adaptive neural network controller for nonlinear highway ... nonlinear behaviour in a...

Jordan Journal of Civil Engineering, Volume 13, No. 2, 2019

- 308 - © 2019 JUST. All Rights Reserved.

Adaptive Neural Network Controller for Nonlinear Highway Bridge Benchmark

Ahmad Y. Rababah 1), Khaldoon A. Bani-Hani 1)* and Wasim S. Baraham 1)

1) Civil Engineering Department, Jordan University of Science and Technology, Irbid, Jordan. * On sabbatical leave at Qatar University.

ABSTRACT

In this paper, a neural network-based active control algorithm is proposed and evaluated for a seismically

excited highway bridge. A nonlinear three-dimensional highway bridge model equipped with 16 active

hydraulic actuators placed orthogonally between the deck-ends and the abutments is employed to demonstrate

and evaluate the developed method. The control strategy proposes a training emulator neural network model

that operates online to generate the training data for the controller. The neural network controller is trained by

the aid of the emulator neural network and by back propagating the control signal error through the emulator

neural network. An H2/LQG control algorithm is designed for the bridge and results are compared to those of

the proposed method. Performance indices for the benchmark bridge response are defined, computed and

compared. The results revealed that the controller was quite effective in seismic response reduction for a wide

range of ground motions. Also, it was robust and stable enough so that it was not sensitive to either sensor noise

or sensor failure.

KEYWORDS: Neural networks, Active control, Highway bridge benchmark, LQG, Bridge, Vibration control, Dynamic system modeling, Acceleration sensor, Smart structures.

INTRODUCTION

Structural vibration induced by an earthquake or

large wind forces can be destructive for structures during

their lifetime. Severe vibrations can cause serious

structural damage and likely ground for different

structural failure scenarios. Consequently, significant

studies focused on the development of self-adjusting

structures through adaptive control systems to alleviate

structural vibrations due to severe environmental forces,

particularly seismic response of buildings and bridges.

Many researchers attempted to improve the

performance of civil structures under severe

environmental loads, such as earthquakes, winds and

blast waves. It has become a vital issue to revolutionize

structural design methods to sustain the increased safety,

reliability and economic considerations. Incorporate

life-like functions of sensing, actuation, control and

intelligence to civil structures have always been an

attractive tempting idea. Developing intelligent, robust

and adaptive systems has motivated civil engineers and

researchers in the past several decades. Such efforts

resulted in substantial advancement toward more

reliable, safer, lighter and more vigorous structures.

There are many proposed and well-defined methods

to accomplish the structural control strategy. These

methods fall into two categories according to their

dynamics and energy requirements: active control and

passive control. Moreover, combinations of these two

categories produced new control methods, such as

hybrid control and semi-active control. Active control

systems are those systems that accomplish the control

objectives with external source powers (actuators, Received on 20/11/2018. Accepted for Publication on 23/1/2019.


- 309 -

motors,… etc.) that apply forces in a selective manner.

James Yao (1972) has introduced the first reliable

study of active control system theories and designs in the

early 1970s, where the first clear vision of the active

control concept and its implementation have been

cultivated for civil engineering applications. After Yao's

prominent system, systematic advances were achieved in

the active control of civil structures inspired by the fast

improvements of sensors, computers and mechanical

systems (Roorda, 1975; Yang et al., 1978; Chang et al.,

1980; Abdel-Rohman et al., 1983; Yang et al., 1979a,

1979b; Abdel-Ghaffar, 1991; Warnitchai et al., 1993).

Interestingly, artificial neural networks are still the

most appropriate candidate among structural control

methods. Neural networks have been depicted as a kind

of dream-like alternative to the conventional control

methods due to their inherent nonlinearity, flexibility,

noise immunity, generalization ability and robustness.

Some researchers have shown interest in the use of

neural networks in structural control applications. The

objective was to find practicable solutions to

conventional methods. As a result, neural networks have

been found to be well suited for complex linear and

nonlinear control problems. The method of structural

control using neural networks has been proposed and

developed in previous studies by several researchers. A

neurocontrol method based on the inverse transfer

function was developed by Bani-Hani (2007) to control

tall buildings under wind excitation with ATMD. A

neurocontrol method, which utilized an emulator neural

network in its training, was developed and applied in

linear and nonlinear structural control problems

(Ghaboussi, 1994; Ghaboussi and Joghataie, 1995;

Ghaboussi and Bani-Hani, 1996; Bani-Hani and

Ghaboussi, 1998a, 1998b; Bani-Hani et al., 1998c,

1998d). Due to the popularity and importance of

structural control, several related textbooks and review

papers have been presented. Housner et al. (1997)

provided a brief point of departure for researchers and

practitioners and discussed passive, active and semi-

active control systems, in addition to presenting a brief

review of hybrid control systems and exploring the

potential of control theory in structural vibration control.

This study introduces a neural network algorithm for

active control of a seismically excited highway bridge.

The highway bridge was modeled considering the

nonlinear behaviour in a three-dimensional finite element

model with 16 active actuators placed orthogonally

between the deck-ends and the abutments. To compare

the results and evaluate the success of the neurocontrol

method, an H2/LQG control algorithm is designed for the

bridge and the results are compared and discussed.

Structural Model

The highway bridge benchmark that has been

studied is the newly constructed 91/5 overcrossing,

located in Orange County of southern California as

shown in Figs. (1 and 2). The bridge has two-continuous

spans, each 58.5m (192 ft) long, spanning a four-lane

highway and has two abutments skewed at 33⁰. The

width of the deck along the east span is 12.95 m (42.5

ft) and it is 15 m (49.2 ft) along the west direction. The

cross-section of the deck consists of three cells. The

deck is supported by a 31.4 m (103 ft) long and 6.9 m

(22.5 ft) high prestressed outrigger, which rests on two

pile groups, each consisting of 49 driven concrete

friction piles. The highway bridge is a benchmark

problem which was proposed for researchers to

investigate their control strategies. Two phases are

proposed in the benchmark bridge. In phase I, base

isolation is at the abutments only, with four nonlinear

lead-rubber bearings (LRBs) utilized to isolate the deck

at each abutment. In addition, the deck of the bridge is

fixed to the center outrigger and is expected to exhibit

inelastic behaviour under severe ground motions. In

phase II, the model of the highway bridge has been

developed by installing lead–rubber bearings between

the deck and the center outrigger to simulate the

behaviour of a base-isolated highway bridge. This paper

adopted phase I for the neurocontrol method. More

details can be found in the benchmark definition papers

of Agrawal and Tan (2008).

Adaptive Neural… Ahmad Y. Rababah, Khaldoon A. Bani-Hani and Wasim S. Baraham

- 310 -

Figure (1): View of 91/5 highway over-crossing

Figure (2): Elevation and plan views of the benchmark bridge showing controller and sensor locations


- 311 -

16 control devices, 8 for each end of the bridge, are

proposed to be placed orthogonally between the deck-

ends and abutments for the reduction of earthquake

responses of the highway bridge. 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. A full three-dimensional (3D) finite-

element model with 430 degrees of freedom was

developed in (ABAQUS) by Agrawal et al. (2009). A set

of 21 evaluation criteria was developed to evaluate the

effectiveness of different control systems. A summary

of the evaluation criteria is presented in Fig. 3.

Figure (3): Summary of evaluation criteria

After the 3D FEM model was developed, natural

frequencies and mode shapes were performed. The

representative mode shape for the first 6 modes is shown

in Figure 4.


- 312 -

Figure (4): Six mode shapes of the 3D bridge model

Earthquake Excitations

Ground motions are recorded for most events and

classified due to their strengths. Six moderate to severe

earthquakes are used to investigate the performance and

effectiveness of the neural network system. These

earthquakes are:

1. North Palm Springs (1986)

2. TCU084 component of the Chi-Chi earthquake,

Taiwan (1999)

3. El Centro component of (1940) Imperial Valley

earthquake

4. Rinaldi component of Northridge (1994)

earthquake

5. Bolu component of Duzce Turkey (1999)

earthquake

6. Nishi-Akashi component of Kobe (1995) earthquake.

Time histories of these earthquakes are shown in

Fig. 5.

Figure (5): Two components of time histories of earthquake records


- 313 -

State-Space Representation

The classical control methods that have been used

were based on the input-output representation. These

methods do not use any knowledge of the interior

structure and limit the designer to a single-input single-

output system, allowing limited closed-loop control. By

describing a dynamic system as a set of coupled first-

order differential equations in a set of internal variables

known as state variables, the concept of the state of a

dynamic system refers to a minimum set of variables,

known as state variables that fully describe the system

and its response to any given set of inputs. Because of

the nonlinear properties of the system in this case of

study, using state-space representation has been

prepared in order to transform the equation of motion for

a structural system of a second-order differential

equation to a first-order differential equation.

Then, the equation of motion for a structural system

with actuator under earthquake excitation is formulated

as follows:

ƞÜ (1)

This formula can be written as:

ƞÜ (2)

In matrix form:

ẋẍ

0 0M K M C

xx

0M b

F t0ƞ Üg t (3)

To transform second-order to first-order differential

equation, the following formulation can be introduced:

Z ẋẍ (4)

Two state-space equations can be written to describe

the overall controlled system, as follows:

Z AZ BÜg EF (5)

Y CZ DF (6)

The matrices can be written as:

Z XX (7)

A0 eyeM K M C

(8)

B0 0M b ƞ (9)

C eye 0A

(10)

D 0B (11)

Neural Network Controller

Artificial neural network (ANN) is a smart method

inspired by the structure of the nerve cells of the human

brain. ANN is a network of interconnected nonlinear

processing units; the knowledge is stored in the

interconnected units or weights by a process of adaption

to learn from a set of training patterns. Artificial neural

networks have been widely used in the field of structural

engineering in recent years. The backpropagation

method is mostly used to minimize the error of all output

neurons. Backpropagation Through Time Neural

Controller (BTTNC), developed for active control of

structures under dynamic loadings, consists of two

components: (1) the emulator neural network to

represent the structure to be controlled; and (2) a neural

action network to determine the control action on the

structure as discussed by Chen (1995).

The emulator neural network was designed and

trained to learn the system to predict the responses of the

structure from the history of the structural responses.

This system is necessary to be introduced to model the

relationship between control commands from neuro-

controller and the response of the structure. The

responses of a structure subjected to ground acceleration


- 314 -

and control command u(t) can be described by the

following equation:

(12)

where M, C and K are mass, damping and stiffness

matrices’ vector, respectively. Z(t) is the displacement

vector. It can be written in the state-space equation as

follows:

(13)

(14)

The matrices A, B, F and f(t) can be determined as

follows:

0;

0 (15)

0; (16)

The purpose of the emulator neural network is to

provide a path for backpropagation error in the training

of the neurocontroller. In the training process of the

emulator, the weights are first initialized with small

random numbers. Then, the outputs are computed by

feeding forward the inputs through the network. The

error function (E) is calculated from the difference

between the outputs of the emulator network and the

outputs of the structure. By backpropagating the error

function (E) to adjust the weights, the emulator neural

network can be trained to reach the desired accuracy for

modelling the dynamic behaviour of the real structure.

For training the emulator neural network, delayed

responses from history and delayed forces from noise in

volts were used as inputs. Responses without delay from

history were also used as outputs.

Figure (6): Structure and training of BTTNC

Comparison of the nonlinear model or the structure

and emulator neural network responses for different six

earthquakes (N.Palm Springs, Chi-Chi84, El Centro,

Rinaldi, Turkey and Kobe), respectively, is shown in the

following figures. The figures show two displacement

responses in different locations in addition to

acceleration responses.


- 315 -

Figure (7): Comparison of the nonlinear model and emulator neural network responses for N. Palm Springs earthquake

Figure (8): comparison of the nonlinear model and emulator neural network responses for ChiChi084 earthquake


- 316 -

Figure (9): Comparison of the nonlinear model and emulator neural network responses for

El Centro earthquake


Rinaldi earthquake


- 317 -


the Turkey earthquake

Figure (12): Comparison of the nonlinear model and emulator neural network responses for the Kobe earthquake

The methodology for the training of the neuro-

controller, used in this study, is based on using emulator

neural networks to develop the training dataset for the

controller. In this method, the objective is to find the

appropriate control signal that achieves the control

criterion. However, this knowledge is not available to


- 318 -

the designer. Therefore, with the aid of the emulator

neural network, the control signal is predicted and

estimated. The neuro-control method is based on an

iterative search for the control signal u that achieves the

control criterion at each digitized time step.

First, a particular ground motion with no control

force excites the system. Next, the system response yk is

collected at time step k, which is then modified by a

reduction factor co, to define ∗ as the

reference uncontrolled system response at zero control

signal, uk =0 and time step = k. Next, the challenge is to

find the required control command uk that achieves the

assigned control criteria for the time step k. This can be

done by assuming that the control signal that satisfies

the control criteria exists somewhere between the upper

and lower limits of the control signal (umax and umin).

Consequently, the search for this control signal is

conducted at each time step. For time step k, the control

signal is varied alternately between zero and the upper

and lower limits (umax and umin) by an increment ∆ as

follows:

1 , 0.0, 1, . . . . . , (17)

where m is an integer which determines the number

of control signal trials at .

For the time step k, the previous time history

response yk-1 is available. Then, for each j=1,…., m

increment, the response yk-1 and its history as well as

defined by Equation (17) are fed to the first emulator

neural network, Consequently, the predicted response at

the present time is collected. Next, the predicted

response is assumed to be approximately equal to the

system response at time step k, . In this way,

the emulator neural networks have been used to predict

and collect the system responses from the previous

system response and the control signal .

The estimated equivalent response for the

control signal is used with the collected system

response at time step k ( ) to check the validation of

the control criterion. If the control criterion is met, the

control signal is chosen to be the appropriate control

signal for the time step k; . The control

criterion can be set by the designer to achieve different

control objectives. The following control criterion is

used in this phase of the study:

0

j j

p p

kk ky y and y (18)

The control criterion in Equation (18) should be

satisfied. This procedure is repeated at each time step k

for the duration of the ground motion record chosen to

train the neuro-controller. Clearly, this method has two

nested loops for the control signal increments j=1, ..., m.

The final training data for the neuro-controller is

collected for the control signals that achieved the control

criteria and from the associated system response. The

training pattern generation process is summarized in

Figure 13.

u

uum minmax


- 319 -

BridgeSystem

Response oC

)(..

tX g

EmulatorENN1

Random Signal

ku

Control Criterion

1Z

1Z

1ky

Ku

1ZKgX

..

pky

oky

ky1Z

Emulator Neural Networks

Kup

ky

Figure (13): Schematic diagram showing the training pattern generation process for

the neuro-controller using emulator neural networks

The network architecture was designed to have 38

input neurons representing three acceleration inputs

located at (left deck-end, midspan and up beam-end) and

two displacement inputs located (between end-deck and

abutment at left and right). All were delayed 6 times to

give 30 inputs. These inputs were derived from

structural response and emulator. In addition, two force

inputs located at both ends are delayed 4 times to give 8

inputs; these force inputs were used to simulate the

backpropagation and were derived from the emulator by

using noise voltage. Two hidden layers with 30 and 20

neurons or nodes, respectively, were selected by trial

and error. Finally, the output layer has two nodes that

represent the desired control voltage. Because of the

similarity of outputs in the same directions, two outputs;

one for the x-direction and the other for the y-direction,

have been used. Then, these outputs have been

duplicated eight times for each direction to give 16

commands that are supplied to 16 actuators to produce

the required forces to mitigate seismic loads. The final

neural network architecture is shown in Fig. 14.

Figure (14): Architecture and topology of the neural network model


- 320 -

The results of neuro-controller are shown in the

following figures for Chichi084 and Kobe earthquakes

as samples of all results. The upper figures refer to the

x-direction and the lower figures refer to the y-direction.

Also, the transfer function from ground acceleration to

the midspan displacement and accelerations for

earthquakes can be seen in the same figures. The values

of indices are represented in Table 1. This table makes a

comparison between neuro-controller and H2/LQG

controller. The peak points for uncontrolled, H2/LQG-

controlled and neural network-controlled cases for each

earthquake are shown in Table 2. This table shows a

comparison between the controllers and gives an idea

about the efficiency of the neurocontroller.

Figure (15): Displacement and acceleration responses and transfer function for the ChiChi084 earthquake


- 321 -

Figure (16): Displacement and acceleration responses and transfer function for the Kobe earthquake

Table 1. Results of the neuro-controller compared with (H2/LQG) controller

N. PalmSpr. ChiChi84 El Centro Rinaldi Turkey Kobe

J1 0.929 (0.950)

0.758 (0.877)

0.646 (0.790)

0.788 (0.896)

0.658 (0.912)

0.755 (0.789)

J2 0.648 (0.770)

0.955 (0.966)

0.581 (0.742)

0.954 (0.978)

0.723 (0.978)

0.573 (0.704)

J3 0.698 (0.823)

0.657 (0.799)

0.646 (0.779)

0.749 (0.867)

0.475 (0.746)

0.596 (0.704)

J4 0.823 (0.794)

0.873 (0.875)

0.854 (0.883)

0.879 (0.844)

0.677 (0.798)

1.008 (0.899)

J5 1.025 (0.937)

0.643 (0.803)

0.712 (0.643)

0.758 (0.883)

0.807 (0.714)

0.656 (0.586)

J6 0.648 (0.770)

0.530 (0.743)

0.581 (0.742)

0.648 (0.852)

0.159 (0.463)

0.573 (0.704)

J7 0.000 (0.000)

0.166 (0.512)

0.000 (0.000)

0.472 (0.624)

0.000 (0.332)

0.000 (0.000)

J8 0.000 (0.000)

0.500 (0.667)

0.000 (0.000)

1.000 (1.000)

0.000 (0.333)

0.000 (0.000)

J9 0.680 (0.743)

0.759 (0.885)

0.521 (0.676)

0.786 (0.867)

0.670 (0.894)

0.657 (0.739)

0 5 10 15 20 25 30 35 40 45-0.04

-0.02

0

0.02

0.04Uncontrolled LQG/H2 Controller Neuro-Controller

0 5 10 15 20 25 30 35 40 45-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25 30 35 40 45-10

-5

0

5

10Uncontrolled LQG/H2 Controller Neuro-Controller

0 5 10 15 20 25 30 35 40 45

Time (sec)

-5

0

5


- 322 -

J10 0.589 (0.696)

0.744 (0.834)

0.479 (0.643)

0.858 (0.878)

0.358 (0.532)

0.594 (0.713)

J11 0.631 (0.703)

0.606 (0.784)

0.500 (0.656)

0.700 (0.805)

0.428 (0.607)

0.630 (0.729)

J12 0.798 (0.723)

0.683 (0.791)

0.595 (0.685)

0.736 (0.796)

0.777 (0.795)

0.831 (0.798)

J13 0.480 (0.483)

0.604 (0.784)

0.492 (0.484)

0.722 (0.821)

0.531 (0.521)

0.481 (0.472)

J14 0.589 (0.696)

0.605 (0.648)

0.479 (0.643)

1.091 (0.827)

0.035 (0.239)

0.594 (0.713)

J15 0.014 (0.010)

0.024 (0.024)

0.011 (0.006)

0.024 (0.023)

0.024 (0.015)

0.015 (0.008)

J16 0.986 (0.902)

0.616 (0.769)

0.655 (0.592)

0.690 (0.804)

0.800 (0.708)

0.647 (0.578)

J17 0.065 (0.051)

0.131 (0.109)

0.027 (0.021)

0.130 (0.110)

0.096 (0.066)

0.054 (0.036)

J18 0.015 (0.012)

0.018 (0.015)

0.004 (0.003)

0.018 (0.015)

0.020 (0.014)

0.010 (0.006)

J19 16.000 (16.000)

16.000 (16.000)

16.000 (16.000)

16.000 (16.000)

16.000 (16.000)

16.000 (16.000)

J20 12.000 (12.000)

12.000 (12.000)

12.000 (12.000)

12.000 (12.000)

12.000 (12.000)

12.000 (12.000)

J21 28.000 (28.000)

28.000 (28.000)

28.000 (28.000)

28.000 (28.000)

28.000 (28.000)

28.000 (28.000)

Table 2. Peak points of displacement and acceleration at midspan for the earthquakes

El Centro N. Palm Springs

ChiChi08 Rinaldi Turkey Kobe

Displacement in x direction (m) (m) (m) (m) (m) (m)Uncontrolled 0.014 0.020 0.028 0.030 0.031 0.024

H2/LQG 0.014 0.015 0.024 0.024 0.021 0.020

Neural Network 0.014 0.011 0.021 0.023 0.021 0.019

Displacement in y direction (m) (m) (m) (m) (m) (m)Uncontrolled 0.056 0.058 0.292 0.310 0.181 0.057

H2/LQG 0.044 0.051 0.234 0.269 0.127 0.047

Neural Network 0.037 0.043 0.192 0.233 0.081 0.045

Acceleration in x direction (m/s^2) (m/s^2) (m/s^2) (m/s^2) (m/s^2) (m/s^2)Uncontrolled 2.565 5.900 4.466 6.913 7.629 5.169

H2/LQG 1.772 4.538 4.579 4.677 6.668 3.660

Neural Network 1.996 4.421 4.444 5.810 6.534 3.808

Acceleration in y direction (m/s^2) (m/s^2) (m/s^2) (m/s^2) (m/s^2) (m/s^2)Uncontrolled 3.258 3.599 11.271 9.682 8.485 4.421

H2/LQG 2.361 2.728 8.529 7.276 7.284 4.007

Neural Network 2.107 3.010 8.602 7.774 6.665 4.660


- 323 -

CONCLUSION

An adaptive neural network (ANN) system is used in

this study to develop active controllers. This control

system was designed and trained to represent the

nonlinear behaviour of the highway bridge structure.

The active controller produces the appropriate command

voltage to the actuators to reduce the seismic responses

of the structure.

Neural network controller was considerably able to

reduce the destructive acceleration and displacement

responses, as shown by the results using a set of

performance indices. The results show that

neurocontroller performance was very accurate and

better than that of the H2/LQG controller in reducing

responses. As a prelude for the neuro-controller design,

an emulator based on neural network model has been

also designed. This emulator learned to predict the

response of the structure from the history to model the

relation between the control command and the structural

response. Finally, the controller system was robust

under a wide range of ground motion when subjected to

six different earthquake records. Additionally, the

system is stable enough, so that it is not sensitive to

either sensor noise or sensor failure.

REFERENCES

Abdel-Ghaffar, A.M., and Khalifa, M.A. (1991).

“Importance of cable vibration in dynamics of cable-

stayed bridges”. Journal of Engineering Mechanics, 117

(11), 2571-2589.

Abdel-Rohman, M., and Leipholz, H.H. (1983). “Active

control of tall buildings”. Journal of Structural

Engineering, 109 (3), 628-645.

Agrawal, A., Tan, P., Nagarajaiah, S., and Zhang, J. (2009).

“Benchmark structural control problem for a seismically

excited highway bridge—part I: phase I-problem

definition”. Structural Control and Health Monitoring:

The Official Journal of the International Association for

Structural Control and Monitoring and of the European

Association for the Control of Structures, 16 (5), 509-

529.

Bani-Hani, K., and Ghaboussi J. (1998a). "Nonlinear

structural control using neural networks". J. Engrg.

Mech. Div., ASCE, 124 (3), 319-327.

Bani Hani, K., and Ghaboussi, J. (1998b). "Neural network

for structural control of a benchmark problem-active

tendon system". Earthquake and Structural Dynamics,

27.

Bani-Hani, K., Ghaboussi, J., and Schneider, S.P. (1998c).

“Experimental study of neural network-based structural

control". Proceedings of the Sixth U.S. National

Conference on Earthquake Engineering, Seattle,

Washington.

Bani-Hani, K., Ghaboussi, J., and Schneider, S. P. (1998d).

“Experimental study of structural control using neural

networks." Proceedings of Second World Conference on

Structural Control, Kyoto, Japan.

Bani‐Hani, K.A. (2007). “Vibration control of wind‐induced

response of tall buildings with an active tuned mass

damper using neural networks”. Structural Control and

Health Monitoring: The Official Journal of the

International Association for Structural Control and

Monitoring and of the European Association for the

Control of Structures, 14 (1), 83-108.

Chang, J.C., and Soong, T.T. (1980). “Structural control

using active tuned mass dampers”. Journal of

Engineering Mechanics Division, 106 (6), 1091-1098.

Chen, H.M., Tsai, K.H., Qi, G.Z., Yang, J.C.S., and Amini,

F. (1995). “Neural network for structure control”.

Journal of Computing in Civil Engineering, 9 (2), 168-

176.

Ghaboussi, J. (1994). “Some applications of neural

networks in structural engineering”. Analysis and

Computation, 25-36. ASCE.


- 324 -

Ghaboussi, J., and Bani-Hani, K. (1996). “Neural network-

based nonlinear structural control methods”. In: Proc.

2nd Int. Workshop on Structural Control.

Ghaboussi, J., and Joghataie, A. (1995). “Active control of

structures using neural networks”. Journal of

Engineering Mechanics, 121 (4), 555-567.

Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos,

A.G., Claus, R.O., Masri, S.F., Skelton, R.E., Soong,

T.T., Spencer, B.F., and Yao, J.T. (1997). “Structural

control: past, present and future”. Journal of Engineering

Mechanics, 123 (9), 897-971.

Narasimhan, S. (2009). “Robust direct adaptive controller

for the nonlinear highway bridge benchmark”. Structural

Control and Health Monitoring: The Official Journal of

the International Association for Structural Control and

Monitoring and of the European Association for the

Control of Structures, 16 (6), 599-612.

Roorda, J. (1975). “Tendon control in tall structures”.

Journal of Structural Division, 101 (3), 505-521.

Spencer Jr, B.F., Dyke, S.J., and Deoskar, H.S. (1998).

“Benchmark problems in structural control: part I—

Active mass driver system”. Earthquake Engineering

and Structural Dynamics, 27 (11), 1127-1139.

Tan, P., and Agrawal, A. (2008). “Benchmark structural

control problem for a seismically excited highway

bridge—part ii: phase I-sample control design”. Journal

of Structural Control and Health Monitoring. DOI:

10.1002/stc.300.

Yang, J.N., and Giannopolous, F. (1979). “Active control

and stability of cable-stayed bridge”. Journal of

Engineering Mechanics, 105 (4).

Yang, J.N., and Giannopoulos, F. (1978). “Active tendon

control of structures”. Journal of Engineering Mechanics

Division, 104 (3), 551-568.

Yang, J.N., and Giannopoulos, F. (1979). “Active control of

two-cable-stayed bridge”. Journal of Engineering

Mechanics, 105 (5).

adaptive neural network controller for nonlinear highway ... nonlinear behaviour in a...

Documents