CN103926947A - Semi-active vibration control method of nonlinear system of long-span cable bridge structure - Google Patents

Abstract

Translated fromChinese

本发明是一种大跨度索桥结构非线性系统半主动振动控制方法，包括如下步骤：（1）建立作用在大跨度索桥结构上的抖振力荷载模型；（2）建立大跨度索桥结构振动控制系统模型；（3）选取如式3所描述的复合型性能指标； 式3；（4）对式2和式3实施极大值原理： 式2；（5）求最优振动控制律。本发明考虑了大跨度索桥结构中的非线性动力特征，建立了更加接近现实的控制系统；使用了较为合理的控制器的求解方案，最优振动控制律对于大跨度索桥结构非线性系统具有较好的减振作用，更加适应于大幅值振动的大跨度索桥的控制。

The invention is a semi-active vibration control method for a nonlinear system of a long-span cable bridge structure, comprising the following steps: (1) establishing a buffeting force load model acting on the long-span cable bridge structure; (2) establishing a long-span cable bridge Structural vibration control system model; (3) Select the composite performance index as described in formula 3; Equation 3; (4) Implement the maximum value principle for Equation 2 and Equation 3: Equation 2; (5) Find the optimal vibration control law. The present invention considers the nonlinear dynamic characteristics in the long-span cable bridge structure, establishes a control system closer to reality; uses a more reasonable controller solution scheme, and the optimal vibration control law is for the long-span cable bridge structure nonlinear system It has a good vibration damping effect and is more suitable for the control of large-span cable bridges with large-value vibrations.

Description

Translated fromChinese

大跨度索桥结构非线性系统半主动振动控制方法Semi-active vibration control method for nonlinear system of long-span cable bridge structure

技术领域technical field

本发明涉及一种振动控制方法，尤其是一种大跨度索桥结构非线性系统半主动振动控制方法。The invention relates to a vibration control method, in particular to a semi-active vibration control method for a nonlinear system of a large-span cable bridge structure.

背景技术Background technique

桥梁在交通系统中起着不可替代的作用，在政治、经济、文化以及军事方面具有重要的价值。尽管我国交通系统得到了很大的发展，然而仍然难以满足日益增长的交通需求，交通基础设施承受着巨大的压力。加之最近几年我国频繁出现极端天气和地质灾害，我国交通系统中的桥梁结构频频出现事故和险情。桥梁结构承受各种荷载作用导致结构振动剧烈，桥梁结构的振动不但降低了行车环境质量，还加速了结构的机械损伤和疲劳破坏，降低了桥梁的安全性。结构的长期振动能够导致桥梁倾覆，造成重大的经济损失和不良的社会影响。因此，构建桥梁振动控制系统，减小桥梁的振动，提高桥梁的可靠性和安全性，已经成为亟待解决的问题。Bridges play an irreplaceable role in the transportation system and have important values in politics, economy, culture and military. Although my country's transportation system has been greatly developed, it is still difficult to meet the growing traffic demand, and the transportation infrastructure is under tremendous pressure. Coupled with the frequent occurrence of extreme weather and geological disasters in my country in recent years, accidents and dangers frequently occur in bridge structures in my country's transportation system. The bridge structure is subject to various loads, resulting in severe structural vibration. The vibration of the bridge structure not only reduces the quality of the driving environment, but also accelerates the mechanical damage and fatigue damage of the structure, reducing the safety of the bridge. The long-term vibration of the structure can cause the bridge to overturn, resulting in significant economic losses and adverse social impacts. Therefore, building a bridge vibration control system to reduce bridge vibration and improve bridge reliability and safety has become an urgent problem to be solved.

近年来，国内外许多学者对不同荷载与桥梁之间的动力作用响应以及桥梁的振动控制进行了大量的理论分析与试验研究。随着结构动力学理论和有限元理论的不断完善及高速大容量计算机的广泛运用，车桥振动研究取得极大的发展。现在人们可以建立更加贴近实际工程的车辆和桥梁计算模型，可考虑车体和转向架的质量、阻尼器和弹簧的作用，行车速度，桥梁上部结构、下部结构的质量、刚度、阻尼，车轮和轨道、轨道和梁之间的动力相互作用，轨道不平顺等多种因素的影响，采用数值模拟方法研究车桥时变系统的空间振动控制。In recent years, many scholars at home and abroad have conducted a large number of theoretical analysis and experimental research on the dynamic response between different loads and bridges and the vibration control of bridges. With the continuous improvement of structural dynamics theory and finite element theory and the wide application of high-speed and large-capacity computers, the research on axle vibration has made great progress. Now people can establish a vehicle and bridge calculation model that is closer to the actual engineering, which can consider the mass of the car body and bogie, the effect of dampers and springs, the speed of the vehicle, the mass, stiffness, damping of the bridge superstructure and substructure, wheels and The dynamic interaction between the track, the track and the girder, the influence of various factors such as the track irregularity, the numerical simulation method is used to study the space vibration control of the time-varying system of the vehicle bridge.

在进行桥梁结构的振动控制系统的构建过程中，线性模型适用于小幅值的振动情况，对于结构上作用的强振动荷载(如地震等)，另外，大跨度索桥结构常具有材料非线性、几何非线性等特点，线性模型已不再适用。因此找到一种适应于具有非线性动力特征的振动控制方法已成为亟待解决的问题。In the process of constructing the vibration control system of the bridge structure, the linear model is suitable for small-amplitude vibration. For the strong vibration load (such as earthquake, etc.) acting on the structure, in addition, the long-span cable bridge structure often has material nonlinearity. , geometric nonlinearity and other characteristics, the linear model is no longer applicable. Therefore, finding a vibration control method suitable for nonlinear dynamic characteristics has become an urgent problem to be solved.

有鉴于此，特提出本发明。In view of this, the present invention is proposed.

发明内容Contents of the invention

本发明的目的是提供一种适应于具有非线性动力特征的大跨度索桥结构半主动振动控制方法。The purpose of the present invention is to provide a semi-active vibration control method suitable for a long-span cable bridge structure with nonlinear dynamic characteristics.

为解决上述技术问题，本发明采用技术方案的基本构思是：In order to solve the problems of the technologies described above, the present invention adopts the basic idea of technical solution to be:

一种大跨度索桥结构非线性系统半主动振动控制方法，包括如下步骤：A semi-active vibration control method for a nonlinear system of a long-span cable bridge structure, comprising the following steps:

（1）建立作用在大跨度索桥结构上的抖振力荷载模型：(1) Establish the buffeting force load model acting on the long-span cable bridge structure:

由多个三角函数组成的谐波相互叠加模拟作用在大跨度索桥结构上的抖振力荷载，第j个组成波的作用在大跨度索桥结构上的抖振力可由下列式子给出：The harmonics composed of multiple trigonometric functions are superimposed to simulate the buffeting force load acting on the long-span cable bridge structure. The buffeting force of the jth component wave acting on the long-span cable bridge structure can be given by the following formula :

j=1,2,…,rj=1,2,...,r

其中，A_j，ω_j分别表示第j个组成波的振幅、频率和初相角，Among them, A_j , ω_j respectively represent the amplitude, frequency and initial phase angle of the jth component wave,

令$\stackrel{\‾}{p}\left(t\right)=[\stackrel{\‾}{p_1},\stackrel{\‾}{p_2},\·\·\·,\stackrel{\‾}{p_r}{]}^T,$得make$\stackrel{\‾}{p}\left(t\right)=[\stackrel{\‾}{p_1},\stackrel{\‾}{p_2},\·\&Center\; Dot;\&Center\; Dot;,\stackrel{\‾}{p_r}{]}^T,$ have to

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>=</span>=</span><span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>,</span>,</span>\mathrm{<span><span>j</span>j</span>}<span><span>=</span>=</span>\mathrm{<span><span>1,2</span>1,2</span>}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>,</span>,</span>\mathrm{<span><span>r</span>r</span>}<span><span>,</span>,</span>$

$\begin{array}{c}\stackrel{\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{<span><span>~</span>~</span>}}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\mathrm{<span><span>diag</span>diag</span>}<span><span>\{</span>\{</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>r</span>r</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>\}</span>\}</span>\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\end{array}$

其中，$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&omega;}}_r\}.$令$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&CenterDot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$可得in,$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&omega;}}_r\}.$ make$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&Center\; Dot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$ Available

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\left[\begin{array}{cc}\mathrm{<span><span>0</span>0</span>}& {\mathrm{<span><span>I</span>I</span>}}_{\mathrm{<span><span>r</span>r</span>}}\\ <span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}\end{array}\right]\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$

其中，I_r是单位矩阵，0∈R^r×r是零矩阵；in, I_r is the identity matrix, 0∈R^r×r is the zero matrix;

作用在大跨度索桥结构上的抖振力荷载模型为式1所描述的外系统：The buffeting force load model acting on the long-span cable bridge structure is the external system described by formula 1:

$\begin{array}{c}\stackrel{\&CenterDot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{F}w\left(t\right)\end{array}$   式1，$\begin{array}{c}\stackrel{\&CenterDot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{f}w\left(t\right)\end{array}$ Formula 1,

其中：p(t)为作用在大跨度索桥结构上的抖振力；Where: p(t) is the buffeting force acting on the structure of the long-span cable bridge;

（2）建立大跨度索桥结构振动控制系统模型：(2) Establish the vibration control system model of the long-span cable bridge structure:

抖振力荷载作用下索-桥耦合振动动力系统为：The cable-bridge coupled vibration dynamic system under buffeting force load is:

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>)</span>)</span>\mathrm{<span><span>W</span>W</span>}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}<span><span>,</span>,</span>$

$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>u</span>u</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

其中，W(t)为偏离平衡位置的位移；Y(t)为索端的位移，即桥面沿索方向的位移；ω₁和ω₂为索和桥面的固有频率；ξ为为桥面的阻尼比；a_i(i=1,2,…,6)为系统参数，u(t)是半主动控制力，p(t)是抖振力荷载，由式1产生；Among them, W(t) is the displacement away from the equilibrium position; Y(t) is the displacement of the cable end, that is, the displacement of the bridge deck along the direction of the cable; ω₁ and ω₂ are the natural frequencies of the cable and the bridge deck; ξ is the bridge deck The damping ratio of ; a_i (i=1,2,…,6) is the system parameter, u(t) is the semi-active control force, p(t) is the buffeting force load, which is generated by formula 1;

选取状态变量：Select state variables:

${\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>4</span>4</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$

则大跨度索桥结构振动控制系统模型为:Then the structural vibration control system model of the long-span cable bridge is:

$\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$   式2；其中，$\begin{array}{c}\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$ Formula 2; where,

$\mathrm{<span><span>A</span>A</span>}<span><span>=</span>=</span>\left[\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}& {<span><span>-</span>-</span>\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}& <span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>B</span>B</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>D</span>D.</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>$

$\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>[</span>[</span>\begin{array}{c}\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}\end{array}{<span><span>]</span>]</span>}^{\mathrm{<span><span>T</span>T</span>}}$

（3）选取如式3所描述的复合型性能指标；(3) Select the composite performance index as described in formula 3;

$J=\underset{T\&RightArrow;\&infin;}{\lim }\frac{1}{T}{\&Integral;}_0^T[x^T\left(t\right)\mathrm{Qx}\left(t\right)+{\mathrm{Ru}}^2\left(t\right)]\mathrm{dt},$   式3；$J=\underset{T\&Right\; Arrow;\&infin;}{\lim }\frac{1}{T}{\&Integral;}_0^T[x^T\left(t\right)\mathrm{Qx}\left(t\right)+{\mathrm{Ru}}^2\left(t\right)]\mathrm{dt},$ Formula 3;

（4）对式2和式3实施极大值原理：(4) Implement the maximum value principle for formula 2 and formula 3:

首先构造哈密顿函数如式4：First construct the Hamiltonian function as formula 4:

H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω)   式4；H(·)=x^T Qx₁ +Ru² +λ^T (Ax+Bu+f(x)+Dp_ω ) Formula 4;

进而根据极值条件，把式1在性能指标式3的约束下求控制器u(t)的问题转化为求解下述非线性两点边值的问题，如式5：Furthermore, according to the extreme value condition, the problem of finding the controller u(t) in Equation 1 under the constraint of performance index Equation 3 is transformed into the problem of solving the following nonlinear two-point boundary value, such as Equation 5:

$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$   式5；$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$ Formula 5;

（5）求最优振动控制律：(5) Find the optimal vibration control law:

使用逐次逼近方法求解式5描述的非线性两点边值问题的迭代解，令Using the successive approximation method to solve the iterative solution of the nonlinear two-point boundary value problem described in Equation 5, let

λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)λ(t)=P₁ x(t)+P₂ p(t)+P₃ p_ω (t)+g(t)

其中，g(t)是一个共态向量，对λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)求导，然后把式2的第一式$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$代入其中，得Among them, g(t) is a co-state vector, for λ(t)=P₁ x(t)+P₂ p(t)+P₃ p_ω (t)+g(t) derivative, then the formula The first formula of 2$\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$ Substitute into it, get

$\begin{array}{c}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>A</span>A</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>D</span>D.</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>Sg</span>S\; g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}$

由式2和式5，得From formula 2 and formula 5, get

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>+</span>+</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}\mathrm{<span><span>g</span>g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>f</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}^{\mathrm{<span><span>T</span>T</span>}}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span>\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

考虑到considering

$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\stackrel{\mathrm{<span><span>A</span>A</span>}}{<span><span>=</span>=</span>}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

通过比较的系数，得到矩阵方程组式7以及序列式8和式9，从而可求得式6所描述的最优振动控制律By comparison The coefficients of the matrix equation 7 and the sequence formula 8 and formula 9 are obtained, so that the optimal vibration control law described in formula 6 can be obtained

$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{F}+P_2\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_{1}D\stackrel{\&OverBar;}{F}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{F}+P_3\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=P_{1}D\stackrel{\&OverBar;}{F}\end{array},,$   式7；$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{f}+P_2\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{f}+P_3\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\end{array},,$ Formula 7;

$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式8；$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$ Formula 8;

$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式9；$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&Center\; Dot;\&CenterDot;\&CenterDot;,\end{array}$ Formula 9;

由上式7、8、9得出近似最优振动控制律为：From the above formulas 7, 8, and 9, the approximate optimal vibration control law can be obtained as:

u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]式6；u^(k) (t)=-R^-1 B^T λ^(k) (t)=-R^-1 B^T [P₁ x^(k) (t)+P₂ p(t)+P₃ p_ω (t)+g^(k) (t)] formula 6;

其中，而P₁，P₂和P₃由式7求得：x^(k)及g^(k)(t)由式8和式9求得。in, And P₁ , P₂ and P₃ are obtained by formula 7; x^(k) and g^(k) (t) are obtained by formula 8 and formula 9.

进一步地，在步骤（2）后包括：将式（2）的大跨度索桥结构振动控制系统模型描述的大跨度索桥结构振动控制系统简化为一个单自由度系统。Further, after step (2), it includes: simplifying the structural vibration control system of the long-span cable bridge described by the structural vibration control system model of the long-span cable bridge in formula (2) into a single-degree-of-freedom system.

本发明的有益效果为：本发明考虑了大跨度索桥结构中的非线性动力特征，建立了更加接近现实的控制系统；使用了较为合理的控制器的求解方案，最优振动控制律对于大跨度索桥结构非线性系统具有较好的减振作用，更加适应于大幅值振动的大跨度索桥的控制，使用逐次逼近算法避免了直接求解非线性两点边值问题的解析解，只需要求解一个线性向量差分等式序列的解，把直线迭代整个控制律转位迭代共态向量，计算量大大降低。The beneficial effects of the present invention are as follows: the present invention considers the nonlinear dynamic characteristics in the structure of the long-span cable bridge, and establishes a control system closer to reality; uses a more reasonable solution for the controller, and the optimal vibration control law is suitable for large The nonlinear system of the span cable bridge structure has a better vibration damping effect, and is more suitable for the control of large-span cable bridges with large-scale vibrations. Using the successive approximation algorithm avoids directly solving the analytical solution of the nonlinear two-point boundary value problem, and only needs Solving the solution of a sequence of linear vector difference equations, and iterating the entire control law by transposing the straight line and iterating the common state vector, the amount of calculation is greatly reduced.

附图说明Description of drawings

图1为本发明的流程图；Fig. 1 is a flow chart of the present invention;

图2为拉索和桥面的力学模型图；Fig. 2 is the mechanical model diagram of stay cable and bridge deck;

图3为本发明的大跨度索桥结构的位移曲线图；Fig. 3 is the displacement curve figure of long-span cable bridge structure of the present invention;

图4为本发明的大跨度索桥结构的速度曲线图。Fig. 4 is a velocity curve diagram of the long-span cable bridge structure of the present invention.

具体实施方式Detailed ways

为了使本技术领域的人员更好地理解本发明方案，下面结合附图和具体实施方式对本发明作进一步的详细说明。In order to enable those skilled in the art to better understand the solution of the present invention, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

参照图1，本发明是一种大跨度索桥结构非线性系统半主动振动控制方法，包括如下步骤：With reference to Fig. 1, the present invention is a kind of non-linear system semi-active vibration control method of long-span cable bridge structure, comprises the steps:

S100、建立作用在大跨度索桥结构上的抖振力荷载模型：S100, establish the buffeting force load model acting on the long-span cable bridge structure:

作用在大跨度索桥结构上的抖振力荷载可利用谐波合成法来进行构造。根据谐波合成法，作用在大跨度索桥结构上的抖振力荷载可由多个三角函数组成的谐波相互叠加模拟作用在大跨度索桥结构上的抖振力荷载，第j个组成波的作用在大跨度索桥结构上的抖振力可由下列式子给出：The buffeting force load acting on the long-span cable bridge structure can be constructed by using the harmonic synthesis method. According to the harmonic synthesis method, the buffeting force load acting on the long-span cable bridge structure can be superimposed by harmonics composed of multiple trigonometric functions to simulate the buffeting force load acting on the long-span cable bridge structure, the jth component wave The buffeting force acting on the long-span cable bridge structure can be given by the following formula:

j=1,2,…,rj=1,2,...,r

其中，A_j，ω_j分别表示第j个组成波的振幅、频率和初相角，Among them, A_j , ω_j respectively represent the amplitude, frequency and initial phase angle of the jth component wave,

令$\stackrel{\&OverBar;}{p}\left(t\right)=[\stackrel{\&OverBar;}{p_1},\stackrel{\&OverBar;}{p_2},\&CenterDot;\&CenterDot;\&CenterDot;,\stackrel{\&OverBar;}{p_r}{]}^T,$得make$\stackrel{\&OverBar;}{p}\left(t\right)=[\stackrel{\&OverBar;}{p_1},\stackrel{\&OverBar;}{p_2},\&Center\; Dot;\&Center\; Dot;\&Center\; Dot;,\stackrel{\&OverBar;}{p_r}{]}^T,$ have to

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>=</span>=</span><span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>,</span>,</span>\mathrm{<span><span>j</span>j</span>}<span><span>=</span>=</span>\mathrm{<span><span>1,2</span>1,2</span>}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>,</span>,</span>\mathrm{<span><span>r</span>r</span>}<span><span>,</span>,</span>$

$\begin{array}{c}\stackrel{\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{<span><span>~</span>~</span>}}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\mathrm{<span><span>diag</span>diag</span>}<span><span>\{</span>\{</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>r</span>r</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>\}</span>\}</span>\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\end{array}$

其中，$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&omega;}}_r\}.$令$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&CenterDot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$可得in,$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&Center\; Dot;\&CenterDot;\&CenterDot;,{\mathrm{\&omega;}}_r\}.$ make$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&Center\; Dot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$ Available

$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\left[\begin{array}{cc}\mathrm{<span><span>0</span>0</span>}& {\mathrm{<span><span>I</span>I</span>}}_{\mathrm{<span><span>r</span>r</span>}}\\ <span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}\end{array}\right]\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$

其中，I_r是单位矩阵，0∈R^r×r是零矩阵；in, I_r is the identity matrix, 0∈R^r×r is the zero matrix;

作用在大跨度索桥结构上的抖振力荷载模型为式1所描述的外系统：The buffeting force load model acting on the long-span cable bridge structure is the external system described by formula 1:

$\begin{array}{c}\stackrel{\&CenterDot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{F}w\left(t\right)\end{array}$   式1，其中：p(t)为作用在大跨度索桥结构上的抖振力；$\begin{array}{c}\stackrel{\&Center\; Dot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{f}w\left(t\right)\end{array}$ Equation 1, where: p(t) is the buffeting force acting on the structure of the long-span cable bridge;

S101、建立大跨度索桥结构振动控制系统模型：S101. Establishing the structural vibration control system model of the long-span cable bridge:

为了研究的方便，将桥面简化为作用在索端的集中质量M,桥面刚度用弹簧K模拟,桥面阻尼由C模拟。抖振力荷载作用下索-桥耦合振动可以分解为沿轴向及垂直于索轴线的两种运动,轴向振动为本文的研究对象，拉索和桥面的力学模型如图2所示，其中：L为索的长度。For the convenience of the research, the bridge deck is simplified as the concentrated mass M acting on the cable end, the bridge deck stiffness is simulated by spring K, and the bridge deck damping is simulated by C. The cable-bridge coupling vibration under buffeting force load can be decomposed into two kinds of motions along the axial direction and perpendicular to the cable axis. The axial vibration is the research object of this paper. The mechanical model of the cable and the bridge deck is shown in Figure 2. Where: L is the length of the cable.

由于在拉索和桥面构成的振动系统中，基本模态占主要地位，此处取一阶振动模态，则抖振力荷载作用下索-桥耦合振动动力系统为：Since the fundamental mode plays a dominant role in the vibration system composed of the cables and the bridge deck, the first-order vibration mode is taken here, and the cable-bridge coupled vibration dynamic system under the buffeting force load is:

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>)</span>)</span>\mathrm{<span><span>W</span>W</span>}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}<span><span>,</span>,</span>$

$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>u</span>u</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

其中，W(t)为偏离平衡位置的位移；Y(t)为索端的位移，即桥面沿索方向的位移；ω₁和ω₂为索和桥面的固有频率；ξ为为桥面的阻尼比；a_i(i=1,2,…,6)为系统参数，u(t)是半主动控制力，p(t)是抖振力荷载，由式1产生；Among them, W(t) is the displacement away from the equilibrium position; Y(t) is the displacement of the cable end, that is, the displacement of the bridge deck along the direction of the cable; ω₁ and ω₂ are the natural frequencies of the cable and the bridge deck; ξ is the bridge deck The damping ratio of ; a_i (i=1,2,…,6) is the system parameter, u(t) is the semi-active control force, p(t) is the buffeting force load, which is generated by formula 1;

选取状态变量：Select state variables:

${\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>4</span>4</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$

则大跨度索桥结构振动控制系统模型为:Then the structural vibration control system model of the long-span cable bridge is:

$\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$   式2；$\begin{array}{c}\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$ Formula 2;

其中，in,

$\mathrm{<span><span>A</span>A</span>}<span><span>=</span>=</span>\left[\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}& {<span><span>-</span>-</span>\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}& <span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>B</span>B</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>D</span>D.</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>$

$\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>[</span>[</span>\begin{array}{c}\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}\end{array}{<span><span>]</span>]</span>}^{\mathrm{<span><span>T</span>T</span>}}$

S102、选取如式3所描述的复合型性能指标，用来保证输入的控制力和控制时间达到最小；S102. Select a composite performance index as described in formula 3 to ensure that the input control force and control time are minimized;

   式3； Formula 3;

S103、对式2和式3实施极大值原理：将非线性最优控制问题转化为一个非线性非齐次两点边值S103, implement the maximum value principle on formula 2 and formula 3: transform the nonlinear optimal control problem into a nonlinear non-homogeneous two-point boundary value

首先构造哈密顿函数如式4：First construct the Hamiltonian function as formula 4:

H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω)   式4；H(·)=x^T Qx₁ +Ru² +λ^T (Ax+Bu+f(x)+Dp_ω ) Formula 4;

进而根据极值条件，把式1在性能指标式3的约束下求控制器u(t)的问题转化为求解下述非线性两点边值的问题，如式5：Furthermore, according to the extreme value condition, the problem of finding the controller u(t) in Equation 1 under the constraint of performance index Equation 3 is transformed into the problem of solving the following nonlinear two-point boundary value, such as Equation 5:

$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$   式5；$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$ Formula 5;

S104、求最优振动控制律：S104. Find the optimal vibration control law:

由于非线性两点边值问题的解析解难以求得，使用逐次逼近方法解决式5描述的非线性两点边值问题的迭代解，令Since the analytical solution of the nonlinear two-point boundary value problem is difficult to obtain, the iterative solution of the nonlinear two-point boundary value problem described in Equation 5 is solved by using the successive approximation method, so that

λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)λ(t)=P₁ x(t)+P₂ p(t)+P₃ p_ω (t)+g(t)

其中，g(t)是一个共态向量，对λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)求导，然后把式2的第一式$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$代入其中，得Among them, g(t) is a co-state vector, for λ(t)=P₁ x(t)+P₂ p(t)+P₃ p_ω (t)+g(t) derivative, then the formula The first formula of 2$\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$ Substitute into it, get

$\begin{array}{c}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>A</span>A</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>D</span>D.</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>Sg</span>S\; g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}$

由式2和式5，得From formula 2 and formula 5, get

$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>+</span>+</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}\mathrm{<span><span>g</span>g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>f</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}^{\mathrm{<span><span>T</span>T</span>}}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span>\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

考虑到considering

$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\stackrel{\mathrm{<span><span>A</span>A</span>}}{<span><span>=</span>=</span>}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$

通过比较的系数，得到矩阵方程组式7以及序列式8和式9，从而可求得式6所描述的最优振动控制律By comparison The coefficients of the matrix equation 7 and the sequence formula 8 and formula 9 are obtained, so that the optimal vibration control law described in formula 6 can be obtained

$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{F}+P_2\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_{1}D\stackrel{\&OverBar;}{F}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{F}+P_3\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=P_{1}D\stackrel{\&OverBar;}{F}\end{array},,$   式7；$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{f}+P_2\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{f}+P_3\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\end{array},,$ Formula 7;

$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式8；$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&Center\; Dot;\&Center\; Dot;\&Center\; Dot;,\end{array}$ Formula 8;

$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式9；$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&Center\; Dot;\&Center\; Dot;\&Center\; Dot;,\end{array}$ Formula 9;

由上式7、8、9得出近似最优振动控制律为：From the above formulas 7, 8, and 9, the approximate optimal vibration control law can be obtained as:

u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]  式6；u^(k) (t)=-R^-1 B^T λ^(k) (t)=-R^-1 B^T [P₁ x^(k) (t)+P₂ p(t)+P₃ p_ω (t)+g^(k) (t)] Formula 6;

其中，而P₁，P₂和P₃由式7求得：x^(k)及g^(k)(t)由式8和式9求得。in, And P₁ , P₂ and P₃ are obtained by formula 7; x^(k) and g^(k) (t) are obtained by formula 8 and formula 9.

优选的，在步骤（2）后包括：将式（2）的大跨度索桥结构振动控制系统模型描述的大跨度索桥结构振动控制系统简化为一个单自由度系统。Preferably, after step (2), it includes: simplifying the structural vibration control system of the long-span cable bridge described by the structural vibration control system model of the long-span cable bridge in formula (2) into a single-degree-of-freedom system.

本发明考虑了大跨度索桥结构中的非线性动力特征，建立了更加接近现实的控制系统；使用了较为合理的控制器的求解方案，最优振动控制律对于大跨度索桥结构非线性系统具有较好的减振作用，更加适应于大幅值振动的大跨度索桥的控制，使用逐次逼近算法避免了直接求解非线性两点边值问题的解析解，只需要求解一个线性向量差分等式序列的解，把直线迭代整个控制律转位迭代共态向量，计算量大大降低。The present invention considers the nonlinear dynamic characteristics in the long-span cable bridge structure, establishes a control system closer to reality; uses a more reasonable controller solution scheme, and the optimal vibration control law is suitable for the long-span cable bridge structure nonlinear system It has a good vibration reduction effect and is more suitable for the control of large-span cable bridges with large-value vibrations. It uses the successive approximation algorithm to avoid directly solving the analytical solution of the nonlinear two-point boundary value problem, and only needs to solve a linear vector difference equation For the solution of the sequence, the linear iteration of the entire control law is transposed to iterate the common state vector, and the amount of calculation is greatly reduced.

为了验证该控制律的有效性，图3和图4给出了系统开环和系统在该控制律控制下的位移和速度曲线，从图中可以看出，该控制律对于大跨度索桥结构非线性系统具有较好的减振作用。在该控制律控制下该结构的位移和速度加速度分别下降到系统开环状态的19%及17%。In order to verify the validity of the control law, Fig. 3 and Fig. 4 show the displacement and velocity curves of the system open-loop and the system under the control of the control law. It can be seen from the figure that the control law is suitable for the long-span cable bridge structure Nonlinear systems have a better damping effect. Under the control of the control law, the displacement and velocity acceleration of the structure decrease to 19% and 17% of the system open-loop state, respectively.

以上所述仅是本发明的优选实施方式，应当指出，对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。The above is only a preferred embodiment of the present invention, it should be pointed out that for those skilled in the art, without departing from the principle of the present invention, some improvements and modifications can also be made, and these improvements and modifications are also It should be regarded as the protection scope of the present invention.

Claims (2)

Translated fromChinese

1.一种大跨度索桥结构非线性系统半主动振动控制方法，其特征在于，包括如下步骤：1. a kind of semi-active vibration control method of non-linear system of long-span cable bridge structure, it is characterized in that, comprises the steps:（1）建立作用在大跨度索桥结构上的抖振力荷载模型：(1) Establish the buffeting force load model acting on the long-span cable bridge structure:由多个三角函数组成的谐波相互叠加模拟作用在大跨度索桥结构上的抖振力荷载，第j个组成波的作用在大跨度索桥结构上的抖振力可由下列式子给出：The harmonics composed of multiple trigonometric functions are superimposed to simulate the buffeting force load acting on the long-span cable bridge structure. The buffeting force of the jth component wave acting on the long-span cable bridge structure can be given by the following formula :j=1,2,…,rj=1,2,...,r其中，A_j，ω_j分别表示第j个组成波的振幅、频率和初相角，Among them, A_j , ω_j respectively represent the amplitude, frequency and initial phase angle of the jth component wave,令$\stackrel{\&OverBar;}{p}\left(t\right)=[\stackrel{\&OverBar;}{p_1},\stackrel{\&OverBar;}{p_2},\&CenterDot;\&CenterDot;\&CenterDot;,\stackrel{\&OverBar;}{p_r}{]}^T,$得make$\stackrel{\&OverBar;}{p}\left(t\right)=[\stackrel{\&OverBar;}{p_1},\stackrel{\&OverBar;}{p_2},\&CenterDot;\&CenterDot;\&CenterDot;,\stackrel{\&OverBar;}{p_r}{]}^T,$ have to$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>=</span>=</span><span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>j</span>j</span>}}}<span><span>,</span>,</span>\mathrm{<span><span>j</span>j</span>}<span><span>=</span>=</span>\mathrm{<span><span>1,2</span>1,2</span>}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>,</span>,</span>\mathrm{<span><span>r</span>r</span>}<span><span>,</span>,</span>$$\begin{array}{c}\stackrel{\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{<span><span>~</span>~</span>}}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\mathrm{<span><span>diag</span>diag</span>}<span><span>\{</span>\{</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>,</span>,</span>{{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>r</span>r</span>}}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>\}</span>\}</span>\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\end{array}$其中，$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&omega;}}_r\}.$令$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&CenterDot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$可得in,$\stackrel{\&OverBar;}{\mathrm{\&Omega;}}=\mathrm{diag}\{{\mathrm{\&omega;}}_{1,}{\mathrm{\&omega;}}_2,\&CenterDot;\&Center\; Dot;\&Center\; Dot;,{\mathrm{\&omega;}}_r\}.$ make$w\left(t\right)={\left[\begin{array}{cc}\stackrel{\&OverBar;}{p}\left(t\right)& \stackrel{\stackrel{\&Center\; Dot;}{\&OverBar;}}{p}\left(t\right)\end{array}\right]}^T$ Available$\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\left[\begin{array}{cc}\mathrm{<span><span>0</span>0</span>}& {\mathrm{<span><span>I</span>I</span>}}_{\mathrm{<span><span>r</span>r</span>}}\\ <span><span>-</span>-</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>\&Omega;</span>\&Omega;</span>}}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}\end{array}\right]\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$$\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\left[\begin{array}{cc}{\mathrm{<span><span>I</span>I</span>}}_{\mathrm{<span><span>r</span>r</span>}}& \mathrm{<span><span>0</span>0</span>}\end{array}\right]\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$其中，I_r是单位矩阵，0∈R^r×r是零矩阵；in, I_r is the identity matrix, 0∈R^r×r is the zero matrix;作用在大跨度索桥结构上的抖振力荷载模型为式1所描述的外系统：The buffeting force load model acting on the long-span cable bridge structure is the external system described by formula 1:$\begin{array}{c}\stackrel{\&CenterDot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{F}w\left(t\right)\end{array}$   式1，$\begin{array}{c}\stackrel{\&CenterDot;}{w}\left(t\right)=\stackrel{\&OverBar;}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\&OverBar;}{f}w\left(t\right)\end{array}$ Formula 1,其中：p(t)为作用在大跨度索桥结构上的抖振力；Among them: p(t) is the buffeting force acting on the structure of the long-span cable bridge;（2）建立大跨度索桥结构振动控制系统模型：(2) Establish the vibration control system model of the long-span cable bridge structure:抖振力荷载作用下索-桥耦合振动动力系统为：The cable-bridge coupled vibration dynamic system under buffeting force load is:$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>)</span>)</span>\mathrm{<span><span>W</span>W</span>}<span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}<span><span>,</span>,</span>$$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>W</span>W</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>u</span>u</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$其中，W(t)为偏离平衡位置的位移；Y(t)为索端的位移，即桥面沿索方向的位移；ω₁和ω₂为索和桥面的固有频率；ξ为为桥面的阻尼比；a_i(i=1,2,…,6)为系统参数，u(t)是半主动控制力，p(t)是抖振力荷载，由式1产生；Among them, W(t) is the displacement away from the equilibrium position; Y(t) is the displacement of the cable end, that is, the displacement of the bridge deck along the direction of the cable; ω₁ and ω₂ are the natural frequencies of the cable and the bridge deck; ξ is the bridge deck The damping ratio of ; a_i (i=1,2,…,6) is the system parameter, u(t) is the semi-active control force, p(t) is the buffeting force load, which is generated by formula 1;选取状态变量：Select state variables:${\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>W</span>W</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>W</span>W</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>4</span>4</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>Y</span>Y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>,</span>,</span>$则大跨度索桥结构振动控制系统模型为:Then the structural vibration control system model of the long-span cable bridge is:$\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$   式2；$\begin{array}{c}\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$ Formula 2;其中，in,$\mathrm{<span><span>A</span>A</span>}<span><span>=</span>=</span>\left[\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}& \mathrm{<span><span>0</span>0</span>}& {<span><span>-</span>-</span>\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>4</span>4</span>}}& \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>0</span>0</span>}& \mathrm{<span><span>1</span>1</span>}\\ <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>5</span>5</span>}}& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}^{\mathrm{<span><span>2</span>2</span>}}& <span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>\&xi;</span>\&xi;</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>B</span>B</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>\mathrm{<span><span>D</span>D.</span>}<span><span>=</span>=</span>\left[\begin{array}{c}\mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>0</span>0</span>}\\ \mathrm{<span><span>1</span>1</span>}\end{array}\right]<span><span>,</span>,</span>$$\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>[</span>[</span>\begin{array}{c}\begin{array}{cccc}\mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>& \mathrm{<span><span>0</span>0</span>}& <span><span>-</span>-</span>{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>6</span>6</span>}}{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}^{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}\end{array}{<span><span>]</span>]</span>}^{\mathrm{<span><span>T</span>T</span>}}$（3）选取如式3所描述的复合型性能指标；(3) Select the composite performance index as described in formula 3;$J=\underset{T\&RightArrow;\&infin;}{\lim }\frac{1}{T}{\&Integral;}_0^T[x^T\left(t\right)\mathrm{Qx}\left(t\right)+{\mathrm{Ru}}^2\left(t\right)]\mathrm{dt},$   式3；$J=\underset{T\&Right\; Arrow;\&infin;}{\lim }\frac{1}{T}{\&Integral;}_0^T[x^T\left(t\right)\mathrm{Qx}\left(t\right)+{\mathrm{Ru}}^2\left(t\right)]\mathrm{dt},$ Formula 3;（4）对式2和式3实施极大值原理：(4) Implement the maximum value principle for formula 2 and formula 3:首先构造哈密顿函数如式4：First, construct the Hamiltonian function as formula 4:H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω)   式4；H(·)=x^T Qx₁ +Ru² +λ^T (Ax+Bu+f(x)+Dp_ω ) Formula 4;进而根据极值条件，把式1在性能指标式3的约束下求控制器u(t)的问题转化为求解下述非线性两点边值的问题，如式5：Furthermore, according to the extreme value condition, the problem of finding the controller u(t) in Equation 1 under the constraint of performance index Equation 3 is transformed into the problem of solving the following nonlinear two-point boundary value, such as Equation 5:$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$   式5；$\begin{array}{c}-\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\&lambda;}\left(t\right)+f_x^T\left(x\right)\mathrm{\&lambda;}\left(t\right),\\ \stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\&lambda;}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\&lambda;}(\&infin;)=0\end{array}$ Formula 5;（5）求最优振动控制律：(5) Find the optimal vibration control law:使用逐次逼近方法求解式5描述的非线性两点边值问题的迭代解，令Using the successive approximation method to solve the iterative solution of the nonlinear two-point boundary value problem described in Equation 5, letλ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)λ(t)=P₁ x(t)+P₂ p(t)+P₃ p_ω (t)+g(t)其中，g(t)是一个共态向量，对λ(t)=P₁x(t)+P₂p(t)+P₃pω(t)+g(t)求导，然后把式2的第一式$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$代入其中，得Among them, g(t) is a co-state vector, which is derived from λ(t)=P₁ x(t)+P₂ p(t)+P₃ pω(t)+g(t), and then formula 2 the first formula$\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$ Substitute into it, get$\begin{array}{c}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>A</span>A</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>D</span>D.</span>}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\\ <span><span>+</span>+</span><span><span>(</span>(</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>SP</span>SP</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span><span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>Sg</span>S\; g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>f</span>f</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>+</span>+</span>\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>g</span>g</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>.</span>.</span>\end{array}$由式2和式5，得From formula 2 and formula 5, get$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>+</span>+</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>3</span>3</span>}}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>A</span>A</span>}}^{\mathrm{<span><span>T</span>T</span>}}\mathrm{<span><span>g</span>g</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>f</span>f</span>}}_{\mathrm{<span><span>x</span>x</span>}}^{\mathrm{<span><span>T</span>T</span>}}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span>\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$考虑到considering$\stackrel{<span><span>\&CenterDot;</span>\&Center\; Dot;</span>}{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&CenterDot;</span>\&CenterDot;</span>}{\mathrm{<span><span>w</span>w</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>w</span>w</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>\stackrel{\mathrm{<span><span>A</span>A</span>}}{<span><span>=</span>=</span>}{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>\&omega;</span>\&omega;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span>$通过比较的系数，得到矩阵方程组式7以及序列式8和式9，从而可求得式6所描述的最优振动控制律By comparison The coefficients of the matrix equation 7 and the sequence formula 8 and formula 9 are obtained, so that the optimal vibration control law described in formula 6 can be obtained$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{F}+P_2\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_{1}D\stackrel{\&OverBar;}{F}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{F}+P_3\stackrel{\&OverBar;}{F}{\mathrm{\&Omega;}}^2=P_{1}D\stackrel{\&OverBar;}{F}\end{array},,$   式7；$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\&OverBar;}{f}+P_2\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=-(A^T-P_{1}S)P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\&OverBar;}{f}+P_3\stackrel{\&OverBar;}{f}{\mathrm{\&Omega;}}^2=P_1\mathrm{D.}\stackrel{\&OverBar;}{f}\end{array},,$ Formula 7;$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(D-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{Sg}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式8；$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\&Phi;}\left(t\right)x_0+{\&Integral;}_0^t\mathrm{\&Phi;}(r-t)[(\mathrm{D.}-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\&omega;}}\left(r\right)-{\mathrm{S\; g}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\&Center\; Dot;\&Center\; Dot;\&Center\; Dot;,\end{array}$ Formula 8;$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;,\end{array}$   式9；$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\&Integral;}_t^{\&infin;}{\mathrm{\&Phi;}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\&omega;}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\&Center\; Dot;\&Center\; Dot;\&CenterDot;,\end{array}$ Formula 9;由上式7、8、9得出近似最优振动控制律为：From the above formulas 7, 8, and 9, the approximate optimal vibration control law can be obtained as:u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]   式6；u^(k) (t)=-R^-1 B^T λ^(k) (t)=-R^-1 B^T [P₁ x^(k) (t)+P₂ p(t)+P₃ p_ω (t)+g^(k) (t)] Formula 6;其中，而P₁，P₂和P₃由式7求得：x^(k)及g^(k)(t)由式8和式9求得。in, And P₁ , P₂ and P₃ are obtained by formula 7; x^(k) and g^(k) (t) are obtained by formula 8 and formula 9.2.根据权利要求1所述的大跨度索桥结构非线性系统半主动振动控制方法，其特征在于，在步骤（2）后包括：将式（2）的大跨度索桥结构振动控制系统模型描述的大跨度索桥结构振动控制系统简化为一个单自由度系统。2. The semi-active vibration control method for the nonlinear system of the long-span cable bridge structure according to claim 1, characterized in that, after step (2), it comprises: the vibration control system model of the long-span cable bridge structure of the formula (2) The described structural vibration control system of a long-span cable bridge is simplified to a single-degree-of-freedom system.