## 3 Composite Control Method  
In this section, a composite control method is tailored for the established MHL model, as illustrated in Figure 6. This composite controller consists of two components: a feedforward compensator and a traditional proportional-integral (PI) controller. The feedforward compensator is the inverse of the rate-dependent term \(H_{D}(v(t))\), with a structure similar to that shown in Figure 3, and its output is  
\[u_{ff}(t)=H_{S}^{-1}(y_{d}(t))+\Psi^{\prime}(t,s) \tag{6}\]  
\[\Psi^{\prime}(t,s)=\left(k_{1}^{\prime}y_{d}(t)+b_{1}^{\prime}\right)\odot \frac{1}{k_{2}^{\prime}s+b_{2}^{\prime}} \tag{7}\]  
where \(H_{S}^{-1}(y_{d}(t))\) is the rate-independent inverse model based on [PERSON]'s rules. \(\Psi^{\prime}(t,s)\) is the rate-dependent correction term with a structure similar to Equation (4). \(k_{1}^{\prime}\), \(b_{1}^{\prime}\), \(k_{2}^{\prime}\), and \(b_{2}^{\prime}\) are parameters to be identified. The PI controller is employed in the feedback loop, which can be expressed as  
\[u_{fb}(t)=k_{p}+k_{i}\int_{0}^{t}e(\sigma)d\sigma \tag{8}\]  
where \(k_{p}\) and \(k_{i}\) are the proportional and integral coefficients, respectively, and \(e(t)\) is the tracking error. Since the Hammerstein model incorporates a dynamic term \(G(s)\), it is generally used to describe the high-frequency dynamic behavior of the hysteresis system. However, due to the lack of an inverse model for this linear dynamic term, the Hammerstein model cannot directly implement feedforward compensation. In order to eliminate hysteresis nonlinearity and achieve better control performance, the traditional composite control method, as shown in Figure 7, is commonly applied to the hysteresis system described by the Hammerstein model. This controller typically employs a rate-independent hysteresis inverse model as the feedforward compensator and uses a PI controller with the same structure as Equation (8) for closed-loop regulation. The expression for the output of the feedforward compensator is as follows:  
\[u_{ff}(t)=H_{S}^{-1}(y_{d}(t)) \tag{9}\]  
This compensator can only diminish rate-independent hysteresis nonlinearity, and it is difficult to further expand the compensation range except for achieving input-output linearization in the low-frequency range. The dynamic characteristics of the entire system still need to be handled by the PI controller, which undoubtedly increases its workload. Additionally, for input signals with higher frequencies, the traditional composite control method will not be able to achieve the desired tracking accuracy. Figure 6: Schematic diagram of proposed composite control method. Unlike the Hammerstein model, we introduce a rate-dependent model \(H_{D}(v(t))\) in the MHL modeling process, which often possesses an inverse model that can be directly utilized for feedforward compensation. By replacing the rate-independent inverse model (see Equation (9)) in traditional composite control methods with a rate-dependent inverse model (see Equation (6)), a novel composite control method is established. Such a modification aids in compensating for more system dynamics that can be accurately modeled, thereby expanding the compensation range and enhancing the control performance of the closed-loop system. After compensation using the rate-dependent inverse model, the hysteresis system achieves input-output linearization within a certain frequency range (within 100 Hz), reducing the workload of the feedback controller. The introduction of the rate-dependent term allows for a reduction of up to 40% in the output of the PI feedback controller while maintaining the same closed-loop tracking accuracy. Conversely, with the same PI feedback controller output, the closed-loop tracking error is decreased by up to 80%. Reducing the feedback controller's workload effectively decreases the control output amplitude, avoiding excessive responses or high-frequency oscillations.