Control design of active vibration isolation using mu-synthesis

doi:10.1006/jsvi.5036, available online at http://www.idealibrary.comon

CONTROL DESIGN OF ACTIVE VIBRATION ISOLATION

USING

-SYNTHESIS

M. R. BAI ANDW. LIU

Department of Mechanical Engineering, National Chiao-¹ung ;niversity, 1001 ¹a-Hsueh Road, Hsin-Chu 30050, ¹aiwan, People1s Republic of China. E-mail: [email protected]

(Received 19 July 2001, and in ,nal form 3 January 2002)

Numerical and experimental investigations on active vibration isolation system are presented in this paper. Two con"gurations are implemented for a statically balanced three-mount system. To reduce the in#uence of payload dynamics and coupling among the control actuators, intermediate masses are added to the system. Linear quadratic Gaussian control and
-synthesis are employed in the controller synthesis. The controllers are implemented on the platform of a #oating-point digital signal processor. The results obtained from simulations and experiments indicated that the optimal controllers achieved the desired performance under the constraint of robust stability.

 2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Vibration isolation systemis often required in protecting high-tech equipment such as those used in semiconductor manufacturing industry, where high precision of machining is crucial. Use of passive isolators has been the common practice to isolate vibration from the #oor in that passive systems are easy to design and install. However, in the conventional passive design, a trade-o! exists between the isolation performance in the low- and high-frequency regions. To reduce the response at resonance, damping treatment is often applied at the expense of degradation of isolation at high frequencies [1]. Alternatively, isolation of vibration can be achieved by active means by introducing secondary vibration sources to the original system. This approach has been found to be more e!ective, particularly in low frequencies, than the passive control [1].

There have been many studies devoted to the subject of active isolation system. Karnopp

et al. proposed an inertial or &&skyhook'' damping idea to bypass the aforementioned

trade-o! in relative velocity damping [1]. Beard et al. presented an active hard-mount strategy to achieve desired performance and stability robustness [2]. Sievers and von Flotow developed two methods based on linear quadratic Gaussian (LQG) control: disturbance rejection via disturbance modelling and LQG control with frequency-shaped cost function [3]. Watanabe et al. developed a levitated vibration isolation systemusing

H synthesis and PI control with relative displacement feedback [4]. Kim et al. developed

a decentralized control scheme using the positive real property of the system for active vibration isolation [5].

This paper investigates two con"gurations and two algorithms in the context of active vibration isolation control. The major di!erence between two con"gurations is whether or not intermediate masses are introduced. As shall be seen later, intermediate masses are useful in decoupling the in#uence due to payload dynamics and interactions among active

Figure 1. The model of the vibration isolation system: (a) top view; (b) side view.

mounts, which could destabilize the system. Two control algorithms, the LQG control and the
synthesis, are employed in the controller design. The controllers are implemented on the plateformof a #oating-point digital signal processor (DSP). The results obtained from numerical simulations and experimental veri"cations are also discussed.

2. SYSTEM MODELLING

The model of the vibration isolation system shown in Figure 1 consists of two circular platforms, six passive mounts, and three sets of collocated accelerometers and electromagnetic actuators. The upper platform serves as the isolation table, while the lower platformserves as the vibrating #oor. There are two sources of disturbance acting on the system. One for modelling the disturbance from the payload is located at the center of the upper platform, and the other for modelling the disturbance from the #oor is located at the center of the lower platform.

For the systemdepicted in Figure 1, variational principles are employed in deriving the equations of motion [6]. The symbols used in the derivation are summarized as follows: ¹

, < the kinetic energy and potential energy of the system, m, m the masses of the upper and lower platforms, respectively, I, I the moment of inertias of the upper and lower platforms, respectively, k, c the spring and damping coe$cients of the passive mounts, z, z the displacements of the isolation table and #oor,,
, ,
 the angular displacements of the isolation table and #oor, r the radius of platform, F, F, F the control forces of actuators, F, F the payload disturbance and #oor disturbance, and qH, QH: generalized co-ordinate and generalized force.

De"ne the generalized co-ordinates z, z, , ,
 and
. Assume that the amplitude of vibration is small such that the non-linear terms are negligible. The Lagrange equation

d dt



mzK"!3k(z!z)#F#F#F!F!3cz I
$"!32kr(
!
)# (3 2 r(!F#F)!(3rc
Q I$"!32kr(!
)# r 2(F#F)!rF!2rcQ mzK"3k(z!2z)#F!6cz I
G"kr(
!2
)!2(3rc
Q IG"kr(!2)!4rcQ. (2)

In order for the subsequent use of active control design, the equations of motion are converted into the state-space form

xR "Ax#Bu, y"Cx#Du (3)

where state variables _{x"[z, z, z, z, , Q, , Q,
,
Q,
,
Q]2, control force} u"[F F F F F]2, y is a column vector consisting of the accelerations measured by the sensors, the systemmatrices

0 1 0 0 0 0 0 0 0 0 0 0 ! 3k m ! 3c m 0 0 0 0 3k m 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 !3kr 2I !(3rc I 0 0 0 0 3kr 2I 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 !3kr 2I !2rc I 0 0 0 0 3kr 2I 0 A" 0 0 0 0 0 0 0 1 0 0 0 0 , 3k m 0 0 0 0 0 !6k m !6c m 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3kr 2I 0 0 0 0 0 ! 3kr I ! 2(3kr I 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3kr 2I 0 0 0 0 0 ! 3kr I !4 rc I

0 0 0 0 0 1 m 1 m 1 m ! 1 m 0 0 0 0 0 0 !(3r 2I (3r 2I 0 0 0 B" 0 0 0 0 0 , r 2I r 2I ! r I 0 0 0 0 0 0 0 0 0 0 0 1 m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ! 3k m ! 3c m 3(3kr 4I 3cr 2I ! 3kr 4I ! cr I 3k m 0 ! 3(3kr 4I 0 3kr 4I 0 C" !3k m ! 3c m 3(3kr 4I 3cr 2I 3kr 4I ! cr I 3k m 0 ! 3(3kr 4I 0 3kr 4I 0 , ! 3k m ! 3c m 0 0 3kr 2I 2cr I 3k m 0 0 0 3kr 2I 0 D" 1 m# r I 1 m! r 2I 1 m! r 2I ! 1 m 0 1 m! r 2I 1 m# r I 1 m! r 2I ! 1 m 0 1 m! r 2I 1 m! r 2I 1 m# r I ! 1 m 0 . 3. CONTROLLER SYNTHESIS

The methods for controller synthesis employed in this paper are LQG control and
-synthesis in state-space control theory. The need for the seemingly complicated control strategies is motivated by the fact that large number of modes including structural modes are present in the dynamics and strong coupling exists between each sensor}actuator pair. Although the system is symmetric and driven in the center, the controller is still based on a multiple-input}multiple-output (MIMO) design. This general con"guration is at "rst glance an overkill, but is necessary for experimental veri"cation of the active-control system, where rocking is inevitable due to some practical reasons, e.g., imperfect symmetry. In reality, the &&collocated'' three pairs of frequency response functions would never be identical. We need an automatic means to design the controller because of large number of modes; we also need a controller within a MIMO framework because of the strong coupling. LQG control and
-synthesis are two such methods well suited for these purposes.

z(t) u(t) y(t) Augmented plant G(s) Controller K(s) w(t)

Figure 2. Generalized feedback control structure.

Because LQG control is a well-known control method, its review is omitted here. For details, one may consult Reference [7]. As compared to the LQG method,
-synthesis provides a more robust and comprehensive design that achieves the performance and the stability bounds as well, by choosing appropriate frequency weighting on system perturbations and uncertainties. In this section, the general ideas of
-synthesis will be given, alongside a brief review of H control theory for it is the basis of the former theory. 3.1. H

ROBUST CONTROL

The H theories can be found in much control literature and we present only the key ones needed in the analysis of our problem. The rest are mentioned without proof.

In modern control theory, all control structures can be cast into a generalized control framework, as depicted in Figure 2. The framework contains a controller K(s) and an augmented plant P(s) in which z(t) is a vector signal including all controlled signals and tracking errors, w(t) is a vector signal including noise, disturbances, and reference signal, u(t) is the control signal, and y(t) is the measurement output.















where the submatrices PGH(s) are compatible partition of the augmented plant P(s). Let the transfer function matrix from W(s) to Z(s) be denoted by TXU(s) that can be expressed by linear fraction transformation (LFT)

TXU(s)"FJ(P,K)"P(s)#P(s)K(s)[I!P(s)K(s)]\P(s). (5) The H control problemamounts to "nding a stabilizing controller K that minimizes

TXU(s)" sup \)

S)

 [TXU(j)].

(6) Finding an optimal H controller, however, is generally very di$cult. In practice, it is usually easier to obtain a_{`suboptimala controller. That is, for a given '0, we seek to "nd} a stabilizing controller such that

TXU(s)( . (7)

The details of the synthesis procedure of the H controllers can be found in references [8, 9] and are omitted here for simplicity.

3.2.
-SYNTHESIS

The aforementioned H tends to be conservative since it does not take into account the structure of uncertainties. A less conservative design would be to use
-synthesis. The
-analysis probleminvolves determining whether a control systemremains within the stability and performance bounds and the controlled signals remain small, in the presence of exogenous inputs and norm-bounded perturbations and uncertainties [10]. Assume that the uncertainty model belongs to the set

"diag(II,2,PIIP,,,2,Q): G3C, H3CKH"KG, (8)

B"3())1. (9)

There are two types of uncertainty model: blocks-repeated scalar and full blocks. Two non-negative integers, r and s, represent the number of repeated scalar blocks and the number of full blocks respectively. The dimension of the ith repeated scalar blocks is kP;kP, hile the jth full block is mH;mG. In equation (9),  denotes the maximum singular value. For a system M, the structured singular value
is de"ned as

(M)"

3(): det(I!M)"0



(10) which is essentially a measure of the smallest uncertainty  that may destabilize the closed-loop system. The ultimate goal of the
-synthesis lies in "nding a controller to achieve the so-called robust performance, which implies that the performance speci"cations of the closed-loop system are met for all uncertainty models. The condition of robust performance is the linear fractional transformation FS (G, ) is stable for all 3B and its in"nity normis less than 1FS(G, )(1.

Figure 3(a) illustrates the block diagramof the systemwhich can be rearranged into the
-synthesis framework in Figure 3(b). In the"gure,G is the augmented plant described by the input}output relation

















The term WNCP is a performance weight for disturbance rejection in the desired frequency band. In our case, WNCP is chosen to be a band-pass "lter. The nominal plant P together with uncertainty models
de"nes a set of plants P within which the real physical systemis assumed to lie. In our case, the frequency response measured by the signal analyzer represents the perturbed plant P and the identi"ed model represents the nominal plantP. A frequency-domain multiplicative type of uncertainty is used. The relation between the perturbed model set P and the nominal plant P is

P "(I#)P (12)

or

"P\1

P !I. (13)

The uncertainty weight satis"es

∆in z u z d y K Wc P Wper ∆out 1 0 0 2 K Multiplicative(3) uncertainty Multiplicative(3) uncertainty Performance(3) & Actuator limits(3) Disturbance(1) Control input (3) Measurements Accelerometer(3) Wmul Wc Wper P Augmented plant G Wmult 1 2 ∆ ∆ (a) (b) ∆

Figure 3. Block diagram of the closed-loop system: (a) original block diagram; (b) generalized feedback control structure for
_-synthesis.

To prevent the systemfrominstability, the controller gain should be restricted at high frequencies. In this work, we use a high-pass "lter as the controller weight WA.

The objective of
-synthesis is to"nd a stabilizing controller such that the closed-loop systemachieves robust performance, i.e.,

sup SZ0
$

[G( jw)](1, (15)

where $"diag(
,
),
3B
. Like H control, it is very di$cult to "nd an optimal solution satisfying the criterion in equation (15). A practical remedy to this problem is the so-called D-K iteration technique. This method uses H controller and weighting matrices as the starting point of the iteration procedure. In general, it takes only a few iterations to reach a suboptimal solution. The details of
analysis and synthesis can be found in reference [10].

Figure 4. The simulation results of frequency responses (m/N s) fromdisturnances to acclerometer 1 obtained usin LQG control: (a) payload disturbance; (b) #oor disturbance (**, control o!; } } } }, control on).

4. NUMERICAL AND EXPERIMENTAL INVESTIGATIONS

Numerical simulations and experimental investigations were carried out to verify the aforementioned control methods applied to the vibration isolation problem. First, a numerical simulation is conducted. In the simulation, the isolation table is assumed to be a circular plate with a diameter 1 m, thickness 15 mm, and mass 5 kg. The passive mounts has sti!ness k"4550 N/mand damping c"147 Ns/m. LQG controllers are designed on the basis of the state-space model derived in section 2. The frequency responses (m/N s) fromdisturnances to acclerometer 1 are shown in Figure 4. Attenuation can be observed in two separate bands, 7}12 and 18}40 Hz. The maximum attenuation is found to be 3)5 dB for the #oor disturbance. Some ampli"cation can also be seen in the band 12}18 Hz.To improve the performance, a second order internal model is introduced in the control loop [11]

M(s)" 100s

s#23)88s#14250, (16)

where the resonant frequency is set to be 19 Hz and the damping ratio is set to be 10%. The parameters of the LQG controller are QA"1000I;, RA"I;, QD"600I;_, and

RD"I;_. Figure 5 shows the block diagramof the total system. The frequency responses

(m/N s) fromdisturnances to acclerometer 1 are shown in Figure 6. Attenuation can be observed in a more concentrated band, 11}36 Hz. Furthermore, no adverse ampli"cation is observed in this case. Thus, with the use of the internal model, the control bandwidth tends to be more concentrated without adverse ampli"cation of disturbances.

In addition to the numerical simulations, experiments are carried out to investigate the proposed active vibration isolation system. Two con"gurations of the isolation systemare

+ + LQG controller Internal Model

u Disturbance y K(s) M(s) P1 (s) P2(s) Controller

Figure 5. The block diagramof the LQG control with internal model.

Figure 6. The simulation results of frequency responses (m/N s) fromdisturnances to acclerometer 1 obtained using LQG with internal model: (a) payload disturbance; (b) #oor disturbance (**, control o!: } } } }, control on).

implemented in the experiments. Con"guration 1 is shown in Figure 7, where the isolation table is supported by three passive mounts and three electromagnetic actuators. The isolation table is a circular plate with a diameter 1 m, thickness 15 mm, and mass 5 kg. Three sets of collocated seismic accelerometers (PCB 393A03) and electromagnetic actuators are used in the system. Similar to the principles of loudspeakers and vibration shakers, the actuator operates primarily by means of electromagnetic interactions between a voice coil and a permanent magnet. A Nd}Fe}B cylindrical magnet is used to generate high-#ux-density magnetic "eld. Each actuator has equivalent sti!ness k?"4550N/mand damping ca"147 Ns/m. In addition, two electromagnetic actuators located at the center of the plates serve as the disturbance sources acting on the paylaod and the #oor respectively. Con"guration 2 is shown in Figure 8(a). The experimental set-up is shown in Figure 8(b). In the con"guration, the intermediate masses are square boards of dimensions

Figure 7. The experimental rig of con"guration 1.

Figure 8. The experimental rig of con"guration 2: (a) schematic diagram; (b) photo.

220;200;20 mm. The passive mount above the intermediate mass is a square PU-foam block softer than the passive mount below the intermediate mass. The system dynamics of entire experimental arrangement is apparently more complex than the discrete rigid-body

Figure 9. The experimental result of con"guration 1 obtained using LQG control with internal model (**, control o!, } } } }, control on).

model used in the numerical simulation. Unmodelled dynamics, mostly structural dynamics, of the platforms and the passive mounts have pronounced e!ects on the physical system. Apart from the complexity of structural modes, it is very di$cult if not impossible to identify within acceptable accuracy the meterial properties and transducer dynamics such that the overall system dynamics can be fully accounted for. Instead of structrual dynamic view point, this paper regress to adopt a more control system-oriented approach, eigensystem realization algorithm (ERA). ERA is able to extract the modal parameters, based on the minimum realization of the MIMO sysetm in question [12]. The method per

se is an experimental system identi"cation approach that may provide little physical insight,

yet a practical and e!ective method for capturing the overall dynamics of a realistic system, including the physical system (platforms and passive mounts), sensors (accelerometers), actuators (voice-coil excitors), etc.

4.1. EXPERIMENTAL RESULTS OF CONFIGURATION 1

A LQG controller with internal model is designed and implemented by using a DSP. The parameters of LQG controller are QA"200;I;, RA"I;, QD"150;I;_, and

RD"I;_. The discrete-time transfer function of the internal model is H(z)" 0)1556z!0)1556

z!1)187z#0)8137 (17)

which corresponds to the resonanance frequency 70 Hz and damping ratio 12%. The experimental result is shown in Figure 9. In our control problem, it is in the same way to deal with the disturbance frompayload and #oor. Hereafter, we show only the experimental results of rejection of the disturbance fromthe #oor. Attenuation is found in the frequency

lp ku ku kd kd m2 x1 m1 x2 xb fc1 fc2 p mp,Ip fd xp

Figure 10. The simulation model with intermediate masses.

range 60}92 and 140}185 Hz.The maximum attenuation is 13 dB. However, in frequency range 25}52 and 82}128 Hz, the disturbance is slightly ampli"ed due to the waterbed e+ect [13]. The waterbed e!ect is essentially a control spill-over as well as a physical constraint imposed by non-minimum phase zeros of a system. For a given system con"guration, this constraint is "xed and unavoidable. Nevertheless, one can usually manipulate the type and position of sensors and actuators to alter the zeros of a system, so that the waterbed e!ect can be alleviated. For example, collocation of sensors and actuators is a common approach for mitigating the waterbed e!ect.

4.2. EFFECTS OF THE INTERMEDIATE MASSES

A modi"cation is made to the system of con"guration 1 by introducing intermediate masses to the mounts. The purpose of this approach is two-fold. First, the in#uence of payload dynamics can be reduced. Second, the interactions among the actuators can be decoupled. In a multivariable system as in our case, every input controls more than one output and every output is controlled by more than one input. Because of this phenomenon, which is called coupling or interaction, it is in general very di$cult to control a mulitivariable system. Therefore, one seeks whenever possible to &&decouple'' either electronically or mechanically a multivariable system such that every input controls only one output and every output is controlled by only one input. Consequently, a decoupled systemcan be regarded as consisting of a set of independent single-variable systems and the resulting systemtransfer function matrix becomes nearly diagonal [11].

Consider a simpli"ed systemof Figure 10. The equations of motion are

mN 0 0 0 0 _{IN 0} 0 0 0 _{m 0} 0 0 0 _m xKN $N xK xK # 2kS 0 !kS !kS 0 _{2kSlN !kSlN kSlN} ! kS !kSlN kS#kB 0 ! kS kSlN 0 _kS#kB xN N x x " 1 0 0 0 0 0 0 0 0 _{kB 1 0} 0 _{kB 0 1} FN x@ FA FA . (18)

Figure 11. Frequency responses of the systemin Figure 10: (a) kB"kS; (b) kB"100kS (**, frequency response of x/fc; } }} }, frequency response of x/fA).

Figure 11(a) shows the frequency responses of x/fA and x/fA, where kB is equal to kS. One can see that the level of the_{`crosstalka x/fA is comparable with that of the `drive-pointa} responsex/fA. However, if kB is much lager than kS, the level of x/fA is also much greater than x/fA, as shown in Figure 11(b). This simulation indicates that the coupling among actuators can be reduced by using an upper mount much softer than the lower mount.

On the other hand, the e!ect of payload dynamics is also examined. For the 1-degree-of-freedomsystemof Figure 12(a), the transfer function between the control force

(a) (b) mp Isolaton table mi ka fc i

x

Figure 12. System with payload dynamics: (a) isolation table with payload dynamics; (b) isolation table with intermediate mass and payload dynamics.

fA and the sensor xKG can be written as xKG

f"

s/mG(s#kN/mN)

s#(kN/mN#(k?#kN)/mG)s#k?kN/mNmG (19)

It can be seen that pole}zero pairs are added into the transfer function due to the payload dynamics. If k?<kN, the relation of resonance frequencies between the additional poles and zeros can be approximated as

Figure 13. Frequency responses of xKG/fA. (a) k?"kN; (b) k?"100kN.

As the intermediate mass is added into the system, it becomes the structure as shown in Figure 12(b). The transfer function between the control force fA and the sensor output

xKG turns out to be xKG f" N(s) D(s) (21) where N(s)"mSmNs#[mS(kS#kN)#kSmN]s#kSkNs, D(s)"mSmNmGs#mSmNmG



If k?kN, the peak due to payload dynamics can be cancelled, as shown in Figure 13. Thus, it can be concluded fromthis simulation, the in#uence of payload dynamics could be reduced by introducing intermediate masses to the system.

4.3. EXPERIMENTAL RESULTS OF CONFIGURATION 2

Two methods are employed for controller synthesis of con"guration 2. One is the LQG controller with internal model and the other is the
controller.

4.3.1. ¸QG with internal model

The parameters of LQG controller are QA"2000;I;, RA"I;, QD"600;I;_,

Figure 14. The experimental result of con"guration 2 obtained using LQG control with internal model (**, control o!, } } } }, control on).

transfer function of the internal model is

H(z)" 0)2556z!0)2556

z!0)7528z#0)6429 (22)

which corresponds to the resonance frequency of 180 Hz and the damping ratio of 20%. The experimental result is shown in Figure 14. Attenuation can be observed in the frequency range 165}213 Hz.Maximum attenuation is found to be 12)6 dB. In frequency range 218}300 Hz, the disturbance is ampli"ed because of waterbed e!ect. However, the maximum ampli"cation is only 3)9 dB that is smaller than con"guration 1. It is noted that the LQG design is a somewhat ad hoc method and there is no guarantee whether the stability and performance margins are met. In addition, the design is based on the nominal plant that does not account for the high-frequency dynamics. From our experience, system may become unstable using LQG design, due to un-modelled high-frequency dynamics. To avoid instability, loop gain of the systemshould be restricted outside the control bandwidth, particularly when there is perturbations or uncertainties in the system. Incidentally, the following
-synthesis is such a systematic and robust method able to ful"ll this goal.

4.3.2.
-Synthesis

Prior to
-synthesis, the &&size'' of uncertainties must be characterized as a robustness speci"cation. This is done by subtracting the modelled frequency response functions (by ERA) fromthe measured frequency response functions. The result is shown in Figure 15, where the so-calculated uncertainties (solid lines) are bounded by the associated uncertainty

Figure 15. Spectra of uncertainties and the associated weights (**, uncertainty; } } } }, uncertainty weight).

weights (dash lines). The uncertainty weights are expressed in the following 3;3 matrix:

WKSJR" 1)516e006 s#804s#1)011e006 0)11 0)11 1900s s#1068s#1)141e006 1)617e006 s#804s#1)011e006 0)29 0)26 0)29 1)997e006 s#1068s#1)141e006 , (23)

where the parameter  can be varied in order to gauge the e!ect of the weight on the closed-loop performance and robustness. In this experiment,"1, 0)8, 0)1. Figure 16 shows the experimental results with respect to each value. It is seen that the design with "1 tends to be conservative and the performance is poor, as shown in Figure 16(a). In Figure 16(b), when is reduced to 0)8, the performance is improved. However, Figure 16(c), when  is too small, the closed-loop system is nearly unstable. The robust performance index (the
plot) with "0)8 and 0)1 are shown in Figure 17. It is desired to keep the
values as small as possible such that the control design achieves robust performance. If is too small, the maximum
value becomes much greater than unity.

5. CONCLUSIONS

In this work, a DSP-based active isolation systemhas been developed. Numerical and experimental investigations are carried out to examine two control algorithms and two

Figure 16. Experimental result with various gains of uncertainty weights: (a)"1 (b) "0)8; (c) "0)1 (**, control o!, } } } }, control on).

con"gurations. The results indicate that proper con"guration is very crucial for entire control design. In particular, intermediate masses are shown to be useful in reducing the in#uence of payload dynamics and coupling among the actuators. LQG control and
synthesis are employed in the controller synthesis. In LQG design, system may become unstable due to un-modelled high-frequency dynamics, whereas
-synthesis is able to provide better robustness against uncertainties in the system.

There are some possible extensions of this research. Decoupling of the actuators could be enhanced mechanically or electronically to an extent such that the controller could be designed by a single-input and single-output basis. In the work, only narrowband performance has been achieved, but overall frequency-averaged attenuations are poor. It is suspected that the electromagnetic actuators are unable to deliver su$cient force output within a broad bandwidth. In the future, other types of actuators such as pneumatic or hydraulic actuator should be used for application with heavier payloads. In this work, only vertical motions are of concern. Other types of motions, e.g., in-plane motion should also be accommodated in the future research.

ACKNOWLEDGMENTS

The work was supported by the National Science Council in Taiwan, Republic of China, under the project number NSC 89-2212-E-009-007. The authors would like to thank Prof. Stephen Elliot of ISVR, University of Southampton, U.K., for his helpful suggestions and provision of reference [5].

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