# Target State Optimization: Drivability Improvement for Vehicles with Dual Clutch Transmissions
## 2 Optimization Problem and Objective Functions
### Discomfort Objectives
#### 2.2.2 Clutch Torque/Jerk Objective  
Compared to acceleration, humans are even more sensitive to high jerks (first derivative of the acceleration). High accelerations are related to discomfort, but passenger comfort is also tied to the change of acceleration of a vehicle [49]. [PERSON] et al. [50] detected the JND  
Figure 3: (**a**) Engine speed (red) does not drop, (**b**) engine speed drops; Input-shaft speed is green.  
of the jerk with an experiment in a vehicle. It amounts to 0.53 m/s\({}^{3}\) with a 95% confidence interval from 0.07 m/s\({}^{3}\) to 1 m/s\({}^{3}\).  
The longitudinal acceleration of the vehicle is directly linked to the clutch torque (\(tq_{clu}\)) since the clutch torque is directly transmitted to the wheels during slip phases. Therefore, it is necessary to take care of the clutch torque gradient during the control of the clutch since during clutch engagement every inconsistency is related to discomfort [25]. The objective analyzes the clutch torque during launch for local minima, which indicates an inconsistency in the acceleration signal during launch. A local minimum is found if the first derivative of the clutch torque is zero and the second derivative of the clutch torque is greater than zero. The number of local minima is counted and optimally has to be zero to avoid changes of jerk:  
\[obj_{Torque}=\sum_{i=1}^{n}\left[\dot{t}q_{clu}(t)=0\ \wedge\ddot{t}q_{clu}(t)>0\right] \tag{9}\]  
In contrast to the other objectives the raw value is not normalized, so if an inconsistency occurs the objective value is equal or greater one and eliminates the chance of having a successful launch in case of an objective value different to zero.  
Although the equation above is theoretically correct, an application is difficult since the measured clutch torque in a real vehicle is a noisy signal with several local minima. Therefore, the signal is filtered with a one dimensional gaussian filter [51] for reducing the noise. Despite filtering, examining local minima numerically is still difficult since the measured signal is sample based and finding an exact zero of the derivatives is often not possible. Instead, the derivative of the filtered torque is investigated with the following Algorithm 1.  
```
Begin Set tfg = gradient(torque_filtered) Declare osc Set q = 0 Set i = 1 While i < length(tfg) if (tfg [i] < 0 && tfg[i - 1] \(\geq\) 0)!! (tfg [i] \(\geq\) 0 && tfg[i - 1] \(<\) 0) then Set osc[q] = i  Set q++  end end Set l = 0 Declare lp For Each o in osc if torque_filtered[o - 1] > torque_filtered[o] then Set l[p] = o Set l++  end end Set obj_Torque_ = length(lp) Return obj_Torque
```  
**Algorithm 1** Local minima detection of the clutch torque.  
The derivative of the filtered torque is observed for changes of the sign. If a change of the sign is detected (\(osc\)) either a maximum or minimum is detected. Further, the filtered torque is investigated with the indices of the sign changes. If the value of the found extrema is lower than the value of the former found extrema it is an indication of a local minima. The number of local minima is the value of the objective. The influence of the filtering is illustrated in Figure 4.  
The number of local minima (determined with Algorithm 1) for the unfiltered signal of the test vehicle on the left (Figure 4a) is 25 since the signal has a noisy behavior. The result indicates an uncomfortable driving behavior. Nevertheless, such a behavior with small amplitudes is not noticeable and hence does not affect the assessment negative. In contrast, Algorithm 1 applied on the filtered signal (Figure 4b), indicates a driving behavior without discomfort since no local minima is determined. A typical behavior of a discomfort causing clutch torque is illustrated in Figure 5.  
A local minimum of the clutch torque is indicated with the red marker which would be noticeable as discomfort.