Feedforward and Feedback Vibration Control and Algorithm Design for Cable-Bridge Structure Nonlinear Systems

Abstract

Technique of feedforward and feedback optimal vibration control and simulation for long-span cable-bridge coupled systems is developed. Buffeting loading systems of long-span cable-bridge structure are constructed by weighted amplitude wave superposition method. Nonlinear model of cable-bridge coupled vibration control system is established and the corresponding system of state space form is described. In order to reduce buffeting loading influence of the wind-induced vibration for the structure and improve the robust performance of the vibration control, based on semiactive vibration control devices and optimal control approach, a feedforward and feedback optimal vibration controller is designed, and an algorithm is presented for the vibration controller. Numerical simulation results are presented to illustrate the effectiveness of the proposed technique.

1. Introduction

Bridge plays an irreplaceable role in the transportation system and has an important value in politics, economy, culture, and military. Influenced by winds, vehicles, pedestrians, and seism loads, structural vibration of bridges is inevitable. Sustained and severe vibration of bridge structures not only shortens the service life of bridges but also reduces the security of traffic. Combined with extreme weather situations and geological disasters, accidents and dangerous cases happen frequently to bridge structures. Therefore, in order to decrease bridge vibration and to improve the reliability and safety of bridge structures, constructing vibration control systems has become an urgent problem to be solved.

Most of the studies on bridge structure focus on dynamic behavior [1–4], fatigue damage [4–6], and reliability analysis [7, 8]. Using frequency and time domain methods in [1], dynamic characteristics of a laboratory bridge model are determined by operational modal analysis. According to the actual project, the effects of bridge width and bridge deck pavement thickness on dynamic characteristics of hinge joint voided Slab bridge are researched in [2]. Considering wind, railway, and highway loadings in [3], dynamic stress analysis of long suspension bridges is presented. In [4], the dynamic properties of a decommissioned timber bridge are measured in the laboratory to observe the characteristics of the lowest few mode shapes as the support stiffness is varied to simulate deterioration of the pile supports, and an evaluation method is proposed to quantitatively assess the foundation competence of timber bridges. To avoid heavy interventions for strengthening of bridge deck slabs, in [5], an improved building material is used, fatigue tests for the determination of the fatigue behavior of beams reinforced by this building material are presented, and then a technique is developed to reduce the use costs as well as life cycle costs. The performance of an instrumented concrete bridge deck that has longitudinal cracks along with the entire length of the bridge is presented in [6], and the influence on their fatigue life is monitored. To analyze the reliability of Dongjiang Bridge, in [7], a new analysis method combined with advantages of some common reliability computing methods is put forward. The bridge is modeled as a single span simply supported Euler-Bernoulli beam and the vehicle is modeled as a single degree of freedom system in [8], and they present a reliability analysis of a simply supported bridge deck subjected to random moving and seismic loads. The previous studies are mostly about the vibration response, monitor, and analysis [9–11], but few studies are about vibration control for bridge structure, and the result about optimal control problems [12–15] for nonlinear systems of the bridge structure vibration control is rare.

The remainder of this paper is structured as follows. In Section 2, first, employing weighted amplitude wave superposition method and buffeting loading forces on long-span cable-bridge structure are established. Then, mechanical model and dynamical system of the long-span cable bridge are constructed. In Section 3, optimal vibration problem of cable-bridge systems is presented. In Section 4, first, in order to use successive approximation approach (SAA), two lemmas are introduced. Then, feedforward and feedback optimal vibration controller and algorithm are designed for the long-span cable-bridge system. In Section 5, numerical experiments are presented. Finally, in Section 6 some conclusions are drawn.

2. Cable-Bridge Structure Nonlinear Control System

2.1. Buffeting Loading

As the span of the bridge extends, the stiffness of the bridge structure reduces, and the bridge has much more characteristics of the flexible structure. Therefore, the influence of the wind loading on the bridge structure cannot be neglected in engineering, and many scholars turn their attention to the influence of the wind-induced vibration on the structure. Bridge buffeting is a sort of random vibration of the bridge structure influenced by the fluctuating wind. Fluctuating wind field induced by natural wind can be described by ergodic and stationary Gauss random process and can be regarded as a single variable four-dimensional random field in mathematics.

We employ weighted amplitude wave superposition method to describe buffeting loading forces on long-span cable-bridge structures. According to weighted amplitude wave superposition method, buffeting loading forces on long-span cable-bridge structure can be simulated by superposition of weighted amplitude waves. The buffeting force of the 𝑗th composition wave on long-span cable-bridge structure is as follows: 𝑝_𝑗=𝐴_𝑗sin𝜔_𝑗𝑡+𝜑_𝑗,𝑗=1,2,…,𝑟, (1) in which 𝐴_𝑗 and 𝜔_𝑗 are, respectively, the amplitude and frequency of the 𝑗th wave component and 𝜑_𝑗 is the random phase angles uniformly distributed in 0≤𝜑_𝑗<2𝜋. Let 𝑝(𝑡)=[𝑝₁,𝑝₂,…,𝑝_𝑟]^𝑇; we have 𝑝̈‌_𝑗=−𝜔²_𝑗𝑝_𝑗𝑝̈(𝑡)𝜔,𝑗=1,2,…,𝑟,=−diag₁²,𝜔₂²,…,𝜔_𝑟²𝑝(𝑡)=−Ω²𝑝(𝑡), (2) in which Ω=diag{𝜔₁,𝜔₂,…,𝜔_𝑟}. Let 𝑤(𝑡)=̇𝑝(𝑡)𝑝(𝑡)^𝑇; then 𝑤̇(𝑡)=0𝐼_𝑟−Ω²0𝑤(𝑡)=(𝑡),𝑝(𝑡)=𝐼𝐺𝑤_𝑟0𝑤(𝑡),𝑝(𝑡)=[]𝑝(𝑡)=1,…,11,…,1_𝑟,0,…,0_𝑟𝑤(𝑡)=𝐹𝑤(𝑡), (3) where 𝐹=[1,…,1_𝑟,0,…,0_𝑟], 𝐺=^0𝐼𝑟_−Ω20, 𝐼_𝑟 is the 𝑟-order unit matrix, and 0∈𝑅^𝑟×𝑟 is the zero matrix.

So, the total buffeting loading force acting on the bridge structure can be generated by the exosystem: 𝑤̇(𝑡)=(𝑡),𝑝(𝑡)=𝐺𝑤𝐹𝑤(𝑡). (4)

2.2. Mechanical Model and Dynamical System

Considering cable-bridge structures with semiactive tune mass damper devices, the deck is simplified as the lumped mass influenced by the cable end and is denoted by the parameter 𝑀. And the stiffness and damping of the deck are represented by 𝐶 and 𝐾, respectively. The cable-bridge coupled vibration induced by the buffeting loading is decomposed into the motion along the axial direction and the motion perpendicular to the cable axis. The problem of the axial direction motion is studied in this paper. The mechanical model of the cable and deck is shown in Figure 1.

Figure 1 

The mechanical model of cable and deck.

In order to simplify the problem and reflect the essence of the vibration control for the cable-bridge structure, we make some fundamental assumptions as follows:(1)the flexural stiffness, torsional stiffness, and shear stiffness are disregarded;(2)the gravity sag curve is considered as a parabola;(3)the constitutive relation of the deformation for the cable satisfies Hooke’s law and is uniform for each point;(4)the effect of the tower vibration on the cable is disregarded.

Generally, the fundamental mode is in the dominant position; therefore we consider the first mode as the main object to study the cable-bridge coupled vibration control problem. The dynamical system of the cable-bridge coupled vibration induced by the buffeting loading is as follows: 𝑊̈(𝑡)+𝜔²₁+𝑎₃𝑌(𝑡)𝑊(𝑡)+𝑎₁𝑊³(𝑡)+𝑎₂𝑊²(𝑡)+𝑎₄𝑌(𝑡)𝑢(𝑡)(𝑡)=𝑌̈(𝑡)=0,+𝑝+2𝜔₂𝜉𝑌̇(𝑡)+𝜔²₂𝑌(𝑡)+𝑎₅𝑊(𝑡)+𝑎₆𝑊²(𝑡), (5) in which 𝑊(𝑡) is cable displacement from the equilibrium position; 𝑌(𝑡) is the displacement of the cable end, namely, the deck displacement along the axial direction of the cable; 𝑢(𝑡) is control input variable; 𝜔₁ and 𝜔₂ are the intrinsic frequencies of the cable and deck, respectively; 𝜉 is the damping ratio of the deck; 𝑎_𝑖(𝑖=1,2,…,6) are parameters determined by the factors such as the mass per unit length and the dip angle of the cable and the regular function of the deck vibration mode.

Choose state variables and state vector for the cable-bridge coupled vibration control system (5) as follows: 𝑥₁(𝑡)=𝑊(𝑡),𝑥₂(𝑡)=𝑊̇(𝑡),𝑥₃(𝑡)=𝑌(𝑡),𝑥₄(𝑡)=𝑌̇(𝑡),𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)^𝑇, (6) and the system (5) is rewritten in the state-space representation: 𝑥̇(𝑡)(𝑡)(𝑡)(𝑥)(𝑡),𝑥(0)=𝐴𝑥+𝐵𝑢+𝑓+𝐷𝑝=𝑥₀, (7) where 𝑝(𝑡) is external input disturbance, namely, buffeting loading force, and 𝐴=0100−𝜔²₁0−𝑎₄00001−𝑎₅0−𝜔²₂−2𝜔₂𝜉,00010001,𝑓(𝑥)=𝐵=,𝐷=0−𝑎₁𝑥³₁(𝑡)−𝑎₂𝑥²₁(𝑡)−𝑎₃𝑥₃(𝑡)𝑥₁(𝑡)0−𝑎₆𝑥²₁(𝑡)^𝑇. (8)

3. Optimal Vibration Control Problem of the Long-Span Cable-Bridge System

In order to study optimal vibration control problem for long-span cable-bridge, we choose an average performance index for system (7) as follows: 𝐽=lim_𝑇→∞1𝑇^𝑇₀𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑅𝑢²(𝑡)𝑑𝑡, (9) where 𝑄∈ℝ^4×4 are positive semidefinite matrices and 𝑅∈ℝ is a positive definite matrix.

The objective of this paper is to find a control law 𝑢^∗(𝑡) for system (7) and make the value of the performance index (9) a minimum.

Applying the maximum principle to the optimal control problem in (7) and (9), the optimal control law is described by 𝑢^∗(𝑡)=−𝑅⁻¹𝐵^𝑇𝜆(𝑡), (10) where 𝜆(𝑡) is the solution to the following nonlinear two-point boundary value (TPBV) problem: −𝜆̇(𝑡)=𝑄𝑥(𝑡)+𝐴^𝑇𝜆(𝑡)+𝑓_𝑥^𝑇(𝑥)𝜆(𝑡),𝑥̇(𝑡)(𝑡)(𝑡)(𝑥)(𝑡),𝑥(0)=𝐴𝑥−𝑆𝜆+𝑓+𝐷𝑝=𝑥₀,𝜆(∞)=0, (11) where 𝑆=𝐵𝑅⁻¹𝐵^𝑇 and 𝑓_𝑥^𝑇(𝑥)=𝜕𝑓^𝑇(𝑥)/𝜕𝑥 is the Jacobian matrix of 𝑓(𝑥) with respect to vector 𝑥.

Unfortunately, for the nonlinear TPBV problem in (11), with the exception of the simplest cases, there is no analytic solution. Therefore, we try to find the numerical solution to such problem. The main purpose of this paper is to develop SAA [15] in order to find the feedforward and feedback optimal control law for the system described by (7) with average performance index (9). In this approach, the optimal control law is composed of exact linear term and the compensation of the nonlinear term. The compensation of the nonlinear term is the limit of the costate vector iteration sequence. The original optimal control law is approached by the process of solving the iteration sequence, instead of solving the nonlinear TPBV problem.

4. Optimal Vibration Control Law Design

Consider the nonlinear system 𝑧̇(𝑡)(𝑡)𝑧(𝑡)(𝑧(𝑡))(𝑡),𝑧𝑡=𝐺+ℎ,𝑡+𝐹𝑣₁=𝜂, (12) where 𝑧∈ℝ^𝑛 is the state vector, 𝑣∈ℝ^𝑚 is the input vector, ℎ∶ℝ^𝑛×ℝ₊→𝑈, ℎ(0,𝑡)≡0, 𝐺∶ℝ₊→ℝ^𝑛×𝑛, 𝐹∈ℝ^𝑛×𝑚, and 𝜂 is the initial state vector (for 𝑡₁=𝑡₀) or the terminal state vector (for 𝑡₁=𝑡_𝑓). Assume that ℎ satisfies the Lipschitz conditions on ℝ^𝑛×ℝ₊.

In order to use the SAA, we introduce two lemmas [16].

Lemma 1. Define the vector function sequence {𝑧^(𝑘)(𝑡)} as 𝑧⁽⁰⁾(𝑡)=Φ𝑡,𝑡₀𝜂+^𝑡_𝑡0Φ()(𝑟)𝑧𝑡,𝑟𝐹𝑣𝑑𝑟,_(𝑘)(𝑡)=Φ𝑡,𝑡₀𝜂+^𝑡_𝑡0Φ(𝑡,𝑟)ℎ𝑧^(𝑘−1)(𝑟),𝑟+𝐹𝑣(𝑟)𝑑𝑟,𝑘=1,2,…,𝑡∈ℝ_𝑇, (13) where Φ(𝑡,𝑡₀) is the state transition matrix corresponding to 𝐺(𝑡); then the sequence {𝑧^(𝑘)(𝑡)} uniformly converges to the solution of (12) for 𝑡₁=𝑡₀.

Lemma 2. Define the vector function sequence {𝑧^(𝑘)(𝑡)} as 𝑧⁽⁰⁾(𝑡)=Φ𝑡,𝑡_𝑓𝜂+^𝑡_𝑡𝑓Φ()(𝑟)𝑧𝑡,𝑟𝐹𝑣𝑑𝑟,_(𝑘)(𝑡)=Φ𝑡,𝑡_𝑓𝜂+^𝑡_𝑡𝑓Φ(𝑡,𝑟)ℎ𝑧^(𝑘−1)(𝑟),𝑟+𝐹𝑣(𝑟)𝑑𝑟,𝑘=1,2,…,𝑡∈ℝ_𝑇. (14) Then the sequence {𝑧^(𝑘)(𝑡)} converges uniformly to the solution of system (12) as 𝑡₁=𝑡_𝑓.

4.1. Feedforward and Feedback Optimal Controller and Algorithm Design

Then, we design vibration controller for system (7). The optimal control law can be presented in the following theorem.

Theorem 3. Consider the optimal control problem described by system (7) with performance index (9), the feedforward and feedback optimal control law 𝑢^∗(𝑡) exists and is unique. Its form is as follows: 𝑢^∗(𝑡)=−𝑅⁻¹𝐵^𝑇𝑃₁𝑥(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡)+lim_𝑘→∞𝑔^(𝑘)(𝑡), (15) where 𝑃₁ is the unique positive definite solution of the following Riccati matrix equation: 𝐴^𝑇𝑃₁+𝑃₁𝐴−𝑃₁𝑆𝑃₁+𝑄=0. (16)𝑃₂ and 𝑃₃ are the unique solutions of the following Sylvester matrix equations: 𝐴^𝑇−𝑃₁𝑆²𝑃₂𝐹+𝑃₂𝐹Ω²=−𝐴^𝑇−𝑃₁𝑆𝑃₁𝐷𝐴𝐹,_𝑇−𝑃₁𝑆²𝑃₃+𝑃₃Ω²=𝑃₁𝐷. (17) The 𝑘th adjoint vector is found from the following adjoint vector sequences: 𝑔⁽⁰⁾(𝑡)=^∞_𝑡Φ^𝑇(𝑟−𝑡)𝑓_𝑥^𝑇(0)𝑃₂𝑝(𝑟)+𝑃₃𝑝_𝜔(𝑟𝑔)𝑑𝑟,_(𝑘)(𝑡)=^∞_𝑡Φ^𝑇(𝑟−𝑡)𝑃₁𝑓𝑥^(𝑘−1)(𝑟)+𝑓_𝑥^𝑇𝑥^(𝑘−1)(𝑟)×𝑃₁𝑥^(𝑘−1)(𝑟)+𝑃₂𝑝(𝑟)+𝑃₃𝑝_𝜔(𝑟)+𝑔^(𝑘−1)(𝑟)𝑑𝑟,𝑘=1,2,…, (18) and the 𝑘th state vectors 𝑥^(𝑘)(𝑡) satisfy the following state vector sequences: 𝑥⁽⁰⁾(𝑡)=Φ(𝑡)𝑥₀+^𝑡₀Φ(𝑟−𝑡)𝐷−𝑆𝑃₂𝑝(𝑟)−𝑆𝑃₃𝑝_𝜔(𝑟)−𝑆𝑔⁽⁰⁾(𝑟𝑥)𝑑𝑟,_(𝑘)(𝑡)=Φ(𝑡)𝑥₀+^𝑡₀Φ(𝑟−𝑡)𝐷−𝑆𝑃₂𝑝(𝑟)−𝑆𝑃₃𝑝_𝜔(𝑟)−𝑆𝑔^(𝑘)(𝑟)+𝑓𝑥^(𝑘−1)(𝑟)Φ(𝑡)𝑑𝑟,𝑘=1,2,…,=𝑒^{(𝐴−𝑆𝑃1)𝑡},Ω=diagΩ,Ω,𝑝_𝜔(𝑡)=𝑝̇(𝑡)=𝐹𝐺𝑤(𝑡). (19)(20)

Proof. To use the SAA to solve nonlinear TPBV problem (11), we separate the linear part in nonlinear TPBV problem (11); let 𝜆(𝑡)=𝑃₁𝑥(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝̇(𝑡)+𝑔(𝑡), (21) where 𝑔(𝑡) is an adjoint vector introduced to compensate for the effect of the nonlinear term in system (7).
Substituting the first equation of (7) and (10) into the derivative of (21), we get 𝜆̇(𝑡)=𝑃₁𝑥̇(𝑡)+𝑃₂𝑝̇(𝑡)+𝑃₃𝑝̈(𝑡)+𝑔̇(𝑡)=𝑃₁𝐴−𝑃₁𝑆𝑃₁𝑥(𝑡)+𝑃₁𝐷−𝑃₁𝑆𝑃₂𝑝(𝑡)+𝑃₂−𝑃₁𝑆𝑃₃𝑝̇(𝑡)+𝑃₃𝑝̈(𝑡)−𝑃₁𝑆𝑔(𝑡)+𝑃₁𝑓(𝑥)+𝑔̇(𝑡). (22) Note that 𝑝_𝜔(𝑡)=𝑝̇(𝑡),𝑝̈(𝑡)=𝐹𝐺²𝑤(𝑡)=−𝐹Ω²𝑤(𝑡). (23) From (11) and (21), we obtain 𝜆̇(𝑡)=−𝑄+𝐴^𝑇𝑃₁𝑥(𝑡)−𝐴^𝑇𝑃₂𝑝(𝑡)−𝐴^𝑇𝑃₃𝑝_𝜔(𝑡)−𝐴^𝑇𝑔(𝑡)−𝑓^𝑇_𝑥(𝑥)𝜆(𝑡). (24) Substituting (23) into (22) and comparing the coefficients of (22) and (24), we obtain matrix equations: 𝐴^𝑇𝑃₁+𝑃₁𝐴−𝑃₁𝑆𝑃₁𝐴+𝑄=0,_𝑇𝑃₂𝐹+𝑃₁𝐷𝐹−𝑃₃𝐹Ω²−𝑃₁𝑆𝑃₂𝐴𝐹=0,_𝑇𝑃₃+𝑃₂−𝑃₁𝑆𝑃₃=0, (25) and the adjoint vector differential equations: 𝑔̇(𝑡)=−𝐴−𝑆𝑃₁^𝑇𝑔(𝑡)−𝑃₁𝑓(𝑥)−𝑓_𝑥^𝑇(𝑥)𝜆(𝑡),𝑔(∞)=0. (26) From the matrix equations in (25), we obtain 𝐴^𝑇𝑃₁+𝑃₁𝐴−𝑃₁𝑆𝑃₁𝐴+𝑄=0,_𝑇−𝑃₁𝑆²𝑃₂𝐹+𝑃₂𝐹Ω²=−𝐴^𝑇−𝑃₁𝑆𝑃₁𝐷𝐴𝐹,_𝑇−𝑃₁𝑆²𝑃₃+𝑃₃Ω²=𝑃₁𝐷. (27) Substituting (21) into the second equation of TPBV problem (11), we have 𝑥̇(𝑡)=𝐴−𝑆𝑃₁𝑥(𝑡)+𝐷−𝑆𝑃₂𝑝(𝑡)−𝑆𝑃₃𝑝_𝜔(𝑡)(𝑡)(𝑥),𝑥𝑡−𝑆𝑔+𝑓₀=𝑥₀. (28) In order to obtain the solution of TPBV problem (11) from the SAA, according to (28) and (26), we construct the following state equation sequences: 𝑥̇⁽⁰⁾(𝑡)=𝐴−𝑆𝑃₁𝑥⁽⁰⁾(𝑡)+𝐷−𝑆𝑃₂𝑝(𝑡)−𝑆𝑃₃𝑝_𝜔(𝑡)−𝑆𝑔⁽⁰⁾(𝑡),𝑥⁽⁰⁾𝑡₀=𝑥₀,𝑥̇^(𝑘)(𝑡)=𝐴−𝑆𝑃₁𝑥^(𝑘)(𝑡)+𝐷−𝑆𝑃₂𝑝(𝑡)−𝑆𝑃₃𝑝_𝜔(𝑡)−𝑆𝑔^(𝑘)(𝑡)+𝑓𝑥^(𝑘−1),𝑥^(𝑘)𝑡₀=𝑥₀,𝑘=1,2,…, (29) and the adjoint vector differential equation sequences: 𝑔̇⁽⁰⁾(𝑡)=−𝐴−𝑆𝑃₁^𝑇𝑔⁽⁰⁾(𝑡)−𝑓_𝑥^𝑇(0)𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡),lim_𝑇→∞𝑔⁽⁰⁾(𝑇)𝑔̇=0,_(𝑘)(𝑡)=−𝐴−𝑆𝑃₁^𝑇𝑔^(𝑘)(𝑡)−𝑃₁𝑓𝑥^(𝑘−1)(𝑡)−𝑓_𝑥^𝑇𝑥^(𝑘−1)(𝑡)𝜆^(𝑘−1)(𝑡),lim_𝑇→∞𝑔⁽⁰⁾(𝑇)=0,𝑘=1,2,…, (30) where 𝜆^(𝑘)(𝑡)=𝑃₁𝑥^(𝑘)(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡)+𝑔^(𝑘)(𝑡),𝑘=1,2,…. (31) According to Lemmas 1 and 2, the adjoint vector sequence in (18) and the state vector sequences in (19) uniformly converge to the solution of (26) and (28), respectively.
Correspondingly, the control sequence is given in the following form: 𝑢^(𝑘)(𝑡)=−𝑅⁻¹𝐵^𝑇𝜆^(𝑘)(𝑡)=−𝑅⁻¹𝐵^𝑇𝑃₁𝑥^(𝑘)(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡)+𝑔^(𝑘)(𝑡). (32) When 𝑘→∞, we obtain the feedforward and feedback optimal control law 𝑢^∗(𝑡) in (15).
From (15), we can see that the feedforward and feedback optimal control law 𝑢^∗(𝑡) is composed of the state feedback term, the disturbance and its derivative, and the compensation of the nonlinear term of the cable-bridge vibration control system. In order to explain the work of the control law, the control block diagram of the system is drawn simply in Figure 2.

Figure 2 

Control block diagram of the system.

Remark 4. In fact, it is impossible to calculate feedforward and feedback optimal control law in (15). We can find the feedforward and feedback optimal control law by replacing ∞ with 𝑁 in (15): 𝑢_𝑁(𝑡)=−𝑅⁻¹𝐵^𝑇𝑃₁𝑥(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡)+𝑔^(𝑁)(𝑡). (33)
In order to implement the feedforward and feedback control law described in Theorem 3, we design an algorithm in the following.

Algorithm 5. Consider the following.
Step 1. Calculate the matrices 𝑃₁, 𝑃₂, and 𝑃₃ from the matrix equations (16) and (17).
Step 2. Establish the buffeting loading force 𝑝(𝑡) by the model (4) and relevant data from (23).
Step 3. Choose an allowable error 𝜎.
Step 4. Calculate 𝑔⁽⁰⁾(𝑡) and 𝑥⁽⁰⁾(𝑡) by the first equations of (18) and (19), respectively, and let 𝑖=1.
Step 5. Calculate 𝑔^(𝑖)(𝑡) and 𝑥^(𝑖)(𝑡) by the second equations of (18) and (19), respectively.
Step 6. Calculate the 𝑖th feedforward and feedback optimal controller 𝑢^(𝑖)(𝑡) by (32).
Step 7. Calculate 𝐽^(𝑖): 𝐽^(𝑖)=lim_𝑇→∞1𝑇^𝑇₀𝑥^𝑇(𝑖)(𝑡)𝑄𝑥^(𝑖)(𝑡)+𝑅𝑢^2(𝑖)(𝑡)𝑑𝑡. (34) If |𝐽^(𝑗)−𝐽^(𝑗−1)|/𝐽^(𝑗)≤𝜎, 𝑗=𝑖, 𝑖−1, 𝑖−2, 𝑖−3, and 𝑖≥4, stop calculating, and 𝑁 can be chosen from [𝑖−3,𝑖]. The 𝑁th order feedforward and feedback optimal controller 𝑢_𝑁(𝑡) is obtained by (33); else let 𝑖=𝑖+1 and go to Step 5.

5. Numerical Simulations

In this section, we apply the proposed feedforward and feedback optimal vibration controller to a road suspension cable-bridge. The cable-bridge with three towers located in Yangtze River, whose span arrangement is 360 m + 1080 m + 1080 m + 360 m = 2880 m, whose stiffening beam of 3.5 m height and 38.5 m width with a tuyere consisting of a top and a bottom inclined plates is closed flat steel box beam, whose cable bent towers are 178.3 m high, whose two main cables crosswise space between is 35.8 m, whose main part of tower is filled by concrete and is 𝐻 model, and whose middle tower is supported by expanded triangle. The values of system (7) and performance index (9) are as follows: 𝜔₁=0.470 Hz, 𝜔₂=0.793 Hz, 𝜉=0.02, 𝑎_𝑖=1.15(𝑖=1,2,…,6), 𝑄=𝐼, and 𝑅=1.

Employing Matlab software, numerical experiments are carried out for the proposed optimal vibration controller. Figure 3 shows the buffeting loading force curve.

Figure 3 

Buffeting loading force curve.

The main purpose of vibration control of long-span cable-bridge is to reduce the deck displacement which indicates the limit of the deck motion and to reduce the deck velocity which ensures the road holding ability for vehicles and pedestrians. So, to evaluate effectiveness of the proposed control strategy, the deck displacement and velocity are considered. Then, the corresponding curves of open loop system, feedback optimal control system, and the system controlled by the proposed optimal vibration controller are compared and shown in Figures 4 and 5. Then, in Figure 6, the control forces of the feedback controller represented by the red line and proposed controller denoted by the green line are shown.

Figure 4 

Displacement curves.

Figure 5 

Velocity curves.

Figure 6 

Control force curves.

The curves of displacement are shown in Figure 4 and velocities are in Figure 5, in which blue lines represent the open loop results of the deck of long-span bridge systems, red lines denote the results controlled by feedback optimal controller, and green lines describe the results of the long-span bridge systems controlled by the proposed control strategy. It can be seen from Figures 4 and 5 that the proposed optimal controller is efficient, real-time, and robust in reducing displacement and velocity of the deck, thereby ensures safety of the long-span bridge, and enhances passing vehicle ride comfort. Moreover, in Figure 6, it is demonstrated that it needs less control forces than feedback optimal vibration control and therefore it needs less energy. In Figure 5, the system controlled by the proposed controller (green line) responds much quicker than that controlled by feedback controller (red line), but, as it is shown in Figure 6, the output of two controllers is synchronous in terms of changing time; this is the trait and merit of optimal control. In fact, we can change the weighting matrix 𝑄 and the weighting coefficient 𝑅 to balance the control effect and control energy.

6. Conclusions

The effect of the wind-induced vibration on the long-span cable-bridge structure cannot be neglected in construction phase or in operation stage. The influence of buffeting loading and nonlinear factor for long-span cable-bridge structures is considered in this paper. Based on the semiactive vibration control devices, optimal vibration controller and control algorithm are designed for the long-span cable-bridge. Numerical simulation results show that the proposed strategy is efficient, real-time, and robust in reducing the vibration induced by buffeting loading.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Natural Science Foundation of China (nos. 11301009, U1204402, and 61374003) and Natural Science Foundations of Henan Province Education Department (nos. 12A120001, 13A520018, 14B520018, and 13A110022).