigure 2.

4. Implementation of the controller on the

MRI tomograph

Optimal LQ controller with additional dynamics isimplemented for the vibration control (in the sense of

vibration magnitudes reduction) of the funnel-shapedshell structure, the inlet part of the Siemens� MRItomograph (figure 3). The aim is reduction of trans-mitted vibrations from the cylindrical body of thetomograph to the funnel-shaped inlet using piezoelectricactuators and sensors. The vibration control problemof a cylindrical shell used in MRI equipment wasaddressed in (Qiu and Tani 1995, Tani et al. 1995,Qiu and Tani 1996). This paper represents originalresults for the vibration control of the complex funnel-shaped structure, which is excited by transmittedvibrations from the cylindrical-shell-shaped body ofthe tomograph. Control of these transmitted vibra-tions contributes to the overall vibration suppression

Figure 3. Magnetic resonance tomograph.

Figure 2. Optimal LQ tracking system with additional dynamics and state estimator.


of the medical device and therefore plays an impor-tant role.The experimental rig includes the funnel-shaped con-

trolled structure with piezoelectric actuators and sensorsglued to the surface of the funnel and placed as shownin figure 4. They are denoted as 1L, 2L, 3L for theleft-hand side actuators and sensors and 1R, 2R, 3Rfor the right-hand side ones.The funnel is a part of the Hardware-in-the-Loop

system with dSPACE� implemented as represented bythe scheme of the control experimental rig in figure 5. Theperiodic excitation is generated by Matlab/Simulink�

and dSPACE� and exerted by the shaker acting atdefined point on the funnel, as shown in figure 5.

Applying the FE based approach for the analysis ofthe funnel behaviour and numeric model development(using the finite element software COSAR�), the eigen-frequencies were determined and on the basis of thecomparison with results of the experimental modalanalysis, a reduced order state-space model of thefunnel was adopted for the control of the funnel eigen-modes in the frequency range up to 35HZ, wherethe eigenfrequencies of interest are f1¼ 9.573Hz,f2¼ 23.333Hz and f3¼ 31.439Hz. Comparison of the

Figure 4. Funnel-shaped inlet of the MRI tomograph with actuator/sensor placement.

Figure 5. Scheme of the experimental rig: HiL system with dSPACE� and the funnel.


modal analysis results shows good agreement betweenthe eigenvalues obtained with the finite element softwareCOSAR� and experimentally. For the controllerimplementation purposes the state space model of thefunnel was identified using the subspace based approachas well.

4.1. Optimal LQ controller with additional dynamicsbased on the FE model

The modally reduced state space model of the funnel oforder 20 is initially obtained on the basis of the first 10eigenmodes. Since the control is aimed at the first threelowest eigenfrequencies, the state space model is furtherreduced to order 6 resulting in a controllable andobservable realisation necessary for the controllerimplementation. Different periodic excitations wereconsidered. The results are represented for theactuator/sensor pair A2R�S1R.Control system is first designed assuming the sine

excitation with the frequency f1. The design modelorder is then equal to eight and the weighting matricesfor the optimal LQ tracking system are selected asQ¼ I8�8 and R¼ 10. In this way the first eigenfrequencyis controlled. Time response of the sensor S1R and thecontrol effort obtained experimentally with this controlsystem in the presence of the sine excitation withfrequency f1 is shown in figure 6. The period without

control is clearly indicated by the zero control inputand obviously greater vibration magnitudes of thesensor response. The same control system exhibitsbehavior presented in figure 7 when used in the presenceof the excitation obtained as a sum of the sinusoidswith frequencies f1, f2, f3.

The control system is now designed in order tocontrol simultaneously the first three eigenfrequenciesassuming the excitation as a sum of the sinusoids withfrequencies f1, f2, f3. Order of the design model is 12and the weighting matrices are selected as Q¼ I12�12and R¼ 10. According to expectations this controlsystem provides better vibration suppression (figure 8)in comparison with figure 7 using a slightly greatercontrol effort.

The magnitude of the Fast Fourier Transform (FFT)of the sensor signal is taken as the measure of thevibration suppression in the frequency domain. Resultsobtained from the simulated and measured responsesare presented in figure 9.

Finally, the results in the time domain when designedcontrol system is used to control the first and the secondeigenfrequency as well as only the first eigenfrequencyare presented in figures 10 and 11 respectively.

Presented results for the considered single-input singleoutput system modeled using the FE approach showefficiency of the controller through the vibrationmagnitudes reduction. According to expectations thecontrol system designed taking into account three

Figure 6. Sensor response and control in the presence of the sine excitation with the frequency f1 (FE model based control systemdesigned to control the first eigenfrequency).

Figure 7. Sensor response and control with excitation containing three eigenfrequencies (FE model based control system designedto control the first eigenfrequency).


different frequencies of the periodic excitation performsbetter in the presence of excitations obtained as a sum ofthree sinusoidal signals then the controller with addi-tional dynamics which takes into account only oneeigenfrequency in the presence of the same excitation.

4.2. Optimal LQ controller with additional dynamicsbased on the identified model

This section presents the implementation results for theoptimal LQ controller with additional dynamics when

the state space model used for the controller design

was obtained using the subspace based identification.

For the actuator/sensor pair A2R–S1R a good approx-

imation of the measured frequency response function is

obtained with the identified state space model of the

order 60. Vibration magnitudes reduction and control

in time domain for the actuator/sensor pair A2R–S1R

are represented in figures 12 and 13. Weighting matrices

for the optimal LQ controller design are Q¼ I62�62 � 0.1

and R¼ 100.For the multiple-input multiple-output (MIMO)

control, two cases were considered: one actuator – two

Figure 9. FFT magnitude of the sensor signal: (a) simulated; (b) measured.

Figure 8. Sensor signal and control with excitation containing three eigenfrequencies (FE model based controller designed for

simultaneous control of the first three eigenfrequencies).


Figure 9. Continued.

Figure 11. Sensor signal and control with excitation containing the first eigenfrequency (FE model based controller designed forsimultaneous control of the first three eigenfrequencies).

Figure 10. Sensor signal and control with excitation containing the first two eigenfrequencies (FE model based controller designedfor simultaneous control of the first three eigenfrequencies).


sensors and two actuators – two sensors. In the case

‘‘one actuator A2R – two sensors S1R, S2L’’, the model

identified using subspace identification has order 50.

Good agreement between the frequency responses

obtained from measurements and from simulations

based on the identified model was the criterion for the

selection of the model order. Figure 14 represents the

magnitudes of the frequency response in the range

up to 35HZ corresponding to the frequency range of

interest. Dashed lines represent frequency response

obtained as a ratio of the Fast Fourier Transforms

(FFT) of the measured output-to-input (sensor-

to-actuator) signals. Solid lines represent the same

ratio, but obtained from the simulated responses based

on the identified model. Based on the identified model,

optimal LQ tracking system with additional dynamics

is designed. The weighting matrices of the LQ controller

are: Q¼I54�54 � 100 and R¼ 100. Additional dynamics

takes into account the first eigenfrequency of the

funnel. Designed system successfully reduces oscillation

amplitudes at sensors locations in the presence of

the sine excitation with the frequency f1 as well as in

the presence of the excitation containing the first three

eigenfrequencies (figure 15).Optimal LQ tracking system with additional

dynamics was also designed in the MIMO case for

the actuators A2L, A2R and sensors S1R, S2L.

Additional dynamics, which takes into account the

first eigenfrequency of the funnel, is replicated two

times due to two outputs. Control system is designed

using the weighting matrices Q¼ I54�54 � 0.1 andR¼ I2�2 � 100. Time responses of the sensors and con-trol signals for two excitation cases (sine excitationwith the frequency f1 and the excitation containingthe first three eigenfrequencies) are represented infigure 16.

In this subsection the state space model used for thecontroller design was obtained using the subspacebased identification method. The method is charac-terized by efficient model development based on themeasured input and output signals resulting in astate space realization which is convenient for thecontroller design. Designed controller in turn performssuccessful vibration control shown by the obviousvibration magnitudes reduction in comparison withthe uncontrolled case.

5. Conclusions

In this paper the optimal LQ tracking system withadditional dynamics was proposed as a solution forthe control of vibrations caused by excitations ordisturbances, in the sense of the vibration magnitudessuppression. The controller is aimed at active vibrationcontrol of structures with distributed piezoelectricactuators and sensors, but it can be widely used fordifferent control tasks. In vibration control aspecial class of periodic excitations/disturbances withfrequencies corresponding to the eigenfrequencies of

Figure 12. Actuator/sensor pair A2R–S1R: sensor response and control in the presence of the sine excitation withfrequency f1.

Figure 13. Actuator/sensor pair A2R–S1R: sensor response and control with excitation containing three eigenfrequencies.


Figure 14. Frequency responses of the identified model up to 35Hz: (a) actuator/sensor pair A2R–S1R; (b) actuator/sensorpair A2R–S2L.


the controlled structure is of interest, due to the need forthe exclusion of the resonant states. Such excitations/disturbances can be successfully handled using theproposed control strategy which combines tracking

system with additional dynamics, optimal LQ regulatorand Kalman filter. Incorporating the a priori knowledgeabout the frequency of the excitation (eigenfrequencyof the structure) in the design model for the optimal LQ

Figure 16. Actuators A2L, A2R – sensors S1R, S2L: sensor responses and control in the presence of the sine excitation f1 (left)and with excitation containing three eigenfrequencies (right).

Figure 15. Actuator A2R – sensors S1R, S2L: sensor responses and control in the presence of the sine excitation f1 (left) and withexcitation containing three eigenfrequencies (right).


controller design in terms of the excitation poles used to

form additional dynamics, control problem reduces to

a regulation problem for the design model.Since the controller design is model based, two

approaches to state space model obtaining have been

considered in this paper: FE approach based modeling

and subspace identification. The first approach is con-

venient in the early development phases when no real

structure is available and controller development is

based upon the FE modeling, simulation and modal

analysis. When the real structure is at disposal, the sub-

space based identification is convenient for the state

space model development.Proposed controller is tested on the funnel shaped

structure, the inlet part of the MRI tomograph. Using

the state space models obtained by both mentioned

approaches the optimal LQ tracking system with addi-

tional dynamics was designed and tested with periodic

excitations having the frequencies corresponding to the

eigenfrequencies of the funnel in the frequency range

up to 35Hz. The HiL structure with dSPACE� was

used for the real-time simulations. The results showed

considerable reduction of the vibration magnitudes

during the time when the controller is switched on.

It was noted that controller designed with additional

dynamics which takes into account the first eigenfre-

quency can be used for the vibration magnitudes reduc-

tion in the presence of excitations obtained as a sum of

three sinusoidal signals with frequencies corresponding

to the eigenfrequencies of the funnel. Nevertheless,

much better results are obtained in the presence of

such excitations when the additional dynamics takes

into account all the frequencies present in the excitation

signal. In comparison with feedback control or the LQ

regulator without additional dynamics implemented as

a part of the design model, proposed controller results

in better vibration suppression.Closely related to this issue of better vibration

suppression is one of the main contributions of this

work. Although the use of optimal LQ controllers is a

well-known topic in the control theory, this work treats

the problem of the optimal LQ controller design in

combination with the additional dynamics implemented

as a part of the design model. According to authors’

knowledge, such an approach, which combines at the

same time additional dynamics, optimal LQ control

and Kalman estimator, has not been treated and imple-

mented in the literature, especially not for the vibration

suppression applications yet. The advantage of the

approach can be seen from the fact that the incorpo-

rated additional dynamics which take into account

information about the most critical resonant states,

takes care for better vibration suppression in compari-

son with standard LQ controller in the presence of the

excitations/disturbances which can cause such resonantstates.

Further, the paper shows the feasibility of thenumerically obtained state space model for the con-troller design, which has been demonstrated not onlythrough simulations, but also experimentally. Thefact that the state space model obtained using theFE approach well represents the object under investi-gation, justifies the application of the LQ control as amodel based controller design technique. Besides theadvantage that based on the model the controlledsystem behavior can be investigated in the earlyphases before the prototype has been built, a relativelysimple control law (in terms of a state feedback)represents another advantage, since it is known thatpractical applications often require a straightforwardway to the solution of vibration suppression problems.Controller design requires estimation of the state vari-ables, which are in case of the FE based state spacemodel modal coordinates and therefore cannot bemeasured. In the case of model identification, thestates of the identified model are not measurableeither. Nevertheless due to its relative simplicity theapplication of the LQ controller in combination withthe Kalman estimator is viewed by the authors asan efficient and advantageous control technique forthe considered class of vibration suppression problems(with periodic excitations especially) as opposed tosome other alternatives which would not requirestate estimation (like for example direct model refer-ence adaptive control or adaptive filters), but on theother hand do require estimation or even onlineidentification of the model parameters, through oftentime consuming estimation algorithms. Kalman esti-mator in turn represents a means for overcoming thespillover problem (Preumont 1997, 1988) which mayoccur due to truncation of the higher modes in themodel order reduction procedure.

Experimental implementation on a complex structuredemonstrated practically the vibration suppression andthe effectiveness of the proposed control technique,which makes the experimental results attractive sincein this area validations have mostly been performed bysimulation only.

Further application is directed to implementationof the controller for solving acoustic problems. Insuch cases the acoustic fluid interacts with the smartstructure itself, which has to be taken into accountin the control design technique. It can be shown thatnot all vibration modes of the structure have thesame influence to the sound pressure in the acousticfluid. So, it is required to estimate such behavior orto take into account additional sensors, such as micro-phones in the surrounding fluid. Based on the finiteelement method or by model identification a coupled


fluid-electro-mechanical model can be developed, whichcan be used, in a similar manner as presented in thepaper, for a model based controller design approach(Lefevre and Gabbert 2004). The feasibility of the devel-oping models in combination with the proposed controlhas to be proven experimentally through the furtherinvestigations. On the other hand, practical require-ments impose the need for solving the acoustic problem,e.g. in MRI tomography, which is still the subject ofongoing researches as an attempt to reduce the noiseaccompanying the medical treatment of patients.Vibration suppression taking into account the influenceof the surrounding fluid is a first step towards the solu-tion of the noise reduction and further investigationsin this field are required.

Acknowledgements

This work has been partially supported by thepostgraduate program of the German Federal Stateof Saxony-Anhalt. It was motivated and supported bya cooperation with the Siemens� company in the frameof the German industrial Research project ‘Adaptronik’supported by the German Ministry for Education andResearch. These supports are gratefully acknowledged.

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