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Benchmark Structural Control Problem for a Seismically Excited Highway Bridge

Part II: Sample Control Designs

Ping Tan and Anil K. Agrawal

ABSTRACT

This paper presents sample control system designs for the three-dimensional

benchmark structural control problem for seismically excited highway bridge. Three

types of sample control systems, namely nonlinear viscous dampers, ideal hydraulic

actuators and magnetorheological (MR) fluid dampers, are designed and presented

for comparison by participants in the study. For each of the three sample control

system, a total of 16 control devices are considered to be placed orthogonally

between the deck-ends and abutments for the reduction of earthquake induced

vibrations of the highway bridge. An H2/LQG control algorithm is selected for the

active case and a clipped optimal control algorithm is chosen for the semi-active case.

To facilitate the controller design, an eigenmode reduction method is used to reduce the

number of degrees of freedom of the initially elastic model to obtain a reduced-order

model. However, the evaluation model to simulate the performance of control strategies

is the full-order nonlinear finite element model. A Kalman filter is used to estimate

states of the reduced-order model required for the applications of controllers for both

active and semi-active controllers using selected acceleration and displacement

measurements. The modeling and sample control system designs presented in this

paper are for illustration purposes only, and are not intended to be competitive.

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Participants of this benchmark study are expected to employ more competitive

control designs for their own control strategies. These control strategies may be

passive, active, semi-active or a combination thereof.

1. INTRODUCTION

The potentially detrimental effect of spatial variation of seismic ground motion

on the responses of highway overcrossing or bridge has been recognized for some

time. The condition of highway bridges in transportation infrastructure is a critical

factor influencing national productivity and ability to compete in the international

economy. Thus, a higher level of performance with less structural damage is required

for seismic designs of these lifeline bridges. To deal with seismic risk to bridges,

seismic upgrading of critical highways is under way by various state and federal

agencies, and considerable attention has been paid to the research and development

of smart protective structural control systems.

The technology for response control of structures against natural hazards, such as

earthquakes and strong winds, has progressed from passive and active control systems

to smart and effective semi-active systems with recent advances in microprocessor,

sensor and actuator technologies (Housner et al. 1997; Spencer and Nagarajaiah 2003).

A passive control system utilizing the local motion at a point where the control system

is connected to the structure to produce control forces is well understood and widely

accepted worldwide. However, passive systems are unable to adapt to changes in

structural properties and stochastic nature of external excitations. In contrast to

passive control systems, active control systems can adapt to a wide range operating

conditions and structures, but their ability to input mechanical energy into the

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structural system results in significant increase in hardware costs and reliabilities

issues, both because of uncertainty of external power supply and capability of these

systems to destabilize structural systems in the presence of sensor/actuator

malfunction. Semi-active control systems achieve a compromise between fully active

and passive control systems by combining the reliability of passive systems and the

adaptability of active systems without requiring the large external power sources, and

they are inherently stable.

Hybrid control systems, in which passive control systems such as passive

isolation bearings are used in combination with passive, semi-active or active control

systems, have significant practical potentials for highway bridges because of their

reliability and effectiveness. The seismic isolation bearings, which usually replace

conventional bridge bearings, decouple the superstructure from piers and abutments

during strong earthquakes, thereby significantly reducing the seismic forces induced

in the bridge structure and lowering the strength and ductility demands on the bridge.

There have been several studies on the seismic design of bridges using different

isolation systems. Turkington et al (1989) showed that the lead-rubber bearings are

most effective when used in conjunction with stiff substructures and can be used to

redistribute seismic forces between piers and abutments. Constantinou et al. (1992)

and Kartoum et al. (1992) investigated the performance of sliding isolation systems.

Zayas e al (1996) studied the effectiveness of friction pendulum bearings. Dicleli

(2002) found that a hybrid seismic isolation system employing laminated elastomeric

and friction pendulum bearings are most appropriate for the seismic design of the

bridges and are also effective in reducing wind-induced vibrations. However,

nonlinear behavior of the bridge columns or excessive displacement of bearings

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during near-fault ground motions may result in significant damage to the bridge with

isolation bearings only. It has been demonstrated by several researchers that passive,

active and semi-active control systems installed in parallel with isolation bearings are

capable of reducing excessive displacement of bearings or significant damage to

bridge piers due to nonlinear behavior (Yang et al. 1995, Nagarajaiah et al. 1993,

Kawashima and Unjoh 1994, Pagnini and Solari 1999, Feng et al 2000, Saharabudhe

et al. 2004). Hence, it is important to investigate the relative merits of various

protective systems in reducing response quantities of highway bridges.

The objectives of this paper are to present sample control designs using passive,

active and semi-active control systems for the newly developed benchmark highway

bridge model (Agrawal et al 2004). The benchmark highway bridge model is based on

the recently constructed 91/5 highway bridge in southern California to provide

systematic and standardized means by which competing control strategies, including

devices, control algorithms and sensors, can be evaluated. The benchmark package

consists of the MATLAB based 3-D finite element model of the highway bridge,

designs of sample control systems, prescribed ground motions and a set of evaluation

criteria. This paper presents three sample control system designs, namely nonlinear

viscous dampers, ideal hydraulic actuators and magnetorheological (MR) fluid

dampers. A total of 16 devices are placed orthogonally between the deck and

abutments for each sample control system design to reduce earthquake-induced

vibrations of the highway bridge. An H2/LQG control algorithm is selected for the

active control system and a clipped optimal control algorithm is chosen for the

semi-active control system.

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In both the active and semi-active cases, the controller design is based on a

reduced-order controller design model. In previous benchmark problems, reduced-

order models obtained by static condensation have also been used as evaluation models

to simulate the controller performance. The controller design model were obtained by

further condensing reduced-order evaluation models using state reduction approaches,

e.g., the balanced truncation method or state order reduction method (Spencer et al.

1998a, b, 1999; Ohtori et al. 2004; Yang et al. 2004; Dyke et al. 2003; Narasimhan et

al. 2003, Nagarajaiah and Narasimhan 2003). In this benchmark highway bridge

model, the full finite element model with 430 degrees of freedoms is used as evaluation

model and the reduced-order model is used only for the controller design. This reduced

order controller design model is obtained by the eigenmode reduction method, which is

a more general method and is more accurate in representing the dynamics of full-order

bridge model (Kim and Spence, 2004 a, b).

The proposed sample control strategies are not meant to be competitive, but are

intended to illustrate some constraints and challenges of control system designs. In

all cases, the participants need to compare the results of their control designs with the

results of the sample control designs. The evaluation model of the benchmark

highway bridge, MATLAB files used for the sample control designs and the

simulation model are available on the benchmark web site:

http://www-ce.engr.ccny.cuny.edu/People/Anil%20Kumar%20Agrawal.htm.

2. REDUCED-ORDER CONTROLLER DESIGN MODEL

Since the evaluation model involves a large number of degrees of freedom and

high-frequency dynamics, a reduced-order model of the system is developed for the

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design of active and semi-active controllers. The reduced-order model has the same

output as the evaluation model, and is assumed to retain the dominant characteristics of

the full-order evaluation system.

Both semi-active and active controllers are designed by the H2/LQG algorithm by

assuming that the system remains linear since the initially elastic model of the

benchmark highway bridge is used to derive a reduced-order controller design model.

However, some components of the highway bridge, e.g., the bearing and bent

columns, will enter the nonlinear regions during a severe earthquake. To obtain a

reduced-order controller design models in previous benchmark problems, several

different approaches have been used. For example, Guyan reduction approach for the

third generation seismic-excited benchmark building model (Ohtori et al 2004),

state-order reduction approach for wind-excited tall building benchmark model and

base-isolated building benchmark model (Yang et al 2004; Narasimhan et al. 2003,

Nagarajaiah and Narasimhan 2003), static condensation and balanced realization in

the cable-stayed bridge benchmark model (Dyke et al 2003). However, in the

highway bridge benchmark model, all above approaches failed to give a

reduced-order model representing dominant dynamics of the full-order elastic system

because of highly non-proportional damping of modes caused by dashpots used to

model soil-structure interaction effects. A new approach called eigenmode reduction

method (Kim and Spencer, 2004a, b), which is an improved version of critical-mode

reduction method (Yang and Lin, 1982), has been found to be effective in dealing with

this problem and is used to obtain the reduced-order controller design model in the

present research. This eigenmode reduction method is a modal decoupling approach by

transforming the original state equation into a diagonal canonical form using the modal

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coordinates as the states. The initially linear elastic full-order evaluation system can be

expressed in state space as

(t)uEBu(t)Az(t)(t)z g&&& ++= (1)

where z(t) is the state vector; nn×A is the system matrix; rn×B is the controller location

matrix; E is the excitation influence matrix; u(t) is the r-dimension control force vector;

and (t)ug&& is the earthquake ground acceleration vector. Regulated output (t)yz and

measurement output (t)ym for the system in Eq. (1) can be written as

(t)uFu(t)Dz(t)C(t)y gzzzz &&++= (2)

(t)uFu(t)Dz(t)C(t)y gmmmm &&++= (3)

where zC , zD , zF , mC , mD and mF are associated matrices of appropriate

dimensions. Equations (1) to (3) can be converted into canonical form as follows

(t)uEu(t)BxAx gEEE &&& ++= (4)

(t)uFu(t)Dx(t)C(t)y gzE

zE

zE

zE &&++= (5)

(t)uFu(t)Dx(t)C(t)y gmE

mE

mE

zE &&++= (6)

where

ATTA -1E = , BTB -1

E = , ETE -1E = , TCC zz

E = , TCC mmE = (7)

zzE DD = , mm

E DD = , zzE FF = , mm

E FF =

In Eq. (4), x is the modal coordinate vector; EA is a diagonal matrix of eigenvalues,

and T is the eigenvector matrix. The state vector x can be partitioned into critical

modes rx and residual modes dx as follows

grrd

E12r

E11r uEuBxAxAx &&& +++= (8)

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gddd

E22r

E21d uEuBxAxAx &&& +++= (9)

where matrices, EA , EB and EE in Eq. (1) are partitioned as follows:

⎥⎦

⎤⎢⎣

⎡=

E22E21

E12E11E AA

AAA , ⎥

⎦

⎤⎢⎣

⎡=

d

rE B

BB , ⎥

⎦

⎤⎢⎣

⎡=

d

rE E

EE (10)

Solving Equation (9) for dx by setting dx =0 and substituting for dx in Eq. (8), we

obtain the resulting system as

grrr

rr uEuBxAx &&& ++= (11)

gzr

zr

rzrz uFuDxCy &&++= (12)

gmr

mr

rmrm uFuDxCy &&++= (13)

where

E211

E22E12E11r AAAAA −−= , E21

E22E12E1r BAABB −−= (14)

E211

E22E2E1r AACCC −−= , E21

E22E2Er BACDD −−= (15)

E21

E22E12E1r EAAEE −−= , E21

E22E2Er EACFF −−= (16)

Figure 1 shows a comparison between transfer functions of the accelerations at

the midspan for the full-order and the reduced-order systems. It is observed that the

dynamics of the full-order system are represented quite reasonably in the

reduced-order system for all the dominated modes retained in Eq. (8).

3. SAMPLE PASSIVE CONTROL

In a passive system, the control force applied to the structure only depends on the

local motion of the structure between two points the damper is connected. A

nonlinear viscous damper is used as the sample passive control device. The damper

is assumed to be ideal, i.e., effects of device dynamics and heating in the device are

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not considered. A block diagram of the sample passive control system is shown in Fig.

2. The general form of a nonlinear viscous damper can be expressed as

)sgn()( iii ddctu && α

α= (17)

where id& is the piston velocity across the ith damper and αc is the experimentally

determined damping coefficient with the units of force per velocity raised to the α

power. In Eq. (17), α is a constant controlling damper nonlinearity with its range

between 0 and 1, and )sgn(⋅ is a signum function. When 1=α , Eq. (17)

becomes ii dctu &1)( = , which represents a linear viscous damper. Nonlinear viscous

dampers are generally less dependent on velocity as compared to the linear dampers.

The nonlinear force equation in Eq. (17) limits the peak damper forces at large

structural velocities while providing sufficient supplemental damping. When 0=α ,

)sgn()( 0 ii dctu &= represents a pure friction damper, whose force is independent of

the magnitude of the velocity.

By equating the energy dissipated by the nonlinear viscous damper in Eq. (17) with

the energy dissipated by an equivalent linear viscous damper, the equivalent

supplemental nonlinear viscous damping ratio dξ is obtained as (Terenzi, 1999; Pekcan

et al., 1999):

2

1 22

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

a

e

aNd T

Dm

C ππλ

ξ (18)

where m is the structural mass, D is the displacement of the structure at equivalent linear

viscous damping ratio, Ґ(.) is the gamma function, Te is the effective structural period

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and )a2(

)2/a1(242

a

+Γ+Γ

⋅=λ .

4. SAMPLE ACTIVE CONTROL

The active devices used in this study are modeled as ideal hydraulic actuators.

The ideal actuator is assumed to have the ability to instantaneously and precisely

supply the force commanded by the control algorithm. For the control design,

actuator dynamics are neglected and no actuator-structure interaction is considered

although these will occur in the physical system (Dyke et al., 1995).

A block diagram of the sample active system is shown in Fig. 3. Sensors,

controller and control device blocks are all required for an active control system.

Note that there are no connection inputs to the control devices because the actuator

dynamics are neglected and the control device model does not require any direct

input from the evaluation model.

4.1 Sensors

In the control of civil engineering structures, absolute acceleration measurements

are readily available. Additionally, measurements of the displacement across control

devices or the control forces are typically available. Eight accelerometers and four

displacement sensors are employed in the sample control system. Four

accelerometers are located at both deck-ends, and two are located at mid span of the

deck to measure the absolute accelerations in two horizontal directions, and other two

are located at both ends of the bent beam to measure the accelerations in transverse

direction. The natural frequencies of the selected accelerometers are assumed to have

a value that is at least an order of magnitude higher than the highest natural frequency

of the modeled dynamics. Hence, the sensor dynamics has been neglected. Each

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accelerometer is modeled as having a constant magnitude and phase. Four

displacement sensors are positioned between the mass center of deck and both

abutments (node pairs (1, 125), (32, 126)) to measure the displacements in

longitudinal and transverse directions of the bridge. Figure 4 shows the locations and

directions of all sensors installed on the bridge, which are used for feedback to the

control algorithm.

To ensure that the accelerations and displacement measured on the bridge are

within the range of the A/D converters, the sensitivity of each accelerometer is

defined as Ga = 10 V/g (i.e., 10Volts = 9.81 m/s2). The sensitivity of each

displacement sensor is Gd = 30 V/m (i.e., 10 V = 0.33 m). Thus, in state space form,

the sensor model can be written as

0xs =& (19)

vs += msyDy (20)

where sy is a vector of the measured absolute accelerations and device displacements

in Volts, my is the vector of measured continuous-time absolute accelerations and

device displacements in physical units (i.e., [m/sec2] for accelerations and [m] for

displacements), v is the measurement noise, and

⎥⎦

⎤⎢⎣

⎡=

×

×

44

88s I0

0ID

d

a

GG

(21)

The sensor block is represented in the SIMULINK block shown in Fig. 5. The gain

block in Fig. 5 converts the continuous-time acceleration measurements from

physical units to Volts. Finally, noise with an RMS value of 0.03 Volt, as is specified

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in the control constraints in the companion paper, is added to the acceleration and

displacement signal.

4.2 Control Devices

In the active case, hydraulic actuators are placed between the deck and abutments

at both ends of the bridge to control the benchmark highway bridge. The active

control forces produced by the actuators can be modeled as

uKf f= and uDy df = (22)

where dD is the gain of actuators, and dD =100 kN/V (i.e., 10Volts = 1000 kN). Fig. 6

shows the SIMULINK control device block. The actuator gain block converts the

input voltages to physical forces. fK is the matrix that accounts for multiple

actuators placed at the same device location and forces applied by control devices on

the bridge.

Control devices for these sample control strategies are placed orthogonally

between the deck-ends and both abutments of the bridge. There are total of 16

devices, 8 at each end of bridge, are employed to reduce the earthquake-indeuced

vibrations of the benchmark highway bridge for each of the three sample control

designs, as shown in Fig. 4.

The sensors and devices locations used for the sample control design are for

illustrative purpose only. Participants are encouraged to select their own number of

control devices and locations of sensors, depending on control algorithm and device

requirements.

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4.3 Control Algorithm

In the active case, the H2/LQG control algorithm has been selected for the sample

active control design. As described in previous section, a reduced-order control

design model, which is developed using the initially elastic evaluation model, is

employed for controller design. In this algorithm, the ground excitation is taken to be

a stationary white noise, and an infinite horizon performance index is chosen that

weights the regulated output vector, i.e.,

{ } ⎥⎦

⎤⎢⎣

⎡+++= ∫∞→

τ

τ τ 0

Tzr

rzr

Tzr

rzr Ruuu)DxC(Qu)DxC(1lim dtEJ (23)

The weighting matrices Q and R, which are used to appropriately weight the

regulated outputs and calculate the controller, are considered as follows

I10R -4= ⎥⎦

⎤⎢⎣

⎡=

I00I

Qa

d

qq

(24)

in which dq weights the midspan displacements and deformation of bridge bearings,

and aq weights midspan accelerations. Further, the measurement noise is assumed to

be identically distributed, statistically independent Gaussian white noise process with

25S/Siigg vvuu == &&&&gγ .

The separation principle allows the control and estimation problems to be

considered separately. Minimization of the performance index in Eq. (18) results in a

linear controller of the form (Spencer, et al., 1994)

rx̂Ku u−= (25)

where rx̂ is the Kalman filter estimate of the state vector, and uK is the full state

feedback gain matrix.

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The Kalman filter optimal estimator is given by

u)DxCL(yuBx̂Ax̂ mr

rmrmrr −−++= rr (26)

where L is the observer gain matrix of the stationary Kalman filter.

For implementation on a digital computer, the controller is converted through the

bilinear transformation (Antoniou, 1993；Quast et al. 1995) into the following

compensator

sk

ckc

ck yBxAx c1 +=+ (27)

sk

ckk yDxCu cc += (28)

Calculations to determine uK , L and the discrete time compensator are performed

using the control toolbox in MATLAB.

The SIMULINK block shown in Fig. 7 is used to represent the sample control

algorithm in the simulation. To represent the hardware used to implement this

algorithm on a digital computer, the input signal passes through a model of an

analog-to-digital converter (A/D) and the output control signal passes through a

model of a digital-to-analog converter (D/A). The model consists of a quantizer and a

saturator as described in the Control Implementation Constraints and Procedures in

the companion paper.

5. SAMPLE SEMI-ACTIVE CONTROL

Semi-active devices have been shown to possess the advantages of active control

devices without requiring the associated large power sources, and are inherently

stable (Fujino, et al., 1996; Spencer, et al., 1996). The semi-active control device

used in this sample semi-active design is the MR fluid damper. A clipped–optimal

control algorithm is implemented in this study because of its successful application in

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previous study (Dyke, et al., 1996, 1998, 1999; Yi et al., 2001) and an H2/LQG

controller is used as a primary controller to calculate the desired control force. The

block diagram of the sample semi-active control system is shown in Fig. 8. The

control devices and sensors are interfaced to the structural evaluation model though

measurement and device connection outputs, designated ym and yc, respectively.

5.1 Sensor Model

In the semi-active sample control case, the number and location of

accelerometers and displacement sensors employed are the same as those of the

active sample control system design. Additionally, the clipped optimal control

algorithm, described subsequently, requires measurement of each of the control

forces applied to the structure. Thus, 16 force transducers are installed to measure

each of the damper control forces. The sensitivity of the force transducers are Gf =

0.01 V/kN (i.e., 10 V = 1000 kN). Thus in the semi-active case, the sensor gain Ds in

Eq. (15) can be rewritten as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

×

×

×

1616f

44d

88a

s

I000I000I

DG

GG

(29)

The corresponding sensor block is represented in the SIMULINK block shown in Fig.

9. The sensor noises are included in all the measured responses.

5.2 Control Device

MR fluid dampers are used as the sample semi-active control devices. To

accurately predict the behavior of the controlled structure, adequate modeling of the

control device is essential. The phenomenological model of MR dampers is based on

the Bouc-Wen hysteretic model in parallel with a dashpot added for a nonlinear

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“roll-off” effect as shown in Fig. 10. This simple mechanical model has been shown

to predict well the behavior of the prototype shear-mode MR damper over a wide

range of inputs in a set of experiments (Dyke et al. 1999; Yi et al. 2001). The

equations governing the force produced by this model of MR damper are given as

zxc α+= &0f (30)

xAzx-βzzxγz nn-&&& +−= 1 (31)

where x is the displacement of the device; z is the evolutionary variable, and

An,,,βγ are parameters controlling the linearity in the unloading and the

smoothness of the transition from the pre-yield to the post-yield region. The

functional dependence of the device parameters on the command voltage cu is

expressed as

cbac uu αααα +== )( ; cbac uccucc 0000 )( +== (32)

In addition, the resistance and inductance present in the circuit introduce

dynamics into this system. These dynamics are accounted for by the first order filter

on the control input given by

)( acc uuu −−= η& (33)

where η is the time constant associated with the first order filter and au is the

command voltage applied to the current driver. The following parameters of the MR

damper were selected so that the device has a capacity of 1000 kN, as follows: aα =

1.0872e5 N/cm, bα = 4.9616e5 N/(cm V) , ac0 = 4.40 N sec/cm, bc0 = 44.0 N sec/(cm

V), n= 1, A= 1.2, γ = 3 cm-1, β = 3 cm-1, and η =50 sec-1. These parameters are based

on the identified model of a shear-mode prototype MR damper tested at Washington

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University (Yi et al. 2001) and scaled up to have maximum capacity of 1000 kN with

maximum command voltage = 10V. According to the device manufacturer’s

expectations, the device described here is assumed to require a maximum power of

50 Watts.

Typical force-velocity and force-displacement hysteretic loops for this device

model are shown in Fig. 11. Here, the device response is shown for various constant

voltages applied to control the input to the MR damper, and a 1.0 Hz sinusoidal

displacement with an amplitude of 5 cm.

Fig. 12 shows the SIMULINK control device block of MR dampers. Note that in

the semi-active sample control design, the device velocities are connected to the

control devices to calculate the control force, as the device dynamics are considered

in the controller design.

5.3 Clipped Optimal Control Algorithm

The clipped optimal control algorithm is used to calculate required control input

signal to the MR damper (Dyke, et al. 1996; Yi et al. 2001). In the clipped optimal

controller, the desired control forces, cF are calculated based on the measured

structural response vector and the measured control force vector

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

−= −

mc

1c F

)(KFy

LsL (34)

where {}⋅L is the Laplace transform operator, and )(Kc s is the selected primary

controller.

Because the force generated in the MR damper is dependent on the local

responses of the structural system, the MR damper cannot always produce the desired

optimal control force. Only the control voltage can be directly controlled to increase

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or decrease the force produced by the device. To induce the MR damper to generate

approximately the corresponding desired optimal control force, the command signal

is selected as follows. When the ith MR damper is providing the desired optimal

force (i.e., cii ff = ), the voltage applied to the damper should remain at the present

level. If the magnitude of the force produced by the damper is smaller than that of the

desired optimal force and the two forces have the same sign, the voltage applied to

the current driver is increased to the maximum level so as to increase the force pro-

duced by the damper to match the desired control force. Otherwise, the commanded

voltage is set to zero. This algorithm for selecting the command signal for the ith MR

damper is graphically represented in Fig. 13 and can be stated as (Dyke et al., 1996)

{ } )(max iicii fffHVv −= (35)

where maxV is the voltage to the current driver associated with saturation of the MR

effect in the physical device, and )(⋅H is the Heaviside step function.

6. EVALUATION OF SAMPLE CONTROL DESIGNS

The values of evaluation criteria for three sample control designs are reported in

Tables 1 to 3. The performance of each control system presented in Table 1-3

depends on the designs of the devices as well as control parameters used. For

nonlinear viscous dampers, we choose αC = 800 kN (m/s)-0.6 with α = 0.6 so that the

maximum control forces for all the six earthquakes do not exceed the limit of 1000

kN. Similarly, the weighting parameters for the active and semi-active control

designs are determined to be 8105.2 ×=dq , and aq = 0.1qd in this study. For the

active and semi-active case, the resulting control design model has 28 states.

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Table 1 shows the evaluation criteria for the sample passive control strategy. It is

observed from Table 1 that the nonlinear viscous dampers are capable of reducing all

response quantities effectively for El Centro, Chichi and Northridge earthquakes. For

N. Palm Springs earthquake, peak base shear and peak midspan acceleration increase

by 22.4% and 29.6%, respectively, and corresponding normed quantities increased

by 3.1% and 1.7% respectively. The peak ductility for this earthquake decreases by

41%. For Turkey earthquake, all response quantities decrease except that the normed

mid-span acceleration remains unchanged. For Kobe earthquake, peak and normed

mid-span acceleration increase by approximately 7% and 6% respectively. Although

the performance of nonlinear viscous dampers varies during different earthquakes,

their overall performance is quite reasonable. From safety point of view, only the

increase in peak base shear during N. Palm Springs earthquake by 22% may be

significant.

For the active and semi-active cases in Table 2 and 3, respectively, both the peak

and normed evaluation criteria are smaller than 1 for all earthquake records. Hence,

active and semi-active control systems are capable of reducing the highway bridge

responses for a wide variety of earthquake records. The hydraulic actuators and MR

dampers have the ability to adapt to different load conditions. In comparing the

performance of the active and semi-active control, it is observed that the overall

performance of the semi-active control system is generally similar or slightly

superior to that of the active control system. As the clipped optimal algorithm only

applies the maximum or zero voltage, larger forces may be produced by the

semi-active control system when the structure is subject to moderate or small

earthquakes compared to the active system. Notice that although larger control forces

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are required for semi-active systems, they do not require significant power associated

with the active control system.

It is interesting to note that the application of three sample control systems can

result in the reduction of the ductility factor (i.e. peak (J6) or normed (J14))

significantly. Further, in all of the cases in which plastic connections form in the

uncontrolled structure, dissipated energy of the curvatures at the end of bent columns

and the number of plastic connections are greatly reduced when control devices are

installed. Thus, damage in the bridge is significantly minimized.

The time history responses of the actively and semi-actively controlled bridge are

compared to those of the uncontrolled bridge for the El Centro earthquake as shown

in Fig. 14. It can be seen that all three sample control systems achieve significant

performance. The passive control system shows a slight improvement over the active

and semi-active control systems in reducing the base shear force and the midspan

displacement, while achieving a lower reduction in acceleration at the expense of

much more control forces applied than the active and semi-active controllers. The

semi-active control system achieves a performance very similar to that of the active

control system except the required control force for semi-active system is higher.

7. CONCLUDING REMARKS

In this paper, design for three sample control systems, namely nonlinear viscous

dampers, ideal actuators and semi-active magnetorheological (MR) fluid dampers,

are presented for the benchmark highway bridge. Each of three sample control

systems employs a total of 16 control devices located between the deck and abutment

to apply forces in the two horizontal directions. To facilitate the controller design, the

21

eigenmode reduction method is used to reduce the orders of the evaluation model.

Participants of this benchmark study are expected to use more competitive control

design for their own control strategy. These control strategies may be passive, active,

semi-active or a combination thereof.

The evaluation model of the benchmark highway bridge, MATLAB files used for

the sample control designs and the simulation model are available on the benchmark

web site: www-ce.engr.ccny.cuny.edu/People/Anil%20Kumar%20Agrawal.htm. If

you cannot access the World Wide Web or have questions regarding the benchmark

problem, please contact Dr. Agrawal at agrawal@ccny.cuny.edu. Participants in this

benchmark problem will be expected to submit their control designs and supporting

MATLAB files electronically for inclusion on the benchmark homepage.

ACKNOWLEDGMENT

The work was supported primarily by the National Science Foundation Grant

Number CMS 0099895 and in part by the Earthquake Engineering Research Center

Program of the National Science Foundation under NSF Award number

EEC-9701471. Any opinions, findings and conclusions or recommendations

expressed in this material are those of the authors and don’t necessarily reflect those

of National Science Foundation. The writers would like to thank Dr. Kim Saang Bum

of Smart Infra-Structure Technology Center at the Korea Advanced Institute of

Science and Technology for helping with the details of the eigenmode reduction

method. The authors sincerely acknowledge the support of the ASCE subcommittee

on benchmark problems and feedback from members of ASCE committee on

structural control and ASCE subcommittee on benchmark problems.

22

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7. Dyke, S. J., Spencer Jr., B. F., Sain, M. K., and Carlson, J. D. (1998). “An Experimental Study of MR Dampers for Seismic Protection.” Smart Materials and Structures: Special Issue on Large Civil Structures, Vol. 7, No.5, pp. 693–703.

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22. Pekcan, G, Mander, J. B., and Chen, S. S. (1999). “Fundamental consideration s for the design of non-linear viscous dampers”. Earthquake Eng. Struct. Dyn., Vol. 28, No. 11, pp.1405-1425.

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24

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25

10-1

100

101

102

-40

-20

0

20

Mag

nitu

de(d

B)

10-1

100

101

102

-60

-40

-20

0

20

Mag

nitu

de(d

B)

10-1

100

101

102

-300

-200

-100

0

Frequency (Hz)

Phr

ase(

deg)

10-1

100

101

102

-600

-400

-200

0

Frequency (Hz)

Phr

ase(

deg)

full-order systemreduced-order system

full-order systemreduced-order system

(a) (b)

Figure 1 Transfer Functions of Mid-span Accelerations: (a) Ground Acceleration in x-direction to Mid-span Acceleration in x-direction; (b) Ground Acceleration in y-direction to Mid-span Acceleration in y-direction

26

Figure 2 Sample Passive Control System SIMULINK Block

27

Figure 3 Sample Active Control System SIMULINK Block

28

Displacement transducer

Accelerometer

Control device

Figure 4 Locations and Directions of Control Devices and Sensors in Bridge

29

Acceleration.

Displacement1ys

SensorNoise

K*u

SensorGain2

K*u

SensorGain1

UU(E)

UU(E)

1ym

Figure 5 Sensor Block in Sample Active Control System Design

30

Number ofDevices

2yf

1f

K*u K*u

ActuatorGain

1u

Figure 6 Control Device Implementation in Sample Active Control

System Design

31

1u

y(n)=Cx(n)+Du(n)x(n+1)=Ax(n)+Bu(n)

Discrete Controller

u(k)u(t)

D/A Converter

y (t)y (k)

A/D Converter

1ys

Figure 7 SIMULINK Block: Control Algorithm

32

Figure 8 Sample Semi-active Control System SIMULINK Block

33

Acceleration

Displacement

Force

1ys

SensorNoise

K*u

SensorGain3

K*u

SensorGain2

K*u

SensorGain1

UU(E)

UU(E)

UU(E)

2yf

1ym

Figure 9 Sensor Model Implementation in Sample Semi-active Control

System Design

34

Figure 10 Mechanical Model of the MR Damper

35

-40 -30 -20 -10 0 10 20 30 40-1.5

-1

-0.5

0

0.5

1

1.5 x 106 F

orce

(N)

Velocity (cm/sec)

0.0volt1.0volt5.0volt10.0volt

-6 -4 -2 0 2 4 6-1.5

-1

-0.5

0

0.5

1

1.5 x 106

For

ce (N

)

Displacement (cm)

0.0volt1.0volt5.0volt10.0volt

Figure 11 Typical Responses of Employed MR Damper: (a) Force-Velocity Hysteresis Loop; (b) Force-Displacement Hysteresis Loop.

36

Velocity

Number ofDevices

3fm

2yf

1f

Mux

Memory(apply 1 step lag)

v elocity

v oltage

f orce

MR-damperModel

-1

Gain UU(E)

K*u

2u

1yc

Figure 12 Control Device Implementation in Sample Semi-active Control System Design

37

Figure 13 Graphical Representation of Clipped-Optimal Control Algorithm

38

0 5 10 15 20 25 30 35 40-1

0

1x 107

Time (sec)

Bas

e sh

ear (

N)

0 5 10 15 20 25 30 35 40-0.1

0

0.1

Time (sec)

Mid

span

Dis

p (m

)

0 5 10 15 20 25 30 35 40-5

0

5

Time (sec)

Mid

span

Acc

. (m

/s/s

)

0 5 10 15 20 25 30 35 40-5

0

5x 10

5

Time (sec)

Con

trol f

orce

(N)

UncontrolledPassiveActiveSemiactive

UncontrolledPassiveActiveSemiactive

UncontrolledPassiveActiveSemiactive

PassiveActiveSemiactive

(a)

(c)

(d)

(b)

Figure 14 Simulated response due to El Centro Earthquake: (a) base shear

forces; (b) midspan displacements; (c) midspan accelerations; (d) control forces

39

Table 1: Evaluation Criteria for the Passive Control Strategy

Criterion N.

Palm Springs

Chichi El Centro Northridge Turkey Kobe Aver.

J1 (peak base shear) 1.2241 0.7625 0.6364 0.7751 0.7760 0.8597 0.8390 J2 (peak base moment) 0.6334 0.9592 0.5757 0.9603 0.8792 0.5494 0.7595 J3 (peak midspan disp.) 0.6418 0.7133 0.6526 0.7034 0.5865 0.6319 0.6549 J4 (peak midspan acc.) 1.2959 0.9542 0.9403 0.9012 0.9366 1.0723 1.0168 J5 (peak bearing deformation) 0.3971 0.6705 0.3444 0.6611 0.5416 0.2750 0.4816 J6 (peak ductility) 0.6334 0.5977 0.5757 0.5853 0.1936 0.5494 0.5225 J7 (peak dissipated energy) 0 0.2265 0 0.3330 0 0 0.0933 J8 (plastic connection) 0 0.6667 0 0.7500 0 0 0.2361 J9 (normed base shear) 1.0313 0.7474 0.5176 0.7091 0.7383 0.7456 0.7482 J10 (normed base moment) 0.5294 0.7467 0.3933 0.7182 0.3662 0.5149 0.5448 J11 (normed midspan disp.) 0.5567 0.6451 0.4098 0.6504 0.4415 0.5439 0.5412 J12 (normed midspan acc.) 1.0174 0.7653 0.7425 0.7772 1.0082 1.0589 0.8949 J13 (normed bearing deformation)

0.2514 0.6165 0.2475 0.6160 0.2914 0.1957 0.3698

J14 (normed ductility) 0.5294 0.6570 0.3933 0.9941 0.0353 0.5149 0.5207 J15 (peak force) 0.0119 0.0238 0.0098 0.0217 0.0172 0.0133 0.0163 J16 (peak device stroke) 0.3822 0.6420 0.3167 0.6021 0.5369 0.2711 0.4585 J17 (peak power) - - - - - - J18 (total power) - - - - - - J19 (number of devices) 16 16 16 16 16 16 J20 (number of sensors) - - - - - - J21 (computational resource) - - - - - -

40

Table 2: Evaluation Criteria for the Active Control Strategy

Criterion N.

Palm Springs

Chichi El Centro Northridge Turkey Kobe

Aver.

J1 (peak base shear) 0.9502 0.8775 0.7905 0.8965 0.9121 0.7888 0.8693 J2 (peak base moment) 0.7699 0.9664 0.7425 0.9782 0.9779 0.7040 0.8565 J3 (peak midspan disp.) 0.8231 0.7994 0.7791 0.8669 0.7460 0.7045 0.7865 J4 (peak midspan acc.) 0.7941 0.8753 0.8829 0.8435 0.7983 0.8986 0.8488 J5 (peak bearing deformation) 0.9370 0.8028 0.6433 0.8826 0.7144 0.5862 0.7611 J6 (peak ductility) 0.7699 0.7433 0.7425 0.8516 0.4626 0.7040 0.7123 J7 (peak dissipated energy) 0 0.5119 0 0.6244 0.3317 0 0.2447 J8 (plastic connection) 0 0.6667 0 1.0000 0.3333 0 0.3333 J9 (normed base shear) 0.7426 0.8852 0.6757 0.8673 0.8937 0.7389 0.8006 J10 (normed base moment) 0.6964 0.8338 0.6433 0.8780 0.5316 0.7127 0.7160 J11 (normed midspan disp.) 0.7033 0.7842 0.6563 0.8047 0.6071 0.7293 0.7142 J12 (normed midspan acc.) 0.7233 0.7910 0.6852 0.7956 0.7946 0.7976 0.7645 J13 (normed bearing deformation)

0.4829 0.7837 0.4844 0.8211 0.5210 0.4720 0.5942

J14 (normed ductility) 0.6964 0.6476 0.6433 0.8274 0.2388 0.7127 0.6277 J15 (peak force) 0.0101 0.0238 0.0057 0.0230 0.0147 0.0079 0.0142 J16 (peak device stroke) 0.9019 0.7687 0.5916 0.8039 0.7082 0.5779 0.7254 J17 (peak power) 0.0512 0.1092 0.0213 0.1105 0.0664 0.0356 0.0657 J18 (total power) 0.0119 0.0150 0.0032 0.0150 0.0136 0.0064 0.0109 J19 (number of devices) 16 16 16 16 16 16 J20 (number of sensors) 12 12 12 12 12 12 J21 (computational resource) 28 28 28 28 28 28

41

Table 3: Evaluation Criteria for the Semi-active Control Strategy

Criterion N.

Palm Springs

Chichi El Centro Northridge Turkey Kobe Aver.

J1 (peak base shear) 0.9619 0.8416 0.7792 0.8857 0.9039 0.8174 0.8650 J2 (peak base moment) 0.7476 0.9781 0.7081 0.9790 0.9788 0.6642 0.8426 J3 (peak midspan disp.) 0.8024 0.7852 0.7753 0.8570 0.7166 0.6632 0.7666 J4 (peak midspan acc.) 0.9814 0.8757 0.8956 0.8993 0.8006 0.9858 0.9064 J5 (peak bearing deformation) 0.8121 0.7647 0.5662 0.8532 0.6744 0.5097 0.6967 J6 (peak ductility) 0.7476 0.6959 0.7081 0.8276 0.3717 0.6642 0.6692 J7 (peak dissipated energy) 0 0.4682 0 0.5674 0.2364 0 0.2120 J8 (plastic connection) 0 0.6667 0 1.0000 0.3333 0 0.3333 J9 (normed base shear) 0.7792 0.8457 0.5970 0.8288 0.8402 0.6909 0.7636 J10 (normed base moment) 0.6622 0.7984 0.5594 0.8378 0.5019 0.6560 0.6693 J11 (normed midspan disp.) 0.6825 0.7532 0.5797 0.7772 0.5732 0.6743 0.6734 J12 (normed midspan acc.) 0.7894 0.8074 0.6895 0.8178 0.8087 0.8376 0.7917 J13 (normed bearing deformation)

0.4543 0.7460 0.3929 0.7744 0.4045 0.3679 0.5233

J14 (normed ductility) 0.6622 0.6927 0.5594 0.7713 0.2204 0.6560 0.5937 J15 (peak force) 0.0109 0.0227 0.0084 0.0226 0.0157 0.0096 0.0150 J16 (peak device stroke) 0.7817 0.7322 0.5207 0.7772 0.6685 0.5025 0.6638 J17 (peak power) - - - - - - J18 (total power) - - - - - - J19 (number of devices) 16 16 16 16 16 16 J20 (number of sensors) 28 28 28 28 28 28 J21 (computational resource) 28 28 28 28 28 28