## 2 Problem Formulation
### Controller Design  
It is checkable that System (3) is with input delay. The design of controllers was not the same as in delay-free systems. With this issue in mind, we first developed a controller based on adaptive dynamic programming [11; 27; 28] via online data. Since the online data could be inaccurate due to the influence of measurement noise, we modified the learned controller to improve the performance of the closed-loop system. The goal of controller design was to design the inputs of the four actuators of each lifting point to maintain the output of the system at the desired value. The input of the system satisfied the relationship given in Equation (5), with the corner mark zero representing the initial state:  
\[\left\{ \begin{array}{l}\mathrm{x}_{1(n)}=\sum\limits_{0}^{n}\mathrm{l}_{ \mathrm{v}x1(n)}\mathrm{dt}+\mathrm{x}_{\mathrm{s}(0)}\\ \mathrm{y}_{\mathrm{s}(n)}=\sum\limits_{0}^{n}\mathrm{l}_{\mathrm{v}ys(n)} \mathrm{dt}+\mathrm{y}_{\mathrm{s}(0)}\\ \mathrm{y}_{\mathrm{b}(n)}=\sum\limits_{0}^{n}\mathrm{l}_{\mathrm{v}y(n)} \mathrm{dt}+\mathrm{y}_{\mathrm{b}(0)}\\ \mathrm{z}_{1(n)}=\sum\limits_{0}^{n}\mathrm{l}_{\mathrm{v}x1(n)} \mathrm{dt}+\mathrm{z}_{10}\end{array} \right. \tag{5}\]  
We could control the input of the four controllers by adjusting the speed commands of the four actuators. Before giving the expression of speed commands, we describe the four methods used in this paper. Method 1, saturation:  
The saturation here was to limit the amplitude of a variable and the change of the variable with time at the same time [29]. When 1 was added to the upper left corner of the variable, this expression was used to represent the value of the variable after limiting, as in Equation (6). \[\mathrm{l(v,a)}_{\mathrm{v}(n)}= \begin{cases}\mathrm{v}_{(n)}&\mathrm{v}_{(n) }\leq\mathrm{v}_{\mathrm{max}},\left|\dot{\mathrm{v}}_{(n)}\right|\leq\mathrm{ a}_{\mathrm{max}}\\ \mathrm{a}_{\mathrm{max}}\mathrm{dt}+\mathrm{l}_{\mathrm{v}(n-1)}&\mathrm{v}_{(n )}\leq\mathrm{v}_{\mathrm{max}},\left(\mathrm{v}_{(n)}-\mathrm{l}_{\mathrm{v} (n-1)}\right)/\mathrm{dt}>\mathrm{a}_{\mathrm{max}}\\ -\mathrm{a}_{\mathrm{max}}\mathrm{dt}+\mathrm{l}_{\mathrm{v}(n-1)}^{\mathrm{v}}& \mathrm{v}_{(n)}\leq\mathrm{v}_{\mathrm{max}},\left(\mathrm{v}_{(n)}-\mathrm{l }_{\mathrm{v}(n-1)}\right)/\mathrm{dt}<-\mathrm{a}_{\mathrm{max}}\\ \mathrm{v}_{\mathrm{max}}&\mathrm{v}_{(n)}>\mathrm{v}_{\mathrm{max}}\end{cases}  \tag{6}\]  
Method 2, piecewise constant function:  
The coefficient of a variable is a function of that variable [30]. We added a piecewise constant function with a hysteresis-like structure at the boundary and then limited the amplitude of the result according to method 1 to obtain this function. The expression of the function is shown in Equation (7):  
\[\mathrm{K}= \begin{cases}\mathrm{K}_{1}&\mathrm{e}<\mathrm{e}_{1}-\mathrm{d}_{1 }\\ \mathrm{K}_{1},\mathrm{K}_{2}&\mathrm{e}_{1}-\mathrm{d}_{1}\leq\mathrm{e}\leq \mathrm{e}_{1}+\mathrm{d}_{1}\\ \mathrm{K}_{2}&\mathrm{e}_{1}+\mathrm{d}_{1}<\mathrm{e}<\mathrm{e}_{2}-\mathrm{d }_{2},\mathrm{K}(\mathrm{e})=\mathrm{l}\mathrm{K}\\ \mathrm{K}_{2},\mathrm{K}_{3}&\mathrm{e}_{2}-\mathrm{d}_{2}\leq\mathrm{e}\leq \mathrm{e}_{2}+\mathrm{d}_{2}\\ \vdots&\vdots\end{cases}  \tag{7}\]  
where \(\mathrm{K}_{1}\) is the control gain learned by the adaptive dynamic programing method. In the following text, the coefficients processed according to the function defined here were used in the later PID controller coefficients and other coefficients, unless otherwise specified. We set a width at each switching point. When an independent variable changed near the switching point plus width, the output could be switched only when the independent variable increased. When an independent variable changed near the decreasing width of the switching point, the output was switched only when the independent variable decreased.