## 3 Method
### Vibration-System Simulator
#### 3.2.2. Simulator Established with FIR Filter  
The simulator can be considered as a 5-input/4-output filter. The input signals are denoted as \(\mathbf{x}_{0}(n)\), \(\mathbf{x}_{1}(n)\), \(\mathbf{x}_{2}(n)\), \(\mathbf{x}_{3}(n)\) and \(\mathbf{x}_{4}(n)\), where \(\mathbf{x}_{0}(n)\) stands for the force signal of the vibration exciter, and \(\mathbf{x}_{1}(n)\), \(\mathbf{x}_{2}(n)\), \(\mathbf{x}_{3}(n)\), \(\mathbf{x}_{4}(n)\) stand for the force signals of the 4 electromagnetic actuators. The desired-output signals are denoted as \(\mathbf{d}_{1}(n)\), \(\mathbf{d}_{2}(n)\), \(\mathbf{d}_{3}(n)\), \(\mathbf{d}_{4}(n)\), which stand for the acceleration signals. Moreover, \(\mathbf{y}_{j}(n)\) is the estimate of the desired output (\(\mathbf{d}_{j}(n)\), \(j=1,2,3,4\)).  
Figure 4: Schematic diagram of FIR filter.  
Obviously, there are \(5\times 4=20\) channels between the inputs and desired outputs. Each channel can be expressed by an FIR filter. It is assumed that the symbol \(C_{ij}\) stands for the filter between \(\mathbf{x}_{i}(n)\) and \(\mathbf{d}_{j}(n)\), where \(i=0,1,2,3,4\) and \(j=1,2,3,4\).  
Note that \(\mathbf{d}_{1}(n)\) is the open-loop response under excitations of \(\mathbf{x}_{0}(n),\mathbf{x}_{1}(n),\mathbf{x}_{2}(n),\mathbf{x}_{3}(n)\), \(\mathbf{x}_{4}(n)\). Thus, the estimate of \(d_{1}(n)\) can be formulated as follows:  
\[\begin{array}{c}y_{1}(n)=\begin{array}{c}\mathbf{\omega}_{01}^{T}(n) \mathbf{x}_{0}(n)+\mathbf{\omega}_{11}^{T}(n)\mathbf{x}_{1}(n)+\mathbf{\omega} _{21}^{T}(n)\mathbf{x}_{2}(n)+\\ \mathbf{\omega}_{31}^{T}(n)\mathbf{x}_{3}(n)+\mathbf{\omega}_{41}^{T}(n) \mathbf{x}_{4}(n)\end{array}\end{array} \tag{6}\]  
where \(y_{1}(n)\) is the estimate of \(d_{1}(n)\), and \(\mathbf{\omega}_{i1}(n)\) are the filter weights of filter \(C_{i1}\). It can be found that \(y_{1}(n)\) is the superposition of the outputs of 5 FIR filters. Similarly, the estimates of \(d_{2}(n)\), \(d_{3}(n)\), \(d_{4}(n)\) are formulated as:  
\[y_{2}(n)=\begin{array}{c}\mathbf{\omega}_{02}^{T}(n)\mathbf{x}_{0}(n)+ \mathbf{\omega}_{12}^{T}(n)\mathbf{x}_{1}(n)+\mathbf{\omega}_{22}^{T}(n) \mathbf{x}_{2}(n)+\\ \mathbf{\omega}_{32}^{T}(n)\mathbf{x}_{3}(n)+\mathbf{\omega}_{42}^{T}(n) \mathbf{x}_{4}(n)\end{array} \tag{7}\]  
\[y_{3}(n)=\begin{array}{c}\mathbf{\omega}_{03}^{T}(n)\mathbf{x}_{0}(n)+ \mathbf{\omega}_{13}^{T}(n)\mathbf{x}_{1}(n)+\mathbf{\omega}_{23}^{T}(n) \mathbf{x}_{2}(n)+\\ \mathbf{\omega}_{33}^{T}(n)\mathbf{x}_{3}(n)+\mathbf{\omega}_{43}^{T}(n) \mathbf{x}_{4}(n)\end{array} \tag{8}\]  
\[y_{4}(n)=\begin{array}{c}\mathbf{\omega}_{04}^{T}(n)\mathbf{x}_{0}(n)+ \mathbf{\omega}_{14}^{T}(n)\mathbf{x}_{1}(n)+\mathbf{\omega}_{24}^{T}(n) \mathbf{x}_{2}(n)+\\ \mathbf{\omega}_{34}^{T}(n)\mathbf{x}_{3}(n)+\mathbf{\omega}_{44}^{T}(n) \mathbf{x}_{4}(n)\end{array} \tag{9}\]  
Equations (6)-(9) can be written in the compact form of Equation (10), which is the simulator of the vibration system:  
\[\mathbf{y}(n)=\mathbf{W}(n)\mathbf{X}(n) \tag{10}\]  
where  
\[\begin{array}{c}\mathbf{y}(n)=\begin{bmatrix}y_{1}(n)&y_{2}(n)&y_{3}(n)&y_{4 }(n)\end{bmatrix}^{T}\\ \mathbf{X}(n)=\begin{bmatrix}\mathbf{x}_{0}^{T}(n)&\mathbf{x}_{1}^{T}(n)& \mathbf{x}_{2}^{T}(n)&\mathbf{x}_{3}^{T}(n)&\mathbf{x}_{4}^{T}(n)\end{bmatrix} ^{T}\\ \mathbf{W}(n)=\begin{bmatrix}\mathbf{\omega}_{01}^{T}(n)&\mathbf{\omega}_{11}^{T }(n)&\cdots&\mathbf{\omega}_{41}^{T}(n)\\ \mathbf{\omega}_{02}^{T}(n)&\mathbf{\omega}_{13}^{T}(n)&\cdots&\mathbf{\omega} _{42}^{T}(n)\\ \cdots&\cdots&\cdots&\cdots\\ \mathbf{\omega}_{04}^{T}(n)&\mathbf{\omega}_{14}^{T}(n)&\cdots&\mathbf{\omega} _{44}^{T}(n)\end{bmatrix}\end{array}\]  
The simulator is composed of 20 single-input/single-output filters. The matrix (\(\mathbf{W}(n)\)) contains the filter weights of the 20 filters, where \(\mathbf{\omega}_{ij}(n)\) is the vector of the weights of filter \(C_{ij}\): \(i=0,1,2,3,4\) and \(j=1,2,3,4\).