## 2 Mathematical Modeling and Methods
### The Composition of the Control System
#### 2.4.2 Pump-Motor Hydraulic System Control System  
The variable pump-fixed motor hydraulic system is the main execution system for speed regulation in the HMCVT stage. The input signal of the proportional valve is controlled by the TCU to adjust the pump-motor displacement ratio. The control principle is shown in Figure 6. The control system is a dual-loop parallel control system. The inner loop adopts PID control, and the outer loop adopts model predictive control (MPC). The inner loop is a servo control of displacement ratio. The displacement of the hydraulic cylinder is fed back to the controller for preliminary adjustment, which is controlled by the electric proportional valve; the function of the outer loop control is the adjustment of the motor output speed. The control input signal is adjusted by comparing the output speed of the feedback motor with the expected value so that the output speed of the motor can reach the ideal value. The constant speed control is an important part of pump-motor speed servo control. Due to the complexity of the flow coupling characteristics of the hydraulic system, the most serious difficulty in achieving constant speed control of the motor is the sudden disturbance of the load torque [17]. The PID control is the most common control algorithm [18]. However, as a result of the very obvious load fluctuations during operation, the hydraulic mechanical equipment will be significantly affected when the engine changes speed. It is very difficult to control the speed ratio parameters of the gearbox to ensure that the engine speed responds quickly and remains stable. Therefore, the role of integral control is not obvious. The interference from the mechanical transmission parts with the pump-motor hydraulic system makes the HMCVT transmission ratio prediction inaccurate, which results in unstable tractor speed. However, when the pump-motor hydraulic system is in a steady-state, HMCVT's pump-motor hydraulic system needs a faster response speed and better steady-state robustness [19]. Therefore, based on the above characteristics, this study firstly performs model predictive control (MPC) on the outer loop of the input voltage-output speed of the hydraulic system. The displacement equation of the variable pump is:  
\[T_{P}\dot{q}_{p}+q_{p}=k_{p}u_{p} \tag{10}\]  
where \(T_{P}\) represents the torque of the variable pump; up represents the control voltage; \(k_{p}\) represents variable pump displacement control coefficient. Figure 6: Pump-motor hydraulic control system principle diagram. The kinetic Formula (4) is simplified: the low-pressure side is connected to the fuel tank (\(p_{r}=0\)). The equation \(n_{m}=d\theta_{m}/dt\) can express the motor speed (\(n_{m}\)), and the established space state equation is expressed as follows:  
\[\left\{ \begin{array}{l}\dot{q}_{p}=\frac{1}{T_{p}}(k_{p}u_{p}-q_{p})\\ \dot{p}=\frac{\delta_{p}}{V_{0}}(q_{p}-(c_{im}+c_{em})p-D_{m}n_{m})\\ \dot{n}_{m}=\frac{1}{J_{t}}(D_{m}p-B_{m}n_{m}+T_{L})\end{array} \right. \tag{11}\]  
The state variables are chosen as:  
\[\mathbf{x}=\left[x_{1},x_{2},x_{3},x_{4},x_{5}\right]^{T}= [q_{p},p,n_{m},u_{p},T_{L}]^{T} \tag{12}\]  
The control variables as:  
\[U=[u_{1},u_{2}]^{T}=\left[\dot{u}_{p},\dot{T}_{L}\right] \tag{13}\]  
With (4), (11), (12), (13), the dynamic model can be shown as:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=-a_{1}x_{1}+b_{1}x_{4}\\ \dot{x}_{2}=a_{2}x_{1}-a_{3}x_{2}-a_{4}x_{3}\\ \dot{x}_{3}=a_{5}x_{2}-a_{6}x_{3}+b_{2}x_{5}\\ \dot{x}_{4}=u_{1}\\ \dot{x}_{5}=u_{2}\end{array} \right.\Rightarrow\left\{ \begin{array}{l}a_{1}= \frac{1}{T_{p}},a_{2}=\frac{\delta_{p}}{V_{0}},a_{3}=\frac{\delta_{p}}{V_{0} }(c_{im}+c_{em}),\\ a_{4}=\frac{\delta_{p}}{V_{0}}D_{m}a_{5}=\frac{D_{m}}{J_{t}},a_{6}=\frac{B_{m} }{J_{t}},\\ b_{1}=\frac{k_{p}}{T_{p}},b_{2}=\frac{T_{L}}{J_{t}}\end{array} \right. \tag{14}\]  
The transmission model of the variable pump-fixed motor is obtained. \[\left\{ \begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}\\ \mathbf{Y}=\mathbf{H}\mathbf{x}\end{array} \right.