Lyapunov-Based Control for Suppression of Wind-Induced Galloping in Suspension Bridges

This paper investigates the suppression of galloping in a suspension bridge due to wind loads. The galloping phenomenon can be destructive due to the high-amplitude oscillations of the structure. Two controllers are proposed to generate the control force needed to suppress the vertical galloping in the suspended cables and in the bridge deck. SIMULINK software is used to simulate the controlled system. The simulation results indicate that the proposed controllers work well. In addition, the performance of the system with the proposed controllers is compared to the performance of the system controlled with a tuned mass damper.


Introduction
Suspension bridges are one of the most popular types of bridges.Very long spans of suspension bridges are being planned 1 .Long, suspended steel cables such as those used in suspension bridges are prone to vibrations induced by wind and moving loads 2-6 .When suspended cables are subjected to wind disturbance, the system behaves nonlinearly due to its flexibility and because there is coupling between the motion of the bridge deck and the motion of the suspended cable.
The wind forces exerted on the cables and the bridge deck are complex.The aerodynamic stability of the cables and the bridge deck depends on many parameters including the wind velocity, the shape, and the size of the cross-sections and the angles of attack.When the wind velocity exceeds a certain critical value, the oscillations of the cable and/or the bridge deck will increase until a steady state response is achieved.This can be quite destructive to the structure due to the large amplitude of the oscillations.The aerodynamic instability of the bridge deck could occur simultaneously in both the torsional and vertical vibrational modes; this is called flutter and occurs when the torsional stiffness Mathematical Problems in Engineering is close in magnitude to the flexural stiffness 7 .Aerodynamic instability can also occur exclusively in the vertical mode if the torsional stiffness is much larger than the flexural stiffness, and this form is called galloping.In this paper, we will consider only the galloping instability for a suspension bridge where changing the shape or size of the cross-sections is difficult.
The occurrence of galloping is difficult to predict.The effect of the galloping phenomenon can be quite destructive, due to the high-amplitude oscillations of the structure in the direction perpendicular to the mean wind direction 8 .For structures with low damping such as suspended cables, suspension bridges, or tall buildings, galloping can easily occur at low, steady or unsteady wind speeds 9 .The nonlinearity behavior of the suspended bridge under wind excitation can cause internal resonances which affect the safety of the structures 7, 10, 11 .
Galloping of suspended cables due to resonant harmonic loading has been studied in 10, 11 while galloping due to wind effects was studied in 12 .It has been shown by Abdel-Rohman in 8 that increasing the damping is a very effective factor in increasing the critical wind speed at which galloping occurs.Attempts to introduce passive or semiactive damping into stay cables were made in 13-18 .In this paper, we proposed two active control schemes to suppress the galloping due to wind loading in suspension bridges.The designs of both controllers are based on Lyapunov theory; the proposed controllers guarantee the asymptotic stability of the controlled suspension bridge.
The paper is organized as follows.The dynamic model of the suspension bridge subject to wind forces is presented in Section 2. In Section 3, a SIMULINK model is presented for each subsystem of the suspension bridge.In Section 4, two state feedback controllers are proposed.Simulation results are given and discussed in Section 5. Finally, some concluding remarks are given in Section 6.
In the sequel, we denote by W T the transpose of a matrix or a vector W. We use W > 0 W < 0 to denote a positive-negative-definite matrix W. Sometimes, the arguments of a function are omitted in the analysis when no confusion can arise.

Equations of Motion of the System
The derivation of the basic equations of motion of the suspended cables subject to wind forces can be found in 12, 17, 19, 20 .For ease of presentation, the dynamic model is rederived in this paper.
Using the displacement directions defined in Figure 1, the general equations of motion are where s is the spatial coordinate along the cable curved length which is λ , t is the time, x s is the horizontal coordinate along the cable span, y s is the equation of the cable static profile, U s, t , V s, t , and W s, t are, respectively, the displacements of the cable at location s in the horizontal, vertical, and transversal directions, T o is the static tension, τ is the additional dynamic tension in the cable, c is the damping coefficient, m c is the mass of the cable per unit length, g is the gravitational force, and f v s, t and f w s, t are the external loading per unit length in the vertical and transverse directions, respectively.
The nonlinear strain-displacement relationship during the deformation of the cable 20 is given by where E is the modulus of elasticity, A is the cross-section area of the cable, ds is the deformed cable segment, and ds is the undeformed cable segment.The variables ds and ds are defined such that

2.3
The two-hinged bridge deck is suspended from the cables using vertical hangers located at s s i as shown in Figure 2. L is the length of the bridge deck and u t is the active control force installed at the general location x x u along the bridge.
Based on the assumption of a small curvature regime and neglecting the horizontal motion U s, t which can be safely disregarded because there is zero longitudinal loading can be simplified as follows: where EI is the flexural rigidity of the bridge deck, c b and m b are, respectively, the damping and mass per unit length of the bridge deck, k i is the stiffness of the vertical hangers each located at x i and s i , Z x, t is the vertical response of the bridge deck, Z i and V i indicate the response of the bridge deck and the suspended cables at x x i and s s i , respectively, and the δ • is the Dirac-Delta function which is used to introduce, into the differential equations, the active control force u t .The forces f w , f v , and f b are, respectively, the lateral and vertical wind forces on the suspended cable and the vertical wind force on the bridge deck.Previous studies have shown that locating the control force at the most flexible location of the structure provides the most feasible and optimal performance of the controlled structure as was shown by Abel-Rohman in 21 .Thus, we set x u 0.5L see Figure 2 .
The displacement functions W s, t , V s, t , and Z s, t are considered to be the contribution of the first modes of vibrations and are assumed to take the forms where λ s and φ s are, respectively, the cable first mode shapes in the transversal and vertical directions, which can be determined using linear theory of cables 20 and satisfy the following boundary conditions: where k o is a constant chosen to make φ s 1 at the midspan of the cable, s λ/2 .The parameter Ω is computed from the following relations:

2.7
Here, the terms mgL and H denote the vertical weight on the cable and the horizontal component of the static tension T o , respectively.The force H is obtained from the static equilibrium of the cable as follows: in which y c is the sag in the cable profile of which is given by the equation: For a two-hinged bridge deck, the mode shape η x can be assumed to take the form Substituting 2.5 into 2.4 and applying an integral transformation one obtains the equations of motion for the suspended cable in the transverse and vertical directions and for the vertical motion of the bridge deck as follows: in which ζ c is the damping ratio in the suspended cable, ω w and ω v are, respectively, the natural frequencies of the cable in W and V directions, ζ b is the damping ratio in the bridge deck, and ω b is the natural frequency of the bridge deck.

Mathematical Problems in Engineering
Expressions of the forces F W t , F V t , and F B t in the above equations are given in 17 as follows: where the coefficients C wi and C vi are functions of the basic wind speed U o .

Simulation of the Uncontrolled Suspension Bridge Model Using SIMULINK
The dynamic model of the uncontrolled suspension bridge is simulated using the SIMULINK software.At first, a SIMULINK model is built for each subsystem of the suspension bridge.
Then, the SIMULINK model subsystems are combined to simulate the overall system.

SIMULINK Model for the Unsuspended Bridge
The response of the bridge alone is obtained by simulating 2.13 with c 8 , c 9 , and, c 10 set to zero.Galloping occurs when c 12 > 2ζ b ω b , and therefore the critical wind speed is calculated to be 10.39 m/sec 17 .The SIMULINK model of the unsuspended bridge is shown in Figure 3.

SIMULINK Model for the Suspended Cable Alone
If the cable is constructed alone, then the transverse and vertical responses of the suspended cable are obtained by simulating 2.11 and 2.12 with d 1 and c 11 set to zero.The critical wind speed of the cables is calculated to be 7.30 m/sec 17 .The SIMULINK model of the suspended cable is shown in Figure 4 while the wind forces are simulated in Figure 5.

Simulation Results of the Uncontrolled Suspension Bridge
In the case of a suspension bridge, the response of the uncontrolled suspension bridge is simulated using 2.11 -2.13 with c 10 and c 11 set to zero.The SIMULINK model of the uncontrolled suspension bridge is shown in Figure 6.Abdel-Rohman and Joseph 17 showed that the critical wind speed is approximately 10.4 m/sec .Therefore, we simulated the response of the uncontrolled suspended bridge at wind speed 12 m/sec which causes galloping as shown in Figure 7.It can be seen that the uncontrolled suspension bridge response has galloping in the vertical direction of both the cable and the bridge deck.
In order to suppress galloping due to wind forces at mean wind speed U o 12 m/sec , in the cable and bridge vertical responses, a vertical cable between the bridge mat deck and the suspended cables is used to install a hydraulic actuator able to generate the required active control force on the bridge deck.The controller will generate two equal and opposite active control forces.In the next section, a controller design based on Lyapunov theory is proposed.Two state feedback controllers are provided: the first one is a nonlinear controller and the second one is a linear controller.

Controllers Design
The suspension bridge model described by the nonlinear differential equations 2.11 -2.13 can be represented in the following state-space form: where In order to insure the controllability of the pair A, B , the element 2,3 of the A matrix is set to a constant α, and, consequently, αx 3 is subtracted from the second element in the g x x t vector.Let g x, t g x x t d t .

4.3
Hence, the equation of the system in 4.1 can be written as ẋ t Ax t Bu t g x, t .

4.4
The objective of this paper is to design control schemes to improve the stability of the system by suppressing the oscillations in the suspension bridge under wind excitation.
Remark 4.1.The simulation results indicate that the nonlinear function g x, t in 4.4 is uniformly bounded, and hence it can be assumed that the nonlinear term g x, t satisfies the following cone-bounding constraint: where μ is a positive scalar and • • • is the Euclidian norm of a vector.
Remark 4.2.It can be checked that the pair A, B in 4.4 is controllable.Hence, the poles of the closed-loop system can be selected such that the response of the linear part of the system i.e., g x, t 0 is as desired.

Design of the First Controller
In this section, a nonlinear controller is used to suppress the oscillations in the suspension bridge due to wind loads.The control law is divided into two parts: a linear part and a nonlinear part.The linear part of the controller is designed by using the standard pole placement technique.The nonlinear part of the controller is designed to guarantee the asymptotic stability of the closed-loop system.Let the matrix A c be such that where K is the feedback gain vector obtained by the standard pole placement technique.Let the symmetric positive definite matrix P 1 be the solution of the following Lyapunov equation 22 : where Proposition 4.3.The control law given by the following equations 4.8 -4.10 when applied to the suspension bridge system described by 4.4 guarantees the asymptotic stability of the closed-loop system: with u L −Kx, 4.9 where ρ 1 is a positive design parameter and sgn is the sign function:

4.11
Proof.Using 4.4 , 4.8 , and 4.9 , it follows that ẋ Ax B −Kx u N g x, t A − BK x Bu N g x, t A c x Bu N g x, t .

4.12
Consider the following Lyapunov function candidate: V 1 x T P 1 x.

4.13
Note that V 1 > 0 for x / 0 and V 1 0 for x 0. Taking the derivative of V 1 with respect to time and using 4.12 , 4.10 , 4.7 , and 4.5 , it follows that V1 ẋT P 1 x x T P 1 ẋ

4.14
where λ min and λ max are the minimum and maximum eigenvalues of a matrix, respectively.Therefore, it can be concluded that V1 < 0 if the matrices P 1 and Q 1 are selected such λ min Q 1 − 2μλ max P 1 > 0. Hence, the control scheme given by Proposition 4.3 guarantees the asymptotic stability of the closed-loop system.

Design of the Second Controller
To simplify the computations involved in the implementation of the first controller, a linear controller is designed for the suspension bridge under wind loads as described by 4.4 .Again, the control law is divided into two parts.The first part of the controller is designed by using the pole placement technique.The second part of the controller is designed to guarantee the asymptotic stability of the closed-loop system.
Define the matrix A c is such that

4.15
Let the symmetric positive definite matrix P 2 be the solution of the following Lyapunov equation: where

4.21
Consider the following Lyapunov function candidate: V 2 x T P 2 x.

4.22
Note that V 2 > 0 for x / 0 and V 2 0 for x 0. Taking the derivative of V 2 with respect to time and using 4.21 , 4.20 , 4.16 , and 4.5 , it follows that The parameters H * and ε are the aerodynamic coefficients which are usually determined experimentally from wind tunnel tests.In this paper, we used the values of these parameters from 23 .
The choice of γ 2 guarantees that μλ max P 2 − γ 2 λ min P 2 BB T P 2 ≤ 0. Therefore, it can be concluded that V2 < 0. Hence, the control scheme given by Proposition 4.4 guarantees the asymptotic stability of the closed-loop system.

Simulation Results
The simulation results of the controlled suspension bridge using the proposed controllers are presented and discussed.The system is simulated using the SIMULINK software.The data for the suspension bridge used in 7 is used here.The details of the bridge deck and the cable and wind parameters are given in Table 1.From this table, the parameters of the model of the suspension bridge can be calculated as given in the appendix.It is worth mentioning that the torsional mode frequency which is equal to 1.276 rad/sec is more than three times greater than the frequency of the vertical bending mode of 0.404 rad/sec.Thus, our assumption of ignoring the torsional frequency is reasonable.
In this example, it is assumed that the wind speed profile U o s, t is a constant and equals to the basic wind speed U o 17 .The coefficients C wi and C vi can then be determined as functions of the basic wind speed U o as given in the appendix.In all simulations, the following initial conditions are used V 0 W 0 B 0 0.1 m and V 0 Ẇ 0 Ḃ 0 0.1 m/sec.

Simulation Results Using the First Controller
The parameters of the first controller are ρ 1 10 6 , the feedback gain vector K 0.1201 2.7694 1.8156 5.5243 −0.5408 3.6275 × 10 6 , and Q 1 I, where I is the identity matrix of appropriate dimension.The feedback gain vector K is computed by MATLAB using the linear part of the system: where clp is the desired closed-loop poles of the linear part of the system.The P 1 matrix which is the solution of the Lyapunov equation in 4.7 is computed by MATLAB as follows: P1 are A − B * K, zeors 6, 6 , eye 6, 6 .

5.2
Any wind speed greater than 10.4 m/sec will cause galloping and can be used for simulation purposes.In this paper, a wind speed of 12 m/sec equivalent to 43.2 Km/hr is used.This wind speed equals the wind closed-loop poles of the linear suspension bridge as shown in Figure 7.The results are shown in Figure 8.
Figure 8 a shows the vertical displacement of the suspended cable, V t .It can be seen that the oscillations in the response decay to zero in about 30 seconds. Figure 8 b shows the vertical displacement of the bridge deck, B t .It can be seen that the oscillations in the response decay to zero in about 30 seconds. of the suspended cable, W t .It can be seen that the response oscillates with a slow decaying envelop as time increases.Hence, it can be concluded that the first control scheme is able to greatly reduce the oscillations in V t and B t .Comparing Figures 7 c and 8 c , we noticed that the controller did not have much of an effect on the amplitude of the transverse displacement oscillations while it reduced the frequency of the oscillations.
It is worth noting that the active control force was introduced only for the vertical displacements, and hence the oscillations of the cable and the bridge deck in this direction was suppressed, but the cable response in the transverse direction oscillates for some time before decaying to a constant amplitude.The transverse response can be improved if the bridge deck is supplemented by a horizontally sliding damper along the mean wind direction.However, for this example, the transverse response is stable irrespective of the value of the mean wind speed U o .Moreover, the plot of the active control force versus time is shown in Figure 9.It can be seen that the force stays within a reasonable range.

Simulation Results Using the Second Controller
The simulation results of the controlled suspension bridge using the second controller are presented in this section.The parameters of the controller are γ 2 10 6 , K used in first controller, and Q 2 I.The wind speed used in testing the performance of the proposed controller is U o 12 m/sec .The results are shown in Figure 10.
Figure 10 a shows the vertical displacement of the suspended cable, V t .It can be seen that the oscillations in the response decay to zero in about 30 seconds.shows the vertical displacement of the bridge deck, B t .It can be seen that the oscillations in the response decay to zero in about 30 seconds. Figure 8 c shows the transverse displacement of the suspended cable, W t .It can be seen that the response oscillates with a slow decaying envelop as time increases.Hence, it can be concluded that the first control scheme is able to greatly reduce the oscillations in V t and B t .Comparing Figures 7 c and

Conclusion
The control of the nonlinear oscillations of suspension bridges due to wind loads is investigated in this paper.In order to control the vertical oscillations of the suspended cables and the bridge deck, a hydraulic actuator can be installed between the bridge deck and the suspended cables.This actuator is used to generate an active control force on the bridge deck.A linear and a nonlinear control schemes are presented to generate the active control forces.These controllers guarantee the asymptotic stability of the closed-loop system.The performance of the controlled system is investigated through simulations using the SIMULINK software.The simulation results indicate that the proposed control schemes work well.Moreover, simulation results indicated that the proposed controllers gave better results when compared to the TMD controller.

Figure 1 :Figure 2 :
Figure 1: Displacement directions of the suspended cables.

Figure 3 :
Figure 3: SIMULINK model of the unsuspended bridge including wind force F B t .

Figure 4 :
Figure 4: SIMULINK model of the suspended cable.

Figure 5 :
Figure 5: SIMULINK model of the wind forces.

Figure 6 :
Figure 6: SIMULINK model of the uncontrolled suspension bridge.

Figure 7 :
Figure 7: Uncontrolled suspension bridge response when U o 12 m/sec : a cable vertical response, b bridge vertical response, and c cable transverse response.

Figure 8 cFigure 8 :
Figure 8: Suspension bridge response when using the first controller: a cable vertical response, b bridge vertical response, and c cable transverse response.

Figure 9 :Figure 10 :
Figure 9: Response of the active control force when applied the first controller.

Figure 11 :Figure 12 :
Figure 11: Response of the active control force when applied the second controller.

Table 1 :
Data for the great belt suspension bridge.

Table 2 :
Norms of vertical displacements of the suspension bridge.