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Application of PSO Technique for Seismic Control
of Tall Building
Article · January 2009
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International Journal of Intelligent Systems and Technologies 4:1 2009
Application of PSO Technique for
Seismic Control of Tall Building
A. Shayeghi, H. Shayeghi , H. Eimani Kalasar
[2,6-8]. In this paper, a TMD system is considered to be
applied to multi-degree-of-freedom structure, without
specifying which mode should be controlled. Thus, there is no
need to transfer the structure to a single-mode model as have
been done in the available research.
Several cost or objective functions have been developed to
meet a specified performance in the optimization process. In
practice, many performance indices can be chosen, as the
objective functions resulted in a different result of the
optimization. In the active vibration control area, there are also
many optimization criteria which have been used by
researchers. The most common ones are LQR, LQG, H2, H∞,
sliding mode control, pole placement, independent model
space control and so on [1,9-13]. It should be noted that in the
active control optimization there is a trade off between the
responses to be minimize the structural responses while
maintaining the control energy to be used, the passive control
optimization is free from balancing the two parameters.
In this paper, the Integral of Time multiplied Absolute
value of Error (ITAE) based on the displacement of all
building's floors is taken as a performance index of the
optimization criteria. Usually, the traditional gradient-based
search methods are used for solution of this problem.
Unfortunately, the optimization requires computations of
sensitivity factors and eigenvectors at its iteration process.
This gives rise to heavy computational burden and slow
convergence. Moreover, there is no local criterion to decide
whether a local solution is also the global solution. Thus,
conventional optimization methods that make use of
derivatives and gradients are, in general, not able to locate or
identify the global optimum, but for real-world applications,
one is often content with a good solution, even if it is not the
best. Consequently, heuristic methods are widely used for the
global optimization problem. Recently, global optimization
techniques like genetic algorithm (GA) have been used for
TMD and Active TMD (ATMD) parameter optimization [1416]. Although, the GA technique seems to be good method for
the solution of TMD parameter optimization problem,
however, when the system has a highly epistemic objective
function (i.e. where parameters being optimized are highly
correlated), and number of parameters to be optimized is large,
then they have degraded efficiency to obtain global optimum
solution and also simulation process use a lot of computing
time. In order to overcome these drawbacks, a Particle Swarm
Optimization (PSO) based TMD (PSOTMD) is proposed for
the control of structural dynamics in this paper. PSO is a novel
population based met heuristic, which utilize the swarm
intelligence generated by the cooperation and competition
between the particle in a swarm and has emerged as a useful
tool for engineering optimization. It has also been found to be
Abstract—In recent years, tuned mass damper (TMD) control
systems for civil engineering structures have attracted considerable
attention. This paper emphasizes on the application of particle swarm
application (PSO) to design and optimize the parameters of the TMD
control scheme for achieving the best results in the reduction of the
building response under earthquake excitations. The Integral of the
Time multiplied Absolute value of the Error (ITAE) based on relative
displacement of all floors in the building is taken as a performance
index of the optimization criterion. The problem of robustly TMD
controller design is formatted as an optimization problem based on
the ITAE performance index to be solved using the PSO technique
which has a story ability to find the most optimistic results. An 11story realistic building, located in the city of Rasht, Iran is considered
as a test system to demonstrate effectiveness of the proposed method.
The results analysis through the time-domain simulation and some
performance indices reveals that the designed PSO based TMD
controller has an excellent capability in reduction of the seismically
excited example building.
Keywords—TMD, Particle Swarm Optimization, Tall Buildings,
Structural Dynamics.
I. INTRODUCTION
critical aspect in the design of civil engineering structures
is the reduction of response quantities such as velocities,
deflections and forces induced by environmental dynamic
loads (i.e., wind and earthquake). Structural control methods
are the most recent strategies for this purpose, which can be
classified as active, semi-active, passive, and hybrid control
methods [1]. In the last three decades or so, the reduction of
structural response, caused by dynamic effects, has become a
subject of research, and many structural control concepts have
been implemented in practice. Tuned mass dampers (TMDs)
are the oldest structural vibration control devices in existence.
The concept of vibration control using a mass damper dates
back to the year 1909, when Frahm invented a vibration
control device called a dynamic corresponding vibration
absorber [2]. Although active tuned mass damper systems
nowadays has received considerable attention from many
researchers [3-5], a passive control technique is still
considered due to its simplicity. Moreover, many passive
control devices such as TMD successfully installed in the real
application to reduce the lateral motion of high rise buildings
A
A. Shayeghi is with the Department of Technical Eng., Islamic Azad
University, Germi Branch, Germi, Iran (e-mail: alishayeghi3@yahoo.com).
H. Shayeghi Eimani Kalasar are with the Department of Technical Eng.,
University of Mohaghegh Ardabili, Ardabil, Iran (corresponding author to
provide phone: 98-551-2910; fax: 98-551-2904; e-mail: hshayeghi@
gmail.com).
28
International Journal of Intelligent Systems and Technologies 4:1 2009
robust in solving problems featuring non-linearity, nondifferentiability and high dimensionality [17-20]. PSO has
been motivated by the behavior of organisms, such as fish
schooling and bird flocking. Unlike the other heuristic
techniques, it has a flexible and well-balanced mechanism to
enhance the global and local exploration abilities. Also, it
suffices to specify the objective function and to place finite
bounds on the optimized parameters.
In this study, the problem of robust TMD design is
formulated as an optimization problem and the TMD
parameters are automatically tuned with optimization the
ITAE based fitness function by the PSO technique to get the
maximum reduction in displacement response of the building's
floors, such that the stability problem is guaranteed and the
time domain specifications concurrently secured. The
effectiveness of the proposed method is tested on an 11-story
realistic building, located in the city of Rasht, Iran, under
different earthquakes loads through the time domain
simulation and some performance indices. The result
evaluation shows that the proposed method achieves good
robust performance aspect response reduction.
II.
TMD
T=n
(4)
(3)
B=1
Fig .1. Example realistic building model and TMD mounted on its
top floor.
⎡( k 1 + k 2 ) − k 2
⎢− k (k + k )
2
3
⎢ 2
⎢
. . .
⎢
. .
⎢
⎢
K=
.
⎢
⎢
⎢
⎢
⎢
⎢
⎣
STRUCTURAL MODEL
Consider an N-story shear structure with mass damper
installed at top floor as shown in Fig. 1. The equations of
motion of the structural system under earthquake load can be
written as:
M x + Cx + Kx = e xg
(1)
Where M, C and K are mass, damping, and stiffness
matrixes, respectively; e and xg are the matrix induced ground
acceleration, and ground acceleration, respectively. If the x in
Eq. (1) is taken as the relative displacement with respect to the
ground, the mass matrix for a tall building structure, with the
assumption of masses lumped at floor levels, is a diagonal
matrix in which the mass of each story is sorted on its
diagonal, as given in the following:
M = diag ( m1 , m 2 ,..., m N , m d )
. .
− kN
X = [ x1 , x 2 ,..., x N , x d ]
(2)
T
(5)
(6)
(7)
Where, xi is displacement of ith floor relative to ground
(i=1,2,…,N); xd is displacement of damper relative to ground
and xg is ground displacement due to earthquake.
The equations of motion can then be converted to statespace realization as follows:
(8)
z = Az + Eug
(3)
2
ωi + ω j
.
e = [− m1 ,− m2 ,...,− m N ,− md ]T
The structural damping matrix [C] is assumed to be
proportional to the mass and stiffness matrices as [21]:
b0 = ξ j
− k3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(k N + k d ) − k d ⎥
⎥
− kd
⎥
kd ⎥
⎥
⎦
Where, ki is stiffness of ith story (i=1,2,…,N); kd is
stiffness of damper.
The e and x vectors are as follows:
Where, mi is mass of ith floor (i=1,2,…,N) in building and md is
mass of damper.
C = a 0 M + b0 K
2ω i ω j
a0 = ξ i
ωi + ω j
h
(2)
(4)
⎡ 0 ( N +1)
A=⎢
−1
⎢⎣− M K
Where,
Where, a0 and b0 are the proportional coefficients; ωi and
ωj are the structural modal frequencies of modes i and j,
respectively; and ξi and ξj are the structural damping ratios for
modes i and j.
The structural stiffness matrix [K] is developed based on
the individual stiffness, ki, of each floor, and is given in Eq.
(5).
I (N +1)
− M −1C
⎡0 ( N +1)×1 ⎤
⎤
⎥ ; E = ⎢ −1 ⎥
⎢⎣ M e ⎥⎦
⎥⎦
Note that by transforming the equations of motion Eq. (1)
to state-space equation, we have transformed the second-order
differential equation to the first-order one. Note also that the
size of the matrixes in the state-space equation is 2 times
larger than that of the ordinary equation of motion.
The tuned mass damper is a classical engineering control
device consisting of a mass, a spring and a viscous damper
attached to a vibrating main system in order to attenuate
undesirable vibration. Usually the natural frequency of a TMD
29
International Journal of Intelligent Systems and Technologies 4:1 2009
is tuned to a frequency close to one of the natural frequencies
of the main system. Thus, there are three main parameters in a
TMD system: TMD mass, TMD stiffness coefficient and
TMD dumping ratio. Consequently, the objective is to find the
optimum value of these parameters that involve in state
matrix, A in Eq. (8). Usually, this problem is converted to a
problem of single-degree-of-freedom (SDOF) structure where
the parameter of the structure is at a specified mode (usually
the first mode) to be chosen. In view of seeking a more
realistic model, this paper is used the multi-degree-of-freedom
(MDOF) model as a structural model without specifying to
special mode. By considering the structure as an MDOF
structure the optimization process is coming more difficult to
solve. Moreover, in case of an inherent damping presence in
the structure, the closed-form solution may not be available.
Thus, only numerical solution could be possible to solve the
problem of MDOF structures with inherent damping. For this
reason, the PSO technique which is a useful tool for
engineering optimization is being used to find optimal
parameter of TMD for reduction of the structural vibrations.
GA, and does not suffer from some of GAs difficulties. It has
also been found to be robust in solving problem featuring nonlinearity, non-differentiability and high-dimensionality. The
PSO is the search method to improve the speed of
convergence and find the global optimum value of fitness
function.
PSO starts with a population of random solutions
‘‘particles’’ in a D-dimension space. The ith particle is
represented by Xi = (xi1,xi2, . . . ,xiD). Each particle keeps track
of its coordinates in hyperspace, which are associated with the
fittest solution it has achieved so far. The value of the fitness
for particle i (pbest) is also stored as Pi = (pi1, pi2, . . . ,piD).
The global version of the PSO keeps track of the overall best
value (gbest), and its location, obtained thus far by any particle
in the population. PSO consists of, at each step, changing the
velocity of each particle toward its pbest and gbest according
to Eq. (9). The velocity of particle i is represented as Vi= (vi1,
vi2. . . viD). Acceleration is weighted by a random term, with
separate random numbers being generated for acceleration
toward pbest and gbest. The position of the ith particle is then
updated according to Eq. (10) [18].
vid = w×vid +c1 ×rand( )×(Pid − xid ) +c2 ×rand( )×(Pgd − xid ) (9)
III. PSO TECHNIQUE AND OPTIMIZATION OF TMD
xid = xid + cvid
PARAMETERS
Particle swarm optimization algorithm, which is tailored
for optimizing difficult numerical functions and based on
metaphor of human social interaction, is capable of mimicking
the ability of human societies to process knowledge [18]. It
has roots in two main component methodologies: artificial life
(such as bird flocking, fish schooling and swarming); and,
evolutionary computation. Its key concept is that potential
solutions are flown through hyperspace and are accelerated
towards better or more optimum solutions. Its paradigm can be
implemented in simple form of computer codes and is
computationally inexpensive in terms of both memory
requirements and speed. It lies somewhere in between
evolutionary programming and the genetic algorithms. As in
evolutionary computation paradigms, the concept of fitness is
employed and candidate solutions to the problem are termed
particles or sometimes individuals, each of which adjusts its
flying based on the flying experiences of both itself and its
companion. It keeps track of its coordinates in hyperspace
which are associated with its previous best fitness solution,
and also of its counterpart corresponding to the overall best
value acquired thus far by any other particle in the population.
Vectors are taken as presentation of particles since most
optimization problems are convenient for such variable
presentations. In fact, the fundamental principles of swarm
intelligence are adaptability, diverse response, proximity,
quality, and stability. It is adaptive corresponding to the
change of the best group value. The allocation of responses
between the individual and group values ensures a diversity of
response. The higher dimensional space calculations of the
PSO concept are performed over a series of time steps. The
population is responding to the quality factors of the previous
best individual values and the previous best group values. The
principle of stability is adhered to since the population
changes its state if and only if the best group value changes
[18, 19]. As it is reported in [17], this optimization technique
can be used to solve many of the same kinds of problems as
(10)
Where, Pid and Pgd are pbest and gbest. Several
modifications have been proposed in the literature to improve
the PSO algorithm speed and convergence toward the global
minimum. One modification is to introduce a local-oriented
paradigm (lbest) with different neighborhoods. It is concluded
that gbest version performs best in terms of median number of
iterations to converge. However, pbest version with
neighborhoods of two is most resistant to local minima. The
PSO algorithm is further improved via using a time decreasing
inertia weight, which leads to a reduction in the number of
iterations [17]. Figure 2 shows the flowchart of the proposed
PSO algorithm.
This new approach features many advantages; it is simple,
fast and easy to be coded. Also, its memory storage
requirement is minimal. Moreover, this approach is
advantageous over evolutionary in many ways. First, PSO has
memory. That is, every particle remembers its best solution
(local best) as well as the group best solution (global best).
Another advantage of PSO is that the initial population of the
PSO is maintained, and so there is no need for applying
operators to the population, a process that is time and
memory-storage-consuming. In addition, PSO is based on
‘‘constructive cooperation’’ between particles.
A. TMD design using PSO
Tuned mass damper consisting of a mass, a damping, and a
spring is an effective and reliable structural vibration control
device commonly attached to the vibrating primary system for
suppressing the undesirable vibrations induced by winds and
earthquake loads. In the proposed method, we must find the
TMD parameters optimally to get maximum reduction in
building responses due to the different earthquake excitations.
Thus, in optimization process the simulated earthquake using
Kanai-Tajimi filter (Eq. (11)) is shown in Fig. 3 by MATLAB
30
International Journal of Intelligent Systems and Technologies 4:1 2009
software [23] is created and used as base earthquake
excitation.
Start
Select parameters of PSO:
N, C1, C2, C and w
Generate the randomly positions
and velocities of particles
Evaluate the fitness of each particle
Update pbest and gbest
Initialize, pbest with a copy of the
position for particle, determine gbest
Satisfying
stopping
criterion
No
Update velocities and positions
according to Eqs. (9) and (10)
Yes
Optimal value of the TMD parameters
End
Fig. 2. Flowchart of the proposed PSO technique.
H (s) = S 0
S 0 = 0.03
(ω g + 4ω g ξ g s 2 )
4
lower the value of this objective function is, the better robustly
the system performance in terms of time domain
characteristics. The design problem can be formulated as the
following constrained optimization problem, where, the
constraints are the TMD parameters bounds.
2
( s 2 − ω g ) + 4ω g ξ g s 2
2 2
ξg g2
2
2
(11)
πω g (4ξ g 2 + 1)
Minimize J subject to
momin ≤ mo ≤ m omax
In this work, ωg and ξg is considered as 37.3 and 0.3, which
are for usual soils, respectively.
β dmin ≤ β d ≤ β dmax
(13)
ζ dmin ≤ ζ d ≤ ζ dmax
N(s)
To improve the overall building response in a robust way
and optimization synthesis, PSO is used to solve the above
optimization problem that search for optimal or near optimal
set of TMD parameters.
Earthquake
D(s)
Band-Limited
White Noise
Kanaji-Tajimi Filter
Base Excitation
IV. CASE STUDY
Fig.3. Base excitation used for tuning of the TMD Parameters
In order to investigate the performance of the proposed
control strategy in reducing the structural responses under
earthquake loads, an 11-story shear building, located in city of
Rasht in the north of Iran, is chosen as an a test system. The
structure represents a tall building. The structural properties of
this building are listed in Table 1 [15].
The TMD frequency ratio to the first model frequency of
the building has been choused βd. The mass of the TMD
system was chosen to be mo percent of total mass of the
building and its damping ratio (ξd) was considered to be
percent of the critical value. These three parameters (mo, βd
and ξd) of TMD are optimized by evaluating the cost function
as given in Eq. (13). In order to acquire better performance,
the number of particle, particle size, number of iteration, c1, c2
and c is chosen as 25, 3, 250, 1.47, 2.2 and 1, respectively.
Also, the inertia weight, ω, is linearly decreasing from 0.9 to
0.4. Results of TMD parameter set values using the proposed
PSO method are given in Table 2. Also, Fig. 4 shows the
minimum fitness value evaluation process.
To acquire an optimal combination, this paper employs
PSO to find the global optimum value of fitness function. It
should be noted that choice of the properly objective function
is very important in synthesis procedure for achieving the
desired level of system robust performance. Because different
objective functions promote different PSO behaviors, which
generator fitness value providing a performance measure of
problem considered [17]. For our optimization problem, the
following objective function based on the system performance
index of the ITAE is used [24].
J = ∑ ITAE i
N
i =1
ITAE i =
∫ t x (t ) dt
t = tsim
(12)
i
0
Where, xi is relative displacement of ith story building
response under base earthquake excitation and is obtained
using the solution of Eq. (8). It is worth mentioning that the
31
International Journal of Intelligent Systems and Technologies 4:1 2009
TABLE I
displacement of floors using the proposed method and
uncontrolled systems. Fig. 7 shows the ITAE performance
index. It can be seen that from the figures and Table, the
response reduction ratio (ratio of the controlled to uncontrolled
response) for maximum displacement of top floor of the 11story example building is about 40% for the PSOTMD. Thus,
PSO-based designed TMD is effective method.
TEST BUILDING STRUCTURAL DATA
5
8
Stories
Mass(kg)×10
Stiffness(N/m) ×10
2.15
4.68
2
2.01
4.76
3
2.01
4.68
4
2.00
4.5
5
2.01
4.5
6
2.01
4.5
7
2.01
4.5
8
2.03
4.37
9
2.03
4.37
10
2.03
4.37
11
1.76
3.12
0.2
TABLE II
OPTIMAL VALUE OF TMD PARAMETERS.
TMD Parameters
PSO
m0
0.06
βd
0.3
ξd
0.13
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0
10
20
30
Time (sec)
40
50
60
Fig.5. Displacement of top floor under El Centro 1940 earthquake
TABLE III
COMPARISON OF THE EFFECTIVENESS CONTROLLER SYSTEM FOR THE EL
CENTER 1940 EARTHQUAKE
41.3
Building
41.25
floor
Fitness Value
Uncon
TMD-PSO
Displacement of top floor (m)
1
Maximum
Controlled to uncontrolled response
uncontrolled ratio percent (Reduction Ratio) using
response (m)
PSOTMD
41.2
41.15
41.1
41.05
41
0
50
100
Iteration
150
200
250
Fig. 4. Minimum fitness convergence by PSO
1
0.0224
44.0260
2
0.0432
43.8906
3
0.0627
42.8707
4
0.0811
41.7516
5
0.0973
40.5694
6
0.1112
39.3886
7
0.1253
39.4371
8
0.1381
39.5509
9
0.1486
39.6833
10
0.1558
39.6494
11
0.1608
39.4321
The effectiveness of the controller system and ITAE in
reducing the response of the example building due to other
earthquakes is also shown for comparison in Figs. 8-13 and
Tables 4-5. Almost the same behavior as for the El Centro
earthquake can be observed for these earthquakes, too.
Moreover, it is seen from Table 4-5 that the Kobe 1995 and
Tabas 1999 earthquakes cause the maximum displacement at
the top floor of the example building, which is a very high
value in comparison with the other earthquakes. This is
expected due to fact that the PGA of the two earthquakes is
very high and almost remains constant for a long period of
time in comparison with the El Centro earthquake.
A. Simulation results
To investigate the effectiveness of the control system for
different disturbances, three different seismic motions are used
in the numerical simulations. These ground acceleration
records are: El Centro 1940, Kobe 1995 and Tabas 1999
earthquakes. The absolute peak ground accelerations (PGAs)
of these earthquake records are 0.3417, 0.8178 and 0.9512 g,
respectively.
The result of controlled displacement response of the
example building top story due to the El Centro NS
earthquake calculated by the PSO-based TMD designed
systems are compared with the corresponding uncontrolled
ones in Fig. 5 and Table 3. Also, Fig. 6 depicted maximum
32
International Journal of Intelligent Systems and Technologies 4:1 2009
11
11
10
10
9
8
Uncon
7
Floors
Floors
9
TMD-PSO
8
7
6
6
5
4
5
3
4
2
Uncon
3
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
2
Maximum Displacement (m)
1 0
Fig. 6. Maximum displacement of floors under El Centro 1940
earthquake.
Uncon
TMD-PSO
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
Maximum Displacement (m)
Fig. 9. Maximum displacement of floors under Kobe 1995
earthquake.
PSOTMD
14
TABLE IV
COMPARISON OF THE EFFECTIVENESS OF THE CONTROLLER SYSTEM FOR THE
KOBE 1995 EARTHQUAKE
Controlled to uncontrolled
Maximum
response ratio percent
Building
uncontrolled
(Reduction Ratio))
floor
response (m)
PSOTMD
12
ITAE
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
Stories
1
0.0596
27.4487
2
0.1184
27.2065
3
0.1771
26.9915
4
0.2361
26.8041
5
0.2919
26.6392
0.3
6
0.3433
26.5216
0.2
7
0.3890
26.4887
0.1
8
0.4289
26.5607
0
9
0.4602
26.5285
10
0.4814
26.4313
11
0.4956
26.3741
Fig. 7. ITAE performance index value for all floors under El Centro
1940 earthquake.
Displacement of top floor (m)
0.5
Uncon
TMD-PSO
0.4
-0.1
-0.2
-0.3
-0.4
-0.5
0
5
10
15
20
25
30
Time (sec)
35
40
45
50
Uncon
PSOTMD
25
Fig.8. Displacement of top floor under Kobe 1995 earthquake
20
IT A E
To demonstrate the robustness effectiveness of the
proposed method, the average reduction ratios (controlled to
uncontrolled displacement and acceleration ratio) for all three
earthquakes on all 11 stories of the example building are
shown in Table 6 and 7, respectively. According to these
Tables, it can be seen that in general the PSOTMD system are
capable of reducing the maximum displacement and
acceleration of the building in each story to about 25.5 and
24.5 of the uncontrolled response, respectively. Thus, based
on average the proposed method is very effective.
15
10
5
0
1
2
3
4
5
6
7
8
9
10
11
Stories
Fig. 10. ITAE performance index value for all floors under Kobe
1995 earthquake.
33
International Journal of Intelligent Systems and Technologies 4:1 2009
Uncon
0.2
PSOTMD
Uncon
TMD-PSO
25
Displacement of top floor (m)
0.15
20
ITAE
0.1
0.05
0
15
10
5
-0.05
0
1
2
3
4
-0.1
0
5
10
15
20
25
30
35
40
Fig. 11. Displacement of top floor under Tabas 1999 earthquake
TABLE V
COMPARISON OF THE EFFECTIVENESS OF THE DIFFERENT CONTROLLER
SYSTEMS FOR THE TABAS EARTHQUAKE
Controlled to
uncontrolled
Maximum
Building
response ratio percent
uncontrolled
floor
(Reduction Ratio)
response (m)
PSOTMD
Kobe earthquake
9
10
11
26.7268
Tabas earthquake
8.7313
Total average
25.4654
1
0.0596
2.127
2
0.1184
3.5263
3
0.1771
5.1041
4
0.2361
6.7949
excitation
5
0.2919
8.5140
Elcentro earthquake
30.2467
6
0.3433
10.1406
Kobe earthquake
32.2242
7
0.3890
11.4235
Tabas earthquake
10.9083
Total average
24.4997
TABLE XI
THE AVERAGE REDUCTION (FOR 11 FLOORS) IN MAXIMUM ACCELERATION
RESPONSE OF THE EXAMPLE BUILDING
8
0.4289
12.0975
9
0.4602
12.2247
10
0.4814
12.1353
11
0.4956
11.9566
Earthquake
A new PSO based TMD controller is proposed for the
reduction of the tall building responses subjected to
earthquake excitations in this paper. This study emphasizes to
design and optimize the parameters of the TMD control
scheme for achieving the best results in the reduction of the
building response under earthquake excitations. The problem
of robustly TMD controller design is formatted as an
optimization problem based on the ITAE performance index
to be solved using PSO technique. The PSO algorithm
proposed in this paper is easy to implement without additional
computational complexity. Thereby experiments this
algorithm gives quite promising results. The ability to jump
out the local optima, the convergence precision and speed are
remarkably enhanced and thus the high precision and
efficiency are achieved. Also, it has simple structure and easy
to implement and it does not depend on the nature of the
function it minimizes and is insensitive to the initial searching
point, there by ensuring quality solution for different trial
runs.
For the numerical study, an 11-story realistic building is
chosen and the problem is solved in state space, as well as a
PSO based TMD controller system are also designed to
9
8
7
6
5
4
Uncon
3
TMD-PSO
2
0.06
0.08
0.1
0.12
0.14
0.16
0.18
reduction percent
V. CONCLUSIONS
10
0.04
Average response
PSOTMD
11
Floors
8
TABLE X
THE AVERAGE REDUCTION (FOR 11 FLOORS) IN MAXIMUM DISPLACEMENT
RESPONSE OF THE EXAMPLE BUILDING`
Average response
Earthquake
reduction percent
excitation
PSOTMD
Elcentro earthquake
40.9318
45
Time (sec)
0.02
7
Fig.13. ITAE performance index value for all floors under Tabas
earthquake.
-0.2
1
6
Stories
-0.15
-0.25
5
0.2
0.22
Maximum Displacement (m)
Fig.12. Maximum displacement of floors under Tabas earthquake.
34
International Journal of Intelligent Systems and Technologies 4:1 2009
[11]
control the building response. The system performance
characteristics in terms of ‘ITAE’ and ‘the average reduction
ratios’ reveal that this control strategy is a promising control
scheme for the getting the maximum reduction in
displacement and acceleration response of the building’s
stories .It is effective and ensures robust performance for
different earthquakes excitations.
[12]
[13]
[14]
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