## 1 Introduction
Unfortunately, the actual vehicle vibration is always affected by some factors such as equipment excitation, structural coupling and, basic interference, and the signal components of vibration source are often very complicated, which may affect system stability. Moreover, when those signals are transferred to the error sensor by the multi-secondary path coupling, the pick-up signal of one sensor often contains the responses of all control signals to this one. Thus, the AVC in the multi-input and multi-output system (MIMO) had no essential differences from SISO, and the traditional algorithms have some problems such as slow convergence, insufficient stability, or even divergence, which caused its engineering applications to be constrained [15; 24]. In addition, for the complicated frequency component in the vibration source, frequency-selective filtering and adaptiveband-stop filtering are often used to realize the narrow-band signal separation, and then the active control is applied. Moreover, for the problem of structural vibration, the single actuator control and error sensor evaluation constantly face difficulties in meeting with the vibration control requirements. Usually, multiple actuators and multiple error sensors are introduced to carry out multi-path control and obtain a better overall control. At this time, the coupling effect of each secondary path will directly affect the convergence performance and control effect, and generally, the centralized or decentralized methods can be used for its compensation [25; 26; 27]. Among them, the update for any control filter of the former requires all error signals, and its calculation amount is very large, which easily reduces the stability and real-time performance [28]. Corresponding to the former, the coupling effect between the actuator and the non-proximity sensor in secondary path is simply ignored by the latter, and if the phase difference between the identification secondary path and the actual secondary physical path is more than 90\({}^{\circ}\), the controller will iterate in the wrong direction and eventually lead to failure [16; 29]. Usually, the phase difference of the former is very large [30], and the latter is prone to frequency mismatches [31], all of which could cause difficulties in engineering applications. Fortunately, studies have shown that [32; 33; 34; 35; 36] wavelet packet transform can quickly convert signals into time-domain signals with different frequency components, which can effectively solve the extraction problems of the complicated signal, and has been applied to signal filtering and denoising such as wavelet neural network fault diagnosis, motor servo control, etc. Therefore, it is necessary to design a more efficient and stable control strategy for an MIMO system with multiple sensors and complicated excitations, which can effectively reduce the radiated noise caused by structural vibration in underwater vehicles. In this paper, based on the traditional M-FxLMS algorithm, the wavelet packet transform is firstly applied to preprocess the complicated reference signal and extract its frequency characteristic, and the feedback compensation factor is introduced to recompense the coupling effect of the secondary path between the actuator and the non-proximity sensors. Then, the independent step-size is also designed for the controller, and the system will be simplified to a parallel control structure of the component subsystem for each frequency band; thus, the WPTDDM-FxLMS algorithm is proposed. Finally, the simulation and experiment also explore the multi-path active control effect in multi-frequency excitation. The remainder of this paper is divided into six sections. Section 2 gives the traditional algorithm. Section 3 carries out the design and analysis for the WPTDDM-FxLMS algorithms. Section 4 presents the numerical simulations for the traditional and the novel algorithms with a complex reference signal and secondary-path coupling. Section 5 conducts the experimental research. Section 6 briefly concludes and discusses the paper.