## 3 Adaptive Rule Based Controller
### Formulation of Adaptive Rule Based Controller  
The DP-based controller is applied to UKBUS6, MANHATTN, NYCC, UDDS, New York Bus, INDIAHWY, EUDC_LOW, and HWFET driving cycles. Urban, suburban, and expressway conditions are included in these driving cycles. In the offline optimization of DP under various driving cycles, the relationship between the output power of the ultra-capacitor and the demand power of the motor can be divided into three regions, except for the UDDS driving cycle, as described in Figure 2. In each region, an approximate linear relationship exists between the output power of the ultra-capacitor and the demand power of the motor. It can be seen from region 1, that when the demand power of the motor is negative (the vehicle is braking), the ultra-capacitor absorbs all of the braking energy. Because of the efficiency of the DC/DC converter, the power of the ultra-capacitor is slightly less than the motor power, and the slope of the linear relationship is less than 1. Region 2 indicates that the ultra-capacitor will not provide the output power when the demand power of the motor is positive and lower than a certain threshold. Under the low demand power of the motor, it is relatively safe for the battery to discharge alone, and the addition of ultra-capacitor, it will reduce the system efficiency of HESS. In region 3, when the demand power of the motor is greater than a threshold, both the battery and ultra-capacitor provide the output power, and the output power of the ultra-capacitor linearly increases with the demand power of the motor. As for the UDDS driving cycle, there is an extra region, region 4. At a few or more seconds after many time points in region 4, a higher demand of power is required and the DP, which is a global optimal algorithm, allocates the electricity of the ultra-capacitor to these points, demanding greater power. Because of the limited electricity of the ultra-capacitor, the time points in region 4 receive relatively low power. As a result of the obvious regularity of the optimal instantaneous power allocation, the average positive power of the various driving cycles and the thresholds of the demand power of the motor in region 2 are counted. Also, the linear relationship in region 3 is fitted and region 4 of the UDDS is ignored because of the individual phenomenon. The results are listed in Table 3.  
From Table 3, the thresholds in region 2 increase with the average positive power under the various driving cycles. The linear relationship in region 3 between the output power of the ultra-capacitor and the demand power of the motor can be expressed by \(f(x)=ax+b\). The slope, \(a\), of each driving cycle is close to the others, which is supposed to around 0.7785 and the values of \(b\) increase with the average positive power of the driving cycle. Thus, by means of the linear fitting of both the relationship between the thresholds in region 2 and the average positive power, and the relationship between the value of \(b\) and the average positive power, the threshold of the demand power of the motor and the linear curves in region 3 can be obtained if the average positive power is received. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Driving Cycle** & **Average Positive Power/kW** & **Threshold in Region 2/kW** & **Fitting Curves in Region 3** \\ \hline UKBUS6 & 3.360 & 3.291 & \(f(x)=0.7647x\) – 3.040 \\ MANHATTAN & 6.172 & 5.132 & \(f(x)=0.7850x\) – 4.366 \\ NYCC & 7.121 & 6.673 & \(f(x)=0.7855x\) – 5.905 \\ UDDS & 10.202 & 6.486 & \(f(x)=0.7772x\) – 6.187 \\ New York Bus & 11.000 & 8.070 & \(f(x)=0.7767x\) – 7.065 \\ INDIAHWY & 11.669 & 10.85 & \(f(x)=0.7892x\) – 8.444 \\ EUDC\_LOW & 11.770 & 13.68 & \(f(x)=0.7872x\) – 11.15 \\ HWFET & 14.251 & 18.67 & \(f(x)=0.7628x\) – 14.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data statistics of each driving cycle. Figure 2: Power allocation under different driving cycles. _UC_—ultra-capacitor; \(P_{dem}\)—demand power of motor.