## 2 Problem Formulation
### Controller Design
This is shown in Figure 2.  
In this way, when an independent variable was the original data collected by the sensor with noise, frequent switching of the function output could be avoided. Method 3, filtering:  
We added the letter \"f\" to the upper left corner of a filtered variable to represent it as Equation (8). The filters used in this paper were Butterworth second-order low-pass filters [31]. \[\mathrm{f_{e_{(n)}}}=\mathrm{f}\Big{(}\mathrm{e_{(n)}},\mathrm{e_{(n-1)}}, \mathrm{e_{(n-2)^{\prime}}},\mathrm{f_{e}}(\mathrm{n-1)},\mathrm{f_{e}}( \mathrm{n-2})\Big{)} \tag{8}\]  
Method 4, lagging suppression:  
This method is used to suppress the system input and output oscillation caused by lagging. In the system described in this paper, it showed a very good effect. We used a simple lagging system model to illustrate this algorithm. As shown in Figure 3, the input of the system is p, which is processed by the algorithm into q. Then, q is the new input for the lagging model, and the output of the model is r. The measured value of r is fed back to the algorithm. Here, the input and the output of the model are required to be the same physical quantity. The algorithm has two expressions: first-order and second-order. For the first-order algorithm, the algorithm output q is a function of the system input p and the model output r, as shown in Equation (9):  
\[\mathrm{q_{(n)}}=\mathrm{f}(\mathrm{p},\mathrm{r}) \tag{9}\]  
The first-order function expression is shown in Equation (10):  
\[\mathrm{f}(\mathrm{x},\mathrm{r})=(1-\mathrm{m})\mathrm{p_{(n)}}+\mathrm{m} \mathrm{r_{(n-1)}} \tag{10}\]  
where m is the constant coefficient between zero and one, \(0<\mathrm{m}<1\); m represents the proportional relationship between the model output and the system input in the algorithm output. A larger m represents a larger proportion of the model output. When m is between zero and one, the total gain of the algorithm can be maintained at 1. The essence of lag compensation is to make the calculated control input of the system closer to the measured control input. When the calculation only involves the current cycle's computed value and measured value, it is called first-order compensation. When the computation and the measured values of the previous control cycle are introduced, it is called second-order compensation. For the second-order algorithm, the algorithm output q is a function of the system input p, the model output r, and the algorithm output q, as shown in Equation (11):  
\[\mathrm{q_{(n)}}=\mathrm{f}(\mathrm{p},\mathrm{q},\mathrm{r}) \tag{11}\]  
The second-order function expression is shown in Equation (12):  
\[\mathrm{f}(\mathrm{p},\mathrm{q},\mathrm{r})=(1-\mathrm{m})\Big{(}(1-\mathrm{ v})\mathrm{x_{(n)}}+\mathrm{v}\mathrm{y_{(n-1)}}\Big{)}+\mathrm{m}\Big{(}(1- \mathrm{w})\mathrm{z_{(n-1)}}+\mathrm{wz_{(n-2)}}\Big{)} \tag{12}\]  
Figure 3: Algorithm for lagging suppression. Figure 2: Piecewise constant function. where m, v, and w are the constant coefficients between zero and one, keeping the total gain at 1. Next, we discuss the speed commands of the four actuators in one lifting point. The system in this paper had three output variables, and we defined three errors that corresponded to the difference between the set values of the three output variables and the measured values of the sensors. The error is described in Equation (13):  
\[\left\{ \begin{array}{l}\mathrm{e}_{\alpha}=\alpha-\alpha^{*}\\ \mathrm{e}_{\mathrm{F}}=\mathrm{F}^{*}-\mathrm{F}\\ \mathrm{e}_{\gamma}=\gamma-\gamma^{*}\end{array} \right.