## 4 Results and Discussion
### Hysteresis Effect of 8 Axle
#### 4.1.2 First-Order Plus Time Delay Model  
There is an obvious hysteresis of the PBS according to the above analysis. In this paper, a simplified model of the system is proposed to benefit the further development of brake controller. Considering the inherent hysteresis characteristics obtained in Section 4.1.1 and [26; 27], the system can be approximately fitted to a first-order plus time delay model. This model relates the opening of the treadle valve and the pressure transient in the brake chamber. The governing equation of the model can be described like this,  
\[T\frac{dP(t)}{dt}+P(t)=Ku(t-\tau), \tag{5}\]  
where \(T\) is the time constant of the system, \(K\) is the gain, and \(\tau\) is the delay time. \(P(t)\) and \(u(t)\) are the output and input of the system. Here, they refer to the pressure transients in the brake chamber and the opening of the brake pedal, respectively. Taking the Laplace transform of the Equation (5), we obtain the transfer function of the system. \[G(s)=\frac{Ke^{-\tau s}}{(Ts+1)}, \tag{6}\]  
Figure 11 shows a schematic description of the first-order plus time delay model of the PBS. Past research on the PBS has shown that the moving displacement of treadle valve plunger has a linear relationship with the desired braking pressure. Therefore, according to Formula (2) we assume that the relationship between pedal opening and the desired braking pressure can be expressed in Formula (7). As for the parameter \(\tau\), it can be defined as Formula (8). \[P_{d}=a\psi+b, \tag{7}\]  
\[\left\{ \begin{array}{l}\tau_{up}=\Delta_{del\_up}-\widetilde{\tau}_{up}\\ \tau_{down}=\Delta_{del\_down}-\widetilde{\tau}_{down}\end{array} \right., \tag{8}\]  
where \(a\), \(b\) are constants, related to the type of the used treadle valve and brake pedal. \(\tau_{up}\), \(\tau_{down}\) are the pure time delay of the system in brake process and brake release process, respectively. \(\widetilde{T}_{up}\), \(\widetilde{T}_{down}\) are the time constants related to \(T\). Where \(\widetilde{T}_{up}\) is the time spent when the pressure in the chamber varies from 0 to 75% of the steady value in brake process, \(\widetilde{T}_{down}\) is the time spent when the pressure in the chamber varies from the steady value to 15% in the brake release process. With the method of system identification, the simulation model of the system can be established. In view of the deviation of hysteresis characteristics of each loop, eight loops correspond to eight different models, as shown in Figure 12, and their relevant parameters are listed in (9). We find that the models approximately coincide with the experimental results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Loop**} & \multicolumn{2}{c}{**K\({}_{1}\)**} & \multicolumn{2}{c}{**K\({}_{2}\)**} & \multicolumn{2}{c}{**W**} & \multicolumn{2}{c}{**R\({}_{2}\)**} \\ \cline{2-9}  & **Up** & **Down** & **Up** & **Down** & **Up** & **Down** & **Up** & **Down** \\ \hline A & 0.597 & 0.290 & \(-\)0.238 & 0.621 & 0.378 & 0.192 & 0.997 & 0.996 \\ B & 0.759 & 0.158 & \(-\)0.423 & 0.746 & 0.356 & 0.109 & 0.998 & 0.985 \\ C & 0.754 & 0.117 & \(-\)0.520 & 0.778 & 0.447 & 0.130 & 0.980 & 0.993 \\ D & 0.393 & 0.321 & 0.026 & 0.497 & 0.308 & 0.287 & 0.975 & 0.990 \\ E & 0.353 & 0.407 & 0.026 & 0.473 & 0.401 & 0.299 & 0.958 & 0.997 \\ F & 0.355 & 0.653 & 0.115 & 0.087 & 0.299 & 0.394 & 0.975 & 0.999 \\ G & 0.397 & 0.533 & 0.081 & 0.298 & 0.316 & 0.324 & 0.971 & 0.999 \\ H & 0.601 & 0.450 & \(-\)0.243 & 0.409 & 0.488 & 0.352 & 0.977 & 0.998 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting parameters. \[\left\{ \begin{array}{l}T=[\quad 0.124\quad 0.118\quad 0.075\quad 0.170\quad 0.114 \quad 0.171\quad 0.169\quad 0.184\quad]\\ \widetilde{T}_{up}=[\quad 0.170\quad 0.160\quad 0.100\quad 0.240\quad 0.150\quad 0.240\quad 0.230\quad 0.250\quad]\\ \widetilde{T}_{down}=[\quad 0.210\quad 0.200\quad 0.130\quad 0.300\quad 0.200\quad 0.310 \quad 0.300\quad 0.330\quad]\end{array}