Min-Max Model Predictive Vehicle Platooning With Communication Delay

Vehicle platooning gains its popularity in improving traffic capacity, safety and fuel saving. The key requirements of an effective platooning strategy include keeping a safe inter-vehicle space, ensuring string stability and satisfying vehicular constraints. To meet these requirements, this paper proposes a distributed min-max model predictive control (MPC). One technical contribution is that the proposed MPC can guarantee input-to-state predecessor-follower string stability, in the presence of vehicle-to-vehicle communication delays and realistic constraints. Another technical contribution is the development of a new concept of input-to-state stability margin for analyzing the platooning system that is nonlinear under MPC. The proposed MPC is applicable to both homogeneous and heterogeneous platoons because only the point-mass vehicle model is needed. The proposed MPC also has reduced communication burden because each vehicle in the platoon only transmits its current acceleration to the adjacent follower. The design efficacy is verified by simulating a platoon composed of five vehicles under different uncertainties and communication delays.

platoon has its own controller using the state information of other vehicles. Distributed MPC has been widely used due to its capability of real-time optimization and explicit constraint handling. This paper adopts the merits of distributed MPC and thus the literature reviewed below focus on MPC designs.
The key requirement of platooning control is ensuring string stability under safety and physical constraints. However, this cannot be met by using the traditional MPC with onlinecomputed nonlinear control policy [6]- [8]. The exceptions are [9]- [11]. The traditional MPC also lacks robustness against disturbance and uncertainty in the platoon. To enhance robustness, a pre-stabilizing MPC is proposed in [12]. It consists of an offline-computed linear control policy to stabilize the platoon, and an online-computed nonlinear control policy to refine the linear policy to satisfy constraints. However, string stability is guaranteed in [12] only when the constraints are inactive. Also, the constant constraints used in [12] are conservative. Following a similar idea as the pre-stabilizing MPC, an event-triggered MPC is developed in [13], which combines a linear quadratic regulator with a tube-MPC that is active only when big disturbances occur. However, using the tube method leads to a conservative design with less control potential to exploit. This paper will develop a less conservative pre-stabilizing MPC with guarantee of string stability.
In order to prove string stability, a performance metric is needed to quantify it. The metrics used in the literature include H ∞ -norm stability [9], [11], [12], p -norm stability [10], [13], input-to-output stability [14], and input-to-state stability [15]. Compared with other metrics, input-to-state string stability is more generic and convenient for both theoretical analysis [15] and implementation [16]. From the aspect of V2V communication network topology, string stability is categorized as leader-follower string stability and predecessor-follower string stability, where the latter is more stringent [11]. This paper will consider the input-to-state predecessor-follower (ISPF) string stability because it is more generic and is scalable to platoons in any size. It has not been discussed in existing MPC designs.
Another requirement of platooning control is allowing some margins of stability for the platoon. With these margins, the platoon can remain stable in the presence of perturbations. Hence, it is important to analyze the stability margin of the platoon using MPC. However, the existing stability margin concept is defined based on linear controllers [3], [17], which is inapplicable to MPC design with nonlinear control policy. This paper will develop a new concept of stability margin to analyze the MPC-based platoon.
A further requirement of platooning control is ensuring good platooning performance under V2V communication delay. Data transmission may be delayed due to channel congestion, contention, signal fading and external radio interference. The delay affects platooning performance and stability [18]. However, this issue has been rarely investigated in the existing MPC designs. This paper will design a MPC capable of handling the communication delay.
A fundamental requirement for MPC is recursive feasibility. This is essential for implementing the platooning control, because recursive feasibility guarantees that the MPC can generate an optimal control policy at every time step. In the literature, recursive feasibility is proved by imposing a terminal constraint set that is either a zero set [13] or a constant set [12]. However, the former may make the MPC infeasible while the latter is conservative. This paper will construct a larger but less conservative terminal constraint set to facilitate MPC implementation and improve platooning performance.
Motivated by the analysis, this paper has the following technical contributions: i) A distributed min-max MPC is proposed for vehicle platooning guaranteeing ISPF string stability: The proposed MPC is in the form of the pre-stabilizing MPC, with control policies determined using min-max optimization based on the zero-sum game theory [19]. The optimization problem uses 2 -norm cost function that directly quantifies the ISPF string stability metric. Hence, the platoon is guaranteed to be ISPF string stable by solving the optimization problem. This paper adopts the robust counterpart technique [20], instead of the tube method [13], to enhance platooning robustness. This allows the MPC to exploit the full control potential. ii) The proposed MPC employs more realistic constraints to reduce design conservativeness: Time-varying constraints are used for the inter-vehicle speed error, which are less conservative than the constant constraints in [12]. A time-varying terminal constraint set is adopted to ensure MPC recursive feasibility. The set depends on the real-time velocity of predecessor and is larger but less conservative than those in [12], [13]. Proving recursive feasibility under such a set is known to be challenging [21]. To address this, the terminal set is constructed using a non-dilating homothetic transformation algorithm. This makes the terminal set robustly positive invariant (RPI) [22]. By confining the platooning errors inside this RPI terminal set in every prediction horizon, recursive feasibility is guaranteed. iii) The new concept of input-to-state stability margin is developed to analyze the platoon: The proposed stability margin is a nonlinear function, rather than a constant as in [3], [17]. Hence, it is applicable for more general platoons with nonlinear vehicle dynamics and/or nonlinear control strategies. It is further shown that both the existing and the proposed stability margins depend on the platoon size and will decay to zero as the size becomes sufficiently large.

iv) The proposed MPC can handle communication delay:
The delay is assumed to be stochastic but its upper bound is known. The min-max MPC design considers the worst delay and ensures stability of the platoon and satisfaction of constraints. The platooning performance under different delays is also investigated through simulations. The proposed distributed min-max MPC also has the advantages discussed below. (i) The proposed design uses the point-mass model applied to any vehicle, making it applicable to a wide range of platoons. The platoon can be homogeneous if all vehicles have identical dynamics [6], [8], [9], [12], [15]. It can also be heterogeneous if the vehicles have different dynamics [10], [11], [23], disturbances and constraints [13], etc. (ii) The proposed design suffers from lower communication burden than the designs in [7], [10], [13], because only the current acceleration of predecessor is shared. (iii) The proposed design offers an opportunity for narrowing the inter-vehicle space to save more fuel with guarantee for safety and stability. As a comparison, the inter-vehicle space can also be narrowed by the MPC in [24] but without string stability guarantee.
The rest of this paper is organized as follows. Section II describes the vehicle platooning problem. Section III provides an overview of the proposed platooning control. Section IV presents the offline linear control design. Section V presents the min-max MPC design. Section VI analyzes the stability and stability margin of the platoon. Section VII describes the simulation study. Section VIII draws the conclusions.
Notation: R a×b is a a × b matrix whose elements are real numbers. Z [a,b] is the set of integer numbers within [a, b]. ⊗ is the Kronecker product. | · | is the absolute value. · is the 2-norm.
is the ∞-norm over the time interval [0, t]. I κ is a κ × κ identity matrix. 1 κ is a κ dimensional vector with all elements being 1. 0 is a matrix whose elements are all zero. The operator col(·, . . . , ·) stacks up its operands as a column vector. diag(·, . . . , ·) is a diagonal matrix with all elements on its main diagonal. s.t. is the abbreviation for subject to. P ( )0 means that the matrix P is positive definite (semidefinite).
for each fixed s, β(·, s) is a K function, and for each fixed r, β(r, ·) is decreasing with β(r, s) → 0 as s → ∞.

II. PROBLEM DESCRIPTION
As in most of the literature, this paper focuses on designing the longitudinal control for each follower to realize platooning. Each follower is assumed to already have a controller ensuring lateral stability and avoid lane departure. Considering this, only the longitudinal dynamics of the vehicles need to be given. Hence, a general platoon with M vehicles can be depicted in Fig. 1, where the dynamics of vehicle i is characterized by the following point-mass model: The leader is controlled to track a velocity profile under velocity and acceleration constraints by any existing method, e.g., a standard MPC tracking controller [25]. This paper focuses on designing controllers for all followers (i.e., vehicles i, i ∈ Z [1,M −1] ) in the platoon to realize four objectives: i) All the followers track the velocity of leader whilst keeping a desired inter-vehicle space d i s , i.e., ii) The platoon is ISPF string stable, i.e., there exists a KL function σ 1 (·, ·), a K ∞ function σ 2 (·), and positive constants c 1 and c 2 such that the platooning error trajectories e i (t) = col(e i p (t) − d i s , e i v (t)) satisfy the following metric [15]: for any e i (0) < c 1 , a i (t) ∞ < c 2 , and i ∈ Z [1,M −1] . iii) All the followers satisfy the given velocity and acceleration constraints, i.e., where v min and v max are the minimal and maximal velocities, respectively; a min and a max are the minimal and maximal accelerations, respectively. iv) All the platooning error trajectories e i (t) = col(e i p (t) − d i s , e i v (t)) satisfy the given performance requirements, i.e., where e p max > 0 and e p min ≤ 0 are the maximal and minimal allowable inter-vehicle space errors, respectively. . This kind of constant spacing policy is widely used in the literature. In general, the value of d i s will be set by the vehicle manufacturer. It is also possible for the manufacturer to embed a functional block into the vehicle control system. The block can then allow the passenger to select a preferred d i s to improve customer satisfaction. However, this is out of the scope of this paper.

A. Platooning Control Without Communication Delay
This paper aims to design a min-max MPC controller for each follower to realize the objectives (2)-(5). For objectives (2) and (3), it is convenient to design the controller using the relative dynamics of each pair of two successive vehicles. Hence, the platooning errors between vehicles i − 1 and i are defined . By using (1), the i-th platooning error system is (6) To facilitate the controller design, (6) is discretized with a sampling time T s and given as Since A, B, D are constant and independent of vehicle characteristics, all the M − 1 platooning error systems can be described in a unified form as where x k ∈ R n ,û k ∈ R m andd k ∈ R q with the dimensions n = 2, m = 1 and q = 1 are the state, control input and disturbance, respectively. By using (8), the controllers for all followers can be designed in the same procedure.
In the absence of V2V communication delay, the controller for vehicle i, i ∈ Z [1,M −1] , is illustrated in Fig. 2(a). The controller has the form ofû The linear controllerû 0 k is to realize the objectives (2) and (3) with the gains determined offline. The nonlinear controllerĉ * k,0 is further designed via online optimization to refineû 0 k to realize the objectives (4) and (5).
As shown in Fig. 2 In practice, there may be communication delay due to channel congestion, contention, signal fading and external radio interference. Hence, at time instance k, vehicle i receives the delayed acceleration a i−1 (kT s − t d ), where t d is the time delay. Implementing the controller (9) with this delayed acceleration will degrade the control performance and affect stability of the platoon [18]. Therefore, it is necessary to consider the delay in control design.

B. Platooning Control With Communication Delay
Assume that the communication delay t d is random in diverse driving environment but satisfying 0 ≤ t d ≤ τ T s with a known integer τ . To realize the objectives (2)-(5) under communication delay, a new controller structure is outlined in Fig. 2(b), where a buffer is used to store the control inputs and the platooning errors Δp i k and Δv i k . At time instance k, the controller applied to vehicle i isû k−τ =û 0 k−τ +ĉ * k−τ,0 . Note thatû k−τ is not simply the τ -step delay ofû k given in Fig. 2(a). It is designed using the available acceleration a i−1 k−τ , platooning errors Δp i k−τ and Δv i k−τ , and the previous control sequence col(û k−2τ , . . . ,û k−τ −1 ). When 0 ≤ k ≤ 2τ , the previous control inputs are set asû k−1 = · · · =û k−2τ =û 0 0 . By implementing the delayed controllerû k−τ , the platooning error system (8) becomes To simplify notation, define Hence,û k−τ can be rewritten as Further definex k = col(x k , u k−τ , u k−τ +1 , · · · u k−1 ) ∈ R n+τm , then the system (10) is augmented as with the system matrices Based on the augmented system (12), the constant gains K x and K d of the linear controller u 0 k are to be determined offline using the approach described in Section IV. At time instance k, by using (12) and the available information of K x , K d and x k , the nonlinear controller c * k,0 is to be designed using the min-max MPC formulation described in Section V. Combining the above designs, at time instance k, vehicle uses the controller The control objectives to be achieved in Sections IV and V are given below, which are the equivalent reformulation of (2)-(5) based on the augmented system (12): whereS and D are constraint sets defined as with the matricesḠ ∈ R r×m ,H ∈ R r×(n+τm) ,b ∈ R r , F ∈ R t×q and h ∈ R t given bȳ This paper focuses on platooning control under communication delay. For the special case when there is no communication delay, the proposed designs in Sections IV and V are directly applicable by setting τ = 0.

IV. OFFLINE LINEAR CONTROL DESIGN
This section describes the offline design of the linear controller u 0 k to realize the objectives (13) and (14). It is equivalent to designing u 0 k (with c * k,0 = 0) to stabilize the system (12) and satisfy the 2 gain property where ( x 0 ) is a non-negative scalar. The signal z k ∈ R n+τm is a performance metric to balance the stabilizing performance ofx k and the control effort u 0 k , defined as The linear controller is designed as where K d d k is the feedforward action for compensating the disturbance d k . The gains K x and K d are determined using Lemma 4.1 based on the zero-sum game theory in [19]. Lemma 4.1: The linear controller (20) ensures that the system (12) is stable and satisfies the 2 gain property (18), if and only if there is a non-negative scalar γ f and a symmetric positive semidefinite matrix P satisfying the conditions: Then the optimal control gains are unique and obtained as The proof of Lemma 4.1 can be found in [19]. According to this lemma, the controller (20) can stabilize the system (12) and thus realize the platooning objective (13). It is shown below that this controller can also realize the objective (14).
Proposition 4.1: If the 2 gain property (18) holds, so does the ISPF string stability metric defined in (14).
The conditions (21) and (22) are feasible if the pair (Ā,B) is stabilizable and the quadruple (Ā,B,C z ,D z ) has no invariant zeros at the unit circle. It can be verified that the augmented system (12) satisfies these requirements. However, solving the discrete-time Riccati equation (22) under the constraint (21) is difficult due the existing indefinite nonlinear term L Q −1 L. This can be addressed by using a recursive method [26], or by converting it into a continuous-time Riccati equation that is easy to solve [27]. To facilitate the implementation, this paper adopts the non-recursive method described in Lemma 4.1 of [27] to solve (21) and (22) for the gains K x and K d .

V. ONLINE NONLINEAR CONTROL DESIGN
This section describes the min-max MPC design based on the linear controller (20). The nonlinear controller c * k,0 is onlinecomputed to refine the linear controller to satisfy the constraints in (15). This ensures that the complete controller (11) can realize all the objectives (13)- (15).

A. Min-Max MPC Problem Formulation
When designing the linear controller u 0 k , the augmented system (12) is completely known. For the MPC design, the prediction of platooning error needs future accelerations of the predecessor over the prediction horizon. However, this future information is assumed to be unavailable in this paper. This requires the proposed MPC to minimize effects of the unknown accelerations of the predecessor. Therefore, at each time instance k, the nonlinear controller c * k,0 is determined online via solving a min-max optimization problem with a prediction horizon N , as formulated below.
Problem 5.1: The nonlinear controller c * k,0 is the first element of the optimal control sequence {c * k+i } N −1 i=0 solving the (zerosum game [19]) min-max optimization problem P N (x k ): x k+N ∈ X f (27) with the cost function where z k+i =C zxk+i +D z u k+i ,C z andD z are given in (19), and P is obtained from Lemma 4.1. The terminal constraint set X f defines the physical constraints that the statex k must satisfy at the end of the prediction horizon N . The method for constructing a suitable X f is described in Section V-B. If the problem P N (x k ) is feasible (which is proved in Section V-C), then applying the controller (11) to the augmented system (12) can stabilizex k , realize the objective (15) and satisfy the finite-horizon 2 gain property where β(x k ) and γ are non-negative scalars. Similar to Proposition 4.1, it can be shown that if (28) holds, so does the ISPF string stability metric (14). Hence, the proposed u k realizes the objectives (13)- (15), and so does the implemented u k−τ . If the predicted accelerations d k+i , i ∈ Z [0,N −1] , of the predecessor is known [10], the cost function J N can be defined as the standard form J N = x k+N Hence, the min-max optimization problem P N (x k ) becomes a minimization problem min J N and the constraint d k+i ∈ D is not needed. In such case, the optimization can be solved following the traditional MPC settings [25]. This paper addresses a more general case when the predicted accelerations are unavailable. It imposes difficulty in solving the problem P N (x k ) because the optimization must be performed considering every disturbance scenario. This challenge will be overcome in Section V-D.

1) Fixed Terminal Constraint Set:
To make the MPC-based vehicle platooning practically applicable, the online optimization problem P N (x k ) must be recursively feasible. This can be achieved by imposing a terminal constraint set on the state [25]. The terminal constraint set is also RPI for the control system and can be designed as the maximal output admissible disturbance invariant set [28] defined below.
Definition 5.1: Consider the system that satisfies the constraints y k ∈ Y and d k ∈ D. A set Ω ∈ R n is output admissible disturbance invariant (OADI) if ∀x 0 ∈ Ω, y k+1 ∈ Y holds for all d k ∈ D. The maximal OADI set Ω ∞ is an OADI set containing every closed OADI set of the system. Substituting the control law u 0 k into (12) gives a system in the form of (29) with The terminal constraint set X f can be constructed using Algorithm 6.1 in [28] and is given as A difficulty in the proposed MPC setting for vehicle platooning is that the constraint setS is time-varying in correspondence to the velocity of predecessor. This can be seen from the definitions in (5) and (16). The terminal constraint set X f is also time-varying as it is constructed usingS. To simplify the MPC implementation, X f is defined as a zero set [13] or the largest possible constant set [12]. However, the former might make the MPC infeasible while the latter is conservative. The use of a time-varying terminal set can reduce the conservativeness and improve MPC performance. Therefore, an algorithm for constructing a time-varying terminal set X k f at each time instance k is designed below.
2) Time-Varying Terminal Constraint Set: The time-varying set X k f can be constructed via running Algorithm 6.1 in [28] at

Algorithm 1: Construction of Time-Varying Terminal Set
Updateb andS using the definition in (16). Determine the current state constraint set where is the Minkowsky difference. 2: Determine the set of scalars β k = {β l k } using [1, g], g is the number of scalars, h i is the i-th row of H g , and v j is the j-th row of V . 3: Determine the homothetic transformation factor α k using 4: Obtain the current terminal constraint set X k f = α k X 0 f . Output:X k f , α k each time instance k. However, it is computationally intensive and may also introduce time delay. To reduce computational burden and facilitate implementation, this paper presents an algorithm to obtain X k f based on the homothetic transformation approach [21]. First, an initial non-zero terminal contraint set X 0 f is determined offline using Algorithm 6.1 in [28], under the initial constraint setS 0 . Second, at each time instance k, the set X 0 f is homogeneously re-scaled online to be a new set X k f which satisfies the current constraintS k .
An issue of the above homothetic transformation is that the obtained sequence {X k f } is not monotonically non-increasing. Hence, by using these terminal constraint sets, recursive feasibility of the MPC is not guaranteed. To address this issue, at each time instance k, the set X k f is defined as the non-dilating homothetic scaling of X k−1 f . This is realized by defining the homothetic transformation factor α k as in (31). It then constructs a monotonically non-increasing set sequence {X k f }. Note that the non-dilating homothetic scaling might lead to loss of a certain degree of controller design flexibility.
The complete procedure for constructing the set sequence {X k f } is summarized in Algorithm 1, where X 0 f needs to be converted from (30) to the following vertex form: Since X 0 f is a closed set containing the origin as its interior and α k ≤ 1, k ∈ Z [0,∞] , the following proposition holds.
Proposition 5.1: The terminal constraint set sequence {X k f } constructed in Algorithm 1 satisfies the inclusion Moreover, the set X k f is RPI for the augmented system (12) under the linear controller (20).

C. Recursive Feasibility of MPC
This section analyzes the recursive feasibility of the proposed min-max MPC. Recursive feasibility means that if the min-max optimization problem P N (x k ) (i.e., Problem 5.1) has a solution at time k ≥ 1, then it also has a solution at time k + 1. Therefore, ensuring feasibility of the initial optimization problem guarantees that an optimal nonlinear controller c * k,0 is always generated to realize the objectives (13)- (15).
To facilitate the analysis, a compact formulation of the problem P N (x k ) is given below. Define the stacked variables: By using the above stacked variables, the constraints (24) and (25) are rewritten compactly as where (1 N ⊗b, 1 s ), A = −HĀ, B = HB, and D = HD, then the constraints (24)- (27) are compactly formulated as a single constraint The cost function J N can also be compactly rewritten as where H x = ΨĀ, H u = ΨB, According to Propositions 1 and 2 in [29], the min-max optimization Problem 5.1 has an optimal solution only when it is convex-concave. Hence, Problem 5.1 needs to be solved with an extra constraint Since H d is a known diagonal block matrix, there always exists a scalar γ > 0 satisfying this constraint. Based on (34)-(36), the min-max optimization problem P N (x k ) is compactly represented as the problemP N (x k , γ): γ) is the set of admissible controller c k defined as Recursive feasibility of the min-max optimization problem P N (x k , γ) in (37) is proved below.
Proof: See Appendix B.
Since the problemP N (x k , γ) is merely a compact formulation of the original problem P N (x k ), Theorem 5.1 also confirms that the problem P N (x k ) is recursively feasible.

D. Computation of Nonlinear Controller
It is difficult to obtain the controller by directly solving the problemP N (x k , γ) in (37), because the constraints Π N (x k , γ) must be robustly satisfied for all disturbance scenarios. To address this difficulty, the problemP N (x k , γ) will be converted into a semidefinite programming problem that is solvable by using off-the-shelf optimization solvers.
The set Π N (x k , γ) in (38) can be equivalently expressed as where max d k ∈D (Dd k ) is the row-wise maximization. The disturbance setD can be compactly represented bŷ can be equivalently represented as its robust counterpart [20]: where the row vector w i ∈ R Nt represents the dual variables associated with the i-th row of the maximization in (39).
Define W = col(w 1 , . . . , w Nr+s ) ∈ R Nt×(Nr+s) . Applying (41) to (39) gives the purely affine constraints: By using (37) and (40), the maximization problem max d k ∈D J N (x k , γ, c k , d k ) can be dually represented by [29]: where y is a column vector y ∈ R Nt . There is no need to include γ 2 I − H d H d 0 as an additional constraint because it is always induced by the first constraint given above. Based on (42) and (43), the problemP N (x k , γ) is reformulated as the following semidefinite programming problem: The computation of the nonlinear controller c * k,0 is summarised below: (i) Construct the terminal constraint set X k f from Algorithm 1. This paper adopts the robust counterpart technique in (42) and (43) to handle the disturbance. The tube technique is used in [13] to tighten the state and input constraints used by MPC. Both techniques can make the MPC robust against the worst-case disturbance. However, the proposed MPC can exploit the full control potential, while the tube-MPC can only exploit part of the control potential.

VI. PLATOON STABILITY AND STABILITY MARGIN
This section provides analysis of the platoon stability and stability margin by implementing the proposed controller.

A. Platoon Stability
Stability of the closed-loop system (12) using the proposed controller (11) is proved below. Theorem 6.1: By using γ ≥ γ f and the terminal constraint set sequence {X k f } constructed in Algorithm 1, the obtained controller (11) ensures the augmented system (12) realize: i) Objective (13) which guarantees convergence of the platooning errors; ii) Objective (14) which guarantees ISPF string stability; iii) Objective (15) which guarantees constraints satisfaction for all k ≥ 1 if the initial state satisfiesx 0 ∈ X N (γ), where X N (γ) = {x k ∈ R n+τm | Π N (x k , γ) = ∅} is the set of state admitting a feasible nonlinear controller, and Π N (x k , γ) is the set of feasible nonlinear controller defined in (38). Proof: See Appendix C. According to Theorem 6.1, it is concluded that applying the delayed controller u k−τ to the original platooning error system (10) also realizes the objectives (13)- (15). Therefore, implementing the proposed controller ensures stability of the follower despite of the disturbance from its adjacent predecessor. Since in this paper a distributed control architecture is adopted where each follower is deployed with a separate controller, the entire platoon is stable despite of the disturbance from the leader. This confirms that the platoon is string stable.

B. Platoon Stability Margin
In the literature (e.g., [3], [17]), stability margin is defined as the absolute value of the real part of the least stable eigenvalue of the closed-loop platoon dynamics. This stability margin characterizes the decay rate of initial errors. It is defined based on using linear platooning control, where the controller of each follower needs the velocities and positions of its neighbouring vehicles (not just the adjacent predecessor as in this paper). Since the proposed controller (11) is nonlinear, the obtained closed-loop platooning system is nonlinear and the eigenvalues cannot be determined. Therefore, the existing stability margin concept is inapplicable in this paper and a new one is needed. Inspired by [30], this paper develops the concept of input-to-state stability margin as below.
Definition 6.1: Consider the system where x k ∈ R n and w k ∈ R q are the system state and disturbance, respectively. The disturbance w k satisfies whereγ(·) and ρ(·) are K ∞ functions. The scalar μ k ≥ 0 describes the fact that w k may not be zero when x k = 0. The functionγ(·) is the input-to-state stability margin of the system (45) if there is a KL function σ 1 (·, ·) and a K ∞ function σ 2 (·) such that (45) is regional input-to-state stable, i.e., where X s is the constraint set of x k . The notion "regional" emphasizes that the system is input-to-state stable whilst satisfying the constraint set x k ∈ X s . Combining together all the platooning error systems under the proposed controller, then the state vector is ). Based on Definition 6.1, stability margin of the proposed platoon is analyzed below. Theorem 6.2: There exist K ∞ functionsγ( X k ) and ρ( d 0 k ) such that the disturbanced k is bounded as where d 0 k is the control input (i.e., acceleration) of the leader. There also exists a KL function σ 1 ( X k , k) and a K ∞ function σ 2 ( d 0 k ) such that the state X k is regional input-to-state stable, i.e., for all (u k , x k ) ∈ S, X k satisfies Then,γ( X k ) is the input-to-state stability margin of the proposed platoon. Moreover, it is size-dependent and will decay to zero as the platoon size N becomes sufficiently large.
Proof: See Appendix D.
Comparisons of the proposed stability margin concept and the one in the literature [3], [17] are made below: i) The stability margin in the literature corresponds to asymptotic stability of linear systems and can quantify the decay rate of initial platooning errors. The proposed one corresponds to regional input-to-state stability of nonlinear systems, and qualitatively characterizes the size of the set in which the evolution of X k is ultimately bounded. The nonlinear nature makes it applicable for more general platoons with nonlinear vehicle dynamics and/or nonlinear control strategies. ii) Both the existing and proposed stability margins are sizedependent and will decay to zero as the platoon size N becomes sufficiently large. The stability margin may be made size-independent by using V2V communication topologies different from the one in this paper, e.g., having a large number of followers connected to the leader [3]. This is out of the scope of this work and left for future research.

A. Vehicle Platooning Without Uncertainty
A platooning system with five vehicles is simulated on MAT-LAB with the parameters listed in Table I for each follower is obtained online via solving the semidefinite programming problem (44) with prediction horizon N = 3 using the tools YALMIP [31] and MOSEK [32]. The initial terminal constraint sets used to solve the MPC for four followers are designed as the same set X 0 f . It is constructed by running Algorithm 6.1 in [28] using the tools YALMIP and MPT [33].
Hence, the initial inter-vehicle space and speed errors for each pair of two successive vehicles are different. The leader is controlled to track the speed reference depicted in Fig. 4 using a standard MPC tracking controller [25]. The use of this speed reference enables validating the proposed platooning system in both the nominal driving and emergency braking cases. The results of vehicle platooning are depicted in Figs. 5-7. It is seen from Fig. 5 that, by using the proposed controller, each follower can track the speed of its predecessor. At the end, all the followers can track the speed of the leader. Meanwhile, as seen from Fig. 6, the inter-vehicle space between each pair of two successive vehicles are controlled to be the desired value d s = 10 m. The results in Figs. 5-7 also show that, due to the  V2V communication delay, the overshoots of inter-vehicle space and speed error become bigger and bigger as the acceleration of leader propagates through the platoon. However, the proposed control guarantees that the inter-vehicle space between vehicles i − 1 and i, i ∈ Z [1,5] , are always within the specified interval The results in Figs. 5-7 also show that the proposed controller has the following advantages: (i) The predicted information is not required in the proposed design, which can reduce the communication burden. Stable platooning can also be achieved by existing MPC designs, where each follower uses the current and predicted accelerations of the predecessor [10], [13] or both the predecessor and leader [7]. (ii) The proposed platooning control can help in increasing the traffic throughput and fuel savings as aerodynamic effects become smaller by reducing inter-vehicle space. Since the proposed control always confines the inter-vehicle space error within the interval [e p min , e p max ] = [−6 m, 6 m], it is possible to reduce the desired inter-vehicle space d s = 10 m to be any value within (6 m, 10 m), whilst keeping the platoon safe. For example, d s can be reduced to be d s = 6.5 m. In such case, by applying the proposed control the inter-vehicle space will always be within (0.5 m, 12.5 m) and the platoon is kept safe.

B. Vehicle Platooning With Different Communication Delays
To evaluate the design efficacy under different V2V communication delays, simulations are carried out with t d = 0.05 s, 0.1 s, 0.15 s, which corresponds to τ = 1, 2, 3, respectively. The controllers in all the three cases are designed using the parameters in Table I. The simulations use the same leader speed reference and initial vehicle state as in Section VII-A. Two types of uncertainties are simulated: (i) Input uncertainty added to the control inputs of all vehicles, which is characterized by a normally distributed random signal w k with |w k | ≤ 0.1; (ii) Internal uncertainties coming from predecessors, including a 1 m increase of the inter-vehicle space between Leader & Follower 1 at 5 s, and a 1 m decrease of the inter-vehicle space between Followers 1&2 at 18 s.
The results of inter-vehicle space between each pair of two consecutive vehicles are depicted in Fig. 9. It is seen that the platoon is stable in the presence of input uncertainty w k . In the presence of the 1 m increase of the inter-vehicle space between Leader & Follower 1 at 5 s, there are deviations in the inter-vehicle space between Followers 1&2, Followers 2&3, and Followers 3&4. However, the deviations are all much smaller than the uncertainty 1 m. This means that the uncertainty is suppressed when propagating through the platoon. The similar phenomena can be observed from the results in the presence of the 1 m decrease of the inter-vehicle space between Followers 1&2 at 18 s. This demonstrates well that the proposed control is robust to unmodelled uncertainties acting on the platoon under the three different communication delays.
However, the deviations of inter-vehicle space become bigger as the delay t d increases. This means that robustness of the platoon is weakened as the delay increases. From Fig. 9, it is observed that under each t d the robustness is also weakened as the platoon size increases. This coincides with the theoretical result in Section VI that the stability margin is size-dependent and will decay to zero as the platoon size increases.

VIII. CONCLUSION
This paper develops a distributed min-max MPC for vehicle platooning with V2V communication delay. The established platoon has negligible platooning errors and is guaranteed to be ISPF string stable under leader velocity disturbances and unmodelled uncertainties. The MPC is rigorously proved to be recursively feasible under realistic time-varying constraints. The new concept of input-to-state stability margin is developed to analyze the platoon. The proposed design has lower communicational requirements because each vehicle only transmits its current acceleration to the adjacent follower. Moreover, the design is applicable to both homogeneous and heterogeneous platoons because it needs only the point-mass vehicle model. The above salient features make the proposed design effective and practically applicable to vehicle platooning.
The simulation results show that the deviations of platooning errors increase with the communication delay. Hence, the platoon may be unstable for a large enough delay. The proposed platoon is stable for delays within 0.2 s. For a general vehicle platoon, it is worth investigating the largest communication delay range within which a control strategy can realize the platooning objectives. It is also worth investigating the effects of communication data loss and designing platooning controls with size-independent input-to-state stability margins. (k+1)(k+2) = 1 to the second term of (51) yields The above inequality is equivalent to It is well-known that for any non-negative scalars a and b and a vector c ∈ R q , the relations √ a 2 + b 2 ≤ a + b and c ≤ √ q c ∞ hold. Hence, the inequality (53) induces with a KL function σ 1 ( x 0 , k) = ¯ ( x 0 ) (k+1)(k+2) and a K ∞ function σ 2 ( d k ∞ . Since x k and d k are bounded, the inequality (54) is in the form of (14). Therefore, if (18) is satisfied, so is the ISPF string stability metric (14).
By using the proof of Proposition 4.1, (64) implies that