## 2 The M-FxLMS Algorithm  
Figure 1 shows the MIMO system schematic diagram of a simplified double-layer vibration isolation platform in underwater vehicles. The upper is the vibration source equipment and platform, which includes the complicated periodic excitation. There are H electromagnetic active and passive integrated vibration isolators installed between the upper and middle rack; these isolators can also be called actuators, where H, an even number, is equal to 4 in Figure 1. The middle platform is a flexible floating raft, fixed on the structure foundation by H rubber vibration isolators in an underwater vehicle. The reference sensor is located at the foot of the vibration source device, and the H error sensors are at the foot of the vibration isolators in the middle platform, where the left front (or rear) and right front (or rear) positions in the Figure 1 are 1\(\#\), 3\(\#\)... (H/2 + 1)\(\#\), 2\(\#\), 4\(\#\)... H\(\#\) error sensors, respectively. These error sensors are used as control targets, and the active control forces are exerted by the H integrated vibration isolators, all of which can achieve global control for the vibration isolation platform. Aimed at the platform in Figure 1, the multi-path algorithm diagram is shown in Figure 2, where the thick lines represent matrix operations, and the multi-path control system only collects one reference signal, which contains all complicated excitation frequency components. Additionally, the platform also contains \(H\) error sensors, \(H\) actuators, and \(H\) error signals, which means the system consists of \(P_{p1},\ P_{p2},\ P_{pi}\cdot\cdot\cdot\cdot\cdot\cdot P_{pH}\) primary paths, and each path \(P_{pi}(i=1,2\ldots H)\) is represented by the \(Q\) order FIR filter. At the same time, it also includes \(H\times H\) secondary paths; each of them is represented by \(S_{i\!h}(n)\,(i=1\cdot\cdot\cdot H,h=1\cdot\cdot\cdot H)\) with the \(M\)order FIR filter, and each path control filter \(w_{i}(n)\ (i=1\cdot\cdot\cdot H)\) is the \(L\) order FIR filter. It is assumed that the reference input signal, the desired signals, the error signals, and the control signals are represented by the matrix as \(\mathbf{X}(n)\), \(\mathbf{D}(n)\), \(\mathbf{E}(n)\) and \(\mathbf{Y}(n)\) at the \(n\) moment, respectively. And the acceleration values of the measuring point #1, #2, #3, #4 are used to evaluate the control effect of the system Thus, the reference signal and the transfer matrix of the secondary path can be expressed as  
\[\mathbf{X}(n)=\left[ \begin{array}{cccc}\mathbf{x}(n)&0&\ldots&0\\ 0&\mathbf{x}(n)&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&\mathbf{x}(n)\end{array} \right] \tag{1}\]  
\[\mathbf{S}(n)=\left[ \begin{array}{cccc}S_{11}(n)&S_{12}(n)&\ldots&S_{1H}(n) \\ S_{21}(n)&S_{22}(n)&\ldots&S_{2H}(n)\\ \ldots&\ldots&\ldots&\ldots\\ S_{H1}(n)&S_{H2}(n)&\ldots&S_{HH}(n)\end{array} \right] \tag{2}\]  
where \(\mathbf{x}(n)=\left[x(n),x(n-1),\cdot\cdot,x(n-L+1)\right]^{T}\), the transfer function \(\mathbf{S}(n)\) is \(\mathrm{H}\times\mathrm{H}\) dimensional matrix, and each secondary path \(S_{i\!h}(n)\) represents the transfer function between the \(h\)-th actuator and the \(i\)-th error sensor. Then, the error matrix can be obtained as  
\[\mathbf{E}(n)=\left[e_{1}(n)\ e_{2}(n)\ \cdot\cdot\cdot e_{H}(n)\right]\ ^{T}= \mathbf{D}(n)+\mathbf{S}(n)\mathbf{Y}(n) \tag{3}\]  
where \(\mathbf{Y}(n)=\left[y_{1}(n),\cdot\cdot\cdot,y_{H}(n)\right]^{T}=\mathbf{X}(n )\mathbf{W}(n)\), and the component-wise of the error signal can be represented as  
\[e_{k}(n)=d_{k}(n)+\sum_{i=1}^{H}S_{i\!i}(n)y_{i}(n) \tag{4}\]  
where \(y_{i}(n)\) is the output of the \(i\)-th control filter, and \(\mathbf{w}_{i}(n)\) is the \(i\)-th weight coefficient of the controller, all of which can be expressed, respectively, as follows:  
\[y_{i}(n)=\mathbf{w}_{i}^{T}(n)\mathbf{x}(n) \tag{5}\]  
Figure 1: The multi-input and multi-output system (MIMO) system schematic diagram of a double-layer vibration isolation system.