## 1 Introduction  
As is well known, active vibration control (AVC) is an ideal way to solve vibration problems and has great significance for reducing radiated noise caused by the vibration of underwater vehicles. It has also been applied in other various fields such as anti-seismic engineering and instrument manufacturing [1; 2]. Currently, this technology has attracted more attention as an important means to control radiation noise, especially in vibration noise control for underwater vehicles [3; 4; 5]. For example, the USA had developed an active noise and vibration control system, which could implement active vibration isolation for all low-frequency vibrations below 100 Hz and has been equipped in the latest Sea Wolf class nuclear submarines to achieve good acoustic stealth performance [6; 7]. At the same time, the research department has also converted the AVC into civilian technology [8]. Australia has also applied it to effectively reduce the radiation noise in the power system of the Collins class submarine [9]. The UK has carried out AVC experiments on minesweepers with a better noise reduction effect [10]. The Defence Research Establishment Atlantic of Canada had also conducted a large number of studies on AVC in submarine [11]. The University of Karlskrona Ronneby in Sweden has developed an active and passive hybrid vibration isolation device and applied it to frigates, which could effectively isolate the acoustic radiation noise caused by the shell structural vibration [12; 13]. The Paulstra Vibrachoc company in France also developed the AVC system for large equipment such as main engines and diesel engines, which have quite a significant damping effect [14]. [PERSON] has retrofitted an active vibration isolation system to a diesel generator set to determine the effectiveness in a tugboat, and the experiments show that the combination of the active control system and the passive isolators is effective in reducing the global vibration and the acoustic pressure at the hydrophone position [15]. In addition, the Institute of Sound and Vibration Research (ISVR) and other institutes have also achieved many fruitful research results with AVC in underwater vehicles. As mentioned above, AVC has been investigated in underwater vehicles research, but there are still many improvements in practical applications that need to be studied urgently, such as multi-path coupling control and processing for complex signals, etc. At present, this technology has been extensively studied in the single-input and single-output system referred to as SISO [3; 16; 17], where an finite impulse response or infinite impulse response (FIR/IIR) filter is used as a structure controller, and its weight coefficient is adjusted by the least mean squares or recursive least squares (LMS/RLS) criterion to minimize the positional response to a certain point. Ein particular, the adaptive filtering algorithm based on the Least Mean Square (LMS), simply called Filtered-x LMS (FxLMS), is commonly used in vibration and noise control [15; 18; 19]. On this basis, [PERSON] [20], [PERSON] [21], [PERSON] [22], and [PERSON] [23] proposed several improved algorithms such as FuLMS, MRFs-Newton, and FX-FONLMS, etc. They also analyzed with these algorithms the calculation amount, execution efficiency, and control effect. Indeed, most of them were improved based on the LMS or RLS algorithms. In particular, the former is widely used due to its simple structure and easy implementation. However, this method mainly controlled a single measuring point; its overall control effect is not obvious to the structural vibration response in underwater vehicles, and it also heavily relies on the stability of the input signal. In this case, when the vibration source excitation is a stable narrow-band signal, the control performance will be much better.