###### Abstract
However, due to the inherent complexity of these mechanisms and the challenges in generalizing them to different materials, physical models have been relatively less studied and reported in the literature. In contrast, phenomenological models employ appropriate mathematical models to describe hysteresis nonlinearity, resulting in behavior similar to physical models, but without explicitly accounting for the physical mechanisms of piezoelectric materials [5; 6; 7; 8]. Phenomenological models such as the Bouc-Wen model [9; 10], the Duhem model [11; 12], the Preisach model [13], the Prandtl-Ishlinskii model [14; 15], the Maxwell model [16], and other models [17; 18] have been developed to describe hysteresis nonlinearity. The Bouc-Wen model and Duhem model belong to differential equation-based models, which use differential equations to model the hysteresis nonlinearity. The Preisach model, Prandtl-Ishlinskii model, and Maxwell model fall into the category of operator-based models, where the weighted superposition of simple hysteresis operators is used to describe hysteresis nonlinearity. A phenomenological hysteresis model based on [PERSON]'s rules is proposed in [19; 20]. Different from the differential equation-based and operator-based models, this method focuses on describing the hysteresis curves directly. It is noted that all the aforementioned models are presented for low-frequency inputs. These models are referred to as static hysteresis models or rate-independent hysteresis models. As shown in Figure 1b, the hysteresis loop gradually widens with the increase in input frequency, making rate-independent hysteresis models unable to describe the variations accurately. Some scholars believe that the change in the shape of the hysteresis loop is caused by the increase in input frequency. Thus, the derivative of the input \(v(t)\) is introduced to rectify the shape change [7; 21; 22; 23]. This type of model, which introduces the input rate correction term into the rate-independent model, is commonly referred to as the rate-dependent hysteresis model. Other scholars consider that the phenomenon of the hysteresis loop shape varying with frequency arises from more complex causes. To account for this, a rate-independent hysteresis in series with a linear dynamic term is employed to describe such variations in the hysteresis loop. This model is also known as the Hammerstein model [24; 25] and has been widely applied in hysteresis modeling. In [26; 27; 28; 29], the Hammerstein-like model is proposed, where the effect of the input time delay is taken into account. Generally, it is difficult to compare rate-dependent models and Hammerstein models. However, to illustrate the main idea of this paper, we attempt to summarize the characteristics of both models. A rate-dependent model often possesses an inverse model, which can  
Figure 1: **(a)** Schematic diagram of FSM. **(b)** Diagram of rate-dependent hysteresis. be directly utilized for feedforward compensation and achieve input-output linearization within a certain frequency range. In contrast, due to the presence of the linear dynamic term in the Hammerstein model, it lacks an inverse model and cannot realize full feedforward compensation (only the rate-independent term can be compensated). However, the Hammerstein model outperforms the rate-dependent model in describing the frequency characteristics of the system, particularly in the high-frequency range. Combining the advantages of both models could potentially yield better results in modeling, feedforward compensation, and closed-loop control, which is precisely the main idea of this paper. The primary contributions of our methods are two-fold:  
1. A modified Hammerstein-like (MHL) hysteresis modeling approach is proposed; it integrates an input time delay, a rate-dependent hysteresis term, and a linear dynamic term in a cascaded structure. As we will see in the experiments, the presented model can effectively capture the dynamic characteristics of an FSM across a wide frequency range;
2.