RESEARCHES ON SUPPORT VECTOR MACHINE BASED SEMI-ACTIVE CONTROL OF STRUCTURES

QING LIU CHUNXIANG LI

Department of Civil Engineering, Shanghai University, No.149 Yanchang Rd Shanghai, 200072, P. R. China

Abstract: The present paper presents the support vector machine (SVM) based semi-active control algorithm with a view to general dampers for multistory structures under earthquakes. Firstly, the LQR controller for the numerical model of a multistory structure formulated using the dynamic dense method is obtained by use of the classic LQR control theory. Then, a SVM model is designed and trained to emulate the performance of the LQR controller. Likewise this SVM model comprises the observers and controllers of the control system. Finally, in accordance with the features of general semi-active dampers, a SVM based semi-active control strategy is put forward. More specifically, an online auto-feedback semi-active control strategy is developed and then realized by resorting to SVM. In order to numerically verify the control effectiveness of the present control strategy, the time history analysis has been implemented to a structure with general dampers designed by the SVM based semi-active control algorithm. In numerical simulations, four seismic waves including El-Centro wave, Hachinohe wave, Kobe wave, and Shanghai artificial wave, whose peak ground accelerations (PGA) are all scaled to 0.1gal, are taken into consideration. Comparative results demonstrate that general semi-active dampers designed using the SVM based semi-active control algorithm is capable of providing higher level of response reduction.

Keywords: Support vector machine (SVM); Semi-active control; LQR; General semi-active dampers; Structures; Earthquakes

1. Introduction

Reducing structural seismic responses, without doubt, can remarkably enhance the buildings’ security. Technologies of controlling the structural vibrations have been widely accepted by most experts and engineers. Among various control devices, the dynamic absorbers or called the tuned mass dampers (TMD) are extensively utilized. But they need one huge mass block which is considered as the greatest shortcoming. While the effectiveness of energy dissipation devices (EDD) most commonly applied for seismic protection of structures, which depends on the spectral contents of earthquake ground motions. On the other hand, although active control is adaptable to real-time external excitations, it must create a large control force. Naturally, the semi-active and hybrid (any combinations of active and passive devices) controls with both low-power and low-cost seem to be the most promising schemes for seismic protection of structures [1-3]. Hence, in this paper, a support vector machine (SVM) based semi-active control algorithm is to be developed.

SVM is a promising statistical learning theory developed by Vapnik in 1995 [4]. SVM provides some special advantages in the fields of small sample issues, nonlinear and high dimensional pattern recognition. Small sample learning and global optimization are both important features. Recently, several researchers have utilized SVM to carry out the structural system identification [5, 6], nonlinear structural response prediction [7], and damage diagnose [8]. However, it is rarely reported that the applications of SVM to

structural vibration control. SVM is introduced in this paper to the field of structural vibration control and consequently, put forward a new strategy of SVM based semi-active control algorithm with a view to general dampers for multistory structures under earthquakes.

2. Theory of SVM

SVM was developed to originally solve classification problems. With the introduction of Vapnik’sε -insensitive loss function, however, SVM has been extended to successfully deal with nonlinear regression estimation problem [4]. The regression model can then be defined as follows:

exfy += )( (1) where x and y represent the input and output, respectively; )(xf denotes the linear regression function defined in the high-dimensional feature space; and apparently, e refers to the independent random error.

Now given n input output sampling pairs { }( , ), 1,2,...,i iG x y i n= = , the SVM approximation

of the linear regression function )(xf can be written in the following general form.

( ) ( )f x x bω= Φ + (2) where )(xΦ is the high-dimensional feature space which is nonlinearly mapped from the input space x . The coefficients ω and b are estimated by minimizing.

2reg

1

1 1( ) ( , ( ))2

N

i ii

R C C L y f xN ε ω

=

= +∑ (3)

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ICCTD.2009.57

411

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ICCTD.2009.57

411

( )( )( , ( ))

( )0y f xy f x

L y f xy f xε

εεε

− >⎧ − −= ⎨ − ≤⎩

(4)

Equation (3) is referred to as the regularized risk function where the first term refers to the empirical error (risk), and the second term, on the other hand, denotes the regularization term. Equation (4) stands for the ε -insensitive loss function which provides the advantage of enabling one to utilize sparse data points (sampling pairs) to represent the decision function given by Equation (1). C represents the penalty factor. More specifically, it is the regularized constant and determines the trade-off between the empirical risk and the regularization term. Increasing the value of the penalty factor results in relative importance of the empirical risk with respect to the regularization term. ε is the maximum allowable error which is named the tube size and equivalent to the approximation accuracy placed on the training sampling pairs.

In order to obtain the estimations of ω and b , Equation (3) is transformed to the primal function by introducing the positive slack variables iξ and *

iξ as follows:

})(21)({

1

*2re ∑

=

++=N

iiiCCRMin ξξω

⎪⎩

⎪⎨

⎧

≥+≤−+Φ

+≤Φ−

0)(

)(.

(*)

*

ξξεω

ξεω

iii

iii

ybxbxy

St (5)

Then introduce Lagrange multipliers and exploit the optimal constraints to Equation (2), resulting in the following equation.

bxxKaaaaxfN

iiiiii +−=∑

=1

** ),()(),,( (6) where ia and *

ia are the Lagrange multipliers. They satisfy 0* =× ii aa ， 0≥ia ， 0* ≥ia , and are obtained with resorting to maximizing the dual function of Equation (5) in the following form.

∑∑==

+−⎩⎨⎧ −=

N

iii

N

iiiiii aaaayaaRM

1

*

1

** )()(),(ax ε

⎭⎬⎫

−−− ∑∑= =

N

i

N

jjijjii xxKaaaa

1 1

** ),())((21

St.

⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤≤≤

=−∑=

CaCa

aa

ii

N

iii

*

1

*

00

0)(

，

(7)

In accordance with the Karush–Kuhn–Tucker (KKT) conditions of quadratic programming, only a certain number of coefficients )( *

ii aa − in Equation (6) are to be assumed not to be equivalent to zero. The data points corresponding to them have the approximation errors equal to or larger than ε , referred to as the support vectors. These are the data points lying on or outside theε -bound of the decision function. According to Equation (6), it is apparent that the support vectors are only those elements of the data points that are used in determining the decision function while the coefficients )( *

ii aa − of other data points are all equal to zero. Generally speaking, the larger theε , the fewer the number of support vectors and thus the sparser the representation of the solution. However, a larger ε can also depreciate the approximation accuracy placed on the training points. In this sense, the ε is a trade-off between the sparseness of the representation and closeness to the data [9].

In Equation (7), ( , )i jK x x is defined as the Kernel function. The Kernel function can transform the problem into solving the linear regression problem in the high-dimensional feature space. The value of the kernel is equivalent to the inner product of two vectors i and j in the feature space ( )ixΦ and ( )jxΦ , that is,

( , ) ( ) ( )i j i jK x x x x= Φ ∗ Φ (8) The purpose of introducing the kernel function is

that one can deal with feature spaces with arbitrary dimensionality while without having to compute the map ( )xΦ explicitly. Any function satisfying Mercer’s condition may all be used as the kernel function [4].

The parameters in the kernel functions should be carefully chosen as it implicitly defines the structure of the high-dimensional feature space and thus controls the complexity of the final solution as well as the accuracy of the result.

3. Structure - semi-active control system

Now, consider a 3-storey shear-type frame structure with dampers, as shown in Figure 1. It is assumed that

kgm i5104 ×= , mNk i /106.1 8×= ( 1, 2, 3i = ).

Likewise the structural damping matrix may be obtained by using of the Rayleigh damping hypothesis. In each floor is one damper installed. Earthquake inputs in this study include the El-Centro, Hachinohe, Kobe, and Shanghai artificial waves.

Establish the motion equations of the structure-semi-active control system as follows:

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UBXMIKXXCXM sg +−=++ (9) Each floor installs one control device (damper).

Employ a state-space representation of the motion equations of the structure-semi-active control system subjected to a base acceleration gX . Given then is the following matrix equation.

gXDBUAZZ ++= (10) In order to implement optimal control, an appropriate cost function incorporating two components, namely both the states to be controlled and control effort, has to be constructed with the weightings on the two parts.

Based on the linear quadratic regulator (LQR) control theory, the optimal control forces of control devices can be calculated by

PZBRU T1−−= (11) For an infinite terminal time, the solution of P to this problem can be obtained through solving the algebraic Riccati equation.

In the present paper, general semi-active dampers for multistory structures under earthquakes are taken into consideration. Based on the optimal LQR control theory, the control algorithm can be can be represented as follows [10]:

000

)sgn()sgn(

)sgn()( max

max

min

max

≥<<><

⎪⎩

⎪⎨

⎧=

ii

idiii

idiii

iid

ii

iid

i

xufuandxufuandxu

xfxu

xftU (12)

in which maxidC and minidC denote the maximum and minimum damping coefficients of the control device in the ith floor; iu represents the obtained optimal control force of the ith floor on the instant time t , by resorting to the optimal LQR control algorithm.

4. Structure-SVM system model

Figure 1 shows the structure-SVM semi-active control system model and implementation flow chart. It is understood from Figure 1 that the support vector machine (SVM) based semi-active control algorithm with a view to general dampers for multistory structures under earthquakes generally includes the following three steps. Firstly, the structure-SVM system model is to conduct the off-line training on the LQR controller, until reaching certain control effectiveness. Then, the trained results are to be integrated with the real-time and then, the structural vibration control can be accomplished. Just as shown in Figure 1, each storey installs one observer (i.e. sensor) and one damper, thus

the system data at time t-1 can be gained from these observers. On the training, the required control forces of the dampers at time t could be predicted. But, it is worth pointing out that the parameters in the kernel function, regression allowable maximum error, as well as penalty factor will determine the control effectiveness of the present control strategy. Likewise, for different earthquake waves, these parameters will vary, thus resulting in different levels of response reduction.

Figure 1 Structure-SVM semi-active control system model and

implementation flow chart

5. Numerical studies

In order to corroborate the control effectiveness of the present algorithm, the two control strategies above-mentioned are programmed by use of the MATLAB language. Then, the comparative numerical study is carried out between the structure-damper system and the structure-SVM system model. In numerical study, four seismic waves, namely El-Centro wave, Hachinohe wave, Kobe wave, and Shanghai artificial wave, are included into consideration, whose peak ground accelerations (PGA) are all scaled to 0.1gal. The seismic responses of structural top storey with the structure-damper system, structure-SVM system, and no control device are shown, respectively, in Figures 2. It is seen from Figure 2 that under El-Centro wave and Shanghai artificial wave, the structure-SVM system model has perfectly learned the control effectiveness of

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the structure-damper system. Additionally, it is detected that at several time points the structural seismic responses are comparatively larger, which implies that the performance of the structure-SVM system model is superior to the structure-damper system. Likewise the change modes for the seismic responses of the structure

with the structure-SVM system model have little difference to those with structure-damper system at most time points. But, the former will provide better effectiveness than the latter at certain extreme points, when the structure is subjected to Kobe wave (see Figure (c)). Likewise, when the structure is subjected to

(a) Under El-Centro wave with PGA=0.1g

- 0 . 0 3

0

0 . 0 3

0 3 6 9 1 2 1 5 1 8

Dis

p. (m

)

S t r u c t u r e - D a m p e r s y s t e m S t r u c t u r e - S V M m o d e l S t r u c t u r e - U n c o n t r o l

- 0 . 2

0

0 . 2

0 3 6 9 1 2 1 5 1 8

Vel.

(m/s

)

- 2

0

2

0 3 6 9 1 2 1 5 1 8T im e ( s )

Acc.

(m/s

2 )

(b) Under Hachinohe wave with PGA=0.1g

- 0 . 0 3

0

0 . 0 3

0 3 6 9 1 2 1 5 1 8D

isp.

(m)

S t ru c t u re - D a m p e r s y s t e m S t r u c t u r e -S V M m o d e l S t ru c t u re - U n c o n t ro l

- 0 .3

0

0 . 3

0 3 6 9 1 2 1 5 1 8

Vel.

(m/s

)

- 2

0

2

0 3 6 9 1 2 1 5 1 8T im e ( s )

Acc.

(m

/s2 )

(d) Under Shanghai artificial wave with PGA=0.1g

- 0 . 0 5

0

0 . 0 5

0 3 6 9 1 2 1 5 1 8

Dis

p. (m

)

S t r u c t u re -D a m p e r s y s t e m S t r u c t u re -S V M m o d e l S t ru c t u r e - U n c o n t ro l

- 0 . 4

0

0 . 4

0 3 6 9 1 2 1 5 1 8

Vel.

(m/s

)

- 2

0

2

0 3 6 9 1 2 1 5 1 8T im e ( s )

Acc.

(m/s

2 )

- 0 .0 6- 0 .0 3

00 .0 30 .0 6

0 3 6 9 1 2 1 5 1 8

Dis

p. (m

) S t ru c tu re -D a m p e r s y s t e m S t ru c t u re -S V M m o d e l S t ru c t u re -U n c o n t ro l

- 0 .6- 0 .3

00 .30 .6

0 3 6 9 1 2 1 5 1 8

Vel.

(m/s

)

- 4- 2024

0 3 6 9 1 2 1 5 1 8T im e ( s )

Acc.

(m/s

2 )

(c) Under Kobe wave with PGA=0.1g

Figure 2 Seismic responses of structural top storey by using general semi-active dampers and SVM based semi-active control algorithm

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05Max. Disp.(m)

Floo

r Lev

el

Structure-UncontrolStructure-Damper systemStructure-SVM model

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05Max. Disp.(m)

Floo

r Lev

el

Structure-UncontrolStructure-Damper systemStructure-SVM model

(a) Under El-Centro wave with PGA=0.1g (b) Under Hachinohe wave with PGA=0.1g

(d) Under Shanghai artificial wave with PGA=0.1g (c) Under Kobe wave with PGA=0.1g

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05Max. Disp.(m)

Floo

r Lev

el

Structure-UncontrolStructure-Damper systemStructure-SVM model

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05Max. Disp.(m)

Floo

r Lev

el

Structure-UncontrolStructure-Damper systemStructure-SVM model

Figure 3 Seismic displacement responses of structural every storey by using general semi-active dampers and SVM based semi-active control

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Hachinohe wave, the seismic responses of the structure with the structure-SVM system model are remarkably reduced. This observation indicates that the structure-SVM system model is significantly better than the structure-damper system, as shown in Figure (b). Compared to the structure-damper system, the structure-SVM system model generally renders higher level of response reduction.

In order to further examine the seismic response reduction of the controlled structure using the present algorithm, the displacement response of every floor under these four seismic waves is shown respectively, in Figure 3. It is seen that under the action of the El-Centro wave, Kobe wave, Shanghai artificial wave, the peak values of the first two floors are similar for the two control strategies. But, in reducing the peak values of top floor displacements, the structure-SVM system model is superior to the structure-damper system. Likewise, under Hachinohe wave, the peak displacement response of every floor, especially the top floor, with the structure-SVM system model is remarkably smaller than that with the structure-damper system. It is verified once again that the proposed structure-SVM system model will render better effectiveness than the structure-damper system.

6. Conclusion

This paper proposes a new semi-active control strategy, referred to as the structure-SVM system model which contains both the observers (sensors) and controllers. The system data at the previous time can be gained from these observers. On training by resorting to SVM, the control forces of the dampers at next time could be predicted, subsequently, implementing the semi-active control. The applicability of the proposed structure-SVM system model has been verified though numerical simulations. Therefore, this study will provide motivation to apply SVM to structural vibration control.

Acknowledgements

The author would like to acknowledge the financial contributions received from the Innovation Fund Project for Graduate Student of Shanghai University (2009).

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