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Decentralized Overlapping Control

Design for a Cable-Stayed BridgeBenchmark

Lubomr Bakule1, Fideliu P

ule -Cr

iniceanu2, Jos Rodellar3

and Josep M. Rossell31Institute of Information Theory and Automation, Academy of Sciences, Prague, Czech Republic2Department of Structural Mechanics, Gh. Asachi Technical University of Iasi, Romania3Department of Applied Mathematics III, Technical University of Catalunya, Barcelona, Spain

ABSTRACT

The paper is focused on overlapping decentralized control. This methodology was chosen tobe suitable for application to an international benchmark problem. After describing the

theoretical approach and the main steps in solving the overlapping control problem, thepaper presents the model of the bridge used for benchmark. A decomposition is proposedand applied. Results are shown in terms of the responses to the external actions,comparison with the non-controlled case and benchmark criteria. The methodology isshown to be reliable and gives results close to or better than the centralized control.

1. INTRODUCTION

In this paper, it is attempted to explore the possibility of applying overlapping decentralizedcontrol tools to the cable-stayed bridge benchmark control problem proposed in (Dyke et al.2000). This problem deals with a long span cable-stayed bridge with two main towers with

over one hundred cables attached. Potential motivating advantages for using decentralizedcontrol schemes are the reduction of transmission costs within the feedback loop, theincreasing of the reliability of the control operation in case of sensor/actuator/controllerfailures, the reduction of overall computational effort and the ability of parallelimplementation in real time.

The paper constructively describes a procedure in which the overall finite element model(FEM) of the benchmark cable-stayed bridge is decomposed into two overlappingsubsystems. Overlapping decentralized control design has been described for instance in(Bakule et al. 1995; Siljak 1991). By expanding the original LQG problem into a largerspace, the overlapping information sets become disjoint and the expanded LQG problem


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2 DECENTRALIZED CONTROL DESIGN FOR A CABLE-STAYED BRIDGE BENCHMARK

can be solved by standard decentralized methods. This design is made first by performing amodel reduction for each subsystem in expanded space. In this study the effectiveness of theoverlapping decentralized control approach is tested by numerical simulations. To do this,the designed decentralized controllers are implemented into the original overall FEMmodel. And, to measure the performance, closed-loop eigenvalue analysis, calculation ofevaluation criteria given in the benchmark problem, and analysis of time history responsefor selected earthquake excitations are used.

2. PROBLEM STATEMENT

Consider the cable-stayed bridge illustrated in Figure 1. It is composed of two towers and128 cables. The bridge is independently excited by three earthquakes longitudinalacceleration: ElCentro NS 1940, Mexico City NS 1985 and Gebze NS 1999. These actionshave maximum values: 3.4170 m/s2, 1.4068 m/s2, and 2.5978 m/s2, respectively. Fiveaccelerometers and four displacement sensors are used to supply feedback information forthe control, which is produced by 24 hydraulic actuators located between the deck and thetowers and the end supports acting to apply longitudinal forces on the deck. A completephysical description of the bridge, a finite element model and a MATLAB/SIMULINKsimulation framework are given in (Dyke et al 2000) as a benchmark for control design. Acentralized LQG control design is also presented in (Paulet-Crainiceanu et al. 2001).

The objectives of this study are the following:

1. To propose a convenient overlapping decomposition of the bridge structure withoverlapping subsystems.2. To design an overlapping decentralized LQG active control strategy.3. To perform simulations to assess the dynamic behavior of the benchmark bridge

model when using the implemented decentralized control.4. To assess the performance of the overlapping decentralized control by checking

eigenvalues for the closed-loop system, calculating benchmark evaluation criteria andanalyzing dynamic responses under selected benchmark earthquake excitations incomparison with results obtained with the sample centralized control design.

3. SOLUTION

This section is divided into three parts: Overlapping decomposition, Overlappingdecentralized control design and Simulation results

3.1. Overlapping decomposition

We propose the decomposition of the bridge model into two subsystems. Each subsystemcorresponds to one of the towers, the cables attached to it and the part of the deck where thecables are attached. This is illustrated in Figure 1. Both subsystems are interconnected


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BAKULE, PAULET AND RODELLAR 3

through the center of the bridge. It corresponds with the part of the deck where no cable isattached. The overlapping decomposition procedure considers the towers stronglyconnected via deck in the part where no cable is attached. When the original model isextended, it defines a state space model in a larger space with the structure of disjointsubsystems and interconnections.

More precisely, the overall original FEM model consists of 838 states. By properly re-arranging the components of the state, input and output vectors, the overall model can besplit into two overlapping subsystems. These subsystems have 412 and 432 states. Theoverlapping subsystem has 6 states. The overlapping common part includes one sensor butno actuator.

subsystem 1 subsystem 2

Figure 1. Bridge model and its overlapping decomposition structure

3.2. Overlapping decentralized control design

First, the benchmark sample LQG design has been selected as a reference. Further, thecontrol subsystems have been defined with the same locations and models of sensors and

actuators as in the reference case. The global model has 8 control inputs and 13 measuredoutputs. The expanded system has two subsystems with 412 and 432 states, 4 and 4 controlinputs, and 7 and 7 measured outputs, respectively. There are 14 measured outputs in theexpanded space because the overlapped part includes a sensor that is also expanded. Itmeans that the commonly shared states and the sensor are doubled in the expanded space toobtain disjoint subsystems.

A decentralized control law is proposed for each free subsystem by combining its modelreduction and the LQG design on the reduced order subsystems. Model reduction firstforms a balanced realization and then condenses out the states with relatively smallcontrollability and observability grammians.


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The following algorithm describes this overlapping LQG control design process.

3.3. Algorithm

1. Expand the original LQG problem with identified overlapping subsystems into alarger expanded space.

2. Perform model reduction for each subsystem. Select a minimal order of thesubsystems states ensuring the stability of the reduced-order models.

3. Perform the LQG with preselected weighting matrices for reduced order subsystems.4. Contract and implement the local controllers into the original overall FEM model and

run simulations.5. Evaluate the results by computing the given benchmark evaluation criteria, the closed-

loop system eigenvalues and the dynamic responses, all in comparison to thecentralized control design reference case.

6. Tune the control laws by repeating the simulations for different weighting matricesuntil acceptable results are reached.

3.4. Simulation results

For the overlapping decomposition of the FEM model described above, the model reductionresults in reduced-order stable subsystems with a minimal dimension of 34 states for eachsubsystem. The decentralized LQG control design has been performed with the weightingon the state defined by the identity matrix multiplied by a scalar q1. Some results are

summarized in the following.

0 10 20 30 40

-3

-2

-1

0

1

2x 10

4

Time(s)

Shearforce(kN)

TOWER SH.FORCE, GEBZE

uncontrolledcontrolled

0 10 20 30 40 50 60-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time(s)

Displacement(m)

TOP TOWER DISPL. EL CENTRO

uncontrolledcontrolled

Figure 2. Tower base shear forces Figure 3. Tower top displacements

For one of the towers, Figures 2 and 3 show the base shear force and the topdisplacement in comparison with the uncontrolled cases. The excitations are theacceleration of Gebze and El Centro earthquake, respectively.


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BAKULE, PAULET AND RODELLAR 5

From Figures 2 and 3, it is observed that the improvement achieved in reducing thetower base shear force is obtained in exchange for an increase (within an acceptable range)in the top tower displacement response with respect to the case without control.

In order to compare the results obtained with the overlapping decentralized control withthose given in the centralized reference case, Table 1 gives the first ten eigenvalues of theclosed-loop system modes. The overlapping decentralized case corresponds to the chosentuning parameters.

Table 1: Eigenvalues comparison (Hz)

Freq. no.No-control

(hinged)

Open loop

(non-hinged)

Closed loop

centralized

Closed loop

overlapping1 0.2899 0.1619 0.2400 0.26832 0.3699 0.2667 0.2665 0.28683 0.4683 0.3725 0.2668 0.36464 0.5158 0.4547 0.3633 0.45475 0.5812 0.5017 0.3710 0.50176 0.6490 0.5652 0.4547 0.56807 0.6687 0.6190 0.5017 0.61908 0.6970 0.6489 0.5641 0.64849 0.7102 0.6968 0.6190 0.6968

10 0.7203 0.7097 0.6487 0.7098

The first modes of vibration are changed by the proposed feedback controllers,

approaching the hinged (non-controlled) case. It can be observed that the overlappingstrategy is bringing the structure closer to the hinged case, keeping the advantage of areduction in tower base shear force (as the non-hinged structure would do).

1 2 3 4 5 6 7 81

1.5

2

2.5

3

3.5

4CRITERION NO.6

trial no.

El CentroMexico CityGebze

1 2 3 4 5 6 7 8

200

400

600

800

1000

MAXIMUM FORCE IN ACTUATOR NO.7

trial no.

Force(kN)

El CentroMexico CityGebze

Figure 4: Benchmark criterion No.6 Figure 5: Actuator No.7 maximum force

The benchmark evaluation criterion No. 6 (Dyke et al. 2000) is presented in Figure 4 forthe overlapping decentralized LQG control design during the tuning trials. Directcomparisons with the benchmark sample centralized LQG control design case given by


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Table 4 in (Dyke et al 2000) are included trough horizontal lines in all graphs. Thiscomparison is made for the three different earthquakes provided by the benchmark andshows satisfactory fulfillment of the control objectives. Criterion No. 6 is the maximumvalue of the displacement at the end of the bridge in the case of control over the same in thecase without (Dyke et al 2000).

Figure 5 shows the variation of one of the actuators maximum force during the tuningprocess. The maximum forces in the four actuators groups were finally set from 932 to 934kN. This setting was essential in establishing the LQG control parameters for eachsubsystem.

4. CONCLUSIONS

The paper has presented simulation results of the overlapping decentralized LQG design forthe cable-stayed bridge benchmark performed in a SIMULINK/MATLAB scheme. Themain motivating ideas and objectives of the paper have been to begin to explore potentialusefulness of overlapping decentralization issues for active control of large structure as theprototype bridge under study.

As effective measures for performance assessment and comparison, the benchmarkevaluation criteria and the analysis of time histories of selected bridge response variablesfor different prototype earthquakes have been adopted. Additionally, the eigenvalues of anumber of significant modes have been checked for the closed loop obtained whenimplementing the overlapping decentralized controller into the global finite element model.

The results look promising and confirm expectations. They are close to the sample

centralized case and lie within acceptable ranges. They satisfy also the requirements oncable tensions that are known a-posteriori.

ACKNOWLEDGEMENTS

This work was supported in part by the ASCR under Grant A2075802, by the MEC underGrant DGESIC-SB99 00787598, and by the CICYT under Project TAP99-1079-C03.

REFERENCES

Bakule, L. and Rodellar, J. (1995) Decentralized control and overlapping decomposition ofmechanical systems. Part I: system decomposition. Part II: decentralizedstabilization. International Journal of Control, 61: 559-587.

Dyke, J. S., Caicedo, J.M., Turan, G., Bergman, L.A. and Hague, S. (2000) Benchmark controlproblem for seismic response of cable-stayed bridges. (http:www.nd.edu/~quake).

Paulet-Crainiceanu, F., Rodellar, J. and Monroy, C. (2001) Control settings and performance analysisof an optimal active control method for cable bridges. 2nd European Conference onComputational Mechanics, ECCM-2001, Datacomp, Cracow. (CD-ROM).

iljak, D. D. (1991)Decentralized Control of Complex Systems. Academic Press, Boston, MA.

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