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MIMO随机振动试验控制的逆多步预测模型法

Inverse multi-step prediction model method for MIMO random vibration test control
ZHENG Wei, CHEN Huaihai
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: Multi Input Multi Output (MIMO) random vibration test is of utter importance in product validation. As a type of time domain method, inverse system methods are capable of generating drive signals directly from reference ones (i.e., the desired response signals). However, in practice, these methods are prone to produce unstable results. To address this issue, an inverse multi-step prediction model is induced from the finite difference model of the system, and a MIMO random vibration test control scheme utilizing both the inverse multi-step prediction model and matrix power algorithm are proposed. First, a simulation of double-input-double-output-random-vibration test on a cantilever beam is conducted to show that the inverse multi-step prediction model can generate drive signals needed to satisfy the control requirement. After that, the capability of the proposed MIMO random vibration test control scheme is further validated in a test performed on a three-axis shaker.
Keywords: inverse system    multi-step prediction model    MIMO    random vibration    environmental test

1 逆多步预测系统

 $\begin{array}{l} \mathit{\boldsymbol{y}}\left( k \right) = - \sum\limits_{i = 1}^p {\mathit{\boldsymbol{\alpha }}_i^{\left( 0 \right)}} {B^i}\mathit{\boldsymbol{y}}\left( k \right) + \sum\limits_{i = 1}^p {\mathit{\boldsymbol{\beta }}_i^{\left( 0 \right)}} {B^i}\mathit{\boldsymbol{u}}\left( k \right) + \\ \;\;\;\;\;\;\mathit{\boldsymbol{\beta }}_0^{\left( 0 \right)}\mathit{\boldsymbol{u}}\left( k \right) \end{array}$ （1）

 $\begin{array}{l} \mathit{\boldsymbol{y}}\left( {k + j} \right) = - \sum\limits_{i = 1}^p {\mathit{\boldsymbol{\alpha }}_i^{\left( j \right)}} {B^i}\mathit{\boldsymbol{y}}\left( k \right) + \sum\limits_{i = 1}^p {\mathit{\boldsymbol{\beta }}_i^{\left( j \right)}} {B^i}\mathit{\boldsymbol{u}}\left( k \right) + \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^j {\mathit{\boldsymbol{\beta }}_i^{\left( 0 \right)}} {B^i}\mathit{\boldsymbol{u}}\left( {k + j} \right) \end{array}$ （2）

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\alpha }}_k^{\left( j \right)} = \mathit{\boldsymbol{\alpha }}_{k + 1}^{\left( {j - 1} \right)} - \mathit{\boldsymbol{\alpha }}_1^{\left( {j - 1} \right)}\mathit{\boldsymbol{\alpha }}_k^{\left( 0 \right)}\quad k = 1,2, \cdots ,p\\ \mathit{\boldsymbol{\beta }}_k^{\left( j \right)} = \mathit{\boldsymbol{\beta }}_{k + 1}^{\left( {j - 1} \right)} - \mathit{\boldsymbol{\alpha }}_1^{\left( {j - 1} \right)}\mathit{\boldsymbol{\beta }}_k^{\left( 0 \right)}\quad k = 0,1, \cdots ,p \end{array} \right.$ （3）

j=1, 2, …, q-1 (q为多步预测模型预测响应步数)，即预测系统从kk+q-1时刻的响应，由式(1)~式(3)可以得到系统的多步预测模型为

 ${\mathit{\boldsymbol{y}}_q}\left( k \right) = \mathit{\boldsymbol{T}}{\mathit{\boldsymbol{u}}_q}\left( k \right) + \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{u}}_p}\left( {k - p} \right) \to \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{y}}_p}\left( {k - p} \right)$ （4）

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{y}}_q}\left( k \right) = {\left[ {\begin{array}{*{20}{c}} {{B^0}}&{{B^{ - 1}}}& \cdots &{{B^{1 - q}}} \end{array}} \right]^{\rm{T}}}\mathit{\boldsymbol{y}}\left( k \right)\\ {\mathit{\boldsymbol{u}}_q}\left( k \right) = {\left[ {\begin{array}{*{20}{c}} {{B^0}}&{{B^{ - 1}}}& \cdots &{{B^{1 - q}}} \end{array}} \right]^{\rm{T}}}\mathit{\boldsymbol{u}}\left( k \right) \end{array} \right.$ （5）

 $\mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\beta }}_0^{\left( 0 \right)}}&{\bf{0}}& \cdots &{\bf{0}}\\ {\mathit{\boldsymbol{\beta }}_0^{\left( 1 \right)}}&{\mathit{\boldsymbol{\beta }}_0^{\left( 0 \right)}}& \cdots &{\bf{0}}\\ \vdots & \vdots &{}& \vdots \\ {\mathit{\boldsymbol{\beta }}_0^{\left( {q - 1} \right)}}&{\mathit{\boldsymbol{\beta }}_0^{\left( {q - 2} \right)}}& \cdots &{\mathit{\boldsymbol{\beta }}_0^{\left( 0 \right)}} \end{array}} \right]$ （6）
 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\alpha }}_p^{\left( 0 \right)}}&{\mathit{\boldsymbol{\alpha }}_{p - 1}^{\left( 0 \right)}}& \cdots &{\mathit{\boldsymbol{\alpha }}_1^{\left( 0 \right)}}\\ {\mathit{\boldsymbol{\alpha }}_p^{\left( 1 \right)}}&{\mathit{\boldsymbol{\alpha }}_{p - 1}^{\left( 1 \right)}}& \cdots &{\mathit{\boldsymbol{\alpha }}_1^{\left( 1 \right)}}\\ \vdots & \vdots &{}& \vdots \\ {\mathit{\boldsymbol{\alpha }}_p^{\left( {q - 1} \right)}}&{\mathit{\boldsymbol{\alpha }}_{p - 1}^{\left( {q - 1} \right)}}& \cdots &{\mathit{\boldsymbol{\alpha }}_1^{\left( {q - 1} \right)}} \end{array}} \right]$ （7）
 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\beta }}_p^{\left( 0 \right)}}&{\mathit{\boldsymbol{\beta }}_{p - 1}^{\left( 0 \right)}}& \cdots &{\mathit{\boldsymbol{\beta }}_1^{\left( 0 \right)}}\\ {\mathit{\boldsymbol{\beta }}_p^{\left( 1 \right)}}&{\mathit{\boldsymbol{\beta }}_{p - 1}^{\left( 1 \right)}}& \cdots &{\mathit{\boldsymbol{\beta }}_1^{\left( 1 \right)}}\\ \vdots & \vdots &{}& \vdots \\ {\mathit{\boldsymbol{\beta }}_p^{\left( {q - 1} \right)}}&{\mathit{\boldsymbol{\beta }}_{p - 1}^{\left( {q - 1} \right)}}& \cdots &{\mathit{\boldsymbol{\beta }}_1^{\left( {q - 1} \right)}} \end{array}} \right]$ （8）

 $\begin{array}{l} {\mathit{\boldsymbol{u}}_q}\left( k \right) = {\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{T}}{\mathit{\boldsymbol{u}}_q}\left( k \right) = \\ \;\;\;\;\;\;\;\;{\mathit{\boldsymbol{T}}^{ - 1}}{\mathit{\boldsymbol{y}}_q}\left( k \right) + {\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{A}}{\mathit{\boldsymbol{y}}_p}\left( {k - p} \right) - {\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{u}}_p}\left( {k - p} \right) \end{array}$ （9）

 ${\mathit{\boldsymbol{T}}^ + }\mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_r}}&{\bf{0}}\\ {\bf{0}}&{\bf{0}} \end{array}} \right]$ （10）

 ${\mathit{\boldsymbol{u}}_l}\left( k \right) = \mathit{\boldsymbol{\bar T}}{\mathit{\boldsymbol{y}}_q}\left( k \right) + \mathit{\boldsymbol{\bar B}}{\mathit{\boldsymbol{y}}_p}\left( {k - p} \right) - \mathit{\boldsymbol{\bar A}}{\mathit{\boldsymbol{u}}_p}\left( {k - p} \right)$ （11）

 ${\mathit{\boldsymbol{u}}_l}\left( k \right) = {\left[ {\begin{array}{*{20}{c}} {{B^0}}&{{B^{ - 1}}}& \cdots &{{B^{1 - l}}} \end{array}} \right]^{\rm{T}}}\mathit{\boldsymbol{u}}\left( k \right)$ （12）
 $\begin{array}{l} \mathit{\boldsymbol{\bar T}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_{\mathit{l}{\mathit{n}_i}}}}&{\bf{0}} \end{array}} \right]\mathit{\boldsymbol{T}}_r^ + \\ \mathit{\boldsymbol{\bar B}} = \mathit{\boldsymbol{\bar TA}},\mathit{\boldsymbol{\bar A}} = \mathit{\boldsymbol{\bar TB}} \end{array}$ （13）

2 随机振动试验算法

 $\mathit{\boldsymbol{Y}}\left( f \right) = \mathit{\boldsymbol{L}}\left( f \right)\mathit{\boldsymbol{P}}\left( f \right)$ （14）

 $\mathit{\boldsymbol{y}}\left( t \right) = {{\cal F}^{ - 1}}\left( {\mathit{\boldsymbol{Y}}\left( f \right)} \right)$ （15）

 图 1 逆多步预测模型驱动信号生成过程 Fig. 1 Drive signal generation using inverse multi-step prediction model

 ${\mathit{\boldsymbol{L}}_{n + 1}} = {{\bf{\Delta }}^\varepsilon }{\mathit{\boldsymbol{L}}_n}\;\;\;\;n = 0,1,2, \cdots$ （16）

 ${\bf{\Delta }} = {\mathit{\boldsymbol{L}}_0}\mathit{\boldsymbol{L}}_y^{ - 1}$ （17）

 图 2 基于逆多步预测模型的矩阵幂次控制 Fig. 2 Matrix power control based on inverse multi-step prediction model
3 仿真算例

 参数 数值 密度/(kg·m-3) 7 850 弹性模量/MPa 720 长度/mm 1 000 截面宽度/mm 50 截面高度/mm 15
 图 3 两输入两输出振动试验悬臂梁 Fig. 3 A cantilever beam for double input double output vibration test

 图 4 参考谱的自功率谱密度 Fig. 4 Auto spectra density of reference spectra

 图 5 响应点1与响应点2的自谱 Fig. 5 Auto spectra at response 1 and 2
 图 6 响应点间相关系数和相位差 Fig. 6 Correlation coefficient and phase between two response points
4 试验

 图 7 MIMO随机振动试验系统的构成 Fig. 7 Configuration of MIMO random vibration test system

 图 8 三轴振动试验参考谱的自功率谱密度 Fig. 8 Auto spectra density of reference spectra for 3-axes vibration test

 图 9 控制前的自谱 Fig. 9 Auto spectra before control
 图 10 控制前的相关系数 Fig. 10 Correlation coefficient before control
 图 11 控制前的相位差 Fig. 11 Phase before control
 图 12 三次迭代后的自谱 Fig. 12 Auto spectra after 3 iterations
 图 13 三次迭代后的相关系数 Fig. 13 Correlation coefficient after 3 iterations
 图 14 三次迭代后的相位差 Fig. 14 Phase after 3 iterations

 图 15 迭代过程 Fig. 15 Iteration process
5 结论

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http://dx.doi.org/10.7527/S1000-6893.2019.23000

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#### 文章信息

ZHENG Wei, CHEN Huaihai
MIMO随机振动试验控制的逆多步预测模型法
Inverse multi-step prediction model method for MIMO random vibration test control

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 223000.
http://dx.doi.org/10.7527/S1000-6893.2019.23000