﻿ 结构瞬时AMD最优控制减震效果计算研究

# 结构瞬时AMD最优控制减震效果计算研究Research about Structural Vibration Mitigation Efficiency for Instantaneous AMD Controlling Systems

Abstract: To evaluate the damping effect of instantaneous optimal AMD control, a theoretical analysis model about instantaneous optimal active control of a 20-storey structure with AMD system was present-ed, and it was solved with software prepared with Matlab language. Results show that an instanta-neous control algorithm could exhibit more notable vibration mitigation efficiency. It is recognized that expecting vibration mitigation efficiency could be reached with an optimized combination of Q and R values, the weight ratios of state variables and controlling force vectors respectively. When Q = 1e8 and R = 0.1, the controlling force on top storey could reach 7.9% of storey gravity load. The same seismic mitigation efficiency and time history of dynamic response could be exhibited both from the two types of control algorithm. While the efficiency for LQR algorithm will not be affected by sample time step, the instantaneous optimizing algorithm will do. Moreover, effect from time delay of outer distribution will make dynamic responses amplified.

1. 引言

2. 建立运动方程

2.1. 瞬时最优主动控制系统状态方程及控制目标函数

$M\stackrel{¨}{z}\left(t\right)+C\stackrel{˙}{z}\left(t\right)+Kz\left(t\right)=-M\left\{I\right\}{\stackrel{¨}{x}}_{g}\left(t\right)+Hu\left(t\right)$ (1)

$x\left(t\right)=\left[z\left(t\right)\text{\hspace{0.17em}}\stackrel{˙}{z}\left(t\right)\right]$，则(1)式可转化为如下的状态方程 [12]：

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+d\left(t\right)\\ x\left({t}_{0}\right)={x}_{0}\end{array}$ (2)

$A=\left[\begin{array}{cc}0& I\\ {M}^{-1}K& {M}^{-1}C\end{array}\right]$, $B=\left[\begin{array}{c}0\\ {M}^{-1}H\end{array}\right]$, $d\left(t\right)=-\left[\begin{array}{c}0\\ \left\{I\right\}\end{array}\right]{\stackrel{¨}{x}}_{g}\left(t\right)$

$J\left(t\right)={x}^{\text{T}}\left(t\right)Qx\left(t\right)+{u}^{\text{T}}\left(t\right)Ru\left(t\right)$ (3)

$u\left(t\right)=-\left(\frac{\Delta t}{2}\right){R}^{-1}{B}^{\text{T}}Qx\left(t\right)$ (4)

$x\left(t\right)={\left[I+{\left(\frac{\Delta t}{2}\right)}^{2}B{R}^{-1}{B}^{\text{T}}Q\right]}^{-1}\left[TD\left(t-\Delta t\right)+\left(\frac{\Delta t}{2}\right)d\left(t\right)\right]$ (5)

$u\left(t\right)=-\left(\frac{\Delta t}{2}\right){R}^{-1}{B}^{\text{T}}Q{\left[I+{\left(\frac{\Delta t}{2}\right)}^{2}B{R}^{-1}{B}^{\text{T}}Q\right]}^{-1}\left[TD\left(t-\Delta t\right)+\left(\frac{\Delta t}{2}\right)d\left(t\right)\right]$ (6)

2.2. 瞬时主动最优控制地震动力反应计算

3. 计算实例与结果分析

Figure 1. Structural plane layout

Figure 2. 1st, 2nd and 3rd for modal structural

Table 1. Control mode vs seismic dynamic response

Figure 3. Time histories of structural seismic dynamic response

Figure 4. Q vs structural seismic dynamic response

Figure 5. R vs structural seismic dynamic response

Figure 6. Q and R vs. structural seismic dynamic response

Figure 7. Comparison of the ideal instantaneous and the actual instantaneous optimal control seismic dynamic response

Figure 8. Comparison of the ideal instantaneous and the actual instantaneous optimal control seismic dynamic response

4. 结论

1) 主动瞬时最优控制有明显的减振效果，可以通过调整权重参数Q、R达到目标减振效果。

2) 主动瞬时最优控制，顶层控制力最大，当Q = 1e8、R = 0.1减振效果较好时，顶层控制力幅值为楼面重力荷载7.9%。

3) 瞬时最优控制及LQR控制可达到同样效果，动力反应几乎相同，但LQR控制不受步长影响。瞬时控制结果受步长影响。瞬时控制受滞后效应的影响，动力反应有所放大。

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