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

Authors: Naif B. Almutairi; Mohamed Zribi; Mohamed Abdel-Rohman
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

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.


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## Body

## 1. 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 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 Section2. 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 byWT 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.


## 2. 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 Figure1, the general equations of motion are(2.1)∂∂s[(To+τ)∂(x+U)∂s]=mc∂2U∂t2,∂∂s[(To+τ)∂(y+V)∂s]=-mcg+mc∂2V∂t2+c∂V∂t+fv(s,t),∂∂s[(To+τ)∂W∂s]=mc∂2W∂t2+c∂W∂t+fw(s,t),
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, To is the static tension, τ is the additional dynamic tension in the cable, c is the damping coefficient, mc is the mass of the cable per unit length, g is the gravitational force, and fv(s,t) and fw(s,t) are the external loading per unit length in the vertical and transverse directions, respectively.Figure 1
Displacement directions of the suspended cables.The nonlinear strain-displacement relationship during the deformation of the cable [20] is given by(2.2)τEA=ds′-dsds,
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 variablesds′ and ds are defined such that(2.3)ds′2=(dx+∂U)2+(dy+∂V)2+(∂W)2,ds2=dx2+dy2.The two-hinged bridge deck is suspended from the cables using vertical hangers located ats=si 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=xu along the bridge.Figure 2
Suspension bridge model.Based on the assumption of a small curvature regime and neglecting the horizontal motionU(s,t) (which can be safely disregarded because there is zero longitudinal loading [17]), the equations of motion from the static equilibrium position of the cable and the bridge can be simplified as follows:(2.4)∂∂s[(To+τ)∂W∂s]=mc∂2W∂t2+c∂W∂t+fw(s,t),∂∂s[(To+τ)∂(y+V)∂s]=-(mc+mb)g+mc∂2V∂t2+c∂V∂t+fv(s,t)-∑i=1N1ki(Zi-Vi)δ(s-si)+u(t)δ(x-xu),EI∂4Z∂x4+cb∂Z∂t+mb∂2Z∂x2=fb(x,t)-∑i=1N1ki(Zi-Vi)δ(x-xi)-u(t)δ(x-xu),
where EI is the flexural rigidity of the bridge deck, cb and mb are, respectively, the damping and mass per unit length of the bridge deck, ki is the stiffness of the vertical hangers (each located at xi and si), Z(x,t) is the vertical response of the bridge deck, Zi and Vi indicate the response of the bridge deck and the suspended cables at x=xi and s=si, 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 fw, fv, and fb 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 xu=0.5L (see Figure 2).The displacement functionsW(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(2.5)W(s,t)=λ(s)W(t),V(s,t)=ϕ(s)V(t),Z(x,t)=η(x)B(t),
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:(2.6)λ(s)=sin(πsl),ϕ(s)=koΩ2(1-tan(Ω2)sin(Ωsl)-cos⁡(Ωsl)),
where ko 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)tan(Ω2)=Ω2-4λ2(Ω2)3,λ2=L(EAHl)×(mgLH)2.Here, the termsmgL and H denote the vertical weight on the cable and the horizontal component of the static tension To, respectively. The force H is obtained from the static equilibrium of the cable as follows: (2.8)H=mgL28yc
in which yc is the sag in the cable profile of which is given by the equation:(2.9)yc=mgL22H(xL-x2L2).For a two-hinged bridge deck, the mode shapeη(x) can be assumed to take the form(2.10)η(x)=sin(πxL).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:(2.11)Ẅ+2ζcωwẆ+ωw2W+c5WV+c6WV2+c7W3=FW(t),(2.12)V̈+2ζcωvV̇+ωv2V+c1V2+c2W2+c3V3+c4VW2+FV(t)=d1B+d2V+c11u(t),(2.13)B̈+2ζbωbḂ+ωb2B=FB(t)+c8B+c9V-c10u(t)
in which ζc is the damping ratio in the suspended cable, ωw and ωv are, respectively, the natural frequencies of the cable in W and V directions, ζb is the damping ratio in the bridge deck, and ωb is the natural frequency of the bridge deck.Expressions of the forcesFW(t), FV(t), and FB(t) in the above equations are given in [17] as follows:(2.14)FW(t)=Cwo+Cw1Ẇ+Cw2Ẇ2+Cw3V̇+Cw4ẆV̇+Cw5V̇2+Cw6V̇3+Cw7V̇3Ẇ+Cw8V̇3Ẇ2+Cw9V̇3Ẇ3+Cw10V̇4+Cw11V̇4Ẇ+Cw12V̇4Ẇ2+Cw13V̇4Ẇ3+Cw14V̇5+Cw15V̇5Ẇ+Cw16V̇5Ẇ2+Cw17V̇5Ẇ3+Cw18V̇6+Cw19V̇6Ẇ+Cw20V̇6Ẇ2+Cw21V̇6Ẇ3+Cw22V̇7+Cw23V̇7W+Cw24V̇7Ẇ2+Cw25V̇7Ẇ3,FV(t)=Cvo+Cv1Ẇ+Cv2Ẇ2+Cv3V̇+Cv4ẆV̇+Cv5V̇2+Cv6V̇3+Cv7V̇3Ẇ+Cv8V̇3Ẇ2+Cv9V̇3Ẇ3+Cv10V̇4+Cv11V̇4Ẇ+Cv12V̇4Ẇ2+Cv13V̇4Ẇ3+Cv14V̇5+Cv15V̇5Ẇ+Cv16V̇5Ẇ2+Cv17V̇5Ẇ3+Cv18V̇6+Cv19V̇6Ẇ+Cv20V̇6Ẇ2+Cv21V̇6Ẇ3+Cv22V̇7+Cv23V̇7W+Cv24V̇7Ẇ2+Cv25V̇7Ẇ3,FB(t)=c12Ḃ+c13ḂB,
where the coefficients Cwi and Cvi are functions of the basic wind speed Uo.


## 3. 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.


### 3.1. SIMULINK Model for the Unsuspended Bridge

The response of the bridge alone is obtained by simulating (2.13) with c8, c9, and, c10 set to zero. Galloping occurs when c12>2ζbωb, 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.Figure 3
SIMULINK model of the unsuspended bridge including wind forceFB(t).


### 3.2. 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 d1 and c11 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.Figure 4
SIMULINK model of the suspended cable.Figure 5
SIMULINK model of the wind forces.


### 3.3. 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 c10 and c11 set to zero. The SIMULINK model of the uncontrolled suspension bridge is shown in Figure 6.Figure 6
SIMULINK model of the uncontrolled suspension bridge.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.Uncontrolled suspension bridge response whenUo=12 (m/sec): (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)In order to suppress galloping due to wind forces at mean wind speedUo=12 (m/sec), in the cable and bridge vertical responses, a vertical cable between the bridge 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.


## 3.1. SIMULINK Model for the Unsuspended Bridge

The response of the bridge alone is obtained by simulating (2.13) with c8, c9, and, c10 set to zero. Galloping occurs when c12>2ζbωb, 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.Figure 3
SIMULINK model of the unsuspended bridge including wind forceFB(t).


## 3.2. 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 d1 and c11 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.Figure 4
SIMULINK model of the suspended cable.Figure 5
SIMULINK model of the wind forces.


## 3.3. 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 c10 and c11 set to zero. The SIMULINK model of the uncontrolled suspension bridge is shown in Figure 6.Figure 6
SIMULINK model of the uncontrolled suspension bridge.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.Uncontrolled suspension bridge response whenUo=12 (m/sec): (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)In order to suppress galloping due to wind forces at mean wind speedUo=12 (m/sec), in the cable and bridge vertical responses, a vertical cable between the bridge 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.


## 4. 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: (4.1)ẋ(t)=Ax(t)+Bu(t)+gx(x(t))+d(t),
where(4.2)x(t)=[x1(t)x2(t)x3(t)x4(t)x5(t)x6(t)]=[W(t)Ẇ(t)V(t)V̇(t)B(t)Ḃ(t)],A=[010000-ωw2-2ζcωwα00000010000d2-ωv2-2ζcωvd1000000100c90c8-ωb2-2ζbωb],B=[000c110-c10],d(t)=[0FW(t)0-FV(t)00],gx(x(t))=[0-αx3-c7x13-(c5+c6x3)x1x30-(c2+c4x3)x12-(c1+c3x3)x3200].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, αx3 is subtracted from the second element in the gx(x(t)) vector.Let(4.3)g(x,t)=gx(x(t))+d(t).Hence, the equation of the system in (4.1) can be written as(4.4)ẋ(t)=Ax(t)+Bu(t)+g(x,t).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 functiong(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:
(4.5)‖g(x,t)‖≤μ‖x(t)‖,
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.


### 4.1. 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 matrixAc be such that(4.6)Ac=A-BK,
where K is the feedback gain vector obtained by the standard pole placement technique. Let the symmetric positive definite matrix P1 be the solution of the following Lyapunov equation [22]:(4.7)AcTP1+P1Ac=-Q1,
where Q1=Q1T>0.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:
(4.8)u=uL+uN
with
(4.9)uL=-Kx,(4.10)uN=-ρ1sgn⁡(BTP1x),
where ρ1 is a positive design parameter and sgn is the sign function:
(4.11)sgn⁡(σ)={+1ifσ>0,0ifσ=0,-1ifσ<0.Proof.
Using (4.4), (4.8), and (4.9), it follows that
(4.12)ẋ=Ax+B(-Kx+uN)+g(x,t)=(A-BK)x+BuN+g(x,t)=Acx+BuN+g(x,t).
Consider the following Lyapunov function candidate:(4.13)V1=xTP1x.
Note thatV1>0 for x≠0 and V1=0 for x=0. Taking the derivative of V1 with respect to time and using (4.12), (4.10), (4.7), and (4.5), it follows that
(4.14)V̇1=ẋTP1x+xTP1ẋ=(Acx+BuN+g(x,t))TP1x+xTP1(Acx+BuN+g(x,t))=xT(AcTP1+P1Ac)x+2xTP1BuN+2g(x,t)TP1x=-xTQ1x+2xTP1BuN+2g(x,t)TP1x=-xTQ1x+2xTP1B(-ρ1sgn⁡(BTP1x))+2g(x,t)TP1x≤-λmin⁡(Q1)‖x‖2+2μλmax⁡(P1)‖x‖2-2ρ1xTP1BBTP1x|BTP1x|=-λmin⁡(Q1)‖x‖2+2μλmax⁡(P1)‖x‖2-2ρ1|BTP1x|≤-(λmin⁡(Q1)-2μλmax⁡(P1))‖x‖2,
where λmin and λmax are the minimum and maximum eigenvalues of a matrix, respectively.
Therefore, it can be concluded thatV̇1<0 if the matrices P1 and Q1 are selected such (λmin(Q1)-2μλmax(P1))>0. Hence, the control scheme given by Proposition 4.3 guarantees the asymptotic stability of the closed-loop system.


### 4.2. 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 matrixAc is such that(4.15)Ac=A-BK.Let the symmetric positive definite matrixP2 be the solution of the following Lyapunov equation:(4.16)AcTP2+P2Ac=-Q2,
where Q2=Q2T>0. Also, let a positive design parameter γ2 be such that(4.17)γ2≥λmax⁡(P2)λmin⁡(P2BBTP2)μ.Proposition 4.4.
The control law given by (4.18)–(4.20) when applied to the suspension bridge system described in (4.4) guarantees the asymptotic stability of the system:
(4.18)u=uL1+uL2
with
(4.19)uL1=-Kx,(4.20)uL2=-γ2BTP2x.Proof.
Using (4.4), (4.18), and (4.19), it follows that
(4.21)ẋ=Ax+B(-Kx+uL2)+g(x,t)=(A-BK)x+BuL2+g(x,t)=Acx+BuL2+g(x,t).
Consider the following Lyapunov function candidate:(4.22)V2=xTP2x.
Note thatV2>0 for x≠0 and  V2=0 for x=0. Taking the derivative of  V2 with respect to time and using (4.21), (4.20), (4.16), and (4.5), it follows that
(4.23)V̇2=ẋTP2x+xTP2ẋ=(Acx+BuL2+g(x,t))TP2x+xTP2(Acx+BuL2+g(x,t))=xT(AcTP2+P2Ac)x+2g(x,t)TP2x+2xTP2BuL2=-xTQ2x+2g(x,t)TP2x+2xTP2BuL2=-xTQ2x+2g(x,t)TP2x+2xTP2B(-γ2BTP2x)≤-xTQ2x+2μ‖P2x‖‖x‖-2γ2xTP2BBTP2x≤-xTQ2x+2μλmax⁡(P2)‖x‖2-2γ2xTP2BBTP2x≤-xTQ2x+2μλmax⁡(P2)‖x‖2-2γ2λmin⁡(P2BBTP2)‖x‖2=-xTQ2x+2(μλmax⁡(P2)-γ2λmin⁡(P2BBTP2))‖x‖2≤-xTQ2x.
The choice of γ2 guarantees that (μλmax(P2)-γ2λmin(P2BBTP2))≤0.
Therefore, it can be concluded thatV̇2<0. Hence, the control scheme given by Proposition 4.4 guarantees the asymptotic stability of the closed-loop system.


## 4.1. 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 matrixAc be such that(4.6)Ac=A-BK,
where K is the feedback gain vector obtained by the standard pole placement technique. Let the symmetric positive definite matrix P1 be the solution of the following Lyapunov equation [22]:(4.7)AcTP1+P1Ac=-Q1,
where Q1=Q1T>0.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:
(4.8)u=uL+uN
with
(4.9)uL=-Kx,(4.10)uN=-ρ1sgn⁡(BTP1x),
where ρ1 is a positive design parameter and sgn is the sign function:
(4.11)sgn⁡(σ)={+1ifσ>0,0ifσ=0,-1ifσ<0.Proof.
Using (4.4), (4.8), and (4.9), it follows that
(4.12)ẋ=Ax+B(-Kx+uN)+g(x,t)=(A-BK)x+BuN+g(x,t)=Acx+BuN+g(x,t).
Consider the following Lyapunov function candidate:(4.13)V1=xTP1x.
Note thatV1>0 for x≠0 and V1=0 for x=0. Taking the derivative of V1 with respect to time and using (4.12), (4.10), (4.7), and (4.5), it follows that
(4.14)V̇1=ẋTP1x+xTP1ẋ=(Acx+BuN+g(x,t))TP1x+xTP1(Acx+BuN+g(x,t))=xT(AcTP1+P1Ac)x+2xTP1BuN+2g(x,t)TP1x=-xTQ1x+2xTP1BuN+2g(x,t)TP1x=-xTQ1x+2xTP1B(-ρ1sgn⁡(BTP1x))+2g(x,t)TP1x≤-λmin⁡(Q1)‖x‖2+2μλmax⁡(P1)‖x‖2-2ρ1xTP1BBTP1x|BTP1x|=-λmin⁡(Q1)‖x‖2+2μλmax⁡(P1)‖x‖2-2ρ1|BTP1x|≤-(λmin⁡(Q1)-2μλmax⁡(P1))‖x‖2,
where λmin and λmax are the minimum and maximum eigenvalues of a matrix, respectively.
Therefore, it can be concluded thatV̇1<0 if the matrices P1 and Q1 are selected such (λmin(Q1)-2μλmax(P1))>0. Hence, the control scheme given by Proposition 4.3 guarantees the asymptotic stability of the closed-loop system.


## 4.2. 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 matrixAc is such that(4.15)Ac=A-BK.Let the symmetric positive definite matrixP2 be the solution of the following Lyapunov equation:(4.16)AcTP2+P2Ac=-Q2,
where Q2=Q2T>0. Also, let a positive design parameter γ2 be such that(4.17)γ2≥λmax⁡(P2)λmin⁡(P2BBTP2)μ.Proposition 4.4.
The control law given by (4.18)–(4.20) when applied to the suspension bridge system described in (4.4) guarantees the asymptotic stability of the system:
(4.18)u=uL1+uL2
with
(4.19)uL1=-Kx,(4.20)uL2=-γ2BTP2x.Proof.
Using (4.4), (4.18), and (4.19), it follows that
(4.21)ẋ=Ax+B(-Kx+uL2)+g(x,t)=(A-BK)x+BuL2+g(x,t)=Acx+BuL2+g(x,t).
Consider the following Lyapunov function candidate:(4.22)V2=xTP2x.
Note thatV2>0 for x≠0 and  V2=0 for x=0. Taking the derivative of  V2 with respect to time and using (4.21), (4.20), (4.16), and (4.5), it follows that
(4.23)V̇2=ẋTP2x+xTP2ẋ=(Acx+BuL2+g(x,t))TP2x+xTP2(Acx+BuL2+g(x,t))=xT(AcTP2+P2Ac)x+2g(x,t)TP2x+2xTP2BuL2=-xTQ2x+2g(x,t)TP2x+2xTP2BuL2=-xTQ2x+2g(x,t)TP2x+2xTP2B(-γ2BTP2x)≤-xTQ2x+2μ‖P2x‖‖x‖-2γ2xTP2BBTP2x≤-xTQ2x+2μλmax⁡(P2)‖x‖2-2γ2xTP2BBTP2x≤-xTQ2x+2μλmax⁡(P2)‖x‖2-2γ2λmin⁡(P2BBTP2)‖x‖2=-xTQ2x+2(μλmax⁡(P2)-γ2λmin⁡(P2BBTP2))‖x‖2≤-xTQ2x.
The choice of γ2 guarantees that (μλmax(P2)-γ2λmin(P2BBTP2))≤0.
Therefore, it can be concluded thatV̇2<0. Hence, the control scheme given by Proposition 4.4 guarantees the asymptotic stability of the closed-loop system.


## 5. 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.Table 1
Data for the great belt suspension bridge.
Bridge deck parametersCable parametersWind parametersSpan,L2500 mSuspended length,l2573 mAir density,ρ1.25 Kg/m3Deck mass,mb14908 Kg/mCable mass,mc4396 Kg/mH
*1.7Width,BB27 mCable sag,yc265 mε100Flexural rigidity,EI2.4×1011N·m2Cross-sectional area,A0.56 m2Stiffness of vertical hangers,ki106 N/m (placed at every 10 m)Young’s modulus of elasticity,E2.1×1011N/m2Damping ratio,ζb0.01Damping ratio,ζc0.001The parametersH* 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].In this example, it is assumed that the wind speed profileUo(s,t) is a constant and equals to the basic wind speed Uo [17]. The coefficients Cwi and Cvi  can then be determined as functions of the basic wind speed Uo as given in the appendix. In all simulations, the following initial conditions are used V(0)=W(0)=B(0)=0.1 m and V̇(0)=Ẇ(0)=Ḃ(0)=0.1 m/sec.


### 5.1. Simulation Results Using the First Controller

The parameters of the first controller areρ1=106, the feedback gain vector K=[0.12012.76941.81565.5243-0.54083.6275]×106, and Q1=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:(5.1)≫K=place(A,B,clp),
where  clp  is the desired closed-loop poles of the linear part of the system. The P1 matrix which is the solution of the Lyapunov equation in (4.7) is computed by MATLAB as follows:(5.2)≫P1=are(A-B*K,zeors(6,6),eye(6,6)).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 Figure7. The results are shown in Figure 8.Suspension bridge response when using the first controller: (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)Figure8(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. 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 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 speedUo. 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.Figure 9
Response of the active control force when applied the first controller.


### 5.2. 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 areγ2= 106, K used in first controller, and Q2=I. The wind speed used in testing the performance of the proposed controller is Uo=12 (m/sec). The results are shown in Figure 10.Suspension bridge response when using the second controller: (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)Figure10(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 10(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. 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 10(c), we noticed that the controller did not have much of an effect on the amplitude of the transverse displacement oscillations.The plot of the active control force versus time is shown in Figure11. It can be seen that the force stays within a reasonable range.Figure 11
Response of the active control force when applied the second controller.Although we have almost the same performance when applying the two controllers, the second controller (which is linear) is much simpler in the implementation than the first controller. Thus, one can suggest applying the second controller.


### 5.3. Comparison with a TMD Controller

In this section, a comparison with the TMD controller is presented. A tuned mass damper with a weight 5% of that of the first mode mass of the bridge deck, a natural frequency ofωd=0.97ωb, and a damping ratio of 20% is used. The tuned mass damper TMD is added to the bridge deck at xu=L/2. The wind speed used in testing the performance of the three controllers is Uo= 12 (m/sec). The results are shown in Figures 12 and 13.Figure 12
Vertical displacement of the suspended cable withUo=12(m/sec) when using first controller, second controller, and TMD controller.Figure 13
Vertical displacement of the bridge deck withUo=12 (m/sec) when using first controller, second controller, and TMD controller.It can be seen from these figures that the proposed controllers gave better results when compared to the passive TMD controller. The proposed controllers have smaller oscillation amplitudes and faster decay rates. For further comparison, the norms of the vertical displacements of the suspended cable and the bridge deck are computed and listed in Table2.Table 2
Norms of vertical displacements of the suspension bridge.
Controller typeNonlinearLinearTMD∥V(t)∥0.74200.66330.8566∥B(t)∥0.65540.60930.7605It can be seen that the linear controller gives the smallest norm among the three controllers. This gives more advantage to the second controller.


## 5.1. Simulation Results Using the First Controller

The parameters of the first controller areρ1=106, the feedback gain vector K=[0.12012.76941.81565.5243-0.54083.6275]×106, and Q1=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:(5.1)≫K=place(A,B,clp),
where  clp  is the desired closed-loop poles of the linear part of the system. The P1 matrix which is the solution of the Lyapunov equation in (4.7) is computed by MATLAB as follows:(5.2)≫P1=are(A-B*K,zeors(6,6),eye(6,6)).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 Figure7. The results are shown in Figure 8.Suspension bridge response when using the first controller: (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)Figure8(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. 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 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 speedUo. 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.Figure 9
Response of the active control force when applied the first controller.


## 5.2. 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 areγ2= 106, K used in first controller, and Q2=I. The wind speed used in testing the performance of the proposed controller is Uo=12 (m/sec). The results are shown in Figure 10.Suspension bridge response when using the second controller: (a) cable vertical response, (b) bridge vertical response, and (c) cable transverse response.
(a)(b)(c)Figure10(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 10(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. 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 10(c), we noticed that the controller did not have much of an effect on the amplitude of the transverse displacement oscillations.The plot of the active control force versus time is shown in Figure11. It can be seen that the force stays within a reasonable range.Figure 11
Response of the active control force when applied the second controller.Although we have almost the same performance when applying the two controllers, the second controller (which is linear) is much simpler in the implementation than the first controller. Thus, one can suggest applying the second controller.


## 5.3. Comparison with a TMD Controller

In this section, a comparison with the TMD controller is presented. A tuned mass damper with a weight 5% of that of the first mode mass of the bridge deck, a natural frequency ofωd=0.97ωb, and a damping ratio of 20% is used. The tuned mass damper TMD is added to the bridge deck at xu=L/2. The wind speed used in testing the performance of the three controllers is Uo= 12 (m/sec). The results are shown in Figures 12 and 13.Figure 12
Vertical displacement of the suspended cable withUo=12(m/sec) when using first controller, second controller, and TMD controller.Figure 13
Vertical displacement of the bridge deck withUo=12 (m/sec) when using first controller, second controller, and TMD controller.It can be seen from these figures that the proposed controllers gave better results when compared to the passive TMD controller. The proposed controllers have smaller oscillation amplitudes and faster decay rates. For further comparison, the norms of the vertical displacements of the suspended cable and the bridge deck are computed and listed in Table2.Table 2
Norms of vertical displacements of the suspension bridge.
Controller typeNonlinearLinearTMD∥V(t)∥0.74200.66330.8566∥B(t)∥0.65540.60930.7605It can be seen that the linear controller gives the smallest norm among the three controllers. This gives more advantage to the second controller.


## 6. 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.



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*Source: 310938-2011-07-14.xml*