Distributed MPC of vehicle platoons with guaranteed consensus and string stability

Control of vehicle platoon can effectively reduce the traffic accidents caused by fatigue driving and misoperation, reduce air resistance by eliminating the inter-vehicle gap which will effectively reduce fuel consumption and exhaust emissions. A hierarchical control scheme for vehicle platoons is proposed in this paper. Considering safety, consistency, and passengers’ comfort, a synchronous distributed model predictive controller is designed as an upper-level controller, in which a constraint guaranteeing string stability is introduced into the involved local optimization problem so as to guarantee that the inter-vehicle gap error gradually attenuates as it propagates downstream. A terminal equality constraint is added to guarantee asymptotic consensus of vehicle platoons. By constructing the vehicle inverse longitudinal dynamics model, a lower-level control scheme with feedforward and feedback controllers is designed to adjust the throttle angle and brake pressure of vehicles. A PID is used as the feedback controller to eliminate the influence of unmodeled dynamics and uncertainties. Finally, the performance of longitudinal tracking with the proposed control scheme is validated by joint simulations with PreScan, CarSim, and Simulink.

Vehicle platoon modeling. Suppose that the leader vehicle is uncontrolled, cf., its position and velocity are given as q 0 (t), v 0 (t) . For following vehicle i, i ∈ 1, 2, · · · , M, describe its position and velocity as q i (t), v i (t) , and define the reference position and velocity are where q des is the desired inter-vehicle gap.
In this paper, the constant distance policy 37 is adopted with q 0 > 0. According to the current position and reference position of the vehicle, the state error is denoted as The leader vehicle's acceleration of a 0 (t) is a sort of "reference" for the vehicle i ≥ 1 since the value of a 0 is already known a priori by vehicle to vehicle communication.

Objective of vehicle platoon control.
Definition 1 27 (Predecessor-leader following string stability): Assume that at some time instant t, if the desired velocity of the leader vehicle changes, the state of (5) asymptotically converges to its equilibrium, and the intervehicle gap error of following vehicles satisfies accordingly and Note that for any vehicle i , if there exists a constant i ∈ (0, 1) , such that (7) and (8) are satisfied, then the vehicle platoon is string stable as shown in Fig. 2.
The objectives of the control of a platoon are summarized as follows: The inter-vehicle gap should maintain a desired safe distance, and the velocity of the vehicles should keep the same: Furthermore, to guarantee that the vehicle platoon maintains steady formation driving, the following constraints should be satisfied.
(1) Minimum safety distance: The distance between any front and rear vehicles should maintain a minimum safe distance to avoid collisions, where q i,ma and q i,mi are the maximum and minimum inter-vehicle gap error. (2) Consistency: The relative velocity deviation of vehicles has to be satisfied, where v i,mi and v i,ma are the minimum and maximum velocity errors.
(3) Passenger comfort: During acceleration or deceleration, the control input needs to be within an admissible region: where u i,mi and u i,ma are the allowed minimum and maximum control input.

Controller design
A hierarchical control framework is employed to achieve the vehicle platoon driving. The hierarchical control framework is illustrated in Fig. 3, where an upper-level DMPC is designed to achieve vehicle platoon control. A feedforward controller in the lower-level controller adopts feedback linearization technology to realize the adjustment of the driving and braking, and a PID controller to eliminate the influence of unmodeled dynamics and uncertainties. In Fig. 3q j is the position of adjacent vehicles; v j is the velocity of adjacent vehicles; p bdes,i is the desired brake pressure; α des,i is the desired throttle angle.
DMPC algorithm with guaranteed string stability. Denote the prediction horizon as N p , sampling time T s > 0 . The updated time for each vehicle is denoted as , define three types of control inputs sequences: • u p i p; t δ : the predicted control input sequence; • u * i p; t δ : the optimal control input sequence; • û i p; t δ : the assumed control input sequence; Accordingly, define three types of output sequences: • y p i p; t δ : the predicted output sequence; • y * i p; t δ : the optimal output sequence; • ŷ i p; t δ : the assumed output sequence, which is transmitted to neighboring vehicles through communication.
At time instant t δ , the maximum position deviation in the prediction horizon and the maximum position deviation within one sampling instant are defined as: At time instant t δ+1 , the assumed control input sequence is: For each vehicle i ≥ 1 , the sequence of control inputs is defined at time instant t δ First, a local optimization problem at time instant t 0 is designed.
Problem 0 where and Q i , F i , G i , R i and W i are weighting matrices. Note that �x i � 2 P i = x T i P i x i with P i ∈ R n×n and P i > 0 for a vector x i ∈ R n . Since the leader vehicle is uncontrolled, the term G 1 = 0 . The term �(y p i (p; t δ ) −ŷ i (p; t δ ))� 2 F i is the penalty of the error of the sequence of the i th vehicle and its assumed output sequence; the term �(y p i (p; t δ ) −ŷ i−1 (p; t δ ))� 2 G i is the penalty between the predicted and the assumed output sequence from the www.nature.com/scientificreports/ communication vehicle; the terms ε i , c i , ρ i ∈ (0, 1) and ̟ i (δ) are the parameters to be determined to ensure string stability of vehicle platoons.
The distributed model predictive control scheme to ensure string stability is as Algorithm 1.
Remark 3 A synchronous distributed model prediction controller is presented for the vehicle platoon, where the following vehicle solves its optimization problem synchronously. Since each vehicle does not know the predicted output sequence of other vehicles, the assumed output sequences are used to replace the actual predicted output sequences in the optimization problems.

Remark 4
A qualitative analysis of the performance of longitudinal tracking with the proposed control scheme is performed in this paper, whereas other important issues including communication delay and packet loss, parameter uncertainty, and measurement noise of sensors will be our future research direction. At the time instant t δ+1 , since u p i p; t δ+1 =û i p; t δ+1 is a feasible control sequence (but suboptimal) for Problem 1, the sum of objective function is bounded According to (16) and (19), one has In terms of (24) and (27), the following inequality is yielded where Due to the triangle inequality, www.nature.com/scientificreports/ Therefore, the asymptotic consensus of Algorithm 1 is guaranteed 38 .

String stability
Remark 5 If a vehicle platoon's communication network is exactly reliable, i.e. there is no communication delay and no data packet loss, string stability with the leader-follower (LF) communication topology is examined. Suppose there exists a velocity change for the leader vehicle, according to (6), if all vehicles are homogeneous, i.e., According to the definition of the assumed trajectory, and (31), the following inequality is yielded Combining (38), (39) and (40), the position deviation of adjacent vehicles at the time instant t 1 can be obtained www.nature.com/scientificreports/ In terms of q * i p; t 1 ∞,T s ≤ q * i p; t 1 ∞ , (41) can be rewritten as In terms of (38) and (39), for each vehicle i at the time instant t 2 , Combining constraints (18d), (20) and (41), Similarly to (42) The values of {ρ i , ̟ i , ̟ i−1 } that satisfy (50) are shown in Fig. 4. where i ∈ N [2, M] , then string stability of vehicle platoons with the LF communication topology is guaranteed.
Since the proof of Corollary 1 is similar to the proof of Theorem 2, it is omitted. where K s,i is the braking coefficient, and www.nature.com/scientificreports/ the term p b,i is the braking pressure, T bf ,i and T br,i are the braking torques of the front and rear wheels, respectively. According to (57), the relationship between braking pressure and acceleration is After obtaining the current desired throttle angle and braking pressure, a PID controller is used to correct the error, i.e., where K P , K I , and K D are parameters of the PID controller.
3) Throttle-brake switching logic: To improve fuel economy and passenger comfort, and to avoid the frequent switching of drive and brake, a threshold-based throttle switching strategy is implemented in this paper 10 . First, the vehicle velocity v i,(0) and maximum acceleration a i,(0) without throttle angle and brake pressure are calibrated, which is shown in Table 1.
A throttle-brake switching logic is designed according to Table 1, which is shown in Fig. 7 as well. Set the transition belt with the width of 2h, where h = 0.1 41 .
(i) When the desired acceleration a des,i is above upper switching line, i.e., a des,i ≥ a i(0) + h , the throttle control is triggered; (ii) When the desired acceleration a des,i is below lower switching line, i.e., a des,i ≤ a i,(0) − h , the brake control is launched; (iii) When the desired acceleration a des,i is inside the transition belt, i.e., a i,(0) − h ≤ a des,i ≤ a i,(0) + h , neither throttle control nor brake control is carried out.

Remark 6
The vehicle driving equation (52) and the brake equation (56) are consistent according to (1).

Simulation and result analysis
A vehicle platoon consists of five vehicles, i.e., one leader vehicle, and four following vehicles. A joint simulation platform with PreScan, CarSim, and Simulink is constructed shown in Fig. 8, where Prescan provides the road environment information, CarSim provides the vehicle dynamics, and Simulink is employed to design and implement of the controller. All vehicle parameters in the joint simulation are the same except for the vehicle mass m i , i.e., C d,    Table 2, and the parameter values of the controller are provided in Table 3. The sampling time is chosen as T s = 0.2s , and the prediction horizon is set as N p = 6 . In addition, choose the parameters of c i = ̟ i , In the joint simulation, a platoon with five vehicles is interconnected by the LF communication topology and PLF communication topology, respectively. A constant distance strategy is employed, i.e., q des = 15m . The leader vehicle in the platoon is running along a given straight road. Set the initial feasible state of the vehicles as [ q i v i ] = [0 0] , i = 1, 2, 3, 4 , respectively. When the leader vehicle accelerates, set the initial state of the leader vehicle as q 0 (t) = 100m , v 0 (t) = 15m/s and the desired velocity trajectory is given by   www.nature.com/scientificreports/ When the leader vehicle decelerates, set the initial state of the leader vehicle as q 0 (t) = 100m , v 0 (t) = 20m/s and the desired velocity trajectory is given by The proposed DMPC algorithm with string stability constraints is implemented in Matlab. The joint simulation performance with the LF communication topology is shown in Figs. 9, 10 Fig. 15 shows that the leader vehicle accelerates, and the following vehicles can track the leader vehicle and maintain consistency with the velocity of the leader vehicle. Figs. [16][17] show the inter-vehicle gap errors and velocity errors of vehicles platoons. It can be found that the inter-vehicle gap error is attenuated as it propagates downstream with the proposed DMPC algorithm. As a comparison, a DMPC without string stability constraints, and with the same controller parameters is implemented, and the results of the joint simulation are shown in Figs. [18][19]. It can be seen that the inter-vehicle gap error is amplified as it propagates downstream.       www.nature.com/scientificreports/ Remark 7 Note that the performance of the proposed control scheme should be assessed by high-fidelity tests 42 . However, the current experimental conditions of the Hardware-in-the-loop or small-scale vehicle are not yet available, and we will consider the experiment with Hardware-in-the-loop or small-scale vehicles in the future.   www.nature.com/scientificreports/

Conclusion
In this paper, a hierarchical control structure was designed for communication vehicles in the platoon. Firstly, a synchronous DMPC algorithm was proposed as the upper-level controller, in which each vehicle in the platoon solves its local optimization problem synchronously to obtain the control sequence, and then transmits its assumed output sequence to neighbouring vehicles. By introducing the assumed output sequence instead of the actual predicted output sequence, the computational efficiency is improved. By adding string stability constraints and terminal equality constraints in the local optimization problem, thereby both the asymptotic consensus and string stability of vehicle platoons are guaranteed. Additionally, the sufficient condition that guarantees asymptotic consensus and string stability of vehicle platoons were given, respectively. Then, a lower-level controller was designed, where the desired control input determined by the upper-level DMPC was first transformed into the desired throttle angle and brake pressure through an inverse longitudinal dynamics model of vehicles. A PID feedback controller was employed to eliminate the influence of unmodeled dynamics and uncertainties so as to achieve the desired control performance. Finally, performance was verified by a joint simulation platform based on PreScan, CarSim and Simulink.

Data availability
Due to space limitation, this paper only shows partial results. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.  www.nature.com/scientificreports/