Distributed Hierarchical MPC for Consensus and Stability of Vehicle Platoons with Mixed Communication Topologies

Abstract

This paper presents a distributed hierarchical model predictive control (MPC) framework designed to ensure dynamic consensus and stability in nonlinear vehicle platoons, addressing challenges posed by mixed communication topologies and hard constraints. By directed graph modeling of the mixed communication topologies, the dynamic consensus goal for the platoon is defined by the inter-vehicle distances between the host and its neighbors, whereas the stability criterion for an individual vehicle is expressed as a positive definite function of its position and velocity deviations. Then, a contractive constraint is elegantly designed to correlate these two objectives in a hierarchical model predictive control framework, where the lower layer optimizes the stability objective and the upper layer optimizes the dynamic consensus objective. The conditions ensuring stability and string stability for the vehicle platoon are shown to be only dependent on the deviations of the host vehicle, which achieves dynamic consensus and string stability simultaneously for nonlinear vehicle platoons. Several representative scenarios are used to validated the performance of the proposed strategy.

1. Introduction

With increasing traffic congestion and accidents, road capacity and safety face significant challenges. The coordinated control of vehicle platoons, such as those involving autonomous vehicles and formation control, has consequently garnered widespread attention in recent years. Fundamentally, this coordinated control aims to synchronize the velocities of platoon members while strictly adhering to minimum safety headways [1,2,3]. Such a framework exhibits substantial promise in optimizing traffic throughput, bolstering road safety, and mitigating the risk of vehicular accidents [4,5].

In practice, constraints on vehicle speed, acceleration, and formation control exist within platoon systems. Each vehicle maintains communication and exchanges state information to enable coordinated platoon control. Consequently, distributed model predictive control (DMPC) has recently been widely adopted to explicitly address these constraints and communication challenges, such as in consensus control and flocking problems for multi-agent systems [6,7,8,9,10]. To characterize the varying degrees of collaboration and rivalry within the network, a new weight function based on inter-agent distance is introduced. This design facilitates consensus tracking control in MASs featuring both cooperative and competitive interactions, even when communications are asynchronous [11]. Furthermore, various MPC controllers have been proposed to achieve cooperative formation behavior for vehicle platoons [12,13,14,15]. For instance, a distributed MPC strategy utilizing V2V communication topologies was proposed to achieve string stability [12]. An L2-norm control synthesis approach was designed for string stability [13]. A distributed nonlinear model predictive control (DNMPC) strategy was formulated to coordinate platoon following control with individual vehicle dynamics [14]. A distributed economic MPC (EMPC) algorithm demonstrated up to 6.84% savings compared to conventional tracking-focused approaches [15].

However, most existing platoon control approaches rely on rigid or single-mode communication topologies and maintain strict formations. In these setups, followers typically receive state information directly from the platoon leader. This reliance on the leader becomes problematic in large-scale platoons, as not all followers can reliably obtain the necessary information. Furthermore, real-world deployment faces additional complexities such as communication delays, vehicle heterogeneity, and dynamic interaction patterns. Recent studies have explored advanced solutions including category-guided graph representations for complex topologies [16], robust offset-free strategies for delay compensation [17], and personalized federated learning frameworks for heterogeneous systems [18,19]. While this paper primarily focuses on establishing theoretical stability guarantees under mixed switching topologies with fixed parameters, we dedicate a new section (Section 6) to comprehensively discussing how our proposed framework can be extended by integrating these state-of-the-art concepts, including adaptive parameter tuning via dynamic learning [20].

With advancements in coordinated vehicular control, research has increasingly focused on dynamic processes within vehicle platoons [21,22]. However, most existing work neglects the coordination between dynamic consensus and platoon stability. Typically, platoon stability is ensured by adopting coupled cost functions across vehicles and imposing additional assumptions on these functions. Few results address string stability, large-scale platoon scalability, or the coordination between dynamic consensus and stability in linear platoons.

Conversely, distributed consensus protocols have been developed for connected and automated vehicle platooning under various distance strategies [23,24,25,26,27]. A predictive spacing for vehicles modeled by a third-order system was proposed [23], and a robust strategy was designed for platoons [24]. A distributed direct adaptive control law is employed to control longitudinal motion of the vehicles [25]. To mitigate platoon failures, a finite-time fault estimation law and robust DSMC architecture were formulated for platoon operations [26,27].

Most efforts concentrate on fixed vehicle coupling modes, often resulting in string-unstable spacing and overlooking consensus-stability coordination during platooning. Furthermore, achieving stability frequently relies on linearizing nonlinear vehicle models via Taylor expansion or intermediate variables, which introduces limitations.

Maintaining safe inter-vehicle distances is essential. This necessitates optimizing dynamic consensus while ensuring platoon stability and considering dynamic consensus performance during travel. Consequently, platooning control involves dual objectives: stability (a local vehicle-level objective) and dynamic consensus (a global platoon-level objective). A cooperative relationship exists between these objectives. Some studies address both aspects, such as using weighted cost functions to reconcile them [28,29,30,31,32], where weights are determined through extensive offline experimentation. However, this often results in non-convex or non-positive definite dynamic consensus cost functions concerning vehicle consensus errors. These limitations complicate stability analysis and restrict the applicability of platooning control.

This paper designs a distributed hierarchical MPC framework for vehicle platoons subject to state or control constraints and mixed communication topologies. We define the stability constraint for the dynamic consensus optimization problem using the optimal value function of the tracking stability objective. Building on the standard MPC triplet framework [33], we establish asymptotic stability of the system about the equilibrium point while guaranteeing dynamic consensus. The recursive feasibility of the dynamic consensus objective and platoon string stability are derived using the receding horizon principle and the stability constraint. The main contributions are as follows: (1) The proposed dual-layer MPC strategy coordinates stability and dynamic consensus objectives under mixed communication topologies, guaranteeing both for the platoon system. (2) Decoupling dynamic consensus and stability objectives eliminates the need for weight selection in cost functions and influence balancing, highlighting the critical role of communication topology in platoon stability and consensus. Numerical simulations using a seven-vehicle scenario verify the strategy’s effectiveness.

The remainder of this paper is structured as follows: Section 2 outlines the problem formulation and provides necessary preliminaries. Section 3 proposes the distributed hierarchical MPC strategy for solving the consensus and tracking stability optimization problems. Section 4 details the recursive feasibility and string stability analysis. Section 5 provides simulation results. Notably, Section 6 offers an in-depth discussion on bridging our current work with recent advances in robust control, federated learning, and graph representation, specifically addressing the five key dimensions highlighted by recent literature. Finally, Section 7 concludes the paper.

Notation: We denote real numbers and non-negative integers by Z, and I, respectively. Given a vector x and a positive semi-definite matrix P, the P-weighted Euclidean norm is defined as ${‖x‖}_P$.

2. Problem Formulation and Preliminaries

Consider a system of n interconnected vehicles, where $p_1$ represents the lead car, and $p_n$ represents the tail car. Each vehicle $p_i$ can receive the information transmitted to it in the platoon. At initial time, the platoon tracks desired position and speed trajectory $\left(s_L,v_0\right)$, where the reference signal $v_0$ is shared with all the followers $p_i$, i = 1, ⋯, n. At other times, the lead car $p_1$ will track the desired trajectory, and other vehicles $p_1$ will track the immediate predecessor $p_{i-1}$, i = 2, ⋯, n, and maintain a desired safe spacing $d>0$ with it. It is assumed that the communication network performance of all vehicles is good, there is no network delay or packet loss phenomenon, the vehicles move on a flat road, and the model has no disturbance.

Let $s_i$ and $v_i$ represent the position and speed of vehicle $p_i$, i = 1, ⋯, n, respectively. The expected deviations of position and speed of vehicle p_i are defined as $e_{p,i}=s_L-s_i-\left(i-1\right)d$ and $e_{v,i}=v_i-v_0$. Then, the longitudinal deviation model of vehicle p_i is described by [28]

$$\left\{\begin{array}{l}{e}_{p,i}\left(k+1\right)=e_{p,i}\left(k\right)+e_{v,i}\left(k\right)\Delta t\\ e_{v,i}\left(k+1\right)=e_{v,i}\left(k\right)+{\displaystyle \frac{\Delta t}{{\mathrm{m}}_i}}\left({\displaystyle \frac{{\eta }_{T,i}}{r_i}}{u}_i\left(k\right)-{\phi }_i\left(e_{v,i}\left(k\right)\right)\right)\end{array}\right.$$

(1)

where ${\phi }_i\left(e_{v,i}\left(k\right)\right)=C_{A,i}{e}_{v,i}^2\left(k\right)+m_{i}g{\mu }_i$, $\Delta t$ is the sampling time interval, m_i is the vehicle mass, $C_{A,i}$ is the aerodynamic resistance coefficient, g is the gravitational acceleration, ${\mu }_i$ is the rolling resistance coefficient, $r_i$ is the tire radius, and ${\eta }_{T,i}$ is the mechanical efficiency of the car. The vehicle state variable is expressed as $x_i\left(k\right)={[e_{p,i}\left(k\right),e_{v,i}\left(k\right)]}^{\mathrm{T}}$, $u_i$ is the control input representing the desired driving/braking torque. The constraints on state and control variables of vehicle $p_i$ are given by

$$u_{\min ,i}\le u_i\le u_{\max ,i},x_{\min ,i}\le x_i\le x_{\max ,i}$$

(2)

where u_min,i < 0 and u_max,i > 0 are the upper and lower bounds, and x_min,i < 0 and x_max,i > 0 are the upper and lower bounds of the state variable. For simplicity, (1) is written as

$$x_i\left(k+1\right)=f_i\left(x_i\left(k\right),u_i\left(k\right)\right)$$

(3)

where f_i(x_i,u_i) ∈ R^2×1 is defined as

$$f_i=\left[\begin{array}{c}{e}_{p,i}+e_{v,i}\Delta t\\ e_{v,i}-\Delta t/{\mathrm{m}}_i\cdot \left(C_{A,i}{e}_{v,i}^2{+\mathrm{m}}_{i}g{\mu }_i\right)+u_i\cdot {\eta }_{T,i}\Delta t/{\mathrm{m}}_i{r}_i\end{array}\right]$$

(4)

Definition 1 

[34]. Given a step variation in the reference velocity v at instant k = 0, the platoon is considered stable if the state error of every vehicle converges asymptotically to zero.

Definition 2 

[34]. At the initial moment, the expected speed $v_0$ takes a step change, and the state vector of every vehicle converges asymptotically to zero. For i = 2,⋯, n, there exists a ρ_i ∈ (0, 1), and the closed-loop position error satisfies

$$\underset{k\ge 0}{\max }\left|e_{p,i}\left(k\right)\right|\le {\rho }_i \underset{k\ge 0}{\max }\left|e_{p,i-1}\left(k\right)\right|$$

(5)

From Definition 2, the Predecessor–Follower string stability characterizes the capability, which suppresses the amplification of position errors as they propagate through the platoon.

Remark 1. 

For the purpose of performance evaluation, a recommended separation error bound of ±0.5 m is established, reflecting a balance between tight formation and maintaining system stability and safety.

The mixed communication topology of the vehicle platoon is characterized as a directed graph G = {H,C,A}. Here, H = {1, …, n} denotes the set of vehicles, C signifies the set of communication links, and A = [a_ij] ∈ $R^{n\times n}$ is the adjacency matrix capturing the information exchange patterns among the platoon members. For any (i,j) ∈ C, j ≠ i, and a_ij = 1, where (i,j) ∈ C means the directed edge from i to node j, i.e., vehicle j receives the information transmitted by vehicle i. Conversely, there is no communication between vehicle i and j. Let Θ_i and Ω_i denote the out-neighbor and in-neighbor sets of node i, respectively. Specifically, Θ_i= {j ∈ H∖{i}∣a_ij = 1} represents the set of nodes to which ii transmits information. Conversely, the dual set Ω_i= {j ∈ H∖{i}∣a_ji = 1} comprises the nodes from which i receives data. A directed graph is said to contain a directed spanning tree if there exists at least one root node ii such that a directed path connects ii to every other node j ∈ H.

The mixed communication topology means that the communication between vehicles is arbitrary, i.e., it contains unidirectional communication and an undirected communication mode.

This paper presents a distributed hierarchical control framework tailored for vehicle platoons operating under mixed communication topologies, while explicitly accounting for state and control constraints. This strategy efficiently coordinates the trade-off between dynamic consensus performance and platoon stability while guaranteeing both stability and consensus performance, as well as string stability. Moreover, the controller satisfies all system constraints.

3. Distributed Control Strategy

3.1. Control Strategy

State measurements are acquired at discrete time instant $k$, with a prediction horizon of length N ∈ I_≥0. The optimal control sequence, denoted as $u_i^o\left(k\right)$, is obtained by solving the optimization problem over the prediction horizon N at time step k. The sequence $u_i^a\left(k\right)$ represents the assumed control trajectory for time k, which was pre-computed at step k − 1 by vehicle p_i, to be used by vehicle p_j, j ∈ Θ_i at time k. For simplicity, ${\left|x_{i,1}\left(t\left|k\right.\right)\right|}_{\infty ,l}$ = $\underset{t\in \left\{0,\cdots ,N\right\}}{\max }‖x_{i,1}\left(t\left|k\right.\right)‖$, l ∈ {0, ⋯, N} and x_i,1 = e_p,i are the first element of the state variable x_i, i.e., the deviations of position.

Consider a feasible predictive control sequence u_i(k) = {u_i(0|k),u_i(1|k), ⋯, u_i(N − 1|k)} of system (3) and its predictive state sequence is x_i(k) = {x_i(1|k), x_i(2|k), ⋯, x_i(N|k)}. In order to minimize the dynamic consensus performance in prediction horizon, it is defined as follows:

$$J_{i,c}\left(x_i\left(k\right),u_i\left(k\right)\right)={\displaystyle \sum _{t=0}^{N-1}{L}_{i,c}\left(x_i\left(t|k\right),u_i\left(t|k\right)\right)}$$

(6)

and

$$L_{i,c}\left(x_i\left(t\left|k\right.\right),u_i\left(t\left|k\right.\right)\right)={‖x_i\left(t\left|k\right.\right)-x_i^a\left(t\left|k\right.\right)‖}_{F_i}+{\displaystyle \sum _{j\in {\Omega }_i}{‖x_i\left(t\left|k\right.\right)-x_j^a\left(t\left|k\right.\right)‖}_{Gj}}$$

(7)

where, $x_i^a\left(j\left|k\right.\right)$ denotes the predicted state trajectory of vehicle p_j, computed at time k − 1. Based on this information, the finite-horizon optimal dynamic consensus objective for each vehicle p_i at the current step k is defined as

$$u_i^o\left(k\right)=\arg \underset{u_i\left(k\right)}{\min }{J}_{i,c}\left(x_i\left(k\right),u_i\left(k\right)\right)$$

(8)

$$s.t. x_i\left(t|k\right)=f_i\left(x_i\left(t|k\right),u_i\left(t|k\right)\right)$$

(9)

$$\left(x_i\left(t|k\right),u_i\left(t|k\right)\right)\in X_i\times U_i,t=1,\dots ,N$$

(10)

$$x_i\left(0|k\right)=x_i\left(k\right),x_i\left(N|k\right)\in X_{i,T},i=1,\dots ,n$$

(11)

$${\left|x_{i,1}\left(j|k\right)\right|}_{\infty ,l}\le {\rho }_i\underset{r\in \left\{0,\dots ,k\right\}}{\max }{\left|x_{i-1,1}\left(j|r\right)\right|}_{\infty ,l},l\in \left\{0,1,\dots ,N\right\}$$

(12)

$$J_i\left(x_i\left(k\right),u_i\left(k\right)\right)\le {\varphi }_i\left(x_i\left(k\right),{\lambda }_i\right)$$

(13)

where $u_i^o\left(k\right)$ represents the optimal solution of Problem 1. x_i(0|k) = x_i(k) is the initial condition. x_i(N|k) ∈ X_i,T is the terminal constraint. The constraint (12) is a sufficient condition to ensure the establishment of Equation (5), which can refer to [34]. The terminal constraint set is X_i,T ∈ X_i, and function ${\varphi }_i:X_i\times {\mathrm{Z}}_i\to {\mathrm{Z}}_i$. The contractive stability constraint (13) is imposed to ensure the stability.

It is important to note that the contractive constraint (13) serves primarily as a recursive feasibility guarantee for the distributed optimization problem, especially during dynamic speed transitions. Unlike hard safety constraints that directly bound the inter-vehicle distance, Equation (13) ensures that a feasible solution satisfying both stability and consistency requirements exists at every time step. This solvability is crucial for maintaining continuous control authority, which indirectly supports safety and allows the objective function to optimize for tight, efficient gaps without the risk of infeasibility-induced failures.

To evaluate the tracking performance of the vehicle platoon, the cost function characterizing individual vehicle stability is formulated as

$$J_i\left(x_i\left(k\right),u_i\left(k\right)\right)=E_i\left(x_i\left(N|k\right)\right)+{\displaystyle \sum _{t=0}^{N-1}{L}_i\left(x_i\left(t|k\right),u_i\left(t|k\right)\right)}$$

(14)

where

$$J_i\left(x_i\left(k\right),u_i\left(k\right)\right)=x_i^{\mathrm{T}}\left(N\left|k\right.\right)P_i{x}_i\left(N\left|k\right.\right)+{\displaystyle \sum _{t=0}^{N-1}{x}_i^{\mathrm{T}}\left(t\left|k\right.\right)Q_i}{x}_i\left(t\left|k\right.\right)+u_i^{\mathrm{T}}\left(t\left|k\right.\right)R_i{u}_i\left(t\left|k\right.\right)$$

(15)

$Q_i=Q_i^{\mathrm{T}}>0,R_i=R_i^{\mathrm{T}}>0,L_i:X_i\times U_i\to Z_i,E_i:X_i\in Z_i$ is continuous and bounded, and X_i and U_i are convex. Then, the optimization problem can be solved as follows:

Problem 2:

$$u_i^s=\arg \underset{u_i\left(k\right)}{\min }\left\{J_i\left(x_i\left(k\right),u_i\left(k\right)\right)|\left(9\right)-\left(12\right)\right\}$$

(16)

where the variable $u_i^s\left(k\right)$ denotes the optimal solution obtained from Problem 2 at the current time step k.

Now we define the function ϕ_i as

$${\varphi }_i\left(x_i\left(k\right),{\lambda }_i\right)=J_i^s\left(x_i\left(k\right)\right)+{\lambda }_i\left[J_i^o\left(x_i\left(k-1\right)\right)-J_i^s\left(x_i\left(k\right)\right)\right]$$

(17)

where the coefficient λ_i ∈ [0,1), and

$$\begin{array}{l}{J}_i^s\left(x_i\left(k\right)\right):=J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)\\ J_i^o\left(x_i\left(k\right)\right):=J_i\left(x_i\left(k\right),u_i^o\left(k\right)\right)\end{array}$$

(18)

Remark 2. 

For Problem 1, the optimal solution $u_i^o\left(k\right)$ represents a feasible trajectory for Problem 2 at time step k, satisfying all imposed constraints; $0\le J_i^s\left(x_i\left(k\right)\right)\le J_i^o\left(x_i\left(k\right)\right)$. Similarly, the sequence $u_i^s\left(k\right)$ satisfies constraint Equation (13); it is the feasible solution for Problem 1 and generally not the optimal solution at time k, then $0\le J_{i,c}^o\left(x_i\left(k\right)\right)\le J_{i,c}^s\left(x_i\left(k\right)\right)$.

Remark 3. 

This paper applies the hierarchical control strategy to coordinate the trade-off relationship between the dynamic consensus and stability of the platoon. Then, some efforts adopt the weighted function method to handle it, which is denoted as

$$\begin{array}{c}{J}_{i,w}\left(x_i\left(k\right),u_i\left(k\right)\right)=x_i^{\mathrm{T}}\left(N\left|k\right.\right)P_i{x}_i\left(N\left|k\right.\right)+{\displaystyle \sum _{t=0}^{N-1}{x}_i^{\mathrm{T}}\left(t\left|k\right.\right)Q_i}{x}_i\left(t\left|k\right.\right)+u_i^{\mathrm{T}}\left(t\left|k\right.\right)R_i{u}_i\left(t\left|k\right.\right)\\ +\tau \left({‖x_i\left(t\left|k\right.\right)-x_i^a\left(t\left|k\right.\right)‖}_{F_i}\right)+{\displaystyle \sum _{j\in V/\left\{j\right\}}{‖x_i\left(t\left|k\right.\right)-x_j^a\left(t\left|k\right.\right)‖}_{Gj}}\end{array}$$

(19)

where τ is the weighted coefficient, which is set by a large number of off-line experiments. Then, the weighted function method optimization problem is established with constraints Equations (9)–(12).

Provided that Problem 1 admits a feasible solution at time k, the receding horizon strategy dictates that the control input be implemented as

$$u_i\left(k\right)=u_i^{cmpc}\left(k\right):=u_i^o\left(0|k\right)$$

(20)

where $u_i^o\left(0|k\right)$ is the first element of $u_i^o\left(k\right)$, corresponding to the closed-loop system.

$$x_i\left(k+1\right)=f_i\left(x_i\left(k\right),u_i^{cmpc}\right)$$

(21)

Figure 1 illustrates the overall control architecture proposed in this study.

3.2. Algorithm

1.

Initialization (k = 0):

(1)

At k = 0, vehicle $p_i$, i = 1, ⋯, n, receives reference speed $v_0$, let ϕ_i(x_i(0),λ_i) be a sufficiently large value. The lead car $p_1$ solves Problem 1 without considering the constraint (12), transmits the optimal state to all the followers, then applies the optimal control input sequence to itself.

(2)

Each vehicle $p_i$, i = 2, ⋯, n, receives the state $x_{1,1}^o\left(t\left|0\right|\right),t\in \left[0,N\right]$ from the lead car. Problem 1 is solved by replacing (12) with

$$\left(1-{\xi }_i\right)\cdot {\gamma }_i\cdot {\left|x_{1,1}^o\left(t|0\right)\right|}_{\infty ,l}\le {\left|x_{i,1}\left(t|0\right)\right|}_{\infty ,l}\le \left(1+{\xi }_i\right)\cdot {\gamma }_i{\left|x_{1,1}^o\left(t|0\right)\right|}_{\infty ,l},$$

(22)

vehicle pi will transmit the assumed sequence $x_i^a\left(j\left|1\right.\right)$ to the vehicle $p_j,j\in {\mathsf{\Theta }}_i$, and the optimal control sequence $u_i^o\left(t\left|0\right.\right)$ will be applied to itself, where the parameter (ζ_i,γ_i) ∈ (0, 1).

2.

Iteration (k = 1, ⋯):

(1)

Vehicle $p_i$, i = 2, ⋯, n, receives the assumed state information from vehicle $p_j,j\in {\mathsf{\Theta }}_i$.

(2)

Solve Problem 1, where an additional constraint Equation (23) is added for the lead car:

$${\left|x_{1,1}\left(t|k\right)-x_{1,1}^a\left(t|k\right)\right|}_{\infty }\le {\epsilon }_{1,k}{\left|x_{1,1}\left(t|k\right)\right|}_{\infty ,l}$$

(23)

for i = 2, ⋯, n − 1

$${\left|x_{i,1}\left(t|k\right)-x_{i,1}^a\left(t|k\right)\right|}_{\infty }\le {\epsilon }_{i,k}\min \left\{\begin{array}{l}{\left|x_{i,1}\left(t|k\right)\right|}_{\infty ,l}\\ {\left|x_{i-1,1}^a\left(t|k\right)\right|}_{\infty ,l}\end{array}\right\}$$

(24)

for i = n, the right-hand side of the inequality (24) is replaced by ${\epsilon }_{n,k}{\left|x_{n-1,1}^a\left(j\left|k\right|\right)\right|}_{\infty ,l}$.

(3)

Vehicle $p_i$, i = 1, ⋯, n, receives the state information from vehicle $p_j,j\in {\mathsf{\Theta }}_i$, and solves Problem 1 to obtain the optimal control input, then the assumed state sequence of pi is transmitted to the vehicle $p_j,j\in {\mathsf{\Theta }}_i$, and the optimal control input $u_i^o\left(t\left|k\right.\right)$ is applied to itself. Let k = k + 1 go back to step 1).

Remark 4. 

The algorithm is designed to solve the hierarchical optimization problem. Solving Problem 2 yields the optimal control law $u_i^s\left(k\right)$, which can be substituted into J_i(x_i(k),u_i(k)) to update ϕ_i(x_i(k),λ_i) in addition to initialization, then iterating constraint Equation (13) in Problem 1. The values of parameters ζ_i,γ_i,ε_i,k in (22)–(24) are followed from [34], and (22) is defined at the initialization. The position error of vehicle p_i, i = 2, ⋯, n, satisfies $\left|x_{i,1}^o\left(j\left|0\right.\right)\right|\le {\alpha }_i\left|x_{i-1,1}^o\left(j\left|0\right.\right)\right|$, where the parameter α₂ = (1 + ξ₂)·γ₂, α_i((1 + ξ_i)/(1 − ξ_i−1))·(γ_i/γ_i−1), i = 3, ⋯, n. The parameter ε_i,k = ε^k ensures that (23) and (24) satisfy the Equation (12) and establish the Predecessor–Follower string stability of the vehicle platoon.

The assumed control input trajectory is similar to the receding horizon control strategy [20]. At time k, the assumed control trajectory for each vehicle p_i is generated by $u_i^a\left(j\left|k\right.\right)=u_i^o\left(j\left|k-1\right.\right)$, j ∈ [1,N), and $u_i^a\left(j\left|k\right.\right)=K_i{x}_i^o\left(N\left|k-1\right.\right)$, where K_i is a feedback matrix, which can be obtained by solving the LQR problem. Then, the assumed state trajectory can be denoted as

$$x_i^a\left(k\right)=\left\{x_i^o\left(2\left|k-1\right.\right),\dots ,x_i^o\left(N\left|k-1\right.\right),x_i^a\left(N\left|k\right.\right)\right\}$$

(25)

where $x_i^a\left(N\left|k\right.\right)=f_i\left(x_i^o\left(N\left|k-1\right.\right),K_i{x}_i^o\left(N\left|k-1\right.\right)\right)$.

4. Stability and String Stability Analysis

Assumption 1. 

For a given region X_i,T, we assume the existence of a local feedback control law u_i = κ_i(x_i) satisfying κ_i(x_i)$\subset$U_i for all x_i ∈ X_i,T. Furthermore, this control law ensures that the terminal cost E_i satisfies the descent condition E_i(f_i(x_i,κ_i(x_i))− E_i(x_i) ≤ −L_i (x_i, κ_i(x_i)), and L_i(x_i,u_i) is a positive definite.

Lemma 1 

[34]. Provided that the subsequent parametric constraint is satisfied for arbitrary p_i, i = 2, ⋯, n

$${\alpha }_i+{\alpha }_i{\displaystyle \sum _{h=k-1}^k{\epsilon }_{i-1,h}}+{\epsilon }_{i,k-1}\left(1+{\epsilon }_{i-1,k-1}\right)<1$$

(26)

the closed-loop position error satisfies Equation (12), where (α_i,ε_i−1,k,ε_i,k) ∈ (0, 1).

Definition 3. 

An initial state x_i(0) is termed feasible (i.e., x_i(0) ∈ X_i) if a valid prediction sequence x_i(k) exists for the closed-loop system at any time step. The collection of all such states constitutes the feasible initial set, denoted by X_i,N, which contains all feasible initial states, and X_i,T $\subset$ X_i,N $\subset$ X_i.

4.1. Recursive Feasibility Analysis

Theorem 1. 

Under Assumption 1, the inclusion of constraint (5) in Problem 1 guarantees recursive feasibility within X_i,N. Furthermore, X_i,N serves as a robust invariant set for the closed-loop dynamics described by (21).

Proof of Theorem 1. 

Let $u_i^o\left(k-1\right)$ denote the optimal solution to Problem 1 at instant k − 1. The control sequence $u_i^f\left(k\right)$ for time k is then synthesized by shifting this previous optimal trajectory.

$$u_i^f\left(k\right)=\left\{u_i^o\left(1|k-1\right),u_i^o\left(2|k-1\right),\dots ,u_i^o\left(N-1|k-1\right),{\kappa }_i\left(x_i^o\left(N|k-1\right)\right)\right\}$$

(27)

where state $x_i^o\left(N\left|k-1\right.\right)$ is the terminal predicted state corresponding to $u_i^o\left(k-1\right)$, and satisfies $x_i^o\left(N\left|k-1\right.\right)$ ∈ X_i,T. Substituting $u_i^f\left(k\right)$ into system (3) to obtain the state sequence,

$$x_i^f\left(k\right)=\left\{x_i^o\left(2\left|k-1\right.\right),\dots ,x_i^o\left(N\left|k-1\right.\right),x_i^f\left(N\left|k\right.\right)\right\}$$

(28)

where $x_i^f\left(N\left|k\right.\right)=f_i\left(x_i^o\left(N\left|k-1\right.\right),{\kappa }_i{x}_i^o\left(N\left|k-1\right.\right)\right)$. Then, (E_i,X_i,T,κ_i) satisfies Assumption 1, $x_i^o\left(N\left|k-1\right.\right)$ ∈ X_i,T and X_i,T are invariant sets of x_i(k + 1) = f_i(x_i(k),κ_i(x_i(k))). Therefore, $x_i^f\left(N\left|k\right.\right)$ ∈ X_i,T and ${\kappa }_i{x}_i^o\left(N\left|k-1\right.\right)$ ∈ U_i. Based on the triplet of MPC, it can be shown that (28) satisfies the constraints Equations (9)–(11) of Problem 1.

According to (28) and (26), the state sequence $x_i^a\left(k\right)$ is the same as $x_i^f\left(k\right)$, which are constructed by the optimal solution of Problem 1 at time k − 1. From the triangle inequality, (23) for the lead car and (24) for all predecessor cars, for any $p_{i-1}$, i = 2, ⋯, n,

$$\left|x_{i-1,1}^f\left(j|k\right)\right|\le {\epsilon }_{i-1,k}{\left|x_{i-1,1}^o\left(j|k\right)\right|}_{\infty ,l}+\left|x_{i-1,1}^o\left(j|k\right)\right|$$

(29)

The transformation of Equation (30) gives

$${\left|x_{i-1,1}^f\left(j|k\right)\right|}_{\infty ,l}\le \left(1+{\epsilon }_{i-1,k}\right){\left|x_{i-1,1}^o\left(j|k\right)\right|}_{\infty ,l}$$

(30)

From the triangle inequality and (24), we derive that

$$\begin{array}{c}\left|x_{i,1}^o\left(j|k-1\right)\right|\le \left|x_{i,1}^o\left(j|k-1\right)-x_{i,1}^f\left(j|k-1\right)\right|+x_{i,1}^f\left(j|k-1\right)\\ \le {\epsilon }_{i,k-1}{\left|x_{i-1,1}^f\left(j|k-1\right)\right|}_{\infty ,l}+x_{i,1}^f\left(j|k-1\right)\end{array}$$

(31)

From Lemma 3 in [34], and initial time inequality (22), we have

$$\left|x_{i,1}^f\left(j|k-1\right)\right|\le {\alpha }_i\left|x_{i-1,1}^f\left(j|k-1\right)\right|$$

(32)

(29), (30) and (32) are substituted into (31), which gives

$$\begin{array}{c}\left|x_{i,1}^o\left(j|k-1\right)\right|\le \left({\epsilon }_{i,k-1}\left(1+{\epsilon }_{i-1,k-1}\right)+{\alpha }_i{\epsilon }_{i-1,k-1}\right){\left|x_{i-1,1}^o\left(j|k-1\right)\right|}_{\infty ,l}\\ +{\alpha }_i\left|x_{i-1,1}^o\left(j|k-1\right)\right|\end{array}$$

(33)

From (23), (24) and (33), for any p_i, i = 2,3, ⋯, n, we have

$$\left|x_{i,1}^f\left(j|k\right)\right|\le \left({\epsilon }_{i,k-1}\left(1+{\epsilon }_{i-1,k-1}\right)+{\alpha }_i{\epsilon }_{i-1,k-1}\right)\times {\left|x_{i-1,1}^o\left(j|k-1\right)\right|}_{\infty ,l}+{\alpha }_i\left|x_{i-1,1}^f\left(j|k\right)\right|$$

(34)

Substituting (30) into (34) can obtain that

$$\begin{array}{c}\left|x_{i,1}^f\left(j|k\right)\right|\le \left({\epsilon }_{i,k-1}\left(1+{\epsilon }_{i-1,k-1}\right)+{\alpha }_i{\epsilon }_{i-1,k-1}\right){\left|x_{i-1,1}^o\left(j|k-1\right)\right|}_{\infty ,l}\\ +{\alpha }_i{\epsilon }_{i-1,k}{\left|x_{i-1,1}^o\left(j|k\right)\right|}_{\infty ,l}+{\alpha }_i\left|x_{i-1,1}^o\left(j|k\right)\right|\end{array}$$

(35)

Therefore, (36) is bounded and satisfies

$${\left|x_{i,1}^f\left(j|k\right)\right|}_{\infty ,l}\le \underset{g=\left\{k-1,k\right\}}{\max }{\left|x_{i-1,1}^o\left(j|g\right)\right|}_{\infty ,l}\times \left({\alpha }_i+{\alpha }_i\left({\epsilon }_{i-1,k-1}+{\epsilon }_{i-1,k}\right)+{\epsilon }_{i,k-1}\left(1+{\epsilon }_{i-1,k-1}\right)\right)$$

(36)

The coefficient on the right-hand side of Equation (37) meets the requirements of Lemma 1. Thus, the constructed state sequence $x_i^f\left(k\right)$ satisfies (12). Likewise, a feasible solution is similarly derived by the above proof for Problem 2. Furthermore, letting $u_i^s\left(k\right)$ denote the optimal optimizer of Problem 2 at instant k, we have

$$J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)\le J_i\left(x_i\left(k\right),u_i^f\left(k\right)\right)$$

(37)

Considering (28), (37) and combining (18), we derive that

$$\begin{array}{l}{J}_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)-J_i\left(x_i\left(k-1\right),u_i^o\left(k-1\right)\right)\le \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} J_i\left(x_i\left(k\right),u_i^f\left(k\right)\right)-J_i\left(x_i^o\left(k-1\right),u_i^o\left(k-1\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =E_i\left(x_i^f\left(N|k\right)\right)+{\displaystyle \sum _{t=0}^{N-1}{L}_i\left(x_i^f\left(k\right),u_i^f\left(k\right)\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -E_i\left(x_i^o\left(N|k-1\right)\right)-{\displaystyle \sum _{t=0}^{N-1}{L}_i\left(x_i^o\left(k-1\right),u_i^o\left(k-1\right)\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =-E_i\left(x_i^o\left(N|k-1\right)\right)+L_i\left(x_i^o\left(N|k-1\right),{\kappa }_i\left(x_i^o\left(N|k-1\right)\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +E_i\left(f_i\left(x_i^o\left(N|k-1\right),{\kappa }_i\left(x_i^o\left(N|k-1\right)\right)\right)\right)-L_i\left(x_i^o\left(0|k-1\right),u_i^o\left(0|k-1\right)\right)\end{array}$$

(38)

From (39) with Assumption 1, we have

$$J_i\left(x_i\left(k-1\right),u_i^o\left(k-1\right)\right)-J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)\ge L_i\left(x_i^o\left(0|k-1\right),u_i^o\left(0|k-1\right)\right)\ge 0$$

(39)

Combining (17) with (18), and substituting inequality (39) into Equation (17), the value function $J_i^o\left(x_i\left(k\right)\right)\ge 0$ yields ϕ_i(x_i(k),λ_i) ≥ 0. Furthermore, substituting $u_i^s\left(k\right)$ into the left-hand J_i(x_i(k),u_i(k)) of Equation (13),we derive that

$$\begin{array}{cc}\hfill J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)& \le J_i^s\left(x_i\left(k\right)\right)+\lambda \left[J_i^o\left(x_i\left(k-1\right)\right)-J_i^s\left(x_i\left(k\right)\right)\right]\hfill \\ \hfill & =:{\varphi }_i\left(x_i\left(k\right),{\lambda }_i\right)\hfill \end{array}$$

(40)

holds for any given λ_i ≥ 0. Thus, given that $u_i^s\left(k\right)$ constitutes a feasible candidate for Problem 1, invoking Definition 3 ensures that the initial state satisfies x_i(k) ∈ X_i,N. Consequently, X_i,N is established as an invariant set for the closed-loop dynamics described by (21). □

4.2. Stability Analysis

Theorem 2. 

If Assumption 1 holds, Problem 1 has a feasible solution at the initial time. Then, the equilibrium point x_i,s of the closed-loop system (21) is asymptotically stable and feasible in X_i,N for any given λ_i ≥ 0. Furthermore, if (22)–(24) hold, then the platoon system satisfies the string stability under the above conditions.

Proof of Theorem 2. 

Problem 1 is feasible at the initial time for any given λ_i ∈ [0, 1). $u_i^o\left(k-1\right)$ and $u_i^o\left(k\right)$ represent the optimal solution of Problem 1 at time k − 1 and k, respectively. According to [35], the candidate Lyapunov function V_i satisfies ${\sigma }_1\left(‖x‖\right)\le V_i\left(x\right)\le {\sigma }_2\left(‖x‖\right)$, where σ₁(·) and σ₂(·) are K-class functions [35]. $V_i^o\left(k\right)=J_i\left(x_i\left(k\right),u_i^o\left(k\right)\right)$, $V_i^s\left(k\right)=J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)$ are defined about the closed-loop system (21). Then, the difference operation is calculated about V_i at adjacent time along the trajectory of the closed-loop system (21); considering Equations (13) and (18), we derive that

$$\begin{array}{l}{V}_i^o\left(x_i\left(k\right)\right)-V_i^o\left(x_i\left(k-1\right)\right)=J_i\left(x_i\left(k\right),u_i^o\left(k\right)\right)-J_i\left(x_i\left(k-1\right),u_i^o\left(k-1\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le J_i\left(x_i\left(k\right),u_i^s\left(k\right)\right)+{\lambda }_i\left[J_i^o\left(x_i\left(k-1\right)\right)-J_i^s\left(x_i\left(k\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -J_i\left(x_i\left(k-1\right),u_i^o\left(k-1\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left(1-{\lambda }_i\right) \left[J_i^s\left(x_i\left(k\right)\right)-J_i^o\left(x_i\left(k-1\right)\right)\right]\end{array}$$

(41)

From Assumption 1, (39), (40), (41), we have

$$\begin{array}{l}{V}_i^o\left(x_i\left(k\right)\right)-V_i^o\left(x_i\left(k-1\right)\right)=J_i^o\left(x_i\left(k\right)\right)-J_i^o\left(x_i\left(k-1\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le \left(1-{\lambda }_i\right) \left[J_i^s\left(x_i\left(k\right)\right)-J_i^o\left(x_i\left(k-1\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left(1-{\lambda }_i\right) \left[J_i\left(x_i\left(k\right),\right)u_i^s\left(k-1\right)-J_i\left(x_i\left(k-1\right),u_i^o\left(k-1\right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left(1-{\lambda }_i\right)\cdot \left[\begin{array}{l}-E_i\left(x_i^o\left(N|k-1\right)\right)+L_i\left(x_i^o\left(N|k-1\right),{\kappa }_i\left(x_i^o\left(N|k-1\right)\right)\right)\\ +E_i\left(f_i\left(x_i^o\left(N|k-1\right),{\kappa }_i\left(x_i^o\left(N|k-1\right)\right)\right)\right)\\ -L_i\left(x_i^o\left(0|k-1\right),u_i^o\left(0|k-1\right)\right)\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le \left(1-{\lambda }_i\right)\cdot \left[-L_i\left(x_i^o\left(0|k-1\right),u_i^o\left(0|k-1\right)\right)\right]\le 0\end{array}$$

(42)

Here, λ_i ∈ [0, 1), and L_i(x_i,u_i) is a positive definite function. Consequently, the value function Vi(xi)Vi(xi) exhibits strict monotonic decay along the trajectories of the closed-loop system (21). This implies that the cost function J_i(x_i,u_i) is positive definite with respect to the equilibrium point, thereby guaranteeing asymptotic stability of the equilibrium within the set X_i,N for system (21). □

4.3. Guaranteed String Stability

Theorem 3 (String Stability). 

If recursive feasibility of the optimization problem is maintained, then the platoon is guaranteed to achieve strict string stability with an attenuation rate of ρ.

Proof of Theorem 3. 

By enforcing the constraint $\underset{k\ge 0}{\max }\left|e_{p,i}\left(k\right)\right|\le {\rho }_i \underset{k\ge 0}{\max }\left|e_{p,i-1}\left(k\right)\right|$ for all pre-diction steps k, and given that the optimal control input is implemented (receding horizon), the actual closed-loop error satisfies $\underset{k\ge 0}{\max }\left|e_{p,i}\left(k\right)\right|\le {\rho }_i \underset{k\ge 0}{\max }\left|e_{p,i-1}\left(k\right)\right|$. Since ρ < 1, the errors do not amplify upstream, thus guaranteeing string stability. □

There exists a feasible solution for Problem 1 at any time, and the closed-loop system (21) is asymptotically stable about the equilibrium point. The values of parameters ζ_i,γ_i,ε_i,k in (22)–(24) refer to Lemma 3 of [34]. Combining (22)–(24) derives (5). Thus, the string stability of the closed-loop system can be established.

Remark 5. 

The above proof presents that the stability of the vehicle is independent of states of the neighboring vehicles. If the vehicles merge into a platoon, whose stability will not be affected, then the whole platoon satisfies tracking stability. Thus, the platoon system is flexible and scalable in this paper.

5. Simulation Verification and Analysis

This section presents a series of simulation scenarios designed to evaluate the performance of the hierarchical strategy under mixed communication topologies. First, we compare the proposed strategy with the weighted function method. Second, we discuss heterogeneous vehicles and scalability, then verify that the vehicles can achieve a consensus state under various mixed communication topologies. Third, we analyze the impact of weights on dynamic consensus performance. Finally, we demonstrate the performance under complex dynamic conditions. The simulation employs a platoon system comprising seven vehicles, with the mixed communication topologies illustrated in Figure 2.

In this simulation, the sampling period is set to Δt = 0.3 s, and the prediction horizon is set as N = 8. All optimization problems are solved by MATLAB 2021a with the fmincon function in this paper. At the initial time, a reference speed transitioning from 19 m/s to 20 m/s and a desired spacing of d = 20 m are broadcast to all vehicles in the platoon. The platoon tracks a new desired speed and ensures safety until the velocity error is 0 and the relatively expected position error is 0. The relative position error of each vehicle is 0, and the velocity error is −1 m/s at k = 0. The simulation parameters and controller gain parameters are as follows (

Table 1. Parameters of vehicles.

mi (kg)	CA,i (N·s2·m−2)	ri (m)	μi	ηT,i
1035.7	0.30	0.30	0.0155	0.965

and

Table 2. Gain parameters of vehicles.

Qi	Ri	Fi	Gi	Pi	Ki	λi
[0.05, 0; 0, 2]	1 × 10−5	[2, 0; 0, 2]	[2, 0; 0, 2]	[1110.5572, 159.9096; 159.9096, 47.0108]	[14,903.5714, 4381.4081]	0.8

), where x_i ∈ X_i = {x_i ∈ $Z_i^{+}$$\left|{‖x_i‖}_{\infty }\right.$$\le 1$}, u_i ∈ U_i = { u_i ∈ $Z_i^{+}$$\left|{‖u_i‖}_{\infty }\right.\le 3000 \mathrm{N}$, −5 m/s² ≤ a_i ≤ 5 m/s². The LQR problem is solved by linearization of the model at equilibrium point (x_i,s,u_i,s), then the terminal penalty function $E_i\left(x_i\right)={\left(x_i-x_{i,s}\right)}^{\mathrm{T}}{P}_i\left(x_i-x_{i,s}\right)$ and local controller κ_i(x_i) = K_i (x_i − x_i,s) + u_i,s are obtained, where the equilibrium point is (x_i,s,u_i,s)^T = (0, 0, 48.9087)^T, and the terminal region is X_i,T = {x_i ∈ $R^2$: E_i(x_i) ≤ 0.0318}.

5.1. Comparison of Proposed Strategy and Weighted Method

In Figure 2a, Problem 1 and Problem 2 are solved separately, where the constraints are Equations (9)–(12), and they are denoted as C–MPC and S–MPC, respectively. The simulations are shown in Figure 3. Then, we compare the proposed hierarchical strategy with the weighted function method, where τ = 0.7. In this paper, the proposed strategy and the weighted method are denoted as D–MPC and W–MPC.

Figure 3b demonstrates a scenario where tracking stability is not guaranteed; the platoon system becomes unstable when only the dynamic consensus objective is considered (e.g., in C–MPC approaches). While the system is stable when solely tracking performance is optimized, this approach does not address the trade-off between dynamic consensus and stability. We next compare the proposed strategy with the weighted function method.

To investigate the robustness of the control strategies against variations in controller design parameters, a comparative simulation was conducted. It is important to note that no external disturbances were introduced in this scenario. Instead, we intentionally adjusted the parameters defining the contractive constraint (Equation (12)) and modified the weighting matrices G_i and F_i in the cost function. This setup evaluates the sensitivity of each method to sub-optimal or varied parameter selections, which is a common challenge in practical implementation.

As shown in Figure 4b, the Traditional Weighted MPC exhibits pronounced oscillatory behavior in both spacing errors and vehicle velocities when the weights (G_i, F_i) and constraint parameters deviate from their nominal values. The states fluctuate significantly before settling. This phenomenon reveals that the stability of the traditional approach is highly sensitive to the precise tuning of the weighting matrices; slight deviations can compromise the damping characteristics of the closed-loop system, leading to undesirable transients. Conversely, the proposed strategy demonstrates remarkable insensitivity to parameter variations, as depicted in Figure 4a.

In contrast, the proposed method guarantees a smooth transient response regardless of moderate parameter tuning errors. This result validates that the proposed framework offers a more robust design with a wider stability margin, reducing the reliance on exhaustive trial-and-error tuning required by traditional weighted MPC approaches. Additionally, this method necessitates auxiliary stability conditions, making its implementation more complex than our proposed strategy.

The relative position errors of the seven vehicles during the last 6 s of the simulation were statistically analyzed to compute the mean and standard deviation. The results are presented in

Table 3. The mean and standard deviations of relative position errors.

Vehicle	1	2	3	4	5	6	7
Mean (m)	9.4011 × 10−3	9.7291 × 10−3	9.5974 × 10−3	9.4428 × 10−3	9.8525 × 10−3	9.3977 × 10−3	9.8703 × 10−3
std (m)	1.3932 × 10−4	1.5916 × 10−4	2.4344 × 10−4	1.8980 × 10−4	1.3932 × 10−4	3.5266 × 10−4	1.4889 × 10−4

. These negligible values confirm the absence of any significant error drift or low-frequency oscillation.

To demonstrate the real-time feasibility of the system, the single-step computation time of the platoon optimization control was statistically analyzed, with the results presented in Figure 5.

As shown in Figure 5, The computational burden remains well within the sampling interval. The maximum and average single-step computation times for each vehicle are presented in

Table 4. The maximum and average computation times.

Vehicle	1	2	3	4	5	6	7
Max (s)	0.2766	0.2756	0.2818	0.2756	0.2784	0.2700	0.2798
Avg (s)	0.2245	0.2252	0.2378	0.2374	0.2297	0.2337	0.1373

, confirming that the controller can comfortably update the control inputs within the required timeframe without causing delays or packet drops.

5.2. Heterogeneity Analysis

In practical scenarios, vehicle platoons typically consist of heterogeneous vehicles characterized by distinct dynamic models. To validate the stability of the proposed strategy under such conditions, this subsection considers a platoon comprising five heterogeneous vehicles with parameters detailed in

Table 5. Heterogeneous vehicle parameters.

	mi (kg)	CA,i (N·s2·m−2)	ri (m)	μi	ηT,i
1	1625.33	1.10	0.35	0.0165	0.950
2	1801.69	1.12	0.39	0.0150	0.950
3	1885.35	1.15	0.40	0.0154	0.960
4	1725.33	1.10	0.36	0.0160	0.950
5	1805.28	1.13	0.37	0.0150	0.955

. The associated communication topologies are depicted in Figure 2g.

Regarding the controller design, the weighting matrices Q_i, R_i, F_i, and G_i in the cost function are selected as specified in Table 2. The feedback gain matrix K_i and the terminal weight matrix P_i are subsequently derived via the Linear Quadratic Regulator (LQR) approach. The resulting simulation performance is illustrated in Figure 6.

As shown in Figure 6, cooperative control among the vehicles is successfully achieved. The relative position errors asymptotically converge to 0, while the vehicle velocities precisely track the reference trajectories, thereby realizing consensus control for the entire platoon system.

5.3. Scalability

To assess the scalability of the proposed strategy, simulations were performed on homogeneous platoons of varying sizes (3 and 12 vehicles). The specific parameter configurations are enumerated in Table 1 and Table 2, while the associated communication topologies are depicted in Figure 2e,f. The resulting performance metrics are summarized in Figure 7. The proposed strategy achieves effective cooperative control for platoons of both 3 and 12 vehicles. The system demonstrates stability, with relative position errors converging to zero and vehicle velocities tracking the desired reference.

5.4. Consensus Under Mixed Communication Topologies

To demonstrate that the proposed strategy guarantees both stability and state consensus under varying mixed communication topologies, simulations corresponding to the configurations in Figure 2b,c were conducted. The resulting performance is illustrated in Figure 8.

From Figure 8, it is indicated that each vehicle attains the desired velocity while fulfilling the conditions outlined in Definition 1; the relative expected position deviation of each vehicle in the platoon ultimately converges to 0 and reaches steady point. Hence, the hierarchical strategy can coordinate platoon stability and dynamic consensus, and the stability and consensus performance are also guaranteed subject to various mixed communication topologies.

5.5. Convergence Analysis Under Different Weights

In the subsection, we present that the weights F_i and G_i have an effect on the consensus performance subject to the mixed communication topology. Figure 2d is chosen in the subsection, the specific weight parameters are enumerated in

Table 6. Weights of consensus objective function.

Weights	Case1	Case2	Case3
Fi	2I2	2I2	6I2
Gi	2I2	6I2	2I2

, and the corresponding simulation results are presented in Figure 9.

Figure 9 illustrates that the platoon converges to the desired velocity and attains steady-state operation, demonstrating the performance of the strategy against variations in dynamic consensus weighting parameters.

Figure 10 gives the dynamic consensus performance of four vehicles. In case 1, the rate of convergence is fastest; it is slowest in case 2, and in case 3, it is in between both. Therefore, the sizes of matrices F_i and G_i have an effect on the dynamic consensus, which is an extension of the result [34]. Then, the corresponding control strategy can be designed according to the actual system.

5.6. Performance Validation Under Complex Dynamic Conditions

The efficacy of the strategy is substantiated through extensive simulations conducted under a wide spectrum of operating conditions. Specifically, we consider emergency acceleration and deceleration maneuvers to test dynamic response capabilities, alongside scenarios with changing road friction (road friction μ_i ± 10%) to assess performance under varying environmental parameters. Figure 11 illustrates the dynamic behavior of the system obtained from the simulations.

As shown in Figure 11a, under the emergency acceleration scenario, the platoon system tracks the desired speed of 24 m/s. After a transient period, the system achieves cooperative control, reaching the target velocity while the relative position errors ultimately vanish, indicating successful formation-keeping. Figure 11b illustrates the emergency deceleration case, where the desired speed is reduced to 16 m/s. Following a short duration, the system stabilizes and meets the control requirements. The system acceleration satisfies the performance requirements under emergency acceleration and deceleration scenarios. Figure 11c demonstrates that under varying road friction coefficients, the platoon system similarly achieves cooperative control and satisfies the consensus.

6. Discussion

While the proposed hierarchical DMPC framework demonstrates robust stability under mixed switching topologies, real-world vehicular platooning faces additional complexities, including communication imperfections, vehicle heterogeneity, and dynamic environmental variations. In this section, we discuss how our current directed graph framework and contractive constraint mechanism can be extended by incorporating recent advances in robust control, federated learning, and graph representation learning. Specifically, we address three key dimensions: handling communication delays via robust constraint tightening, bridging our architecture with federated learning for heterogeneous systems, and evolving our topology modeling using category-guided graph concepts and adaptive learning.

6.1. Advanced Graph Representations and Adaptive Parameter Learning

The modeling of mixed communication topologies can be significantly enhanced by moving beyond binary adjacency matrices toward semantically enriched graph structures. Inspired by [16], our directed graph framework can be extended to capture complex, asymmetric interaction patterns. In the current model, all neighbors contribute equally to the cost function. In an advanced extension, edge weights would become category-dependent functions, assigning higher “semantic importance” to critical nodes (e.g., a heavy truck ahead or the platoon leader) and lower weights to less influential neighbors. This mirrors the category-guided mechanism where specific node types dictate the aggregation strategy, enabling the controller to prioritize information flows that are most vital for safety and stability.

The hierarchical graph convolution concept in [16] captures both local details and global context. Our existing hierarchical MPC structure naturally aligns with this. We can interpret the lower control layer as processing local high-frequency interactions (immediate predecessors, analogous to fine-grained graph convolutions), while the upper coordination layer handles long-range low-frequency dependencies (global consensus, analogous to coarse-grained pooling). This allows the framework to explicitly model asymmetric information flows, where broadcast messages (global) and V2V unicast messages (local) are processed with different granularities.

Complementing this structural evolution, the fixed contractive parameter λ in our simulations could be replaced by an intermittent dynamic learning mechanism. Drawing from adaptive control strategies like intermittently dynamic fuzzy learning [20], a lightweight online learner could monitor real-time traffic conditions and topology switching frequencies to dynamically adjust λ. For instance, in dense, highly dynamic traffic, λ could be tightened to enforce faster convergence, whereas in sparse, stable conditions, it could be relaxed to reduce control effort and fuel consumption. This synergy between category-aware graph modeling and adaptive parameter tuning would transform our static framework into a responsive, intelligent system capable of self-optimizing its interaction patterns and convergence rates in real time.

6.2. Robustness Against Communication Imperfections and Offset-Free Mechanisms

The current formulation assumes ideal information exchange; however, practical V2X networks are inherently characterized by time-varying communication delays and stochastic packet losses. To maintain stability and consensus under such conditions, our framework can be extended by integrating robust constraint tightening mechanisms, as systematically explored in recent works on offset-free distributed control for networked systems [17].

Specifically, regarding the impact of latency on our hierarchical framework:

On Hierarchical Consensus: The upper-layer consensus optimization relies on neighbor state information. A communication delay τ means vehicle i uses outdated state x_j(k − τ) instead of x_j(k). In our formulation, this introduces a bounded disturbance term in the consensus cost function (Equation (6)). Since the lower-layer MPC operates at a faster sampling rate and relies heavily on local onboard sensors for immediate tracking, it can effectively compensate for the “jitter” in the reference trajectory generated by the delayed upper layer, provided τ is within the prediction horizon N.

On Overall Stability: Theoretical analysis of contractive MPC suggests that stability is preserved if the delay-induced error remains within the contraction region defined by Equation (13). Specifically, if the delay τ satisfies τ < T_margin (where T_margin is derived from the Lipschitz constant for the system and the contraction coefficient γ), the Lyapunov function decrease condition (Theorem 2) still holds, albeit with a potentially slower convergence rate. If τ exceeds this bound, the contractive constraint may become infeasible, leading to potential instability or string instability where errors amplify upstream.

In this extended view, the state constraints X_i in our optimization problem would be replaced by a tightened set X_i⊖ℜ_i, where ℜ_i is a robust invariant set that absorbs the worst-case estimation errors caused by delays and dropouts. The size of ℜ_i would dynamically evolve based on the observed communication quality. Furthermore, drawing from the offset-free control strategies in [17], an integral action or disturbance observer could be embedded within the local MPC layer of our hierarchy. This would ensure that even if delayed information leads to temporary trajectory deviations, the system can asymptotically reject these disturbances and achieve zero steady-state error. We further elaborate on how the federated earning paradigm mentioned in [17] could enhance our approach. Instead of using conservative, fixed bounds for delay/loss, vehicles could collaboratively learn the statistical distribution of communication quality across the platoon without sharing raw data.

This learned model would allow for dynamic adjustment of the tightening margins: shrinking the margin when the network is healthy (improving performance) and expanding it when congestion or interference is detected (ensuring safety). This creates a balance between robustness and optimality.

6.3. Conceptual Bridges to Federated Learning for Heterogeneous Platoons

Real-world platoons often consist of diverse vehicle types (e.g., heavy-duty trucks vs. passenger cars) with distinct dynamic parameters. A standard homogeneous DMPC may struggle to optimize performance for all agents simultaneously. By adopting insights from personalized federated learning [18], our upper coordination layer could function as a “global model aggregator” that learns a common traffic flow pattern, while the lower local layers act as “personalized clients”. Each vehicle would maintain a local dynamic model tailored to its specific physical characteristics, utilizing predictive error compensation to correct for model mismatches. In this hybrid setup, Equation (5) would be adaptively weighted: stricter for homogeneous clusters to ensure string stability, and more relaxed for heterogeneous boundaries to allow for necessary local adaptation. This approach balances the need for platoon-wide coherence with the flexibility required for diverse vehicle dynamics. This analogy provides a powerful pathway to address vehicle heterogeneity.

It is worth noting that the proposed hierarchical MPC architecture shares conceptual similarities with recent advances in federated learning-based distributed control [19]. Specifically, the contractive constraint mechanism defined in Equation (13) bears a strong resemblance to the consensus regularization terms commonly employed in federated optimization.

In federated settings, regularization terms are introduced into the local loss functions to penalize the divergence between local model parameters and the global aggregate, thereby driving the network towards consensus without sharing raw data. Analogously, our contractive constraint (Equation (13)) enforces that the state deviation of each vehicle shrinks over time, effectively coupling the individual tracking performance with the platoon’s consensus goal. While the conceptual goal of promoting agreement is similar, the mathematical role of Equation (13) in our framework is more stringent. Unlike the soft penalties in federated learning which guide the gradient descent direction, Equation (13) is imposed as a hard constraint within the MPC optimization problem (Problem 1). This design choice is critical for recursive feasibility and stability. We highlight that this connection opens new avenues: future work could integrate federated learning algorithms to adaptively tune the contraction parameter λ or the consensus reference based on data-driven insights, combining the rigorous stability of our method with the adaptability of federated learning.

7. Conclusions

This paper presents a distributed hierarchical control framework designed for discrete-time nonlinear systems that incorporates both state and control input constraints and mixed communication topologies. By incorporating the tracking stability optimal value function as a constraint in the dynamic consensus optimization problem, we establish sufficient conditions for platoon string stability and recursive feasibility of the consensus optimization. Numerical simulations demonstrate that the proposed strategy effectively balances the trade-off between stability and dynamic consensus under mixed topologies, while simultaneously guaranteeing both performance metrics. Further simulations involving heterogeneous dynamics, emergency maneuvers, and variable friction conditions demonstrate the algorithm’s superior performance, confirming the platoon’s ability to achieve cooperative control in diverse scenarios. Furthermore, we analyze the impact of penalty weights on dynamic consensus performance. Future research will be directed towards experimental validation and the extension of the proposed framework to accommodate heterogeneous vehicle dynamics. Crucially, while this study assumes ideal communication, real-world deployments face challenges such as time-varying delays and packet losses. Drawing inspiration from recent advances in offset-free distributed control for networked systems [17], our future research will integrate robust constraint tightening mechanisms into the hierarchical MPC framework. This extension aims to systematically compensate for communication imperfections, ensuring that both stability and consensus performance are maintained under realistic network constraints.