## 4 Results and Discussion
### The 1 LCM for the Vehicle with a Driver Assistance System  
#### 4.4.1 Examples of the Steering Wheel Angle Control Systems for the Lane Change Maneuver  
The systems for the automatic controlling of the steering wheel angle during the lane change have been described in many publications, e.g., [6; 7; 8; 9; 11; 12; 13; 17].  
In the control strategy of the system described in [9], the lane change process is decomposed into time-related phases, which is in line with the driving practice of experienced drivers. In the first phase, the controlling is performed in a partly open system (\"blindly\", \"just fast\") by generating an appropriate turn of the steering wheel. The precision of this phase of the maneuver is ensured by the previously identified reference model of the vehicle dynamics. During this phase of the control process, corrective controlling also takes place. The course of changes in the steering wheel angle is corrected on the principle of regulation based on comparing the current should-be values of the variable that describes the lateral vehicle displacement according to the reference model with the current values of this variable, which are actually measured. In [6], the lane-changing maneuver is also analyzed as a multi-stage process carried out by controlling the steering angle of the steered wheels. This work highlights the benefits of using MPC (model predictive control) and SMC (Sliding Mode Control) algorithms and the use of neural networks to adjust the parameters of the control algorithm.  
In publication [11], a model for controlling the steering system in a closed loop has been presented, where controllers of two types, i.e., proportional-integral (PI) and proportional-derivative (PD) ones, and a low-pass filter with the delay time at a level of 0.1 s, were used. In that work, a straight-line braking maneuver, with a possibility of changing it into a lane change when the driver notices a sudden change in the road surface grip, was successfully simulated.  
Figure 9: Vehicle trajectories during a single lane change on a road section with a length of Lm = 75 m at the final lateral displacement of H = 3.5 m. Analytical model 3 (“M3”) by [PERSON] [34]. The result of a simulation \(\mathrm{y_{OC}(t)}\) (“Simulation”) for the wet concrete road surface at a speed of V = 90 km/h.  
In the work reported in [12], the object of control and regulation was a vehicle model with 9 degrees of freedom and a driver model was added to it. Preliminarily, the vehicle trajectory for the maneuver of keeping the lane at a constant speed was defined. The model of the controller was developed to control the steering wheel angle. The proposed structure of the controlling in the major loop of the controller model is a combination of the proportional gain control (P) with the adaptive control, using the fuzzy logic to control the yaw angle. In [7], a dual PID controller is considered to regulate the deviation of lateral displacement and yaw angle. The PID controller is also used in [8], where an assistant system called ACAS (adaptive collision avoidance strategy) was proposed, enriched with additional adaptive fuzzy-logic systems. The system is intended to ensure operation in conditions of variable speed and wheel adhesion to the road surface.  
Other publications, e.g., [13; 44; 45] deal with V2V (vehicle-to-vehicle) and connected and automated vehicles (CAV) technology.  
As it can be seen, various control methods, including very advanced and complex ones, are employed to address the issues of controlling the vehicle movement during the lane change maneuver. For this purpose, models of proportional-integral-differential (PID) controllers are used because of their relative simplicity, tried, proven, and documented operational reliability, and known methods of identification of their parameters [46; 47; 48]. In most cases, the predefined and planned vehicle trajectory is used as the reference.