- Email: [email protected]
Modeling and Vibration Control for a Flexible Building Structure with Multiple Vibration Modes Using H-infinity Based Robust Control
IFAC
Copyright Cl IFAC Mechatronic Systems. California. USA. 2002
c
0
C>
Publications www.elsevier.comlIocalelifac
MODELING AND VIBRATION CONTROL FOR A FLEXIBLE BUILDING STRUCTURE WITH MULITPLE VIBRATION MODES USING H-INFINITY BASED ROBUST CONTOL Seto, K.
e. ,Kar, L N.
·2
and Matsumoto, Y. * •
• } Department of Mechanical Engineering. Col/ege of Science and Technology. 1-8 Kanda Surugadai. Chiyoda-ku, Tokyo 101-8308. Japan ·2 Department ofElectrical Engineering, Indian Institute of Technology, New Delhi - 110016. India.
Abstract: This paper proposes an experimental method of controlling the vibration of a flexible structure using Hinfinity control theory. For this purpose, two reduced-order lumped-mass models, expressed by 5-DOF systems each, are derived. These models are derived by considering the five vibration modes along the x- and y-direction respectively and neglecting all the other higher frequency modes. Neglected vibration modes are regarded as unstructured uncertainties and taken care of during the design process. The dynamic output feedback controller is designed based on the reduced order model and the approximate knowledge of the unmodeled system uncertainties. The first five vibration modes of the structure along the x- and y-directions are well suppressed using two newly proposed actuators (iynamic vibration absorbers) without any spill over proble m. The control algorithm is verified through simulation and experimental studies. Copyright © 2002 IFAC Key Words : Flexible Structure, Vibration Control, Dynamic Absorber, Modeling, Robust Control
I.
control algorithm. Many authors (Cui and Nonami, 1993, Nishimura, 1999) have reported the H-infinity based control scheme to reduce vibration. To the author's knowledge, most of the active protective scheme for civil structures are concerned with the first few modes (say below three modes). This is not only because the first few modes usually dominate structure defonnations, but because of some technical difficulties such as modeling and robustness issues. However, as the tendency of constructing more reliable and higher structures increases, it becom:s insufficient to consider the first mode only, since the vibration of higher modes also raises problems of structural strength. In most of the research work, the reduced order model is derived for controller design by considering the bending modes of the structure along one direction. However, a building-like structure may have bending modes along the Ji- and y-
INTRODUCTION
In recent years, attention has been made to reduce the vibration of various flexible structures such as vibratory platfonns, tall building, bridge tower structures etc (Seto 1992, Spencer and Sain, 1997, Merovitch. 1990, Wodek,1996, Seto, 1999, Crassisdis , 1994, Salas, 1998. Wei, 1992). The inherent damping of these structures is low and unable to withstand the vibration caused by disturbances like moderate earthquake motion or strong winds or even traffic flow . In this context, feedback control schemes are useful to produce sufficient control force for suppressing the vibration modes . Spencer and Sain (1997) have reported in detail the application of feedback control to a novel domain: civil structures especially buildings. However, the implementation of feedback control scheme in real structures may cause spill over instability due to the excitation of neglected high frequency vibration modes (Balas, 1978, Seto, 1992). As a result, the reliability of the structure may be severely damaged. So it is necessary to take a precautionary measure to avoid the spill over problem in the design stage itself. Also the suitable actuators and their placement are of great concern in implementing the control scheme. In this context, H-infinity based control is suitable for developing the
In this paper, attempt has been made to derive the two models of a scale building structure by considering its motion in the Ji-direction and the y-direction. These models are derived in physical state space where the states are directly measurable for implementing feedback scheme. Based on these two models, the H-infinity based controller is developed. Two
147
actuators are constructed based on the mechanism of horizontal type pendulum. The main contributions of this paper are (I) Development of two reduced order models in physical state space domain ; (II) Design of a control scheme to suppress the first five vibration modes along x and y direction; (III) The development of two actuators and their placement to suppress vibration in both direction; (IV) Design and validation of control algorithms through simulation and experimental studies.
........- ~
:;;;-
--=3mm-
5mm-
2.
7
-
.-
...-
.-
...--.
.-
.-
.-
--y I-" f-"
-
---
z
The paper is organized as follows . In section 2, the reduced order model of the structure is derived. In section 3, H-infinity based control algorithm has been discussed. Section 4 discusses the construction of a special type of ~ctuator, which is shown to be very effective in reducing the vibration. Section 5 reports all the simulation and experimental results and section 6 concludes the paper.
300
•
I I
~I
I
:J
DERIVATIONOFREDUCED-ORDERMODELD
In this section, the reduced order model of a 15 story building structure, as shown in fig . I , is discussed. This model will be used for the purpose designing a feedback controller and simulation. The dimensions of the structure are indicated in the figure . The floor size is 200mm times 300mm and the thickness is 3mm. The radius of the pipe is 5 mm. Figure 2 shows the first to eighth mode shapes and the corresponding natural frequencies of the structure . In Fig .2, vibration modes 2,3,5,6 and 8 are concerned in the x-direction and 1,3,4,6 and 7 are in the y-direction. Modes 3 and 6 are common for both the directions, as these are torsional modes. In this study, we consider the first five modes to be controlled at both directions therefore a five-degree-of-freedom (5DOF) reduced' order model should be established along the x-direction and y-direction (by neglecting all the higher modes). Using these mode shapes, the model parameters are calculated. The details of the modeling procedure in physical state space for a general structure are discussed in (Seto, Doi and Ren, 1999, Kar" Seto and Doi, 2000, Seto and Mitsuta, 1992, Seto, Ren. and Doi, 1998).. Here a brief introduction of this procedure is introduced
1" mode(y) (4.88 Hz)
2nd mode(x) (5.13 Hz)
3rd mode (8.13 Hz)
4th mode(y) (14.88 Hz)
6th mode 7111 mode(y) 8111 mode (x) 5th mode(x) (24.38 Hz) (25.13 Hz) (26.75 Hz) (15.75 Hz) Fig. 2 Vibration mode shapes
2.1 BriefIntroduction ofSeto sprocedure
if more controlled modes are concerned, unless large number of sensors or distributed sensors are used (Fuller and Elliott, 1990, Bai, and Shieh,1984). In view of this situation, it is considered in this research that if the concerned modes could be represented by a discrete model in physical domain, then the states could be directly measurable and the modal estimation such as "modal filter" and "inverse modal filter" used in modal space methods would be unnecessary.
First, Seto's Method for deriving a reduced order model is schematically presented by using a 5DOF model as an example. The feedback control methods of discrete systems have been well established in theory (for example, 4'-1erovitch, 1990), while actual implementation on distributed systems often presents some difficulties, especially when controlling multiple modes . The usually employed methods for controlling distributed structure are in modal state-space, which results in a simple system matrix in sacrifice of the convenience of measuring states directly. It is not easy to estimate state quantities in modal space accurately
The first five modes in both the directions, including bending and torsional modes, are of our concern.
148
Substituting Eq.(l) into Eq .(2) results in a matrix, 4>tJI, whose off-diagonal positions usually contain, though srnalI, non-zero values. A sensitivity analysis procedure is proposed here to modify the initial matrix 4> in order to make 4>4>T diagonal. Suppose there are p different non-zero components in the off-diagonal positions of the matrix 4>tJI resulted from Eq .(l). we define an error vector £ consistin~ of these components. namely
Ep}
-------
... ... _.-._--"'""""....
;:~
.
--.-.-~
~ Fig. 3 Modeling
/
(4)
'
in which each error E; is a function of some modal shape components of matrix 4> . Suppose there are. in total, q independent modal shape components involved in Eq.(4). then we can write
...... points
Choose five modeling points on the structure, and assume proper lumped masses and stiffness at and between these points, a discrete model can be created. Figure 3 shows the 5 mass points of the reduced order model. Evidently, the modeling points should be able to reflect the motion of these five modes as welI as possible. The normalized modal vector components of the structure at the modeling points are presumedly defined as the mode vector components of the lumped 5DOF model as folIows,
where t/J is a new defined q x 1 vector consisting of 91j ( j = 1 - q. here the subscript j is only a sequence number and has nothing to do with the mode order or the position label). Our purpose is to modify these modal shape components tPj in order that the errors El approach zero. Define the modification vector of t/J as &P. which is to change the errOr vector from its initial value e to O. then we can start the correcting procedure from the following equation
as~umed
9111 9112 9113 9114 9115 9121 91 22 91 23 91 24 91 25 4>= 9131 9132 tP33 91 34 9135 11'41
11'42
11'43
11'44
9151
11'52
tP5 3
91 54 9155
(I)
[~:]&p = 0-£ =-£
11'45
where [ae! at/J] is a p x q matrix called sensitivity matrix whose elements are partial differentiating of the errors. namely
where the subscripts i and j of the mode vector components 91i} mean the number of mass points and mode orders, respectively. On the other hand, for a lumped parameter system, the physical mass matrix M and stiffness matrix K are related to the normalized modal matrix 4> through folIowing equations
M
=(4>4>T r' .
K = ( 4>T
r' n
24>-'
(6)
a£, a£,
a£,
a~
at/Jq
ae ae
~ at/Jq
a~
[:~]=
(2)
2
~
2
a~
aep aep a~
.(3)
a~
.(7)
aep at/Jq
The modification vector can be obtained directly from Eq.(6) using the generalized inverse matrix as follows .
where, D2 is a diagonal matrix whose nonzero diagonal elements are given by the square of natural frequencies of each mode. Equations (2) and (3) suggest that if the normalized modal matrix of a lumped parameter system could be obtained, the physical parameters of the system could then be dl!termined. Unfortunately, the directly obtained modal matrix from the FEM modal analysis data, as given in Eq.(l), is not guaranteed to satisfy Eq.(2), where both sides should be diagonal matrices.
.(8)
and the new vector is calculated by (9)
149
Usually iterative calculation is necessary in this procedure in order that £ converges to O. After certain iteration that a preset accuracy has been reached, a final modified modal matrix can be obtained which will satisfy the diagonal condition of Eq .(2). Consequently, M and K in case of 5DOF model can be obtained as follows. M =diag[MI
K{ where and M
kij
M2
M3
Ms]
M4
-k' J -k,. -k2) -k 2• C -k)4 D Symm.
-kl2 B
(10) x.
- k 2, -k),
-k"l
(11)
-k" E
,
.
A = kll + kl2 + k n + k'4 + k' 5 B = k' 2 + k22 + k 2) + k 2• + k2j
mix = 1.508,
C = kl) + k2) + k )) + k" + k ),
m4x = 3.420. ms":'= 5.709
~J"
\
m2x = 1.508. "
D = k' 4 + k ,. + k )4 + k ... + k"
K Ilx = -3.221, K 12x = 0.317.
E = k" + k21 + k" + k., + k"
KI4x = -13.488, K 23..
means the spring conneted with M j
K 33 ..
j '
22.<
= -3.221
= -13.488, K 24x = 9.862, K 25x = 22.958 = 19.08, K 34x = 6.028, K 35x = 23 .703
(13)
where x6 is the relative displacement
and Xd is the displacement of the mass of the actuator. A Similar type of state equations along the y-directions can be written whose details are not given here.
a mass with Md. actuator with control force id ' supporting spring and damper with kd and cd , respectively. This discrete model will make it easy to design a controller in physical state-space where the states are directly measurable. 2.3 State space equation for the structure
3.
H-INFINITY BASED-FEEDBACK CONTROLLER DESIGN
3.1 Sta~e equation of augment system with dynamic compensator
The state space equations of the system along x-direction are written as (from fig. 4(a»
Xu
K
mdy = 0.158, kdy = 0.2, Cdy = 2 mass and stiffness are [kg] and [kN/m] in unit, respectively Fig. 4 The lumped parameter models of the structure on x-direction
Figure 4 shows the reduced order lumped mass model expressed by 5DOF system of the structure along the x direction. The same reduced order model is described along the y-direction. Using these models. the state space equations are derived. The physical parameters in the x- direction are shown as follows . These parameters in the y-direction are obtained by the same way. Each mass is connected with springs denoted by the spring constant kij which means the spring conneted with Mj and M j . An active dynamic absorber is connected to the mass points I for controlling vibration along x-direction and also another one is connected to the mass point 2 for y-direction . Each active dynamic absorber consists of
In the above, the state vector the symbols in fig 4a as
K 13x = 9.862
K I5x = 22.958,
K 44 ... =' 19.080, K 45x = 23 .703, K 5Sx = 3.886 mdx=0.125, kdx=0.2, cdx=2
2.2 Lumped parameter model of the structure
.i: cx = Acxxcx + Bcxux + B",x w IJt Yx=C cx x cx +D",x W 2x
m 3x = 3.420
In this section, the development of an H-infinity based control scheme is briefly discussed. In the present study, the objective is to design a dynamic output feedback controller based on the reduced order model to control its first five vibration modes and suppress the possible instability due to neglected modes. These objectives can be a(;hieved by solving a mixed sensitivity minimization problem in the framework of H-infinity optimal control (Cui. Nonami and Nishirnura.
(12)
is defined by using
150
1993. Doyle and Glover. I 996). Fig 5 shows a block diagram of a standard H-infinity control problem (using the model in x-direction). The controller design by using the dynamic equation along the x-direction will be discussed in detail and the controller for the model in y-direction is similar in nature and thus omitted.
Acz
A"
=
[
°
B ...bC,.,x
Bu
C2x
o
°
=[:...:x].c Ix =
o
=[Cc.r 0
°o 1
.B lx =
A,.,lx
[D
0
A,.,2.r
°c wx
w2z
[BWX 0
0]O.
°
C wlx
o
0J.D lu = [D;/].D21 =[0
Dwxl
By selecting suitable filters W rx • WSr • the aim is to design a dynamic feedback controller (16)
Fig. 5
such that the H-infinity norm of the transfer function G.,.,x becomes smaller than unity where G:wx is the transfer function from Wx to 4x. In the similar way, the controller K, is designed for the reduced order model along the y-direction.
Block diagram of standard Hoo control problem
In the above figure. Ux is the control input. yx is the measured output. 41x. 42.r are regulated variables. Wlx(s) and W 2z{s) are two weighting functions to shape the sensitivity and complementary sensitivity matrix. In this study. we have one control input and two measurable outputs in x direction. Define two filters WT,,=Wlx and WSr = diag( W 2x" W2.r) for the input and two outputs respectively. Note that WSr bounds sensitivity function and thus sets the disturbance rejection, and Wrz bounds the complimentary-sensitivity function and thus sets the suppression of measurement noise and uncertainty. Here attempt is made to design the H-infinity normed based control subject to these constraints. The state equations of the filters W rz • and WSr are defined as
{
3.2. Selection o/Weighting Filters In the case of multiplicative uncertainty representation. the relation between the actual model and the reduced order model is given by .(17)
where ~ V(J}) is the full order model. P; V(J}) is the reduced order model and 4. is the error function or multiplicative system uncertainty. In the H-infinity framework. (as the exact representation of 4. is difficult). a weighting function is selected to represent the upper envelope of the uncertainty. i.e., (18)
*W/~ = A.,IzXwlx + B WIXUX. C .,lxX wiz + Dw1xu x
1./ x -
*. {
2z
.(14)
where c; (.) is the maximum singular value of (.). The multiplicative uncertainty for both the directions is shown in fig.6. These error functions are computed by using the frequency response data of actual and reduced order model. It has been mentioned earlier that five modes are considered to derive the reduced order model. So ideally. there must not be any sharp peak in the low frequency range. However, the actual system response is obtained experimentally and as a result some anti-resonance peaks appear in the low frequency range of the error function which can be neglected. This is because the anti-resonance peak does not affect the stability conditions. To represent the multiplicative uncertainty, the following filter is used for the x-direction model:
= A .,2zX .,2z + B w2zC wxx cz
1. 2z '"
C .,2z X wlx + D w2z C wzX cz
By combining (12) and (14). the augmented state equations are written as
iX:
j
Axxx+ Blxwx+ B 2x u x
4X - Clxx x + Dl2x"x
.( 15)
y x = C 2.r x x + D 21x W x
where
151
80 - - Kx
(19)
--- Ky
60 40
where
20
r; TI = 0.25, (OnTl = 21tx 25[radlsec), KT = 0.45 r; T2 =0.35, (OnT2 = 21tx37[radlsec).
~
We have also selected another filter It;-2 (s) whose parameters are as follows r; TI = 0.5, {OnTl = 21tX 25[radlsec]. KT = 0.04
,.,
60 40
r; T2 = 0.2, (OnT2 = 21tx 35[radlsec).
20
The frequency response plots of two filters are shown in Fig. 6. It is noticed that the second one does not satisfy the mathematical equation (18) strictly. Later on, it is shown that the controller based on this filter also gives satisfactory results. The reasons may be that the H-infinity based design approach considers the norm bounded uncertainty without any phase information. In this sense, it may result in some over design. So a trial and error procedure is necessary to select a suitable filter for obtaining implementable results (although it may not give logically conclusive results by not satisfying the strict mathematical equations). The weighting filter for y-direction is taken to be same as that of the x-directions.
Control/er using W
0 0
5
Fig. 7
'·2
10 :.1 5 20 25 30 35 !' \Frequency (Hz)
40
45
50
Frequency re&ponse of Hoc> controllers.
-40~----------------------------~
x-direction
-60 -80 -100
iii"-120 :!:!.-140 .~ -40 Cl
'=::!:=================4 without control
-60
2or-------------.------------------, -am.
with control (W,) with control (W
.t
-80
- - l1my
W,.,
o
Controller using W
0
c: 80 '<'6 Cl
1
-100
--W'.2
-120 _140 '--_L__..I..-~__-'-----I.__-,-_Y--,d_ir_e-,c_ti_o.... n,---,
o
-40 1----'1--.- -...
5
10
15
20
25
30 35
40
45
50
Frequency (Hz)
Fig. 8
Simulated frequency response
~O ·L-~--~--~~---L--~~---L--~~
o
Fig. 6
5
10
15
20 25 30 Frequency [Hz)
35
40
45
These observations will be validated by experiment and discussed in the next section. The controller is validated through simulation studies. In the x-direction, an impulse excitation is given at the mass point 1 and the response is measured at the same point.
50
Multiplicative uncertainties and weighting function WJ./' Wj • 2
A low pass filter Ws(s) is used as a weighting function for the output i.e., to minimize the weighted sensitivity function. Here we have selected
0.5
W2(S)=~ s+{On
where
(On
I
=21tx30[radlsec), K s =250.
C Q)
3.3. Simulation Results
~
u
0
-0.5 0.5
.!!! ~
Using the MATLAB function HINFSYN, the feedback controllers K" ,KJ are computed using both the filters It;-I (s) and It;-2 (s). The frequency responses of the controller are shown in the fig .7, which are lowpass in nature. This ensures the reduction of the high frequency component of the control signals, which is good for avoiding spillover.
(5
0
-0.5 L-_ _.L-_ _.L-_ _~_ _~_---'
o Fig. 9
152
234 Time Isec1
Simulated impulse response (x-direction)
5
0.5
E S-
e
Q)
- - with control (W
1·1
to actuators
)
0
-0.5
~0 0.5 to
g. is
0
-0.5 2 3 Time [sec]
0
Fig. 10
4
5
Simulated impulse response (y--direction)
For the Y-direction. excitation is given at mass point 2 and the response is measured at mass point 2. Both frequency and impulse responses are shown in figs (8-10). It is noticed that the vibration modes are suppressed for both cases 4.
Fig. 11
Moving Coil Bearing
EXPERIMENTAL RESULTS
In this section, the simulated results are verified through experiment. Figure 11 shows the experimental setup where the personal computer acts as a controller. The displacement of mass point 1 along the x-direction and mass point 2 along the y-direction are measured by two non-contact laser sensors. Although these laser sensors are useful for the purpose of experiment on a scale model, it may be difficult to use these sensors in actual application building control problem. However, these control schemes can be implemented in real structure by using commercially available velocity sensor.
Permanent Magnet
Fig. 12 Schamatic diagram of actuator -40 . - - - - - - - - - - - - - - - - - - .
x-direction
-60 -80
-100
CD -120 ~ -140
An actuator is constructed for each direction whose schematic diagram is shown in fig. 12. The. actuator is similar to a horizontal type pendulum where a mass is attached at the end of the lever. The feedback control signal changes the current through a pennanent magnetic field and results in the movement of the mass of the actuator.
c
F=d:=d==d=~==~~==~~==~~ without control with control -60
~ -40
(W,.I
-80 -100 -120 -140
During simulation, the continuous time controller is designed. For the purpose of the experiment, we discretize this continuous time controller by using the sampling period of T = O.OOI[sec]. The discrete time controller is given by
L-....L._...l---I._-I-_I-....I.._...l---I._-L...---J
o
5
Fig: 13
10
15
20 25 30 35 Frequency [Hz)
40
45
50
Experimental frequency response
The vibration control effects are evaluated by measuring the transfer function (compliance) of the structure under impulse excitation. The experiment is carried out for the x direction by giving an impulse input at mass point 1 and measuring the response at the same point. For the y-direction, an impulse is given at mass point 2 and the measurement has been made at the same point. Experimental frequency and time responses of the system are shown in figs. 13-14. From these figures, it is clear that all five peaks are
: /k + 1) = Plo;; x(k)+QIo;Y /k), { ux(k) =CIo;; /k)+DIo;Y x(k)
where k is the sampling instant and Plo; = eA •.,T QIo; =
Experimental setup
A~( eA"T - / )BIo;.
153
suppressed compared with the no controlled frequency response. Also the feedback controller suppresses the vibration without any spillover instability which is clear from the impulse response.
E
0.5
- - without control
- - with control
(W,.)
§.
c:
~ B <15
0
~ CS -0.5 2
0
3
5
4
Time [sec)
E
0.5
- - without control
- - with control
(W,.)
§.
c: ~
0
Q)
U
<15
Q.
cS
-0.5 2 3 Time [sec]
0
Fig. 14
4
5
Experimental impulse response
5.
CONCLUSIONS
In this paper. an active control scheme has been designed to suppress the vibration of a flexible building like structure. Two reduced order models have been constructed along the x and y directions to represent the system dynamics of five vibration modes in each direction. Using these models. an attempt has been made to design H-infinity based dynamic output feedback controller to suppress the five vibration modes. These control schemes have been verified through simulation and experimental studies without any spill over problem. In the experiment. only two sensors and two actuators are used. The actuators are constructed based on the mechanism of a horizontal type pendulum and have been shown to be effective to reduce the vibration. REFERENCES
Trans. Japanese society Mechanical engineers. Vo1.59. pp.7 I 4-720. Crassisdis. J. L.. Leo DJ .• Inman D.J .• and Mook D.J .• (1994). Robust identification and vibration suppression of a flexible structure. Journal of Guidance. Control and Dynamics. Vol. 17. No-5. pp. 921-928. Doyle. J.c. and Glover. K..(l996). Robust and optimal control. Prentice Hall. Upper Saddle River. New Jersy. Fuller c.R.. Elliotl S.J .• and Nelson P.A.• (1996). Active control of vibration. Academic. London. Nishimura. H. and Kojima. A. (1999). Seismic Isolation Control for A Building like Structure. IEEE Control System Magazine. pp.38-44. Dec. Kar. I. N.• Seto. K. and Doi. F.. (2000). Multimode vibration control of a flexible structure using H-infinity based robust control :\avv. IEEElASME Transaction on Mechatronics. Vol. 5~.No. l. pp. 23-31 . and Saln.. M.K. (1997). Controlling Spencer. B.F. buildings: A new frontier of feedback. IEEE Control System Magazine. vol. 17. pp.19-35. Dec. Seto. K .• Doi F. and Ren. M .• (1999). Vibration control of bridge towers using a lumped modeling approach. ASME journal of vibration and acoustics. Vol. 121. pp. 95-100. Seto. K.• (1992). Trend s on vibration control in Japan." in Proc. I SI Int. Conf. Motion and Vibration Control (MOVIC). vol.1. pp. 1-1 I. Seto. K. and Mitsuta. S .• (1992). A new method for making a reQuced order model flexible structures using unobservability and uncontrollability and its application in vibration control. in Proc. I SI Int. Conf. Motion and Vibration Control (MOVIC). pp.152-158. Seto. K.• Ren. M. and Doi. F. •. (1998). Feedback vibration control of a flexible plate at audio frequencies by using a physical state space approach." J. Acoust. Soc. Amer.• vo1.103. no.2. pp.924-934. Wodek.G. (1996). Balanced control of Flexible structures. Lecture notes in control and infonnation sciences. 211. Springer. Wei. B. and Marcelo. G .• (1992). Control synthesis for flexible space structures excited by persistence disturbances. Journal of Guidance. Control and Dynamics. Vol. 15. No. I. pp. 73-80.
Bai. M.R .• and Shieh C..(1985). Active noise cancellation by using the linear quadratic gaussian independent modal space control. Journal of Acoustical society of America. Vo1.97. pp.2664-2674 Balas. M. J .•(199O). Feedback control of flexible systems. IEEE Trans. Automat. Contr.• voI.AC-23. pp.673-679. Aug. 1978. Merovitch. L.. Dynamics and Control of Structures. New YorkWUey.. Balas. G J.• (1998.) Synthesis of controllers for the active mass driver system in the presence of uncenainty. Earthquake Engineering and Structural Dynamics. Vol. 27 .• No.l1. pp. 1189-1202. Cui. W.. Nonami. K. and Nishimura. H.. (1993). Experimental study on active vibration control of structures by means of H-infinity and H-2 control.
154
Copyright Cl IFAC Mechatronic Systems. California. USA. 2002
c
0
C>
Publications www.elsevier.comlIocalelifac
MODELING AND VIBRATION CONTROL FOR A FLEXIBLE BUILDING STRUCTURE WITH MULITPLE VIBRATION MODES USING H-INFINITY BASED ROBUST CONTOL Seto, K.
e. ,Kar, L N.
·2
and Matsumoto, Y. * •
• } Department of Mechanical Engineering. Col/ege of Science and Technology. 1-8 Kanda Surugadai. Chiyoda-ku, Tokyo 101-8308. Japan ·2 Department ofElectrical Engineering, Indian Institute of Technology, New Delhi - 110016. India.
Abstract: This paper proposes an experimental method of controlling the vibration of a flexible structure using Hinfinity control theory. For this purpose, two reduced-order lumped-mass models, expressed by 5-DOF systems each, are derived. These models are derived by considering the five vibration modes along the x- and y-direction respectively and neglecting all the other higher frequency modes. Neglected vibration modes are regarded as unstructured uncertainties and taken care of during the design process. The dynamic output feedback controller is designed based on the reduced order model and the approximate knowledge of the unmodeled system uncertainties. The first five vibration modes of the structure along the x- and y-directions are well suppressed using two newly proposed actuators (iynamic vibration absorbers) without any spill over proble m. The control algorithm is verified through simulation and experimental studies. Copyright © 2002 IFAC Key Words : Flexible Structure, Vibration Control, Dynamic Absorber, Modeling, Robust Control
I.
control algorithm. Many authors (Cui and Nonami, 1993, Nishimura, 1999) have reported the H-infinity based control scheme to reduce vibration. To the author's knowledge, most of the active protective scheme for civil structures are concerned with the first few modes (say below three modes). This is not only because the first few modes usually dominate structure defonnations, but because of some technical difficulties such as modeling and robustness issues. However, as the tendency of constructing more reliable and higher structures increases, it becom:s insufficient to consider the first mode only, since the vibration of higher modes also raises problems of structural strength. In most of the research work, the reduced order model is derived for controller design by considering the bending modes of the structure along one direction. However, a building-like structure may have bending modes along the Ji- and y-
INTRODUCTION
In recent years, attention has been made to reduce the vibration of various flexible structures such as vibratory platfonns, tall building, bridge tower structures etc (Seto 1992, Spencer and Sain, 1997, Merovitch. 1990, Wodek,1996, Seto, 1999, Crassisdis , 1994, Salas, 1998. Wei, 1992). The inherent damping of these structures is low and unable to withstand the vibration caused by disturbances like moderate earthquake motion or strong winds or even traffic flow . In this context, feedback control schemes are useful to produce sufficient control force for suppressing the vibration modes . Spencer and Sain (1997) have reported in detail the application of feedback control to a novel domain: civil structures especially buildings. However, the implementation of feedback control scheme in real structures may cause spill over instability due to the excitation of neglected high frequency vibration modes (Balas, 1978, Seto, 1992). As a result, the reliability of the structure may be severely damaged. So it is necessary to take a precautionary measure to avoid the spill over problem in the design stage itself. Also the suitable actuators and their placement are of great concern in implementing the control scheme. In this context, H-infinity based control is suitable for developing the
In this paper, attempt has been made to derive the two models of a scale building structure by considering its motion in the Ji-direction and the y-direction. These models are derived in physical state space where the states are directly measurable for implementing feedback scheme. Based on these two models, the H-infinity based controller is developed. Two
147
actuators are constructed based on the mechanism of horizontal type pendulum. The main contributions of this paper are (I) Development of two reduced order models in physical state space domain ; (II) Design of a control scheme to suppress the first five vibration modes along x and y direction; (III) The development of two actuators and their placement to suppress vibration in both direction; (IV) Design and validation of control algorithms through simulation and experimental studies.
........- ~
:;;;-
--=3mm-
5mm-
2.
7
-
.-
...-
.-
...--.
.-
.-
.-
--y I-" f-"
-
---
z
The paper is organized as follows . In section 2, the reduced order model of the structure is derived. In section 3, H-infinity based control algorithm has been discussed. Section 4 discusses the construction of a special type of ~ctuator, which is shown to be very effective in reducing the vibration. Section 5 reports all the simulation and experimental results and section 6 concludes the paper.
300
•
I I
~I
I
:J
DERIVATIONOFREDUCED-ORDERMODELD
In this section, the reduced order model of a 15 story building structure, as shown in fig . I , is discussed. This model will be used for the purpose designing a feedback controller and simulation. The dimensions of the structure are indicated in the figure . The floor size is 200mm times 300mm and the thickness is 3mm. The radius of the pipe is 5 mm. Figure 2 shows the first to eighth mode shapes and the corresponding natural frequencies of the structure . In Fig .2, vibration modes 2,3,5,6 and 8 are concerned in the x-direction and 1,3,4,6 and 7 are in the y-direction. Modes 3 and 6 are common for both the directions, as these are torsional modes. In this study, we consider the first five modes to be controlled at both directions therefore a five-degree-of-freedom (5DOF) reduced' order model should be established along the x-direction and y-direction (by neglecting all the higher modes). Using these mode shapes, the model parameters are calculated. The details of the modeling procedure in physical state space for a general structure are discussed in (Seto, Doi and Ren, 1999, Kar" Seto and Doi, 2000, Seto and Mitsuta, 1992, Seto, Ren. and Doi, 1998).. Here a brief introduction of this procedure is introduced
1" mode(y) (4.88 Hz)
2nd mode(x) (5.13 Hz)
3rd mode (8.13 Hz)
4th mode(y) (14.88 Hz)
6th mode 7111 mode(y) 8111 mode (x) 5th mode(x) (24.38 Hz) (25.13 Hz) (26.75 Hz) (15.75 Hz) Fig. 2 Vibration mode shapes
2.1 BriefIntroduction ofSeto sprocedure
if more controlled modes are concerned, unless large number of sensors or distributed sensors are used (Fuller and Elliott, 1990, Bai, and Shieh,1984). In view of this situation, it is considered in this research that if the concerned modes could be represented by a discrete model in physical domain, then the states could be directly measurable and the modal estimation such as "modal filter" and "inverse modal filter" used in modal space methods would be unnecessary.
First, Seto's Method for deriving a reduced order model is schematically presented by using a 5DOF model as an example. The feedback control methods of discrete systems have been well established in theory (for example, 4'-1erovitch, 1990), while actual implementation on distributed systems often presents some difficulties, especially when controlling multiple modes . The usually employed methods for controlling distributed structure are in modal state-space, which results in a simple system matrix in sacrifice of the convenience of measuring states directly. It is not easy to estimate state quantities in modal space accurately
The first five modes in both the directions, including bending and torsional modes, are of our concern.
148
Substituting Eq.(l) into Eq .(2) results in a matrix, 4>tJI, whose off-diagonal positions usually contain, though srnalI, non-zero values. A sensitivity analysis procedure is proposed here to modify the initial matrix 4> in order to make 4>4>T diagonal. Suppose there are p different non-zero components in the off-diagonal positions of the matrix 4>tJI resulted from Eq .(l). we define an error vector £ consistin~ of these components. namely
Ep}
-------
... ... _.-._--"'""""....
;:~
.
--.-.-~
~ Fig. 3 Modeling
/
(4)
'
in which each error E; is a function of some modal shape components of matrix 4> . Suppose there are. in total, q independent modal shape components involved in Eq.(4). then we can write
...... points
Choose five modeling points on the structure, and assume proper lumped masses and stiffness at and between these points, a discrete model can be created. Figure 3 shows the 5 mass points of the reduced order model. Evidently, the modeling points should be able to reflect the motion of these five modes as welI as possible. The normalized modal vector components of the structure at the modeling points are presumedly defined as the mode vector components of the lumped 5DOF model as folIows,
where t/J is a new defined q x 1 vector consisting of 91j ( j = 1 - q. here the subscript j is only a sequence number and has nothing to do with the mode order or the position label). Our purpose is to modify these modal shape components tPj in order that the errors El approach zero. Define the modification vector of t/J as &P. which is to change the errOr vector from its initial value e to O. then we can start the correcting procedure from the following equation
as~umed
9111 9112 9113 9114 9115 9121 91 22 91 23 91 24 91 25 4>= 9131 9132 tP33 91 34 9135 11'41
11'42
11'43
11'44
9151
11'52
tP5 3
91 54 9155
(I)
[~:]&p = 0-£ =-£
11'45
where [ae! at/J] is a p x q matrix called sensitivity matrix whose elements are partial differentiating of the errors. namely
where the subscripts i and j of the mode vector components 91i} mean the number of mass points and mode orders, respectively. On the other hand, for a lumped parameter system, the physical mass matrix M and stiffness matrix K are related to the normalized modal matrix 4> through folIowing equations
M
=(4>4>T r' .
K = ( 4>T
r' n
24>-'
(6)
a£, a£,
a£,
a~
at/Jq
ae ae
~ at/Jq
a~
[:~]=
(2)
2
~
2
a~
aep aep a~
.(3)
a~
.(7)
aep at/Jq
The modification vector can be obtained directly from Eq.(6) using the generalized inverse matrix as follows .
where, D2 is a diagonal matrix whose nonzero diagonal elements are given by the square of natural frequencies of each mode. Equations (2) and (3) suggest that if the normalized modal matrix of a lumped parameter system could be obtained, the physical parameters of the system could then be dl!termined. Unfortunately, the directly obtained modal matrix from the FEM modal analysis data, as given in Eq.(l), is not guaranteed to satisfy Eq.(2), where both sides should be diagonal matrices.
.(8)
and the new vector is calculated by (9)
149
Usually iterative calculation is necessary in this procedure in order that £ converges to O. After certain iteration that a preset accuracy has been reached, a final modified modal matrix can be obtained which will satisfy the diagonal condition of Eq .(2). Consequently, M and K in case of 5DOF model can be obtained as follows. M =diag[MI
K{ where and M
kij
M2
M3
Ms]
M4
-k' J -k,. -k2) -k 2• C -k)4 D Symm.
-kl2 B
(10) x.
- k 2, -k),
-k"l
(11)
-k" E
,
.
A = kll + kl2 + k n + k'4 + k' 5 B = k' 2 + k22 + k 2) + k 2• + k2j
mix = 1.508,
C = kl) + k2) + k )) + k" + k ),
m4x = 3.420. ms":'= 5.709
~J"
\
m2x = 1.508. "
D = k' 4 + k ,. + k )4 + k ... + k"
K Ilx = -3.221, K 12x = 0.317.
E = k" + k21 + k" + k., + k"
KI4x = -13.488, K 23..
means the spring conneted with M j
K 33 ..
j '
22.<
= -3.221
= -13.488, K 24x = 9.862, K 25x = 22.958 = 19.08, K 34x = 6.028, K 35x = 23 .703
(13)
where x6 is the relative displacement
and Xd is the displacement of the mass of the actuator. A Similar type of state equations along the y-directions can be written whose details are not given here.
a mass with Md. actuator with control force id ' supporting spring and damper with kd and cd , respectively. This discrete model will make it easy to design a controller in physical state-space where the states are directly measurable. 2.3 State space equation for the structure
3.
H-INFINITY BASED-FEEDBACK CONTROLLER DESIGN
3.1 Sta~e equation of augment system with dynamic compensator
The state space equations of the system along x-direction are written as (from fig. 4(a»
Xu
K
mdy = 0.158, kdy = 0.2, Cdy = 2 mass and stiffness are [kg] and [kN/m] in unit, respectively Fig. 4 The lumped parameter models of the structure on x-direction
Figure 4 shows the reduced order lumped mass model expressed by 5DOF system of the structure along the x direction. The same reduced order model is described along the y-direction. Using these models. the state space equations are derived. The physical parameters in the x- direction are shown as follows . These parameters in the y-direction are obtained by the same way. Each mass is connected with springs denoted by the spring constant kij which means the spring conneted with Mj and M j . An active dynamic absorber is connected to the mass points I for controlling vibration along x-direction and also another one is connected to the mass point 2 for y-direction . Each active dynamic absorber consists of
In the above, the state vector the symbols in fig 4a as
K 13x = 9.862
K I5x = 22.958,
K 44 ... =' 19.080, K 45x = 23 .703, K 5Sx = 3.886 mdx=0.125, kdx=0.2, cdx=2
2.2 Lumped parameter model of the structure
.i: cx = Acxxcx + Bcxux + B",x w IJt Yx=C cx x cx +D",x W 2x
m 3x = 3.420
In this section, the development of an H-infinity based control scheme is briefly discussed. In the present study, the objective is to design a dynamic output feedback controller based on the reduced order model to control its first five vibration modes and suppress the possible instability due to neglected modes. These objectives can be a(;hieved by solving a mixed sensitivity minimization problem in the framework of H-infinity optimal control (Cui. Nonami and Nishirnura.
(12)
is defined by using
150
1993. Doyle and Glover. I 996). Fig 5 shows a block diagram of a standard H-infinity control problem (using the model in x-direction). The controller design by using the dynamic equation along the x-direction will be discussed in detail and the controller for the model in y-direction is similar in nature and thus omitted.
Acz
A"
=
[
°
B ...bC,.,x
Bu
C2x
o
°
=[:...:x].c Ix =
o
=[Cc.r 0
°o 1
.B lx =
A,.,lx
[D
0
A,.,2.r
°c wx
w2z
[BWX 0
0]O.
°
C wlx
o
0J.D lu = [D;/].D21 =[0
Dwxl
By selecting suitable filters W rx • WSr • the aim is to design a dynamic feedback controller (16)
Fig. 5
such that the H-infinity norm of the transfer function G.,.,x becomes smaller than unity where G:wx is the transfer function from Wx to 4x. In the similar way, the controller K, is designed for the reduced order model along the y-direction.
Block diagram of standard Hoo control problem
In the above figure. Ux is the control input. yx is the measured output. 41x. 42.r are regulated variables. Wlx(s) and W 2z{s) are two weighting functions to shape the sensitivity and complementary sensitivity matrix. In this study. we have one control input and two measurable outputs in x direction. Define two filters WT,,=Wlx and WSr = diag( W 2x" W2.r) for the input and two outputs respectively. Note that WSr bounds sensitivity function and thus sets the disturbance rejection, and Wrz bounds the complimentary-sensitivity function and thus sets the suppression of measurement noise and uncertainty. Here attempt is made to design the H-infinity normed based control subject to these constraints. The state equations of the filters W rz • and WSr are defined as
{
3.2. Selection o/Weighting Filters In the case of multiplicative uncertainty representation. the relation between the actual model and the reduced order model is given by .(17)
where ~ V(J}) is the full order model. P; V(J}) is the reduced order model and 4. is the error function or multiplicative system uncertainty. In the H-infinity framework. (as the exact representation of 4. is difficult). a weighting function is selected to represent the upper envelope of the uncertainty. i.e., (18)
*W/~ = A.,IzXwlx + B WIXUX. C .,lxX wiz + Dw1xu x
1./ x -
*. {
2z
.(14)
where c; (.) is the maximum singular value of (.). The multiplicative uncertainty for both the directions is shown in fig.6. These error functions are computed by using the frequency response data of actual and reduced order model. It has been mentioned earlier that five modes are considered to derive the reduced order model. So ideally. there must not be any sharp peak in the low frequency range. However, the actual system response is obtained experimentally and as a result some anti-resonance peaks appear in the low frequency range of the error function which can be neglected. This is because the anti-resonance peak does not affect the stability conditions. To represent the multiplicative uncertainty, the following filter is used for the x-direction model:
= A .,2zX .,2z + B w2zC wxx cz
1. 2z '"
C .,2z X wlx + D w2z C wzX cz
By combining (12) and (14). the augmented state equations are written as
iX:
j
Axxx+ Blxwx+ B 2x u x
4X - Clxx x + Dl2x"x
.( 15)
y x = C 2.r x x + D 21x W x
where
151
80 - - Kx
(19)
--- Ky
60 40
where
20
r; TI = 0.25, (OnTl = 21tx 25[radlsec), KT = 0.45 r; T2 =0.35, (OnT2 = 21tx37[radlsec).
~
We have also selected another filter It;-2 (s) whose parameters are as follows r; TI = 0.5, {OnTl = 21tX 25[radlsec]. KT = 0.04
,.,
60 40
r; T2 = 0.2, (OnT2 = 21tx 35[radlsec).
20
The frequency response plots of two filters are shown in Fig. 6. It is noticed that the second one does not satisfy the mathematical equation (18) strictly. Later on, it is shown that the controller based on this filter also gives satisfactory results. The reasons may be that the H-infinity based design approach considers the norm bounded uncertainty without any phase information. In this sense, it may result in some over design. So a trial and error procedure is necessary to select a suitable filter for obtaining implementable results (although it may not give logically conclusive results by not satisfying the strict mathematical equations). The weighting filter for y-direction is taken to be same as that of the x-directions.
Control/er using W
0 0
5
Fig. 7
'·2
10 :.1 5 20 25 30 35 !' \Frequency (Hz)
40
45
50
Frequency re&ponse of Hoc> controllers.
-40~----------------------------~
x-direction
-60 -80 -100
iii"-120 :!:!.-140 .~ -40 Cl
'=::!:=================4 without control
-60
2or-------------.------------------, -am.
with control (W,) with control (W
.t
-80
- - l1my
W,.,
o
Controller using W
0
c: 80 '<'6 Cl
1
-100
--W'.2
-120 _140 '--_L__..I..-~__-'-----I.__-,-_Y--,d_ir_e-,c_ti_o.... n,---,
o
-40 1----'1--.- -...
5
10
15
20
25
30 35
40
45
50
Frequency (Hz)
Fig. 8
Simulated frequency response
~O ·L-~--~--~~---L--~~---L--~~
o
Fig. 6
5
10
15
20 25 30 Frequency [Hz)
35
40
45
These observations will be validated by experiment and discussed in the next section. The controller is validated through simulation studies. In the x-direction, an impulse excitation is given at the mass point 1 and the response is measured at the same point.
50
Multiplicative uncertainties and weighting function WJ./' Wj • 2
A low pass filter Ws(s) is used as a weighting function for the output i.e., to minimize the weighted sensitivity function. Here we have selected
0.5
W2(S)=~ s+{On
where
(On
I
=21tx30[radlsec), K s =250.
C Q)
3.3. Simulation Results
~
u
0
-0.5 0.5
.!!! ~
Using the MATLAB function HINFSYN, the feedback controllers K" ,KJ are computed using both the filters It;-I (s) and It;-2 (s). The frequency responses of the controller are shown in the fig .7, which are lowpass in nature. This ensures the reduction of the high frequency component of the control signals, which is good for avoiding spillover.
(5
0
-0.5 L-_ _.L-_ _.L-_ _~_ _~_---'
o Fig. 9
152
234 Time Isec1
Simulated impulse response (x-direction)
5
0.5
E S-
e
Q)
- - with control (W
1·1
to actuators
)
0
-0.5
~0 0.5 to
g. is
0
-0.5 2 3 Time [sec]
0
Fig. 10
4
5
Simulated impulse response (y--direction)
For the Y-direction. excitation is given at mass point 2 and the response is measured at mass point 2. Both frequency and impulse responses are shown in figs (8-10). It is noticed that the vibration modes are suppressed for both cases 4.
Fig. 11
Moving Coil Bearing
EXPERIMENTAL RESULTS
In this section, the simulated results are verified through experiment. Figure 11 shows the experimental setup where the personal computer acts as a controller. The displacement of mass point 1 along the x-direction and mass point 2 along the y-direction are measured by two non-contact laser sensors. Although these laser sensors are useful for the purpose of experiment on a scale model, it may be difficult to use these sensors in actual application building control problem. However, these control schemes can be implemented in real structure by using commercially available velocity sensor.
Permanent Magnet
Fig. 12 Schamatic diagram of actuator -40 . - - - - - - - - - - - - - - - - - - .
x-direction
-60 -80
-100
CD -120 ~ -140
An actuator is constructed for each direction whose schematic diagram is shown in fig. 12. The. actuator is similar to a horizontal type pendulum where a mass is attached at the end of the lever. The feedback control signal changes the current through a pennanent magnetic field and results in the movement of the mass of the actuator.
c
F=d:=d==d=~==~~==~~==~~ without control with control -60
~ -40
(W,.I
-80 -100 -120 -140
During simulation, the continuous time controller is designed. For the purpose of the experiment, we discretize this continuous time controller by using the sampling period of T = O.OOI[sec]. The discrete time controller is given by
L-....L._...l---I._-I-_I-....I.._...l---I._-L...---J
o
5
Fig: 13
10
15
20 25 30 35 Frequency [Hz)
40
45
50
Experimental frequency response
The vibration control effects are evaluated by measuring the transfer function (compliance) of the structure under impulse excitation. The experiment is carried out for the x direction by giving an impulse input at mass point 1 and measuring the response at the same point. For the y-direction, an impulse is given at mass point 2 and the measurement has been made at the same point. Experimental frequency and time responses of the system are shown in figs. 13-14. From these figures, it is clear that all five peaks are
: /k + 1) = Plo;; x(k)+QIo;Y /k), { ux(k) =CIo;; /k)+DIo;Y x(k)
where k is the sampling instant and Plo; = eA •.,T QIo; =
Experimental setup
A~( eA"T - / )BIo;.
153
suppressed compared with the no controlled frequency response. Also the feedback controller suppresses the vibration without any spillover instability which is clear from the impulse response.
E
0.5
- - without control
- - with control
(W,.)
§.
c:
~ B <15
0
~ CS -0.5 2
0
3
5
4
Time [sec)
E
0.5
- - without control
- - with control
(W,.)
§.
c: ~
0
Q)
U
<15
Q.
cS
-0.5 2 3 Time [sec]
0
Fig. 14
4
5
Experimental impulse response
5.
CONCLUSIONS
In this paper. an active control scheme has been designed to suppress the vibration of a flexible building like structure. Two reduced order models have been constructed along the x and y directions to represent the system dynamics of five vibration modes in each direction. Using these models. an attempt has been made to design H-infinity based dynamic output feedback controller to suppress the five vibration modes. These control schemes have been verified through simulation and experimental studies without any spill over problem. In the experiment. only two sensors and two actuators are used. The actuators are constructed based on the mechanism of a horizontal type pendulum and have been shown to be effective to reduce the vibration. REFERENCES
Trans. Japanese society Mechanical engineers. Vo1.59. pp.7 I 4-720. Crassisdis. J. L.. Leo DJ .• Inman D.J .• and Mook D.J .• (1994). Robust identification and vibration suppression of a flexible structure. Journal of Guidance. Control and Dynamics. Vol. 17. No-5. pp. 921-928. Doyle. J.c. and Glover. K..(l996). Robust and optimal control. Prentice Hall. Upper Saddle River. New Jersy. Fuller c.R.. Elliotl S.J .• and Nelson P.A.• (1996). Active control of vibration. Academic. London. Nishimura. H. and Kojima. A. (1999). Seismic Isolation Control for A Building like Structure. IEEE Control System Magazine. pp.38-44. Dec. Kar. I. N.• Seto. K. and Doi. F.. (2000). Multimode vibration control of a flexible structure using H-infinity based robust control :\avv. IEEElASME Transaction on Mechatronics. Vol. 5~.No. l. pp. 23-31 . and Saln.. M.K. (1997). Controlling Spencer. B.F. buildings: A new frontier of feedback. IEEE Control System Magazine. vol. 17. pp.19-35. Dec. Seto. K .• Doi F. and Ren. M .• (1999). Vibration control of bridge towers using a lumped modeling approach. ASME journal of vibration and acoustics. Vol. 121. pp. 95-100. Seto. K.• (1992). Trend s on vibration control in Japan." in Proc. I SI Int. Conf. Motion and Vibration Control (MOVIC). vol.1. pp. 1-1 I. Seto. K. and Mitsuta. S .• (1992). A new method for making a reQuced order model flexible structures using unobservability and uncontrollability and its application in vibration control. in Proc. I SI Int. Conf. Motion and Vibration Control (MOVIC). pp.152-158. Seto. K.• Ren. M. and Doi. F. •. (1998). Feedback vibration control of a flexible plate at audio frequencies by using a physical state space approach." J. Acoust. Soc. Amer.• vo1.103. no.2. pp.924-934. Wodek.G. (1996). Balanced control of Flexible structures. Lecture notes in control and infonnation sciences. 211. Springer. Wei. B. and Marcelo. G .• (1992). Control synthesis for flexible space structures excited by persistence disturbances. Journal of Guidance. Control and Dynamics. Vol. 15. No. I. pp. 73-80.
Bai. M.R .• and Shieh C..(1985). Active noise cancellation by using the linear quadratic gaussian independent modal space control. Journal of Acoustical society of America. Vo1.97. pp.2664-2674 Balas. M. J .•(199O). Feedback control of flexible systems. IEEE Trans. Automat. Contr.• voI.AC-23. pp.673-679. Aug. 1978. Merovitch. L.. Dynamics and Control of Structures. New YorkWUey.. Balas. G J.• (1998.) Synthesis of controllers for the active mass driver system in the presence of uncenainty. Earthquake Engineering and Structural Dynamics. Vol. 27 .• No.l1. pp. 1189-1202. Cui. W.. Nonami. K. and Nishimura. H.. (1993). Experimental study on active vibration control of structures by means of H-infinity and H-2 control.
154