## 6 Conclusions and Discussion  
This paper has proposed a novel algorithm to solve the complexity of reference signals and path coupling problems. The wavelet packet transform is used to convert the multi-frequency excitation into the line spectrum signals in multiple non-overlapping frequency bands, and the decoupling feedback compensation factors are also introduced into the identification model in the secondary path. All of them can simplify the multi-path system into parallel single-path systems, which can weaken the multi-path coupling effect with multi-frequency excitation. Overall, several conclusions can be drawn from the simulation and experiment:  
1. When the excitation is 37 Hz, 60 Hz, and 110 Hz line spectrum signals with white noise, sym20 is the best wavelet packet decomposition base, which can provide more a stable and reliable line spectrum decomposition for the WPTDDM-FxLMS algorithm.
2. In multi-frequency excitation and multi-path systems, the traditional algorithm is still inadequate for AVC; some measuring points are basically in a failure state, and it is also much easier to induce other line spectrum signals and the stability is not perfect enough.  
Figure 20: The comparison of the theoretical simulation and experimental verification.  
3. Decentralized decoupling improvement considers the impact of off-diagonal elements on system performance, increases the influence weight of the diagonal secondary paths, and it can also transform a complicated multi-frequency and multi-path coupling problem into a simple single-path control of the multiple parallel line spectrum signals.
4. The WPTDDM-FxLMS algorithm can effectively reduce the reference signal and the multi-path coupling effect on iterative step-size and stability, which has the characteristics of low computation mount, fast convergence speed, and high control precision, and the control effect of the multi-line spectrum is more obvious. Correspondingly, the novel algorithm is more suitable than the traditional one for solving the structural vibration control problems for multi-frequency excitation and multi-path coupling in underwater vehicles.  
Although the algorithm proposed in this paper is well matched in theory and practice, there are still some issues to further research. For example, the choice of wavelet base has an important influence on the decomposition of the reference signal. The method in this paper can be mainly used for a relatively stable excitation signal. If the frequency of the excitation signal changes greatly, it is necessary to explore an adaptive wavelet strategy. On the other hand, if there are more line spectrum excitations and time-varying coupled secondary-paths, the algorithm needs to be further optimized and improved, especially for the vibration control of elastic structures.  
Conceptualization and methodology, L.Y. and W.G.; computer simulation, W.G. and B.Z.; software and validation, L.Y. and L.L.; experimental work, W.G., B.Z., and L.Y.; writing--original draft preparation, L.Y. and L.L.; writing--review and editing, J.Y. and B.Z.; supervision, W.G. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant no.51909267), the project ZR2019 QEE031 by the Shandong Provincial Natural Science Foundation and the Fundamental Research Funds for the Central Universities (Grant no.201965006).  
Not applicable.  
All data included in this study are available upon request by contact with the author [PERSON].  
The authors declare no conflict of interest.