ADAPTIVE FUZZY SLIDING MODE CONTROL ... - Semantic Scholar

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 17, No. 5 (2009) 705−727 © World Scientific Publishing Company

ADAPTIVE FUZZY SLIDING MODE CONTROL FOR SEISMICALLY EXCITED BRIDGES WITH LEAD RUBBER BEARING ISOLATION

CHENG-WU CHEN Department of Logistics Management, Shu-Te University, Yanchao, Kaohsiung, 82445, Taiwan, R.O.C. KEN YEH* Department of Construction Science and Technology, De-Lin Institute of Technology, Tu Cheng, Taipei County, 23654, Taiwan, R.O.C. [email protected] KEVIN FONG-REY LIU Department of Safety, Health and Environmental Engineering, Ming Chi University of Technology, Taishan, Taipei County, 24301, Taiwan, R.O.C. Received 18 January 2008 Revised 7 May 2009 This study examines the feasibility of applying adaptive fuzzy sliding mode control (AFSMC) strategies to reduce the dynamic responses of bridges constructed using a lead rubber bearing (LRB) isolation hybrid protective system. Recently developed control devices for civil engineering structures, including hybrid systems and semi-active systems, have been found to have inherent nonlinear properties. It is thus necessary to develop non-linear control methods to deal with such properties. Generally, controller fuzziness increases the robustness of the control system to counter uncertain system parameters and input excitation, and the non-linearity of the control rule increases the effectiveness of the controller relative to linear controllers. Adaptive fuzzy sliding mode control (AFSMC) is a combination of sliding mode control (SMC) and fuzzy control. The performance and robustness of these proposed control methods are all verified by numerical simulation. The results demonstrate the viability of the presented methods. The attractive control strategy derived there-from is applied to seismically excited bridges using LRB isolation. Keywords: Lyapunov theory; adaptive fuzzy sliding mode control; LRB.

1. Introduction The theory of structural control has been proposed as a means to protect the safety and integrity of structures. Control schemes can be divided into roughly three types, passive, active and hybrid. Hybrid control methods, which possess the advantages of both passive and active control systems, have recently received considerable attention.1 In particular,

*

Corresponding author. 705

706 C.-W. Chen, K. Yeh & K. F.-R. Liu

hybrid protective systems comprised of a combination of passive base-isolation systems and active control devices, as shown in Fig. 1, have been shown to be quite effective in reducing structural responses to strong earthquakes. Several base-isolation hybrid protective systems exist. These include lead rubber bearings (LRBs) and actuators, LRBs and variable dampers, and frictional sliding bearings and actuators.

Deck

abutment Pier control device

bearing

Fig. 1. An isolated two span continuous deck bridge.

Currently, elastomeric bearings are the most widely used of the common LRB isolation systems. The elastomer is made of either natural rubber or neoprene, as shown in Fig. 2. The bearings are formed by vulcanization bonding of sheets of rubber to thin steel reinforcing plates. The bearings are extremely stiff in the vertical direction but highly flexible in the horizontal direction. This approach works by interposing a layer with low horizontal stiffness between the bridge and the foundation which decouples the bridge motion from the horizontal components of the earthquake ground motion.

Lead

Steel Lamination

Rubber Fig. 2. Lead rubber bearings (from Ref.2).

The disadvantage of LRB isolation is the possibility of damage to the bearings or the bridge decks resulting from large lateral displacement. Hybrid control aims to exploit the advantages of both active and passive control systems. The LRB isolation system is used

Adaptive Fuzzy Sliding Mode Control for Seismically Excited Bridges 707

to reduce the inertial loading transmitted by the ground motion to the bridge, while active control devices are used to reduce the response of the bridge. The dynamic behavior of LRB isolation systems can be either highly nonlinear or inelastic. Nonlinear systems require a nonlinear control method. The concept of structural control in civil engineering applications originated in the early 1970s.3 Some commonly used structural control methods include LQR optimal control,4 pole assignment,5 and instantaneous optimal control.6 Recently, other methods such as H2 (Ref. 7) H-infinite8 optimal control, sliding-mode control,9 LQG/LTR,10 fuzzy control,11 and fuzzy sliding mode control32,33 have been introduced to deal with structural control problems. In this study, an adaptive fuzzy sliding mode control method is proposed for the structural control of bridges with LRB isolation. Systems with complex mechanisms, such as are commonly found in the industrial sector, that are non-linear, and/or ill-defined, are difficult to model mathematically, but can be adequately controlled and operated in real world situations. Operator control strategies for such systems, are developed based on intuition and experience, and can be considered as comprised of a set of heuristic decision rules. Fuzzy set theory and fuzzy algorithms can be used to directly and effectively assess such imprecise linguistic statements. However, fuzzy control design still involves several difficulties: (1) the large number of fuzzy rules required for multi-dimension systems make analysis very complex; (2) suitable parameters must be determined for the membership functions using a time-consuming trial and error procedure; (3) no stability analysis tools can be applied to fuzzy control systems.12 In order to solve these problems, Chen et al.,24,25 proposed a stability condition for a nonlinear structural system based on both linear matrix inequality (LMI) transformation and the T-S fuzzy model. Although the controller design problem can be transformed into a solvable LMI problem, the control approach has to be enhanced to be effective for real engineering applications. Here, we consider adaptive fuzzy sliding mode control (AFSMC) strategies for a real bridge structure with an LRB isolation hybrid protective system. Generally, even if the system parameters are difficult to define precisely, the bounds on the uncertain parameters may be known. It is certain that sliding mode control is useful for uncertain and nonlinear dynamic systems.13 This approach can systematically solve the problem of maintaining stability and consistent performance. Yager and Filev14 determined some fuzzy rules based on the sliding mode condition. The sliding surface can dominate the dynamic behaviors of the control system and reduce the number of rules in the fuzzy rule base. Palm demonstrated that fuzzy control can be considered an extension of the conventional sliding mode controller with a boundary layer.19 Adaptive fuzzy control15,23 uses a linear combination of fuzzy basic functions. The consequent parameters are tuned via an adaptive mechanism. The adaptive law for the method of adaptive fuzzy sliding mode control presented in this study is derived from the Lyapunov theory. The adaptive law is used to tune the centers of the consequences of the membership functions. A stable adaptive fuzzy sliding mode control is developed for affine highly nonlinear systems.20 The desired control behavior is achieved by developing an equivalent control using the unknown part of the system dynamics and the fuzzy learning model. Lhee et al.


described sliding mode-like fuzzy logic control with fast self-tuning of the dead-zone parameters given parameter variations in the controlled system.21 Fischle et al. extended the method of stable adaptive fuzzy control to a broader group of nonlinear plants. They achieved this by using an improved controller structure adopted from the neural network domain.22 Their controllers19-22 were designed for application to a high order single output system. However, since civil structures are multi-output systems, the response information from sensors may include a wide variety of data such as displacements, velocities and accelerations. The coefficients of the sliding surface19-22 are selected so that s(t) = 0 is Hurwitz. In this study, the optimal sliding mode method is used to determine the sliding surface. The controller’s sliding surface19-22 can ensure system stability. Notably, the optimal sliding mode method not only ensures system stability, but can also adjust the weighting matrices according to the control objective. The method discussed in this paper is more efficient than other types of controllers.19-22 The aim of this study is thus to develop a systematic AFSMC design procedure capable of controlling the behavior of seismically excited bridges constructed with LRB isolation systems. The effectiveness of the developed algorithm is illustrated using several examples applied to LRB isolated bridges. 2. Equation of Motion for the Structural System The equation of motion for a bridge modeled by an n-degrees-of-freedom system controlled by actuators and subjected to ground excitation ɺxɺg can be expressed as follows: M Zɺɺ (t) + C Zɺ (t) + KZ(t) + H f(t) = B U(t) − M r ɺxɺ , (1) g

T

n

where Z = [x1,x2,…xn] ∈ R = n-vector; and xi denotes the relative displacement of the designed ith element. Matrices M, C, and K = n × n represent the mass, damping, and stiffness matrices, respectively; r = n-vector is the influence of the earthquake excitation; H = n-vector denotes the locations of the isolators; and B = n × m matrix represents the locations of the m control forces. The m-dimensional control force vector U(t) corresponds to the actuator forces (which are generated via an active tendon system or mass damper, for example); and f(t) is the force from the isolators. The hysteretic stiffness of an isolator can be modeled by18 Fsb = akbxb + (1 − a)kbDyv ,

(2)

where Fsb denotes the stiffness of the isolator; a represents the ratio of the post yielding to pre-yielding stiffness; kb is the elastic stiffness of the isolator; xb denotes the isolator displacement; Dy represents the yielding deformation; and v is the hysteretic variable, where

v ' (t ) = D y−1 (α xb' − β xb' v

η −1

η

v − γ xb' v ) .

(3)

Parameter α,β,γ and η determine the scale, general shape, and smoothness of the hysteretic loop, respectively.


In (2), akbxb denotes the linear elastic stiffness that appears in the K matrix of (1). The nonlinear or hysteretic stiffness appearing in the nonlinear or hysteretic, f of (1) is thus expressed by f ( t) = (1− a)kbDyv .

(4)

For this controller design, the standard first-order state equation corresponding to (1) is Xɺ (t) = AX( t) − H f ( t) + BU( t) + L ɺxɺg ,

where XT = [ZT

(5)

Zɺ T ] = 2n vector; and

 0 A=  −1 − M K

I n×n  − M −1C 

 0  B =  −1  M B 

，H =

，L =

 0   −1  M H 

0   . − r 

(6)

3. Adaptive Fuzzy Sliding Mode Control 3.1. Design of the sliding surface For a complete account of the sliding mode control theory see Refs. 16 and 17. This theory is based on the concept that the controller changes its structure according to the position of the state trajectory with respect to a selected sliding surface. The control is designed to force the state trajectory of the system onto the sliding surface and maintain it there. This is achieved with a high speed switching law. This discontinuous component of the sliding control is used to develop fuzzy logic control. The design of the sliding surface is detailed below. The equation of the system has the form Xɺ = AX + BU + F + E, (7) where X( t) denotes an n state vector; A represents an n × n system matrix; B is an n × m controller location matrix; F denotes an n vector containing the system uncertainty and nonlinearity; and E is an n excitation vector. Suppose {x | S(X) = 0} is the selected sliding surface. S(X) = PX, (8) where P is an m × n sliding surface coefficient matrix. Consider the nominal system25 Xɺ = AX + BU,

(9)

From which we obtain the sliding surface of the nominal system. First, the state equation of motion (9) is converted into the so-called regular form via the following transformation: Let Y = JX or X = J

−1

Y,

(10)


where J is a transformation matrix −1 J =  I n − m − B1 B2 

 0

Im



，J

−1

−1   =  I n − m B1 B2  ,

 0

(11)

Im 

and where B1 and B2 are the (n − m)2 m and m × m submatrices obtained by partitioning the B matrix, as in Eq. (9), as follows: B =  B 1  .

(12)

B 2 

Matrix B2 should be nonsingular. With the transformation J, the state equation (7), and the sliding surface (8). Hence Yɺ = A Y + B U

S = P Y, where

Let

(13)

0 P = PJ −1, A = JAJ −1, B =   .  B2  Y  Y =  1 Y2 

； A =  AA

11



21

A12   A22 

； P = [P P ], 1

(14)

2

where Y1 and Y2 are the n−m and m vector, respectively; and A11 , A12 , P1 and P2 , are the (n − m) × (n − m), (n − m) × m, m × (n − m) and m × m matrices, respectively. Substituting (14) into (13) we obtain the equations of motion on the sliding surface Yɺ = A Y1 + A Y2 , (15) 1

11

12

S = P1 Y1 + P2 Y2 = 0.

(16)

For simplicity, P2 is set to be an identity matrix, i.e., P2 = Im and thus, Y2 = − P1 Y1

and

(17)

(

Yɺ1 = A11 − A12 P1 ) Y1 .

(18) T 1

T 2

T

The P1 matrix can be calculated from (18) such that the matrix Y = [ Y Y ] on the sliding surface is stable. The optimal sliding mode method is used to determine the P1 matrix and P is also obtained. The method for obtaining the optimal sliding mode9 is described below. The sliding surface is derived by minimizing the integral of the quadratic function of the state vector ∞

I = ∫0 X T QXdt ,

(19)

where Q denotes a (2n × 2n) positive definite weighting matrix. In terms of the transformed state vector Y, (19) becomes

Adaptive Fuzzy Sliding Mode Control for Seismically Excited Bridges 711 ∞

I = ∫ Y1T 0 where

T  Y1  Y2T  T   dt , Y2 

(20)

T12  T T = (J−1)TQJ−1 ; T =  11 , T21 T22 

(21)

and T11 and T22 are the (2n − m) × (2n − m) and (m × m) matrices, respectively. Minimizing the performance index I subjected to the equations of motion (15) we obtain −1

Y2 = −0.5 T22 ( A12 T Pˆ + 2 T21 ) Y1 ,

(22)

where Pˆ is a (2n− m) × ( 2n− m) Riccati matrix that satisfies the following matrix Riccati equation:

where

Aˆ T Pˆ + Pˆ Aˆ −0.5 Pˆ A12 T22−1 A12 T Pˆ = −2(T11 − T12T22−1T12T) ,

(23)

Aˆ = A11 − A12 T22−1T21 .

(24)

A comparison between (17) and (22) indicates that P1 = 0.5 T22−1 ( A12 T Pˆ +2 T21 ) .

(25)

Finally, the sliding surface is obtained P = P J = [ P1

I m] J .

(26)

3.2. Design of an adaptive fuzzy sliding mode controller The second step is to design the controller. The controllers are designed to drive the state trajectory into the sliding surface S = 0. Define a Lyapunov function V such that T

V = 0.5S S.

(27)

The sufficient condition for the sliding mode S = 0 occurring as t → ∞ is Vɺ = ST Sɺ < −η ||S|| ,

(28)

where η is a positive real value. In (7), F is an n vector containing the system uncertainty and nonlinearity, while E is an n excitation vector. Generally, it is difficult to know system parameters exactly, but the bounds on the uncertainty are knowable.

F ≤ δF

， E ≤δ

W

.

(29)

Let U = Ueq – (γ + η)sgn(STPB)T , where

(30) −1

γ = δ/||Β||, δ = δF + δW , Ueq = −(PB) PAX

Vɺ = STP(AX + BU + F + E)


= STP(AX – B(PB)-1PAX − B (γ + η)sgn(STPB)T + F + E) = STP(−B(γ + η)sgn(STPB)T + F + E) = STPB(−(γ + η)sgn(STPB)T) + STP(F + E) = −η || STPB || − γ || STPB || + STP(F + E) =- η || STPB || − γ || STPB || (1 – ( STP(F + E))/γ || STPB ||) < −η || STPB || . Let K = η + γ (30) and the control force U = Ueq – Ksgn(STPB)T. Stability can be obtained when K ≥ η + δ/ ||Β||,

(31)

where denotes the Euclidean norm. Since PB is a constant matrix, S be used to represent STPB. A disadvantage of the control law given in Eq. (30) is that it is discontinuous and tends to excite the high frequency modes of the plant, also called the controlled system. The problem can be alleviated using a fuzzy inference mechanism. A fuzzy inference mechanism is used to estimate the second part of (30), i.e., uf. The range of u f obtained from (31) is [ −K , K]. The fuzzy rule is.24 If S is PB and S ′ is PB, then S is NB. PB: Positive Big; NB: Negative Big All the rule bases are listed in Table 1. The characteristic U = f ( S ) of the sliding mode controller with a boundary layer is linear, while that of the fuzzy sliding mode controller is nonlinear. The fuzzy sliding mode controller determines different actions for different S regions. For example, the fuzzy sliding mode controller uses slow reaction control for small S values, and quick control for large S values. In existing studies concerning the membership functions of controlled systems, various types of fuzzy numbers are suggested for use, such as trapezoidal, triangular, and Gaussian functions (see Refs. 26−28 and the references therein). For convenience, the triangular membership function is used for each fuzzy number in this paper. Fuzzy output u f can be calculated based on the center of the area of dufuzzification25  c1

l

uf =

[

where v = c1 .....c l

]

wi ci ∑ i =1 l

wi ∑ i =1

=

 w1    ⋯ cl   ⋮  w   l l

wi ∑ i =1

= vTψ ,

(32)

denotes a adjustable parameter vector; c i represents the center of

the consequent part of the membership function; and wi represents the firing strength of the ith rule. Meanwhile, Ψ = [w1 ....wl ] is a firing strength vector. l

∑w i =1 i


From (29), we see that the sliding mode controller requires an upper bound to the uncertainty. When the uncertainty increases, the control cost also increases. However, the optimal value of the uncertainty cannot be precisely obtained owing to a lack of knowledge regarding the structure or system complexity. Therefore, an adaptive fuzzy control is developed to deal with the problem and to estimate the minimum control cost. Table 1. Rule base of the adaptive sliding mode controller.

S ′\S

PB

PM

PS

Z

NS

NM

NB

PB PM PS Z NS NM NB

NB NB NM NM NS NS Z

NB NM NM NS NS Z Z

NM NM NS NS Z Z PS

NS NS NS Z PS PS PS

NS Z Z PS PS PM PM

Z Z PS PS PM PM PB

Z PS PS PM PM PB PB

PB: Positive Big; PM: Positive Medium; PS: Positive Small; Z: Zero; NB: Negative Big; NM: Negative Medium; NS: Negative Small

Assume that there exists a specific uˆ f which achieves the minimum control cost and that uˆ f satisfies the sliding mode condition. From (32), uˆ f can be rewritten as follows:

uˆ f = vˆ T Ψ ,

(33)

where vˆ denotes the optimal vector with which the minimum control cost is achieved. Define the parameter vector as

v~ = v − vˆ .

(34)

Let the Lyapunov function for each controller be V=

1 2 1 ~ T~ ( s + v v ), 2 α

(35)

where α is a positive constant. Now,

1 T Vɺ = s Pbiuf + s P(F + E) + ~ v vɺ

α

= s Pbi(uf − uˆ f ) +

s Pbi uˆ f + s P(F + E) +

= s Pbi(v T − vˆ T)ψ+ s Pbi uˆ f + s P(F + E) + = s Pbi ~ v T ψ + s Pbi uˆ f + s P(F + E) +

1 ~T v vɺ

α

1 ~T v vɺ

α

1 ~T v vɺ

α


=

1 ~T v ( vɺ + α s Pbiψ) + s Pbi uˆ f + s P(F + E)

α

< −η i | s Pbi| ( uˆ f satisfies the sliding mode condition and let vɺ = −α s Pbiψ). Finally, the adaptive law obtained is

vɺ = −α s Pbiψ .

(36)

The adaptive law adjusts the centers of the membership function of the consequent part. This adaptive law is derived from Lyapunov theory, so V → 0 as t → ∞ . IF V → 0, then ~ v → 0. As ~ v → 0, the minimum control cost uˆ f can be achieved. The design procedure for the AFSMC can be briefly summarized as follows:

Step 1: Determine the state and control variables. Step 2: Use the optimal sliding modes method to determine the sliding surface. Step 3: Select thickness of the boundary layer based on the allowable responses. Step 4: Calculate the value of K according to (31). Step 5: Define the fuzzy sets for both the input and output of the fuzzy inference mechanism. Step 6: Perform on-line AFSMC. Some examples are used to illustrate the AFSMC for LRB isolated bridges.

Remark 1: The installment of LRBs on bridges can effectively reduce the energy that might affect the bridge structure in the event of an earthquake. However, there may be negative side-effects in that it renders the upper portion of the structure susceptible to excessive swaying motions. To resolve this problem, an earthquake stop can be installed on the transverse direction of the bridge that will help control lateral motion associated with earthquake events. Nonetheless, longitudinal shift of the bridge owing to the isolation mechanism may cause collisions with the nearby upper pier or abutment structures, even damaging or unseating the bridge. Some problems with bridge isolation have been discussed.30,31 Here we examine the application of AFSMC to prevent extreme earthquake induced oscillations longitudinally along the bridge. The proposed AFSMC can be easily applied in multiple degrees-of-freedom systems. A two degree-of-freedom system involving longitudinal oscillations is simulated to demonstrate the effects discussed in this study. 4. Numerical Simulation and Results The AFSMC is applied to control the bridge with LRB isolators. An isolated two-span continuous deck bridge is illustrated in Fig.1. The nominal values of the bridge deck mass, pier mass, pier stiffness, and damping ratio are 770 ton, 128 ton, 3.1e5 KN/m, and 0.02, respectively. Moreover, the nominal value of LRB elastic stiffness is 9.6e3 KN/m, the LRB yield stiffness is 1.968e3 KN/m and the yielding deformation Dy is 1cm. The optimal sliding mode method is used to determine the sliding surface with a diagonal


weighting matrix Q; Q11 = 1000, Q22 = 10, and Q33 = Q44 = 1. The sliding surface of the bridge with the LRB isolation device can be calculated to be –2.53x1 + 3.39x2 + 9.92e − 3 xɺ1 + xɺ 2 = 0, where x1 denotes the pier displacement relative to the ground; and x2 represents the deck displacement relative to the pier. This fuzzy controller has 49 fuzzy rules, as listed in Table 1. The triangular membership function is used for each fuzzy number. 0 0 1   0 0 0 The system matrices A =  − 1453.13 75.31 − 19.30   1453.13 − 87.83 19.30

0  0  1 , B =  0  ,   0  0     0 1.30e − 3

 0  0  0    H =   , and L =  0  , f( t) are determined by (3) and (4). The parametric  0  − 1     1.30e − 3 − 1

values are as follows: kb = 9.6e3; Dy = 1cm; α = 1; β = γ = 0.5 and η = 3. The SMC and AFSMC discussed in this paper are compared using real earthquake data, consisting of acceleration records from the El Centro (1940), Northridge (1994), and Chi Chi (1999) earthquakes. The acceleration records from the El Centro earthquake are shown in Fig. 3. Figures 4 and 5 indicate the deck displacement of the considered bridge after application of SMC and AFSMC. Excitation is taken from the acceleration records of the El Centro earthquake. Figs. 6−8 show the relationship between the LRB shear force and the deformation with no control, with SMC, and with AFSMC, respectively. The maximum response quantities of the bridge with LRB isolation are listed in Table 2. xd = x1, xp = x2, ɺxɺd = ɺxɺ1 , where Fp, Umax, and σU denote the deck displacement relative to the pier, pier displacement relative to the ground, deck absolute acceleration, shear force of the pier, maximum control force, and standard deviation of the control force, respectively.

Fig. 3. Acceleration records of El Centro.


Fig. 4. SMC controlled and uncontrolled deck displacement history.

Fig. 5. AFSMC controlled and uncontrolled deck displacement history.

Fig. 6. Relationship of LRB shear force and deformation with no control.


Fig. 7. Relationship of LRB shear force and deformation with SMC control.

Fig. 8. Relationship of LRB shear force and deformation with AFSMC control.

Table 2. Maximum response quantities of bridge with LRB isolation.

El Centro earthquake No control

SMC

AFSMC

xd (m )

3.85e-1

9.83e-2

4.19e-2

xp (m)

3.52e-3

2.18e-3

1.54e-3

ɺxɺd (m/s )

1.052

0.668

0.589

Fp (kn)

1092

2

566

476

Umax (kn)

971

771

σu.(kn)

388

225


Fig. 9. Acceleration records for the Northridge earthquake.

Fig.10. SMC controlled and uncontrolled deck displacement history.

The acceleration records of the Northridge earthquake are shown in Fig. 9. Figures 10 and 11 show the deck displacement of the considered bridge after application of SMC and AFSMC using the same acceleration excitation as the Northridge earthquake. The relationship between the LRB shear force and deformation with no control, SMC, and AFSMC, are shown in Figs. 12−14, respectively. The maximum response quantities of the bridge with LRB isolation for Northridge earthquake acceleration are listed in Table 3. Table 3. Maximum response quantities of bridge with LRB isolation.

Northridge earthquake No control

SMC

AFSMC

xd (m )

3.77E-01

1.69E-1

6.48e-2

xp (m)

8.50E-03

4.92E-3

3.66E-3

ɺxɺd (m/s2)

1.002

0.697

0.503

Fp (kn)

1634

785

631

Umax (kn)

1000

897

σu.(kn)

389

245



Fig. 12. Relationship of LRB shear force and deformation with no control.

Fig. 13. Relationship of LRB shear force and deformation with SMC control.


Fig. 14. Relationship of LRB shear force and deformation with AFSMC control.

The acceleration records of the Chi Chi earthquake are shown in Fig. 15. The deck displacement of the considered bridge after the application of SMC and AFSMC is indicated in Figs. 16 and 15. The acceleration is the same as in the Chi Chi earthquake records. The relationship between the LRB shear force and deformation with no control, SMC, and AFSMC, is shown in Figs.18−29, respectively. The maximum response quantities of the bridge with LRB isolation, excited by Chi Chi earthquake acceleration, are listed in Table 4.

Fig. 15. Acceleration records for the Chi Chi earthquake.

Fig. 16. SMC controlled and uncontrolled deck displacement history.



Fig. 18. Relationship of the LRB shear force and deformation with no control.

Fig.19. Relationship of the LRB shear force and deformation with SMC control.


Fig. 20. Relationship of the LRB shear force and deformation with AFSMC control. Table 4. Maximum response quantities of bridge with LRB isolation.

Chi Chi earthquake No control

SMC

AFSMC

xd (m )

3.51E-01

9.85E-02

4.08e-2

xp (m)

4.89E-03

3.75e-3

2.25e-3

ɺxɺd (m/s )

1.007

0.711

0.521

Fp (kn)

1514

2

732

606

Umax (kn)

1000

868

σu.(kn)

337

182

As shown in the above figures, both the SMC and AFSMC controllers suppress the earthquake induced vibrations. The maximum displacement, pier shear force, control force and deck acceleration responses of the considered isolated-bridge (with and without the controllers) are all shown in Tables 2−4. The results show that the AFSMC performs better than the SMC. There are excellent responses shown for different earthquake events as well as more effective control forces. These results indicate that the proposed controller is an effective method for seismic isolation of structures. To examine the robustness of the adaptive fuzzy sliding mode control, the stiffness of all the units of the bridge is varied by ± 40% and a 60 ms time delay is assumed. The El Centro earthquake data scaled to a peak ground acceleration of 0.3g serves as the input excitation. The maximum response quantities of the bridge with LRB isolation are listed in Table 5, where xd = x1, xp= x2, ɺxɺd = ɺxɺ1 , Fp, Umax, and σU denote the deck displacement relative to the pier, pier displacement relative to the ground, deck absolute acceleration, shear force of the pier, maximum control force, and standard deviation of the control force, respectively. Standard deviation of the control force indicates a spread of the density of the mean control force. A look at Table 5 reveals that the adaptive fuzzy


sliding mode control reduces pier shear force, as well as the response of the bridge deck and the amplitude of the bridge deck acceleration. Notably, the performance results show that the AFSMC effectively reduces bridge responses under stiffness uncertainty and time delay conditions. This means that the AFSMC is robust, its maximum control forces are relatively low, all falling below 11% of the superstructure weight. Table 5. Maximum response quantities of bridge with LRB isolation No delay and error No control

control

Time delay No control

control

Stiffness +40% No control

Stiffness −40%

control

No control

control

xd (m )

3.85e-1

4.19e-2

3.85e-1

5.05e-2

3.82e-1

4.13e-2

3.76e-1

4.32e-2

xp (m)

3.52e-3

1.54e-3

3.52e-3

1.51e-3

2.32e-3

1.14e-3

5.61e-3

2.92e-3

ɺxɺd (m/s )

1.052

0.589

1.052

0.755

1.066

0.606

1.049

0.611

Fp (kn)

1092

476

1092

467

1009

495

1043

543

2

Umax (kn)

771

782

771

771

συ.(kn)

225

283

221

231

The effectiveness of this algorithm is further demonstrated by the simulation results for a long-period and a wide-pass white noise artificial earthquake, both scaled to have a peak ground acceleration of 0.3g as the input excitation. The time history of the long-period artificial earthquake and wide-pass white noise artificial earthquake are displayed in Figs. 21 and 23, respectively. The frequencies of these artificial earthquakes are shown in Figs. 22 and 24. The maximum response quantities of the bridges for the simulated earthquakes are listed in Table 6. Compared to the case without control, the deck displacement and the shear force of the pier are significantly reduced. The maximum control forces of the adaptive fuzzy sliding mode control are also relatively low, all less than 11% of the superstructure weight. This phenomenon demonstrates that the adaptive fuzzy sliding mode control works well with different frequency contents.

Fig. 21. Time history of the long-period artificial earthquake.


Fig. 22. Frequency content of the long-period artificial earthquake.

Fig. 23. Time history of the wide-pass white noise artificial earthquake.

Fig. 24. Frequency content of the wide-pass white noise artificial earthquake.


Table 6. Maximum response quantities of bridge with LRB isolation. Long-period artificial earthquake

Wide-pass white noise artificial earthquake

No control

control

No control

control

9.35e-1 6.62e-3 2.47

6.92e-2 1.85e-3 1.42

3.95e-1 4.02e-3 1.08

3.29e-2 1.64e-3 0.52

2053

575 771 304

1245

509 576 194

xd (m ) xp (m)

ɺxɺd (m/s2) Fp (nt) Umax (nt) σF.(nt)

5. Conclusions In this study we develop an efficient adaptive fuzzy sliding mode control (AFSMC) algorithm for stability problems in bridges constructed with lead rubber bearing (LRB) isolation hybrid protective systems. The simulation results indicate that a bridge equipped with an LRB isolation system has reduced pier displacement relative to the ground, the deck absolute acceleration, and the pier shear force. Moreover, AFSMC can reduce the deck displacement and all of the above response quantities. The maximum control forces of the AFSMC are relatively low, all being less than 11% of the deck weight. As can be seen in Table 5, this control method performs well in terms of estimation error and time delay. The proposed method is robust and can be used for structural control in situations involving nonlinearity, uncertainty and time delay. The results in Table 6 reveal that the adaptive fuzzy sliding mode control is not sensitive to different frequency contents. The effectiveness and feasibility of the proposed controller design method is demonstrated using numerical simulations of seismically excited bridges with LRB isolation. The example demonstrates that the proposed methodology can be applied to practical control systems. Besides reducing oscillations that could be considered in the AFSMC approach, some important issues still remain open, such as stability analysis, stabilization problems and the control performance. Here, the focus is on the development of the AFSMC for seismically excited bridges. The proposed control strategies could be extended to time-delay problems and time-varying multi-floor structures in future. Another direction for future research would be to extend the proposed control strategies to time-varying tall building structures.

Acknowledgments The authors would like to thank the National Science Council of the Republic of China, Taiwan, for their financial support of this research under Contract Nos. NSC 96-2628-E-366-004 and NSC 95-2221- E-237-011. The authors are also most grateful for the kind assistance of Prof. B. Bouchon-Meunier, Editor of IJUFKS, and the anonymous reviewers whose constructive suggestions have greatly aided us in the presentation of this paper.


References 1. G. W. Housner, L. A. Bergman, T. K. Caughey et al., Structural control: past, present, and future, J. Eng. Mech. ASCE 123(9) (1997) 897−971. 2. R. I. Skinner, W. H. Robinson and G. H. McVerry, An Introduction to Seismic Isolation (Wiley, 1993). 3. J. T. P. Yao, Concept of structural control, J. Struct. Div. ASCE 98(7) (1972) 1567−1574. 4. J. N. Yang, Application of optimal theory to civil engineering structures, J. Eng. Mech. Div. ASCE 101(6) (1975) 819−838. 5. M. Abdel-Rohman and H.H. Leipholtz, Structural control by pole assignment method, J. Eng. Mech. Div. ASCE 107(7) (1981) 1313−1325. 6. J. N. Yang, A. Akbarpour and P. Ghaemmami, New control algorithms for structural control, J. Eng. Mech. Div. ASCE 113(9) (1987) 1369−1386. 7. J. Suhardjo, B. F. Spencer and A. Kareem, Frequency domain optimal control of wind excited buildings, J. Eng. Mech. ASCE 118(2) (1992) 2463−2481. 8. W. E. Schmitendorf, F. Jabbari and J. N. Yang, Robust control techniques for buildings under earthquake excitation, J. Earthquake Eng. Struct. Dyn. 23(5) (1994) 539−552. 9. J. N. Yang, J. C. Wu, A. K. Agrawal and Z. Li, Sliding mode control for nonlinear and hysteric structures, J. Eng. Mech. ASCE 121(12) (1995) 1330−1339. 10. L. T. Lu, W. L. Chiang and J. P. Tang, LQG/LTR control methodology in active structural control, J. Eng. Mech. ASCE 124(4) (1998) 446−453. 11. K. Yeh, W. L. Chiang and D. S. Juang, Application of fuzzy control theory in active control of structures, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 4(3) (1996) 255−274. 12. J. C. Lo and Y. H. Kuo, Decoupled fuzzy sliding mode control, IEEE Trans. Fuzzy Systems 6(3) (1998) 426−435. 13. S. Hui and S. H. Zak, Robust-control synthesis for uncertain nonlinear dynamic-systems, Automatica 28(2) (1992) 289−29. 14. R. R. Yager and D. P. Filev, Essentials of Fuzzy Modeling and Control (Wiley, NY, 1994). 15. L. X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Systems 1(2) (1993) 146−155. 16. R. A. Decarlo, S. H. Zak and G. P. Matthews, Variable structure control of nonlinear Multivariable systems: a tutorial, Proc. IEE 76(3) (1988) 212−232. 17. V. I. Utkin, Sliding Modes in Control Optimization (Springer-Verlag, NY, 1992). 18. J. N. Yang, Z. Li and S. Vongchavalitkul, A generalization of optimal control theory: linear and nonlinear control, Tech. Rep. NCEER-92-0026 (Nat. Ctr. for Earthquake Res., State Univ. of New York , Buffalo, 1992). 19. R. Palm, Sliding mode fuzzy control, Proc. IEEE Conf. Fuzzy Systems (1992) 519−526. 20. C. L. Hwang and C. Y. Kuo, A stable adaptive fuzzy sliding-mode control for affine nonlinear systems with application to four-bar linkage systems, IEEE Trans. Fuzzy Systems 9(2) (2001) 238−252. 21. C. G. Lhee et al, Sliding mode-like fuzzy logic control with self-tuning the dead zone parameters, IEEE Trans. Fuzzy Systems 9(2) (2002) 343−348. 22. K. Fischle and D. Schroder, An improved stable adaptive fuzzy control method, IEEE Trans. Fuzzy Systems 7 (1999) 27−40. 23. L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis (Prentice-Hall, 1994) Chap 9. 24. C. W. Chen, Stability conditions of fuzzy systems and its application to structural and mechanical systems, Advances in Engineering Software 37(9) (2006) 624−629. 25. F. H. Hsiao, C. W. Chen, Y. W. Liang, S. D. Xu and W. L. Chiang, T-S fuzzy controllers for nonlinear interconnected systems with multiple time delays, IEEE Trans. Circuits & Systems −


Part I 52(9) (2005) 1883−1893. 26. C. W. Chen, W. L. Chiang and F. H. Hsiao, Stability analysis of T-S fuzzy models for nonlinear multiple time-delay interconnected systems, Mathematics and Computers in Simulation 66 (2004) 523−537. 27. C. W. Chen, K. Yeh, W. L. Chiang, et al., Modeling, H ∞ control and stability analysis for structural systems using Takagi-Sugeno fuzzy model, Journal of Vibration and Control 13 (2007) 1519−1534. 28. F. H. Hsiao, W. L. Chiang, C. W. Chen, S. D. Xu and S. L. Wu, Application and robustness design of fuzzy controller for resonant and chaotic systems with external disturbance, Int. J. Uncertainty, Fuzziness and Knowledge-Based System 13(3) (2005) 281−295. 29. K. Yeh, C. Y. Chen and C. W. Chen, Robustness design of time-delay fuzzy systems using fuzzy Lyapunov method, Applied Mathematics and Computation 205(2) (2008) 568−577. 30. K. S. Park, H. J. Jung and I. W. Lee, A comparative study on aseismic performances of base isolation systems for multi-span continuous bridge, Engineering Structures 24(8) (2002) 1001−1013. 31. K. S. Park, H. J. Jung and W. H. Yoon et al., Robust hybrid isolation system for a seismically excited cable-stayed bridge, Journal of Earthquake Engineering 9(4) (2005) 497−524. 32. H. Alli and O. Yakut, Fuzzy sliding-mode control of structures, Engineering Structures 27 (2005) 277−284. 33. H. Alli and O. Yakut, Application of robust fuzzy sliding-mode controller with fuzzy moving sliding surfaces for earthquake-excited structures, Structural Engineering and Mechanics 26(5) (2007) 517−544.