STRUCTURAL PARAMETERS IDENTIFICATIONBASED ON DIFFERENTIAL EVOLUTION AND
PARTICLE SWARM OPTIMIZATION
Wang Yanwei ∗MEE08167

Supervisor : Xue Songtao ∗∗Tang Hesheng ∗∗ ∗ABSTRACT
Civil structures always suffer many kinds of damage caused by long use, strong wind, earthquakes,
and so on, therefore it is very important to detect the damage of structures in advance. The basic theoryof damage is that the changes of the dynamic parameters, such as stiffnesses, of structures are causedby damage. So the structural parameters identification becomes a key method of damage detection.
Based on the research of previous researchers, this thesis uses Differential Evolution (DE) and Particle
Swarm Optimization (PSO) to identify the structural parameters. The two methods belong to
optimization method and minimize the mean square error function of the accelerations of numericalmodel and the identified system to identify parameters. In order to confirm the efficiency and validityof them, a simulation with several cases is performed. The results show that both DE and PSO canidentify structural parameters very well in simulation.
Keywords: Structural parameters identification, Differential evolution, Particle swarm optimization.

1. INTRODUCTION
Structural parameters identification is a method that obtains the dynamics parameters of a structure
based on physical measurements and mathematical analysis. It is an active field of research in the civilengineering, driven by the need of structure health monitoring (SHM). Since P. Cawley and R.D.
Adams (1979) used natural frequency to detect damage, the technologies of structural parameters
identification based on vibration measurements have a tremendous development. These technologies
are currently becoming increasingly common. Hoshiya M. and Saito E. (1984) used the Extended
Kalman Filter to identify the structural stiffnesses. Sato T. and Qi K. (1988) used H∞ filter to
identify the structural stiffnesses. Particle filtering was also successful to identify stiffnesses by Li S etal (2004) and Tang H et al (2005). Differential Evolution (DE) and Particle Swarm Optimization(PSO) appeared in 1990s, as novel optimization algorithms, they have greater advantage than
traditional algorithms, and which developed rapidly and were successfully applied in many fields. H.
Tang (2007, 2008) used the both algorithms to identify masses, stiffnesses and damping ratios andconcluded that the two methods converged fast and had high accuracy. To confirm the performancesof the two methods, this thesis attempts to use DE and PSO to identify the stiffnesses and damping
ratios of Multi-Degree-Of-Freedom (MDOF) system with several cases.
2. BASIC THEORY OF IDENTIFICATION∗

Institute of Engineering Mechanics (IEM), China Earthquake Administration (CEA).

∗∗

Department of Architecture, School of Science and Engineering, Kinki University, Japan.
Research Institute of Structural Engineering & Disaster Reduction, Tongji University, China∗∗∗

1Because most of the buildings belong to MDOF system, MDOF system should be used to discuss. The
dynamic function of MDOF is equation (1).

Mx&& + Cx& + Kx = − Mu&&g(1)
Where M is the mass matrix, K is the stiffness matrix, C is the damping matrix, x is the displacement
vector, and u&&g is the acceleration of earthquake.
It can be said that the responses of MDOF system can be determined by M, K and thedamping ratio ζ , so the identification work of K and ζ becomes to find a MDOF system whose
outputs are the same as the identified system’s under the same M and inputs. Therefore, we can define
the mean square error function as equation (2).

F (θ ) =Whereθ = [ k1 , k2 ,K , kd , ζ 1 , ζ 2 ]1 N∑ &&x ( k ) − &&xˆ ( k )N k =1

2(d is degree of freedom) is the solution vector,

&&
x(k )

(2)
is theacceleration of numerical model calculated by Newmark integration method(β=1/4), and&&x̂ ( k )F (θ )

measured from the unknown system. Whenis the minimum, the values of θ are thestiffnesses and damping ratios of the identified system.
Then the parameters identification becomes the problem of finding the minimum of
so it is a problem of optimization. θ is the solution vector andDE and PSO both can do the search work directly.

F (θ )isF (θ ),

is used as the fitness function,

3. SIMULATION
3.1 The numerical modeA 5 stories mass lumped model (5 DOF system) is chosen as the numerical mode, as shown in the
Figure 1. The masses and stiffnesses of the 5 DOF system are list in Table 1, the first mode dampingratioζ 1 is 0.02 and the second mode ratio ζ 2 is 0.04.

Table 1. Masses and stiffnesses of the 5 DOF system
Story (i)
1
2
3
4
5

ki (N/m)
60
55
50
45
40

mi (kg)
30
25
20
15
10

Figure 1. The 5 DOF system.

An earthquake record, El-centro, is the input of the 5 DOF system, whose sampling rate is
50Hz and duration is about 53 seconds. The Newmark integration method(β=1/4) is used to calculate
the system’s outputs which consist of the acceleration of each story. The interval of the response
calculation is 0.02 second.
In order to check the abilities of identification of DE and PSO, three cases are used in thesimulation:2

Case 1, the outputs don’t have noise;
Case 2, the outputs polluted by different levels of noise;
Case 3, the stiffness of each story is changed at 11th second during the vibration.
In case 2, suppose that the noise is White Gaussian Noise (WGN) and different levels ofnoise are added into the output. The level of noise is defined as equation (3).

L=

σw
× 100%
σr

σ

σ

(3)

Where L is the level of noise, w is the standard deviation of WGN and r is the standard
deviation of acceleration record. There are five levels: 5%, 10%, 15%, 20%, 25%, and 30%.
Table 2. All Stiffness change at 11th second.
In case 3, the masses and damping
ratios of the 5 DOF system keep
constant during vibration, but all
the stiffnesses are changed at 11 th
second. Because the reducing ofstiffnesses is caused by damage,
we can reduce the stiffnesses tosimulate damage. At at 11th second, the damage of each story happens, as shown in the Table 2.
Now suppose that the stiffnesses and damping ratios of the 5 DOF system are unknown inthree cases, the identification work of these parameters is just based on the masses, the input andoutputs, the error of each parameter identified by DE and PSO is defined as equation (4).
Stroy k (N/m) before 11th sec
1st
60
2nd
55
3rd
50
4th
45
5th
40

error =k (N/m)after 11th sec
45
48
46
40
35

( real value ) − ( identified value ) ×100%
real value(4)

3.2 Identification via DE
To use DE to identify the above 5 DOF model, some initial work must be done firstly. The settings ofthe control parameters are: the size of population NP=50, the maximum number of generations G=500,
the weighting factor F=0.8, the crossover probability Cr=0.9, and the search rangebL=0.5*[ k1 , k2 , k3 , k4 , k5 , ζ 1 , ζ 2 ] and bU =2* [ k1 , k2 , k3 , k4 , k5 , ζ 1 , ζ 2 ] .

In case 1, Figure 2 shows the procedure of parameter identification of DE, from which we
can see that the solution vector converges very fast and the fitness value almost equals 0 after its
stabilization. It means that DE identified the parameters without any error.

(a) The stiffnesses(b) The damping ratios(c) The fitness function
Figure 2. The procedure of DE in case 1.

In case 2, we just give the errors (eq. (4)) of the results in Table 3. The stiffnesses can becorrectly identified by DE even the outputs are polluted by 30% noise, but damping ratios have large
errors as the increasing of noise level.

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Table 3. The error of each parameter identified by DE under different noise levels
DE
Noise
10%
15%
20%
25%
30%

k1
error(%)
0.34051.134333
1.412667
3.258833
3.395

k2
error(%)
0.404
1.150364
1.510364
2.706909
0.663818k3
error(%)
0.001
0.002496
0.023166
0.029128
0.027372k4
error(%)
1.132
0.488889
0.144444
0.669556
0.594444k5error(%)
0.4965
0.472
0.32
2.755
1.45475

ζ1
error(%)
10
20
5.5
48
39.5

ζ2
error(%)
2.5
6.75
0.5
2.5
1

In case 3, because the stiffnesses changed during the vibration and we didn’t know when
the change happened, the identification work of DE was repeated every 5 seconds. During theprocedure of identification, the accelerations of every potential solution vector were calculated by the
Newmark integration method(β=1/4), so DE needed the displacements and velocities of each 5
seconds to initialize the Newmark method. From Table 4, we can see that DE can identify correctly thestiffnesses even the stiffnesses change or damage happens during the vibration, but the damping ratios
can’t be identified correctly.
Table 4. The stiffnesses and damping ratios idenfified by DE in case 3.
Time(sec)
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-53

k1
59.993959.673747.64
44.5535
46.8932
44.581344.820544.9988
44.846245.1769k2
54.92254.7686
52.9184
48.038647.147647.443947.699948.14248.11848.3868k3
50.18750.031747.8865
45.8191
45.5701
45.880746.344646.059446.0672
45.8995

k4
45.63744.961840.756240.151339.599339.7995
40.02240.119339.9862
39.8307k5
40.4877
39.969436.66335.0253
34.932934.880435.0047
35.0967
34.972934.9893ζ10.02150.02660.02
0.03120.01760.02570.02050.02250.02190.0176

ζ2
0.0411
0.04190.04360.04250.04250.04150.04
0.04020.04020.03983.3 Identification via PSO
The identification procedure of PSO is similar to that of DE, and some initial work must be done
firstly. The settings of the control parameters are: the size of swarm N=50, the maximum number of
iterations G=500, the inertia weight ω=0.8, acceleration coefficients c1=c2=2, the search range (xmax( 2 * [ k , k , k , k , k , ζ , ζ ] , 0.5* [ k , k , k , k , k , ζ , ζ ])

1
2
3
4
5
1
2
1
2
3
4
5
1
2
and the max velocity is-xmin) is
vmax=0.4*(xmax -xmin).
In case 1, the results of PSO are almost the same as DE’s, from Figure 3 shows theprocedure of parameter identification of PSO, from which we can see that the solution vector
converges very fast and the fitness value almost equals 0 after its stabilization. It means that PSO
identified the parameters without any error.

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(a) The stiffnesses(b) The damping ratios
Figure 3. The procedure of PSO in case 1.

(c) The fitness function

In case 2, the errors (eq. (4)) of each identified parameter in different noise level are listed
in Table 5, from which we find that the damping ratios can not be identified correctly as noise level
increasing, but PSO has a good ability to identify stiffnesses even the output polluted by 30% noise.
Table 5. The error of each parameter identified by PSO under different noise levels

Noise
10%
15%
20%
25%
30%

k1
error(%)
0.757667
1.09552.609
1.9975
1.948667

k2
error(%)
0.850909
1.264364
2.184182
1.049455
2.452

PSO
k3
k4
error(%) error(%)
0.022612
0.712
0.034262 1.038222
0.02341 0.530444
0.024906 4.591333
0.070764 1.883333

k5
error(%)
0.01875
0.00952.19475
1.77
0.0745ζ1error(%)
4.5
7.5
38
50
16

ζ2
error(%)
0.5
1
2
5.25
1.75

In case 3, PSO gave a result of every parameter in each 5 seconds, and the displacements
and velocities of each 5 seconds were both used as initial values of Newmark integration method(β=1/4). From Table 6, we can see that PSO can identify correctly the stiffnesses when the stiffnesseschange or damage happens, but the damping ratios can’t be identified correctly.
Table 6. The stiffnesses and damping ratios idenfified by PSO in case 3.
Time(sec)
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-53

k1
59.993959.673747.64
44.5535
46.8932
44.581344.820544.9988
44.846245.1769k2
54.92254.7686
52.9184
48.038647.147647.443947.699948.14248.11848.3868k3
50.18750.031747.8865
45.8191
45.5701
45.880746.344646.059446.0672
45.8995

k4
45.63744.961840.756240.151339.599339.7995
40.02240.119339.9862
39.83075

k5
40.4877
39.969436.66335.0253
34.932934.880435.0047
35.0967
34.972934.9893ζ10.02150.02660.02
0.03120.01760.02570.02050.02250.02190.0176

ζ2
0.0411
0.04190.04360.04250.04250.04150.04
0.04020.04020.0398

4. CONCLUSIONSIn this thesis, the basic theories of structural parameters identification of MDOF system wereintroduced firstly, and then DE and PSO were used to identify the siffnesses and damping ratios of a 5
DOF system. From the simulant results, some conclusions are summarized in the followings:
1)
DE and PSO almost have the same performance of the structural parameters identification, suchas accuracy and convergence speed.
2)
Both DE and PSO can identify the stiffnesses and damping ratios correctly when the outputs
have no noise and the parameters are constant at all times;
3)
When the outputs are polluted by noise, DE and PSO both can identify the stiffnesses with highaccuracy, but both of them can not identify damping ratios. It means that DE and PSO have theanti-noise abilities for stiffnesses;
4)
When the stiffeness changes during vibration, the stiffnesses can be identified by DE and PSO
with very small errors, but the damping ratios can not be identified by them. It means that DE
and PSO have the abilities of damage detection;
5. RECOMMENDATION
We believe that this thesis contains useful contributions to the structural parameters identification.
Although DE and PSO can identify the parameters of MDOF system in simulation, it is difficult to say
they can be used to identify a real build successfully. Therefore the future research is certainly needed
to use real buildings to confirm the performances of the two methods.
AKNOWLEDGEMENT

I would like to deeply appreciate my adviser Prof. Toshihide Kashima for his guidance, support andencouragement.

REFERENCES
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