# Improved FULMS Algorithm for Multi-Modal Active Control of Compressor Vibration and Noise Reduction
## 4 Improved FULMS Algorithm Active Control  
### Improved FULMS Algorithm Based on Nadam  
The active control algorithm generally adopts the adaptive filtering algorithm, while the classical algorithms are implemented based on FIR filter. Its transmission function is an all zero structure. In practical application, the feedback of the filtered output signal to the reference signal is not considered. However, this factor should be considered. The FULMS algorithm is a least mean square algorithm composed of IIR filter structure. This feedback has been considered in the actual operation process, which improves the classical algorithm to a certain extent. However, compared with NLMS algorithm, the convergence speed of this algorithm is basically the same. However, due to the acquisition and truncation of the feedback signal, the FUNLMS algorithm has a loss of control efficiency. Therefore, this paper optimizes the FUNLMS algorithm based on the Nadam method. The structure of the FULMS algorithm is mentioned in many references [22; 23], and the specific structure of the FULMS algorithm is shown in Figure 5.  
The control system designed in this paper is a multi-channel system, and it includes \(n\) controllers and \(n\) sensors. The meaning of each signal in the figure is as follows:  
1. Term B(k) is the \(n\)-dimensional response vector generated by the external disturbance signal through the external disturbance channel;
2. After the reference signal X(k) with a high degree of correlation with the external disturbance signal is adaptively filtered by the IIR filtering system, an \(n\)-dimensional control vector is generated, corresponding to the signal input of \(n\) actuators respectively, and then the control signal is generated through the control channel \(\hat{\rm Y}\)(k);
3. Term E(k) is the difference between the two signals; when the system has not controlled, there is E(k) = B(k);
4. Terms H\({}_{1}\) and H\({}_{2}\) are the transfer functions of the primary channel and the secondary channel, respectively. Term H\({}_{2}\) is the modeling matrix of the secondary channel. In an ideal case, H\({}_{2}\) = H\({}_{2}\), and they are all \(n\)\(\times\)\(n\)-dimensional arrays. Each array has \(n\)-dimensional vectors, corresponding to the filter length of each filter. 5. Terms \(\hat{\rm X}\)(k) and \(\hat{\rm G}\)(k)are, respectively, X(k) and Y(k) obtained after the secondary channel simulation, which are the control signals updated by the algorithm. The FULMS algorithm generally adopts the gradient descent method for iteration, with a general calculation rate, while the classical momentum method [24] (CM) increases the feedback of gradient change to the iteration, reduces the oscillation of iteration during  
Figure 5: The structure of the FULMS algorithm, LMS—The Least Mean Squares. convergence, and speeds up the convergence process. The process can be expressed as follows:  
\[ \begin{array}{l}m_{t}\ =\ \delta m_{t-1}+\
abla_{t}\\ W_{t+1}\ =\ W_{t}-\eta m_{t}\end{array}  \tag{10}\]  
where \(\
abla_{t}\) represents the gradient at time \(t\), and \(\eta\) is a constant value. At the same time, in order to solve the adaptive problem of gradient in the iterative process, some scholars proposed the gradient adaptive method [25], and its form is as follows:  
\[ \begin{array}{l}n_{t}\ =\ n_{t-1}+\
abla_{t}^{2}\\ W_{t+1}\ =\ W_{t}-\eta\frac{\frac{\
abla_{t}}{\
abla n_{t}+\varepsilon}}{ \sqrt{n_{t}+\varepsilon}}\end{array}  \tag{11}\]  
to prevent division error, set a small parameter \(\varepsilon\). It has little effect on the calculation results. The algorithm is used in each iteration \(\eta\). Compared with the L\({}_{2}\) norm of all previous gradients, it can reduce the learning step of significantly changing coefficients and increase the learning step of slightly changing coefficients, so as to stabilize and balance the change level of each coefficient.