# 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
Based on the above two gradient optimization algorithms, combined with their advantages and iterative properties, an adaptive momentum gradient descent algorithm (Adam) is proposed. Its structure is as follows [26]:  
\[ \begin{array}{l}m_{t}\ =\ \delta m_{t-1}+(1-\delta)\
abla_{t}\\ \hat{m}_{t}\ =\ \frac{m_{t}}{1-\delta^{t}}\\ n_{t}\ =\ vn_{t-1}+(1-\
u)\
abla_{t}^{2}\\ \hat{n}_{t}\ =\ \frac{m_{t}}{1-\
u}\\ W_{t+1}\ =\ W_{t}-\eta\frac{m_{t}}{\sqrt{n_{t}+\varepsilon}}\end{array}  \tag{12}\]  
where \(\delta\), \(\
u\) are constant values less than one. Aiming at the problems of more fluctuations and slow convergence speed in the convergence process of gradient descent method, [PERSON] improved it and proposed [PERSON]'s acceleration gradient algorithm (NAG), which is more convenient than gradient descent method and can be rewritten as an improved momentum algorithm. Its form is as follows [27]:  
\[ \begin{array}{l}g_{t}\ =\ df(W_{t}-\eta\delta m_{t})\\ m_{t}\ =\ \delta m_{t-1}+g_{t}\\ W_{t+1}\ =\ W_{t}-\eta m_{t}\end{array}  \tag{13}\]  
Combining NAG algorithm with Adam, the Nadam algorithm for gradient descent method can be obtained, which can be expressed as [28]:  
\[ \begin{array}{l}g_{t}\ =\ df(W_{t})\\ \hat{g}_{t}\ =\ \frac{g_{t}}{1-\delta^{t}}\\ m_{t}\ =\ \delta m_{t-1}+(1-\delta)g_{t}\\ \hat{m}_{t}\ =\ \frac{m_{t}}{1-\delta^{t}}\\ n_{t}\ =\ vn_{t-1}+(1-\
u)g_{t}^{2}\\ \hat{n}_{t}\ =\ \frac{m_{t}}{1-\
u^{t}}\\ \overline{m}_{t}\ =\ (1-\delta)\hat{g}_{t}+\delta\hat{m}_{t}\\ W_{t+1}\ =\ W_{t}-\eta\frac{\overline{m}_{t}}{\sqrt{n_{t}}+\varepsilon}\end{array}  \tag{14}\]  
Combining the algorithm with FUNLMS can theoretically improve the convergence performance of the algorithm and obtain better control effect. For the FUNLMS algorithm, the gradient can be expressed as the vector product of the error signal and the reference  
Based on the previous modal analysis results, 20 Pa white noise is applied from the lower boundary of the stator to the lower boundary of the cylinder, and the modal energy data corresponding to all characteristic frequencies in the frequency band to be controlled are calculated, as shown in Table 1 below. It can be seen from the table that the energy is mainly concentrated at 3672 Hz and 3469 Hz. These two frequencies correspond to the ninth and seventh modes of the compressor shell, respectively. Therefore, it is mainly necessary to control these two modes during noise reduction. According to the COMSOL simulation results, the modal vibration modes are shown in Figure 6.  
According to the modal shapes, it can be seen that the compressor shell has 4 and 6 peak areas, respectively, under these two modes. Therefore, in order to effectively control these two order vibration modes, six secondary sources are selected. The location configuration of secondary sources needs to consider two problems: configuration criteria and optimization algorithm. In this paper, the controllability/observability criterion is mainly used to determine the actuator/sensor position. The optimization algorithm adopts the particle swarm optimization algorithm and uses the joint simulation of COMSOL and MATLAB to optimize the position configuration scheme and find the best secondary source position, so as to realize the active vibration control of the compressor main shell under the condition of optimal observability/controllability. The sensor position and vibration mode to be controlled are as follows:  
The blue circle in the figure refers to the position of the active controller. According to the above figure, the position of the control point is basically located at the peak of the vibration mode, which has a strong correlation with the modal vibration mode, and the active vibration control based on the vibration mode can be realized. The generalized coordinates of each point are shown in Table 2.