13th World Conference on Earthquake Engineering
Vancouver, B.C., Canada
August 1-6, 2004
Paper No. 3236
NEUTRAL EQUILIBRIUM MECHANISM FOR STRUCTURAL
CONTROL
Wen-Pei SUNG1, Ming-Hsiang SHIH2 and Cheer Germ GO3
SUMMARY
The new control mechanism, Neutral Equilibrium System (NES), will be introduced in this paper. NES is
composed of unstable control mechanism and stable structure frame. It has advantages of all functions; a
merit of active control method and it does not require the extra energy supply. The earthquake record or
the histories of harmonic loading are used as input data to discuss the results of numerical simulation in
investigating the stability, usability and effect of this control mechanism. The results indicate that this
mechanism should be controlled thoroughly. Under appropriate control law, the control state is nearly
perfect. Nevertheless, the external energy is required for control system to operate by using the control
law of related displacement.
INTRODUCTION
New control logic, known as Neutral Equilibrium System (NES) is proposed in this research. It is
a control mechanism with benefits of all functions and merits of active control method. Without requiring
extra energy supply, NES provides effective control effect. NES principal is based on Newton Law of
motion. In the state of unstable equilibrium, the potential energy of structure is placed at the highest point.
When structure is inclined toward unstable state, the potential energy decreases. In view of theorem of
conservation of energy, the potential energy of structure decreases and the kinetic energy increases or
change energy into work. Laborsaving tool can be designed if the energy can be properly changed to
generate active control force. Consequently, the series connection is used to combine unstable and stable
equilibrium system to generate a neutral equilibrium system (NES). This system does not arise
destruction, induced by unstable system, or cause control force for structure.
Active control systems require a large power supply to operate active elements. It provides
control forces to structure based on feedback from sensors and the meant of measurement of the
responses. Assuming power supply does not work, active control system is out of use. Consequently,
active control systems doubt of their reliability. In order to improve shortcomings of active control
1
Associate Professor, Dept. of Landscape Design and Management, National Chinyi Institute of
Technology, Taichung, Taiwan, R.O.C. Email: [email protected]
2
Associate Professor, Dept. of Construction Engineering, National Kaoshiang First University of Science
and Technology, Kaoshiang, Taiwan, R.O.C. Email: [email protected]
3
Professor, Dept. of Civil Engineering, National Chung Hsing University, Taichung, Taiwan, R.O.C.
Email: [email protected]
systems, semi-active control systems are developed. The merits of semi-active control system are the
advantages of passive control, active control system, high-performance, small-energy requirement and
high control stability. There are several semi-active controls elements developed such as: semi-active
hydraulic damper [1-3], magnertorhological dampers [4] and friction damper of semi-active control
systems [5]. The control forces of semi-active control system can be produced by passive control
elements and controlled by on-off switch or micro-adjustment control valve of semi-active damper. Under
this situation, the adjustments of control forces are only able to carry out at absolute value of control force
or the total potential energy of control elements decreasing. The semi-active control systems are
unsatisfied with the required control force of added potential energy of control elements. Namely, the
maximum rate of change of semi-active control force is equal to plus and minus 100% of magnitude
control force. It is the same of control force but the direction of control force is contrary [6]. Because of
above reasons, the new control logic－Neutral Equilibrium System (NES) is proposed in this paper. This
system has the advantages of no energy required of passive control system and whole control capability
of active control system.
NEUTRAL EQUILIBRIUM CONTROL MECHANISM
The potential energy of unstable mechanism is released at the loss of equilibrium. Thus, the NES is a
newly developed mechanism that must meet the requirement of storing energy. The element of linear
spring is satisfied with this idea. The linear spring under compression, shown in Figure 1 in the range of
dash line, is an unstable mechanism. Consequently, the unstable mechanism is added to the statically
determined stable structure and series connection with bracing of structure. The unstable equilibrium of
element of linear spring is destroyed and induced-force is applied to structure under loading of bracing.
This induced –force is used to offset the force of bracing that caused deformation. The relationship of
bracing and linear spring can be shown in free body diagram of Figure 2. Take free body diagrams of
Figure 2 at point 1 and 2, the static equilibrium equations for point 1 and 2 can be shown as follows:
Under Compression
(Neutral Equilibrium)
1
FL
Feq
Fb
Θ
A
2
FL
Figure 1. The interaction between Neutral Equilibrium Mechanism and structure
FS
L0
L
R
L’
1
θ
Fe
Y
2
Fb
R
X
Stiff. of spring Ks
Stiff. Of bracing Kb
FS
Figure 2. The relationship of bracing and linear spring
For point 1:
∑F
∑F
x
=0
R－FLsinθ=0
（1）
y
=0
FLcos-Fs=0
（2）
=0
Fe＋2FLsinθ-Fb=0
（3）
For point 2:
∑F
x
Where:
FL: the internal force of rigid bar;
Fs: the spring force;
R: the reaction force of support;
Fe: internal control force of the proposed mechanism;
Fb: bracing force of idea condition, no relative displacement of Figure 2.
After arrangement of equation (1)~(3), yields
Fe=Fb－2Fstanθ
（4）
Fe is used to balance Fb and Fs, which generally requires energy supply. The values of Fe are less the
better. Using the geometric conditions and Hooke Law to substitute into equation (4) it yields
Fe＝KbL’sinθ -2 Ks（L－L0）tanθ
（5）
Assuming spring S1 and S2 are back to original length L when θ is equal to 900. When θ=00, the
compressive deformation of spring is equal to the length of rigid bar L’. Thus, for arbitrary θ
L’=（L－L0）secθ
（6）
Substituting equation (6) into equation (5) yields
Fe＝（Kb－2Ks）（L－L0）tanθ
If the stiffness of bracing is equal to two times of stiffness of spring
Kb＝2Ks
then the internal control force Fe is equal to zero.
（7）
（8）
Therefore, one set of parameters has generated for NES.
(1) The stiffness of bracing is two times of stiffness of spring;
(2) There is no force in spring under the situation that the proposed mechanism internal angle θ is equal
to 0°.
Thus, the internal deformation di of the proposed mechanism can be defined as follows:
di＝L’sinθ
(9）
The combined force Fd of the proposed mechanism and the theoretical bracing force Fb can use the
parameters of Kb and di to express the following:
Fb＝Kbdi
Fd＝2Fstanθ＝KbL’（1－1＋cosθ）tanθ= Kbdi
Fe＝Fb－Fd
(10)
(11)
(12)
The combined force in X direction of point 2 is composed of the combined force of spring mechanism Fd
and theoretical bracing force Fb and Fe. Therefore, the value of Fe is influenced by the elements of spring
and bracing under change of angle θ.
PRACTICAL CONSIDERATIONS
The actual problems and the hypotheses of NES will be discussed in this section. The idea structure
is defined in figure 3. For this idea situation, assuming that there is no relative displacement of degree of
freedom 1 and 2. The deformation of bracing is affected by the change of angle of connecting level
mechanism. The relative displacement of floor is actually induced by earthquake or wind force. This
relative displacement reflects on the deformation of bracing. Thus, the extra bracing force must apply on
NES. Therefore, equation (4) can be rewritten as follows:
Fe=F*b－2Fstanθ
F*b=Fb－Kbds
(13)
(14)
Where:
Kb is the stiffness of bracing;
F*b is the supply force of bracing which considers the relative displacement of floor.
The assumption of Fe=0 does not exit for the condition of equation (13). However, if the moving direction
of floor and the applying force of bracing for mechanism B are the same, mechanism B can be moved at
the same time that does not need apply work. In figure 3 as an example, we assume that mechanism B
moves dd to right, then the internal force of bracing is shown as follows:
F**b = F*b＋Kbdd = Fb－Kbds＋Kbdd
(15)
Where:
F**b is the force of bracing that considers the displacements of structure and mechanism B.
ds
DOF 2→
ds：Floor displacement of
structure
dd：Total displacement of NES
di：Internal displacement of NES
DOF 1→
dd
di
Figure 3. The interaction relationship of moving direction of floor, bracing and mechanism B
Giving that the directions of ds and dd are the same, the magnitude of ds and dd may be identical. Thus, in
the condition of ds is equal to dd, F**b is equal to Fb, and Fe is equal to zero again. The NES attains the
d (d d )
stands for the moving direction of spring
dt
mechanism in continuous time control system. Assuming the same direction of d& d and F**b, mechanism
•
situation of equilibrium. On the other hand, d d =
B moves automatically and does not need additional energy supply. Contrarily, system requires extra
power supply to move mechanism B and maintains the equilibrium of equation (4). Because there is no
power source for this developed system, the proposed control system is useless in this situation. In this
paper, the useless state of unstable control is defined. Therefore, this proposed control system must meet
the following condition.
•
d d ⋅ Fb > 0
(16)
From the above-mentioned state, we investigate that the deformation of structure is the main reason to
induce instability of NES. Consequently, mechanism of unstable spring should follow the deformation of
structure to move so the value of control force Fe can be decreased as much as possible. At this time, a
brake horsepower with small power and quick response such as server linear control horsepower can be
used to supply internal control force Fe in order to maintain the operation of control system. This control
system almost does not need the extra power supply to operate. Therefore, the choices of control laws are
a key point of successes and failures of this system. This optimal control law should determine the
applying magnitude, timing and direction of control force.
NUMERICAL SIMULATION METHOD
The performance of NES is very well in idea state. Nevertheless, extra bracing force may be induced
by the deformation of structure that causes instability of control or illness of control effect. To solve these
problems, the optimal control method [7], mode control method [8] and instantaneous control method [9]
are used to derive the best control law for multi-degree of freedom structure. Then, this control law is
combined with numerical simulation to investigate the relationship of displacement of structure, bracing
force (Fb, F**b), spring force (Fd) and internal control force (Fe) in conditions of different applying loads
and change of control law. The following steps are proposed to choose the best control law.
The first step: Define a structure and calculate eigenvalues, frequency and damping ratio.
The second step: Choose a disturbing function such as sine wave or record of earthquake wave.
The third step: Substitute the augmented state matrix that is determined by selected control law into
system matrix of structure. Therefore, closed loop feedback system matrix is shown as follows:
−
A = A + BG
（State Feedback）
(17)
or
−
A = A + BGD
（Output Feedback）
(18)
Where A, B, G are in one system and equipped, and control with gain matrix respectively, then D
becomes observation matrix. Thus, program of time domain dynamic analysis is developed. The response
of controlled structure at each time step like state vector z (t) is shown as follows:
•
−
z (t ) = A(t ) z (t ) + E (t ) w(t )
(19)
Where:
E (t) is the equipped matrix of disturbing force;
w(t) is the function of disturbing force.
The forth step: Substitute the response of theoretical analysis into equation (17) or (18) to calculate the
theoretical control force (Fb).
The fifth step: The deserved deformation of db is generated by theoretical control force. The relationship
of actual deformation of bracing and displacement (di,ds and dd)is obtained by equation (15). This related
expression could be shown as follows:
db =
Fb**
= di + d s − dd
Kb
(20)
In order to maintain actual bracing force F**b to be identical to theoretical control force Fb that is
determined by control theorem, the following expression should use equation (16) to judge the
equivalence of displacements of mechanism dd and structure ds under the state of no applying work.
di + ds − dd =
Fb
Kb
Where:
ddi is displacement of NES at ith step;
(21)
dsi is displacement of structure at ith step;
ddi-1 is displacement of NES at i-1th step.
If both of displacements are equivalent, and then let ddi=dsi. Contrarily, let ddi=ddi-1.
The sixth step: Substitute ddi into equation (21) to calculate di and then substitute the result into equation
(10) and (11) to obtain internal control force Fe.
The seventh step: Compare the absolute value of Fe with supply capacity of NES. If absolute value of Fe
is greater than supply capacity of NES, then it represents that the control law of NES is useless.
Contrarily, it represents that the required power supply of internal control force of NES. Moreover, we
define an actual structure model shown in figure 4 to investigate the influence of control law, stiffness of
bracing, type of weighting matrix, weighting multiplier and weighting of velocity type. The parameters
and dynamic characteristics of structure are listed in table 1.
2500
Dof2,u2
2200
Dof1,u1
2200
Rigid Platform
32×100
Figure 4. Two degrees of freedom structure and mathematical model
Table 1. The parameters of structure and dynamic properties
 655.36 − 327.68
K =
 KN/m
− 327.68 327.68 
2 0
M =
 ton
0 2 
Frequency（rad/sec）
Period（sec）
Vector of modal shape
First mode
7.91
0.794
[0.526 0.851]T
Second mode
20.71
0.303
[0.851 -0.526]T
RESULTS OF SIMULATION
5.1 Control Law
In order to keep fine control responses and the minimum required control energy, linear optimal
control theory is applied. The generated performance index is defined as follows:
tf
[
]
J = 12 z T (t f ) Hz (t f ) + 12 ∫ z T (t )Q(t ) z (t ) + u T (t ) R(t )u (t ) dt
t0
Where:
(22)
J is a performance index;
t0 is the start time of control;
tf is the finish time of control;
H and Q are 2n×2n-weighting matrices;
R is r×r -weighting matrix for control force.
Control force of the minimum performance index J is the best control force. Changing weighting matrices
of R and Q influences the performance index, where R is an r×r unit matrix and Q is composed of
stiffness and mass matrices. Matrix Q is defined as follows:
αK
Q=
 0
0 
βM 
(23)
Where:
α and β are the weighting values for stiffness matrix and mass matrix.
Owing to analyze two degree of freedom, the effect of control is dependent on the number of control
forces. The optimal control gain of various weighting values and change of control force number are list
in table2.
Table 2. The optimal control gain of various weighting values and change of control force number.
Weightin Two control forces of Abs. displacement Two control forces of floor displacement
α
）
β
（Gopt）
（Gopt）
10000
10000
1864.2 - 533.8 165.6 - 6.6
- 533.8 1330.5 - 6.6 159 


 1864.2 533.8 208.7 - 62.3
 - 533.8 1330.5 - 62.3 146.4 


10000
0
 1864.2 - 533.8 85.3 - 13.6 
 - 533.8 1330.5 - 13.6 71.7 


 1864.2 - 533.8 85.3 - 13.6
 - 533.8 1330.5 - 13.6 71.7 


0
10000
- 0.00 0.00 141.42 0.00
 0.00 - 0.00 0.00 141.42


 0.00 - 0.00 189.73 - 63.24 
- 0.00 0.00 - 63.24 126.49


100
100
46.99 - 2.64 19.69 - 0.27 
- 2.64 44.34 - 0.27 19.42


46.99 - 2.64 23.74 - 4.95
- 2.64 44.34 - 4.95 18.78 


100
0
46.99 - 2.64 13.70 - 0.39
- 2.64 44.35 - 0.39 13.31 


46.99 - 2.64 13.70 - 0.39
- 2.64 44.34 - 0.39 13.31


0
100
 0.00 - 0.00 14.14 - 0.00
 - 0.00 0.00 - 0.00 14.14 


- 0.00 0.00 18.97 - 6.32
 0.00 - 0.00 - 6.32 12.64 


1
1
 0.50 - 0.00 2.00 - 0.00
 - 0.00 0.50 - 0.00 2.00


 0.50 - 0.00 2.41 - 0.46
- 0.00 0.50 - 0.46 1.94 


g（
1
0
 0.50 - 0.00 1.41 - 0.00
 - 0.00 0.50 - 0.00 1.41


 0.50 - 0.00 1.41 - 0.00
 - 0.00 0.50 - 0.00 1.41 


0
1
 - 0.00 - 0.00 1.41 - 0.00
 - 0.00 - 0.00 - 0.00 1.41


- 0.00 0.00 1.90 - 0.63
 0.00 0.00 - 0.63 1.26 


Considered only the optimal control gain of absolute displacement of floor, the 2DOF of braking
mechanism and relative displacement approach to assimilation. The displacement and velocity part of
control gain become diagonal matrices. In view of the NES is installed between adjoining floors, M
matrix of Q in equation (23) will generate the summation of relative square velocity to obtain the better
control capability. Thus, Q is redefined as follows:
αK
Q=
 0
0 
β M 
(24)
Where:
 4 − 2
M = 
.
− 2 2 
Thus, the optimal control law that considers relative displacements of floor, is list in table 2.
In this paper, the feedback matrix of direct velocity output is used to decide control force except the
optimal control gain. The control force decided by this method is relative to floor velocity. The Rayleigh
Damping that is relative to stiffness matrix is used as control gain. The equation can be written as
follows:
G = γK
(25)
Arranging the coefficient γ of equation (25), the damping action is induced in equation (25) for control
gain. The damping ratio of the first mode of structure is between 0.1 and 1.0. Thus, the value of γ can be
defined as follows:
γ =
2ξ 1
ω
= 0.253ξ 1
(26)
Where:
ξ1 is the expected damping ratio of the first mode.
Therefore, the control gain of direct velocity feedback control is expressed as follows:
G Dir = 0.253ξ 1 K
(27)
Control gain is calculated by the optimal control method and direct velocity feedback method. By
substituting control gain into dynamic analysis program, active control force Fb and internal control force
Fe can be obtained and they are required by control system. To investigate the stability of control system,
the ratio of maximum required internal control force (Fe, max/Fb, max), the ratio of instantaneously maximum
required power (Pe, max/Pb, max) and the ratio of average required power (Pe, avg/Pb, avg) are discussed using
different control laws, stiffness of bracing, weighting multipliers, weighting ratios and definitions of
velocity. The required internal force, instantaneously maximum required power, and the average required
power of the first degree of freedom are shown in figure 5 at each time step. This figure indicates that the
required internal control force Fe of actual internal control force Fb is discontinuous. Fe is near zero at
most of time especially when the minus part of power values can be ignored. Therefore, Fe shows that the
extra energy is not needed to meet the required control.
40
Fb 1(ν =10 0,A )
Fe1(ν =1 00 ,A)
30
Param e ter：
Rela tiv e Velocity W e ig hting
S.R.= 4
Force, KN
20
10
0
-10
-20
-30
0
5
10
15
Time, sec
20
25
30
Figure 5. The required internal force, instantaneously maximum required power, and the average required
power of the first degree of freedom at each time step
5.2 Influence of Control Law
Bar charts are used in figure 6 to compare the maximum internal control force (Fe, max) and the
maximum active control force (Fb, max).
0.00
P e 2, avg /P b , avg
Direct Output Feedback
0 .4 8
ν ( 1 0 0 ),A
0.00
P e1 , a v g/ P b 2 , avg
R e la t iv e V e lo c ity W e ig h tin g
S .R .= 4
0.11
0 .0 0
P e 2 , m ax /P b2 , m ax
P e 1 , m ax /P b1 , m ax
0.08
0.0 0
0 .0 0
0.0 0
F e 2 , m ax /F b2 , m ax
0.36
0.0 0
F e 1 , m ax /F b1 , m ax
0 .0 0
0 .1 2
0.1 0
0.2 0
0.30
0.40
0 .50
0 .60
0 .7 0
0.80
0 .9 0
1 .0 0
R e s p o n s e R a tio
Figure 6. Using bar charts to compare the maximum internal control force (Fe, max) and the maximum
active control force (Fb, max) of the first and second DOF
Figure 6 indicates that the direct velocity feedback has qualification in nearly perfect control
stability. The direct velocity feedback method is satisfied with the initial proposed purpose that the active
control force does not need extra energy supply.
5.3 Influence of Stiffness of Bracing
In order to investigate the relationship between the optimal control of internal control force and
stiffness of bracing, we change the stiffness of bracing. It shows that the required internal force of control
system is linear direct proportion to the stiffness of bracing. Therefore, to acquire the better control effect,
the value of stiffness of bracing must be restricted. On the other hand, the required control power is
directly proportion to stiffness of bracing. It is not a linear relationship because the displacement of
mechanism is inversed to stiffness of bracing and the more stiffness of bracing, the less movement of
mechanism. We suggested that the stiffness of bracing is not suitably too high in order to avoid using too
large control mechanism and to promote the stability of control.
5.4 Influence of Type of Weighting Matrix Q, A, B and C
The different types of weighting matrix Q directly influence the control force of optimal control law.
The more specific gravity of matrix Q and relative submatrices of reflected displacements, the more
internal control force and required power of mechanism are needed. The relative submatrices of reflected
displacements should take the summation of relative square velocity of floor because these submatrices
come close to the type of related-velocity submatrices of augmented matrix derived by direct velocity
feedback method. The performance of required control is similar to direct velocity feedback method. The
comparison between internal control force Fe and required control power are shown in figure 10 based on
various types of weighting matrices under predetermined conditions. From this result, it indicates that the
required maximum instantaneous power can be ignored under the comparison of control condition C and
others control conditions. The reason is that the way the reflected weighting matrix Q is induced by Gopt
matrix and the way Gdir matrix is calculated by direct velocity feedback method are highly similar. Thus,
the performance of this method is nearly perfect as direct feedback control method. Contrarily, if response
displacements only contain in weighting matrix Q such as B, the required control force and required
power of mechanism are higher than other situations. It should be averted to adopt this method in
practical considerations.
5.5 Influence of Weighting Multiplier ν
Except on the ratio of internal displacements and the velocities of weighting matrix Q, the
magnitudes of absolute value of Q matrix affect the control stability tremendously. It indicates the
changed tendency of ratio of internal control force, maximum instantaneous power and average power
when multiplier ν changes regardless each kind of ratios is approaching to zero in large multiplier ν. This
phenomenon attains the purpose of this research.
5.6 Influence of Weighting of Velocity Type
The control stability is affected by weighting of various types of velocity. The influence of velocity type
and absolute velocity type are discussed in this section under the same predetermined assumptions. The
results are shown in figure 7. The bar chart indicates that the required control is about 0.1 of the
maximum control force, maximum instantaneous power and average power no matter if the relative
velocity or absolute velocities are at the first degree of freedom. Nevertheless, the performance of internal
control force is unsatisfactory in the absolute velocity type at the second degree of freedom. Thus, we
suggest that the weighting of relative velocity type should be used as parameter to determine the required
internal control for the demand of high control stability.
the weighting of relative velocity type should be used as parameter to determine the required internal
control for the demand of high control stability.
1.00
：
P a ra m e te r
C o n tr ol C o n d it io n , A
=100
S .R .= 4
0.90
ν
0.80
Response Ratio
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
V r e l.
V abs.
F e 1 ,m a x /
F b 1 ,m a x
F e 2 ,m a x /
F b 2 ,m a x
P e 1 ,m a x /
P b 1 ,m a x
P e 2 ,m a x /
P b 2 ,m a x
P e 1 ,a v g /
P b 1 ,a v g
P e 2 ,a v g /
P b 2 ,a v g
0 .1 2
0 .1 1
0 .3 6
5 .4 5
0 .0 0
0 .0 1
0 .0 8
0 .1 5
0 .1 1
0 .1 0
0 .4 8
0 .5 8
Figure 7. The relationship of control requirement between using relative velocity type and absolute
velocity type as weighting
CONCLUSIONS
A new active energy dissipation control mechanism is proposed in this paper. The numerical results
show that the active control mechanism using the condition of neutral equilibrium meet the purpose of no
control energy supply is needed. The conclusions are discussed as follows:
1. The numerical results show that the perfect control stability is under the condition of direct velocity
feedback control method. The external energy supply is not required to attain the total effect of active
control in the process of control in this mechanism.
2. The usage of control energy is unnecessary for the neutral equilibrium control method. Thus, the
optimal problem can be overlooked.
3. The condition of neutral equilibrium is unconditionally satisfied with the adjustment of internal
stiffness of spring and length.
4. The neutral equilibrium may not be maintained when the deformation of structure is considered. The
brake horsepower is used to adjust position of control mechanism.
REFERENCES
1.
2.
3.
4.
5.
Kurata, N., Kobori, T., 1999, “ Actual Seismic Response Controlled Building with Semi-Active
Damper System”, Earthquake Engineering and Structural Dynamics, 28, 1427-1447.
Michael, D.S. and Michael, C.C., 1997, “ Seismic Testing of a Building Structure with a SemiActive Fluid Damper Control System”, Earthquake Engineering Structural Dynamics, 26, 759777.
Kurata, N., Kobori, T., 2000, “ Forced Vibration Test of a Building with Semi-Active Damper
System”, Earthquake Engineering and Structural Dynamics, 29, 629-645.
Dyke, S.J., Spencer Jr., B.F., Sain, M.K. and Carlson, J.D., 1998, “ An Experimental Study of MR
Dampers for Seismic Protection”, Smart Master Structure, 7, 693-703.
Dupont, P., Kasturi, P. and Stokes, A., 1997, “ Semi-Active Control of Friction Damper”, Journal
6.
7.
8.
9.
of Sound and Vibration, 202(2), 203-218.
Shih, M.H., “ Eine Aktive Regelung zur Schwingungsreduzierung von Bauwerken unter
Seismischen Beanspruchungen”, Dissertation RWTH Aachen Germary, August 1996.
Meirovitch, L., 1989, Dynamics and Control of Structures , by John Wiley & Sons, Inc.
Lu, L.-Y. and Chung, L.-L., 2001, “ Modal Control of Seismic Structures Using Augmented Stste
Matrix”, Earthquake Engineering Structural Dynamics, 30(2), 237-246.
Ribakov, Y., Gluck, J. and Reinhorn, A.M., 2001, “ Active Viscous Damping System for
Control of MDOF Structures”, Earthquake Engineering Structural Dynamics, 30(2),
195-212.References start here