# Research on Variable-Step-Size Adaptive Filter Algorithm with a Momentum Term
## 1 Introduction  
Adaptive filtering algorithms play a crucial role in modern information science and control theory. They are widely applied in fields such as system identification, holographic projection, and engineering control [1; 2]. Among them, the adaptive LMS (Least Mean Square) algorithm, introduced by the American mathematicians [PERSON] and [PERSON] in the 1960s and based on the principles of Wiener and Kalman filters, has been extensively studied and utilized [3; 4; 5; 6]. One distinguishing feature of this algorithm is that it overcomes the drawback of requiring prior knowledge of the input signal to obtain optimal filter coefficients, as observed in the Wiener and Kalman filter principles. Moreover, it has such advantages as simple algorithm principles and structures as well as ease of implementation. Researchers have proposed numerous improvements to address its limitations [7; 8; 9; 10; 11; 12]. For example, [PERSON] et al. [8] introduced a variable-step-size LMS algorithm with the Tanh function as a model. In the early 20 th century, researchers recognized the potential ofkernel methods in extending linear algorithms to the nonlinear field, which led to increased focus on their application in the field of nonlinear adaptive filtering. For example, [PERSON] et al. [13] combined the LMS algorithm with kernel knowledge to propose the Kernel Least Mean Square (KLMS) algorithm. In the complex domain, [PERSON] et al. [7] introduced the Complex Least Mean Square (CLMS) algorithm in 1975. Different from neural network black-box models, adaptive filtering algorithms take full advantages of neural network algorithms and establish white-box data models, providing strong theoretical support for characteristic analyses and the control algorithm design of complex systems. At present, the PID control algorithm is the most widely used and mature control algorithm in the field of engineering technology. However, modern industrial process control is becoming increasingly complex, and traditional mathematical model identification algorithms and PID control algorithms already face difficulties in meeting these ever-changing needs [14; 15; 16; 17]. Moreover, the demand for intelligent control is increasing. With the rapid development of intelligent learning algorithms based on BP neural networks, a large number of intelligent control algorithms have been proposed, which have achieved good results in many studies. However, in practical engineering applications, it is difficult to achieve stable and reliable results. For example, in wind tunnel tests simulating aircraft flight, in order to suppress the vibration of the aircraft in high-speed fluids, PID-algorithm-based control methods were adopted and achieved favorable results. However, for some aircraft with special configurations, their vibration is very severe during wind tunnel testing, and due to the obvious fluid structure coupling effect, the controlled system modal frequency undergoes \"drift\", resulting in poor PID control [18; 19; 20; 21]. The data-driven adaptive control algorithm is an important way to solve the above problems. Although neural networks have performed well in simulation calculations, their stability in engineering is not high, and a large amount of data is required for training in the early stages. These shortcomings make it difficult for neural network algorithms to be applied in wind tunnel tests with high safety requirements. The adaptive filtering algorithm has data fitting capabilities comparable to those of neural networks and does not require a large amount of data for training, saving a great amount of experimental costs. Therefore, the system identification technology based on the adaptive filtering algorithm has been studied in depth, and identifying the controlled object mathematical model more accurately is a key step in achieving adaptive control.