## 3 Robust Controller Design for Variable Mechanism Control System  
There are inconsistencies between the system modeling [30] and the real model since mistakes are often introduced during procedures like simplification and linearization. Additionally, a variety of disturbances often affect the control field. Therefore, the fundamental goal of robust control is to guarantee that the system performs as intended while accounting for these disturbances and errors, as well as to examine the static and dynamic features and other system performance indicators. Since \(H_{\infty}\) control theory [31] has good characteristics, it has been widely studied and the theoretical system is relatively well developed. In this paper, we use the controller to realize the control of the variable mechanism of the axial piston pump;the system block diagram is shown in Figure 5 under  
\(P(s)\) stands for the augmented object model, \(K(s)\) for the controller model, \(z(t)\) for the system's output vector, \(w(t)\) for its input signal, \(u(t)\) for its control vector, and \(y(t)\) for its observation vector. \(P(s)\) can be generally expressed as:  
Its corresponding augmented equation of state is described as:  
The following is an expression for the transfer function between the input signal \(w(t)\) and the output signal \(z(t)\):  
\[T_{zw}(s)=P_{11}(s)+P_{12}(s)[I-F(s)P_{22}(s)]^{-1}F(s)P_{21}(s)\]  
When analyzing the robustness of the system with respect to perturbations, we can check whether the value of \(\|T_{zw}(s)\|\infty\) is less than 1, which indicates that the system is robust and has a strong processing capability for the perturbed signal. \(H_{\infty}\) control theory uses the \"\(H_{\infty}\) parameter\" to describe the performance index of the system to achieve the purpose of resisting perturbations. According to previous analysis [32], the robustness requirement can be achieved by designing the \(H_{\infty}\) controller to meet the requirements of certain metrics. In this paper, a hybrid sensitivity approach is used to design the \(H_{\infty}\) controller in order to accomplish loop formation through the design of system performance metrics via the sensitivity functions \(W_{1}(s)\), \(W_{2}(s)\), and \(W_{3}(s)\). To prevent system errors, the correct \(W_{1}(s)\), \(W_{2}(s)\), and \(W_{3}(s)\) must be selected, As shown in Figure 6.  
Figure 5: Control block diagram. For this system, the optimization strategy is a weighted hybrid sensitivity, as follows:  
\[T_{zw}= \begin{bmatrix}W_{1}S\\ W_{2}R\\ W_{3}U\end{bmatrix} = \begin{bmatrix}W_{1}(I+Gk)^{-1}\\ W_{2}K(I+Gk)^{-1}\\ W_{3}GK(I+Gk)^{-1})\end{bmatrix} \]  
To obtain the ideal \(H_{\infty}\) controller, such that the \(H_{\infty}\) parameter is minimized:  
Step 1. Model the controlled object and analyze the performance requirements of the controlled object to obtain a mathematical description of the controlled problem. Step 2. The weighting function is selected \(W_{1}(s)\), \(W_{2}(s)\), and \(W_{3}(s)\) to determine the shape of the sensitivity function. \(W_{1}(s)\) can generally be made to have a higher low-frequency gain to reduce control errors; \(W_{2}(s)\) has a higher high-frequency gain to enhance stability; \(W_{3}(s)\) can generally be taken as a smaller constant to increase the response speed and bandwidth of the system. Step 3. Solve for the system generalized object \(P(s)\) and its state space implementation, which in turn leads to the robust controller \(K(s)\). Step 4. Verify that the system performance metrics meet the dynamic and static performance requirements set by the resulting controller. If the requirements are met, the controller can be selected as the final result; if not, it is necessary to go back to Step 1 or Step 2 in a loop. Combining the program block diagram of the axial piston pump variable mechanism control system with the data in Table 1, the transfer function of the system can be determined as follows.