Chapter 11. Different methods for platooning control

The vehicle following control law is said to provide individual vehicle stability if the spacing error of the vehicles converges to zero when the preceding vehicle is operating at constant speed. If the preceding vehicle is accelerating or decelerating, then the spacing error is expected to be nonzero. Spacing error in this definition refers to the difference between the actual spacing from the preceding vehicle and the desired inter-vehicle spacing.

Let $x_i$ be the location of the $i^{th}$ vehicle measured from an inertial reference, see Figure 145.

The spacing error for the $i^{th}$ vehicle is then defined as

where $L_{des}$ is the desired spacing and includes the preceding vehicle length. The desired spacing $L_{des}$ must be chosen as a function of variables such as the vehicle speed ${\dot{x}}_i$. The control law is said to provide individual vehicle stability if the following condition is satisfied

If the vehicle following control law ensures individual vehicle stability, the spacing error should converge to zero when the preceding vehicle moves at constant speed. However, the spacing error is expected to be non-zero during acceleration or deceleration of the preceding vehicle. It is important then to describe how the spacing error would propagate from vehicle to vehicle in a string of ACC vehicles that use the same spacing policy and control law.

The string stability of a string of ACC vehicles refers to a property in which spacing errors are guaranteed not to amplify as they propagate towards the tail of the string, see (1), (2). For example, string stability ensures that any errors in spacing between the $i^{th}$ and${(i+1)}^{th}$ cars does not amplify into an extremely large spacing error between cars $j^{th}$ and ${(j+1)}^{th}$ when ($j>i$) further up in the string of vehicles.

In order to evaluate autonomous platoons from control point of view string stability and performance are analyzed. By definition, the platoon is string stable if, for a given $\gamma >0$, there exist $\delta >0$ such that

where $\parallel e_i\left(0\right){\parallel }_{\infty }={\text{sup}}_i\left|e_i\left(0\right)\right|$ and $\parallel e_i{\parallel }_{\infty }={\text{sup}}_t\left|e_i\left(t\right)\right|$. In other words the spacing errors remain bounded along the platoon whenever the initial spacing errors are bounded. It follows from the definition that the spacing errors due to changes in lead vehicle speed do not amplify along the platoon and in time when string stability is satisfied [127].

The following condition will be used to determine if the system is string stable:

where $\widehat{H}{\left(s\right)}_{\infty }$ is the transfer function relating the spacing errors of consecutive vehicles $\widehat{H}{\left(s\right)}_{\infty }={\epsilon }_i/{\epsilon }_{i-1}$.

Platoon performance is commonly defined as either the largest peak spacing error or the total length of the platoon. In the paper, series

are evaluated from which both quantities can be derived. The infinity norm of a scalar signal $x$ is defined by $\parallel x\left(t\right){\parallel }_{\infty }={\text{sup}}_t\left|x\left(t\right)\right|$. From series ${\epsilon }_i$, $i=1,\mathrm{...},n$ one can chose constant safety gaps $L_i$ and inspect the ability of the platoon to attenuate errors. Furthermore, the possibility of actuator saturation can be concluded.

Remark 1.1 It can be shown that an autonomous controller cannot ensure string stability when the constant spacing policy is used. In the constant spacing policy, the desired spacing between successive vehicles is defined by $x_{i,des}=x_{i-1}-L_i$ where $L_i$ is a constant and includes the length $l_{i-1}$ of the preceding vehicle. The spacing error of the i th vehicle is defined as ${\epsilon }_i=x_i-x_{i-1}+L_i$ . When the acceleration of the vehicle can be instantaneously controlled, then a linear control system of the type ${\ddot{x}}_i=-k_p{\epsilon }_i-k_v{\dot{\epsilon }}_i$ yields ${\ddot{\epsilon }}_i={\ddot{x}}_i-{\ddot{x}}_{i-1}=-k_p{\epsilon }_i-k_v{\dot{\epsilon }}_i+k_p{\epsilon }_{i-1}-k_v{\dot{\epsilon }}_{i-1}$ , which leads to the following closed-loop error dynamics ${\ddot{\epsilon }}_i+k_v{\dot{\epsilon }}_{i-1}+k_p{\epsilon }_i=k_p{\epsilon }_{i-1}+k_v{\dot{\epsilon }}_{i-1}$ . This equation shows the the propagation of spacing errors in the platoon of vehicles. The magnitude of the transfer function between ${\epsilon }_{i-1}$ and ${\epsilon }_i$ may be greater than 1 so the autonomous control law is not string stable. Thus, in the case of the constant spacing policy, string stability cannot be ensured by autonomous control.

Two type os control strategies have been proposed:

In the Constant Spacing control strategies, the desired inter-vehicule spacing is independent of the velocity of the controlled vehicle. The tracking requirement is stringent, since every controlled vehicle has to match its position, velocity and acceleration with the vehicle ahead. As a consequence, these strategies require more information to guarantee performance. The achievable traffic capacity is very high in a constant spacing control strategy. There are several solutions for constant spacing strategy:

In the Variable Spacing control strategies the desired inter-vehicle spacing varies with the velocity of the controlled vehicle. The tracking requirement is not as stringent as the previous case. Some of the variable spacing control strategies can be implemented with on-board sensors. A possible solution for variable spacing strategy is the

In the following the main features of the most important three strategies are presented.

Case 1.1 Control with information of lead vehicle

The first method is based on the reference velocity information. Consider the following control law:

where $x_l$ is the position of the lead vehicle in the platoon.

The spacing error dynamics is obtained using the following equation: ${\ddot{\epsilon }}_i={\ddot{x}}_i-{\ddot{x}}_{i-1}=u_i-u_{i-1}$ thus ${\ddot{\epsilon }}_i+k_v{\dot{\epsilon }}_i+k_p{\epsilon }_i=0$. This is the best achievable platoon performance. It is unsafe since it does not take the information of the preceding vehicle into consideration.

Case 1.2 Control with information of preceding vehicle

In this strategy control law is based only on the on-board sensor measurements. The control law is the following

and the spacing Error Dynamics is as follows: ${\ddot{\epsilon }}_i=u_i-u_{i-1}$ thus

for i=1 ${\ddot{\epsilon }}_i+k_v{\dot{\epsilon }}_i+k_p{\epsilon }_i=-{\ddot{x}}_l,i=1$, where ${\ddot{x}}_l$ is the acceleration of the leader. The frequency function:

thus in the frequency domain

Since $H\left(i\omega \right)>1$, the spacing amplifying along the platoon. Consequently, the string stability is not fulfilled.

Case 1.3 Control with information of lead and preceding vehicles

The control strategy is based on lead vehicle acceleration, velocity and position information:

$u=k_a{\ddot{x}}_{i-1}+k_l{\ddot{x}}_l-k_{d\epsilon }{\dot{\epsilon }}_i-k_{\epsilon }{\epsilon }_i-k_v\left(v_i-v_l\right)-k_p\left(x_i-x_l+{\displaystyle \sum }_{j=1}^i{L}_j\right)$

(11)

The string stability is guaranteed. The platoon control can be distributed into two control tasks: longitudinal control for speed tracking and lateral control for trajectory tracking.

In the simulation example four vehicles are traveling along an undulating road and the controller changes the reference velocities, which are tracked by the vehicles accurately, see Figure 3(a). The control signals are the longitudinal control forces, see Figure 3(d), which are realized by the throttle positions and the brake pressures.

Figure 146: Simulation example

Remark 1.2 In the travelling the safety operation must be guaranteed. In the following three possible solutions for collision avoidance are presented.

In another example the platoon is organized with dynamically different vehicles. The leader vehicle followed the target velocity adjusted by the onboard cruise control while saturation occurred in one of the the following vehicles. Vehicles in the platoon having worse mass/performance figures are not able to match the acceleration prescribed by their controller during uphill or heavy acceleration therefore they cannot keep the desired spacing.

Because of the splitting off the following vehicle prescribes bigger acceleration than necessary (due to the growing distance from the leader vehicle), hence the following vehicle can interfere with the saturating vehicle. Figure 6 shows that because if it is notable mass, the desired force related to the prescribed acceleration is too big. Hence saturation occurs at this vehicle, consequently it cannot match the prescribed acceleration and in this manner it splits off from the platoon. The significantly big spacing error with a negative sign shows the split off. Due to this, the fifth vehicle prescribes bigger acceleration than it is necessary, hence it runs into the fourth vehicle.

Figure 148: Simulation results with diverse vehicles