## 4 Results and Discussion
### The 1 LCM for the Vehicle with a Driver Assistance System
#### 4.4.2 The Steering Wheel Angle Control System Adopted for the Lane Change Maneuver  
It was assumed that the car under test would be provided with a lane change assistance system as proposed below. Its operation has been illustrated in Figure 10. If it is ON, then the reference trajectory \(\mathrm{y_{P}(x)}\) is determined at the instant when a lane change is signalled. The trajectory parameters (H and L\({}_{m}\)) depend on: (1) the lane width and the initial vehicle position on the first lane at the instant of signalling the intention to carry out the maneuver (it was assumed that these are determined by means of optical systems similar to those used for the parking and lane keeping sensors); (2) readings of the ambient temperature and rain presence sensors (based on these readings, the road surface condition is assessed, i.e., the question whether the road is dry, wet, or ice-covered is answered); and (3) the current vehicle speed V. Then, the reference trajectory (\(\mathrm{y_{P}(x)}\)) is transformed to the time domain (\(\mathrm{y_{P}(t)}\)) and the system starts the execution of the lane change maneuver. Two proportional-integral-differential (PID) controllers operating in parallel have been provided: for the lateral position \(\mathrm{y_{Cc}}\) and the yaw angle \(\mathrm{\psi_{C}}\). The current values of \(\mathrm{y_{Oc}}\) and \(\mathrm{\psi_{C}}\) are calculated based on the readings of optical sensors (similar to those used in the parking and lane keeping sensors). The input signals are the deviations \(\Delta\mathrm{y}\) and \(\Delta\mathrm{\psi}\) calculated for the lateral position \(\mathrm{y_{Oc}}\) and the yaw angle \(\mathrm{\psi_{C}}\). The resultant value of the steering wheel angle is the weighted average of the components obtained from the two control systems operating in parallel, where the weighting factors are \(\mathrm{w_{y}}\) and \(\mathrm{w_{\psi}}\), as appropriate, according to Equation (5):  
\[\alpha_{\mathrm{k}}(t)=\alpha_{\mathrm{ky}}(t)\cdot\mathrm{w_{y}}+\alpha_{ \mathrm{k\psi}}(t)\cdot\mathrm{w_{\psi}} \tag{5}\]  
The settings of the PID controllers were determined using a step input, where a controlled trajectory \(\mathrm{y_{P}}=0\) was applied for the instant \(t=1\) s, at the initial value of \(\mathrm{y_{P}(0)}=-\mathrm{H}\). The proportional control gains \(\mathrm{K_{py}}\) and \(\mathrm{K_{p\psi}}\) were selected to come close to the stability limit, i.e., to the appearance of non-decaying oscillation. Then, the equivalent delays of the object (vehicle) controlled and the equivalent time-constants of the vehicle under test were determined. Based on them, the controllers' parameter values (\(\mathrm{K_{py}}\), \(\mathrm{K_{iy}}\), \(\mathrm{K_{dy}}\) and \(\mathrm{K_{p\psi}}\), \(\mathrm{K_{i\psi}}\), \(\mathrm{K_{d\psi}}\)) were determined by the Ziegler-Nichols method [46; 47; 48]. Apart from the controllers' parameter values, the weighting factors for the summands \(\alpha_{\mathrm{ky}}(t)\) and \(\alpha_{\mathrm{k\psi}}(t)\) in Equation (5) had also to be determined. The raising of \(\mathrm{w_{y}}\) at the expense of \(\mathrm{w_{\psi}}\) led to reductions of the phase shift of the trajectory (after the transformation to the time domain) relative to the reference one at the expense of the time histories of \(\alpha_{\mathrm{k}}(t)\) becoming more irregular. When the opposite happened, the said phase shift increased but the courses of the steering wheel angle became smoother. Using the trial and error method,the controller parameters were slightly modified to values K\({}_{\rm py}\) = 0.1 rad/m, K\({}_{\rm ly}\) = 38 s, K\({}_{\rm dy}\) = 6 s, K\({}_{\rm pyl}\) = 100 rad/rad, K\({}_{\rm kib}\) = 80 s, and K\({}_{\rm dly}\) = 12 s and the weighting factors equal to w\({}_{\rm y}\) = w\({}_{\rm b}\) = 0.5 were used. In result of that, the upward trend in the phase shift was reduced and the \(\alpha_{\rm k}\)(t) curves obtained were only slightly disturbed.