# Target State Optimization: Drivability Improvement for Vehicles with Dual Clutch Transmissions
## 2 Optimization Problem and Objective Functions
### Optimization Parameters  
During clutch slip phases the gradient of the engine speed is dependent on the difference between the engine torque and the clutch torque. If the clutch torque e.g., is zero and the engine torque is greater than zero, the engine speed is increasing since there is no resistance from the road applied. Generally, if the engine torque is greater than the clutch torque the engine speed is increasing. Accordingly, if the clutch torque is equal to the engine torque the engine speed is remains constant. The engine speed is decreasing if the clutch torque is greater than the engine torque during clutch slip phases. Thus, the engine speed can be controlled by the clutch. Therefore to calibrate the clutch engagement  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Objective** & **Value** & **Unit** \\ \hline Engine Speed Drop & 0.00 & rpm \\ Acceleration Peak & 3.25 & m/s\({}^{2}\) \\ Acceleration Build-up & 6.50 & m/s\({}^{3}\) \\ Reaction Time & 0.50 & s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Target state for the software in the loop environment.  
the engine target speed is shaped [5]. The launch itself is divided into different phases [53]. The three phases are illustrated in Figure 6.  
1. The engine speed is increasing, vehicle accelerating (red).
2. The engine speed is steady until the input-shaft speed is almost equal to the engine speed (blue).
3. The engine speed and its speed gradient are adjusted to the input-shaft speed and its speed gradient (for a smooth clutch engagement (yellow)).  
The objectives of Sections 2.1 and 2.2 are optimized with five calibration parameters. These parameters are dependent on the driver request torque (derived from the accelerator pedal position):  
* Parameter 1--phase 1: percentage of the engine torque which should be used to accelerate the engine speed to phase 2 (low value: quick vehicle acceleration response but the engine speed increases slowly--sluggish vehicle acceleration, high value: fast increase of the engine speed worsened vehicle acceleration response).
* Parameter 2--phase 1: minimum value of the engine torque which should be used to accelerate the engine to phase 2 (active if the value of parameter 1 is to low).
* Parameter 3: P-gain control value to control the behavior of the engine speed regarding the engine target speed.
* Parameter 4--phase 3: time for reducing the slip speed in phase 3.
* Parameter 5--phase 3: engine speed which should be reached at the end of phase 3.  
As mentioned prior, these parameters are dependent on different signals (e.g., driver request torque). An example of such a map is illustrated in Table 2:  
The engine speed error is the deviation of the engine target speed and the measured engine speed. Table 2 illustrates that the value of the P-gain is varied for different engine speed errors (between the nodes the values are interpolated).  
Since not every value in the map is required to be changed to optimize the desired driving maneuver (if the driver request is 150 Nm the nodes of 0 Nm and 300 Nm does not affect the behavior), only the ones which affect the behavior are adjusted. This is exemplary illustrated with the marked row of Table 2. For optimizing the desired driving  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**P-gain**} & \multicolumn{5}{c}{**Engine Speed Error/Rpm**} \\ \hline  & & \(-\)**500** & \(-\)**250** & **0** & **250** & **500** \\ \hline  & 0 & \(z_{11}\) & \(z_{12}\) & \(z_{13}\) & \(z_{14}\) & \(z_{15}\) \\  & 150 & \(\mathbf{z_{21}}\) & \(\mathbf{z_{22}}\) & \(\mathbf{z_{23}}\) & \(\mathbf{z_{24}}\) & \(\mathbf{z_{25}}\) \\  & 300 & \(z_{31}\) & \(z_{32}\) & \(z_{33}\) & \(z_{34}\) & \(z_{35}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Example of a calibration map—Parameter 3: P-gain.  
Figure 6: Three phases of the launch.  
maneuver only, the marked row needs to be adjusted. Therefore, from these five calibration parameters, 22 values are derived which have to be adjusted during optimization.