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Distributed Model Predictive Control for Heterogeneous Vehicle Platoons under Unidirectional Topologies

Yang Zheng, Shengbo Eben Li, Keqiang Li, Francesco Borrelli, J. Karl Hedrick

arXiv:1603.03225v2math.OC

TL;DR

Heterogeneous vehicle platoons must coordinate nonlinear, spatially coupled motion when communication is unidirectional and most followers do not know the desired set point. The paper develops a neighbor-based DMPC algorithm with a neighboring-average terminal constraint and analyzes stability using summed local costs as a Lyapunov function. It proves asymptotic stability under an explicit weight condition and reports stable simulations across four communication topologies, with spacing errors below 1 m.

  • Problem

    Vehicle platoons require global coordination despite nonlinear heterogeneous dynamics, input constraints, and local unidirectional communication that leaves many followers unaware of the desired set point.

    Source: §I · pp. 1–2

  • Method

    The paper assigns each vehicle a neighbor-based local open-loop MPC problem, penalizes predicted-versus-assumed trajectory errors, and imposes a terminal state equal to the average of neighboring states.

    Source: §I · p. 2

  • Results

    The analysis proves asymptotic stability using the sum of local costs as a Lyapunov function under an explicit sufficient condition on cost weights.

    Source: §IV · p. 6, D. Sufficient condition of DMPC… · p. 9, §VI · pp. 10–11

  • Takeaways & Limitations

    The DMPC approach supports heterogeneous platoons with arbitrary unidirectional topologies without requiring every vehicle to know the desired set point beforehand.

    Source: §I · p. 2, §VI · pp. 10–11

    The formulation does not address robustness to model uncertainty and noise or collision-avoidance coupling constraints.

    Source: A. Local open-loop optimal control… · p. 5

Abstract

from arXiv · show

This paper presents a distributed model predictive control (DMPC) algorithm for heterogeneous vehicle platoons with unidirectional topologies and a priori unknown desired set point. The vehicles (or nodes) in a platoon are dynamically decoupled but constrained by spatial geometry. Each node is assigned a local open-loop optimal control problem only relying on the information of neighboring nodes, in which the cost function is designed by penalizing on the errors between predicted and assumed trajectories. Together with this penalization, an equality based terminal constraint is proposed to ensure stability, which enforces the terminal states of each node in the predictive horizon equal to the average of its neighboring states. By using the sum of local cost functions as a Lyapunov candidate, it is proved that asymptotic stability of such a DMPC can be achieved through an explicit sufficient condition on the weights of the cost functions. Simulations with passenger cars demonstrate the effectiveness of proposed DMPC.

I. INTRODUCTION

Platoon control seeks coordinated speed and spacing, but heterogeneous nonlinear vehicles and limited unidirectional communication complicate distributed stabilization. The paper proposes DMPC in which only leader-connected followers know the desired set point while each node uses local neighbor information.

  • Platooning aims to improve traffic efficiency, road safety, and fuel consumption by coordinating vehicle speed and spacing.Source: §I · p. 1
  • Existing consensus-based approaches often use linear dynamics and controllers, without accounting for input constraints and model nonlinearities.Source: §I · p. 1
  • Centralized MPC can explicitly handle nonlinearities and constraints, but computes all control inputs using globally known states.Source: §I · p. 1
  • Many existing DMPC methods assume every agent knows a common desired set point, which is impractical when followers cannot communicate directly with the leader.Source: §I · p. 2
  • The proposed DMPC handles heterogeneous nonlinear vehicles under unidirectional topologies, with only leader-connected followers knowing the desired set point.Source: §I · p. 2, §II · pp. 2–3, A. Nonlinear platoon model for control · p. 3

B. Objective of platoon control

The platoon objective is to track the leader’s speed while maintaining prescribed inter-vehicle gaps. Communication is modeled as a directed graph so each local control problem uses only information available from its incoming neighbors and leader connection.

B. Objective of platoon control

C. Model of communication topology

III. DESIGN OF DISTRIBUTED MODEL PREDICTIVE CONTROL

The DMPC design assigns each vehicle a local optimization problem using neighboring information, with trajectory penalties and terminal constraints supporting coordination without universal knowledge of the desired set point.

B. Algorithm of distributed model predictive control

At each time step, vehicles repeatedly solve local problems, exchange assumed trajectories, update shifted predictions, and apply the first optimal control input.

B. Algorithm of distributed model predictive control

IV. STABILITY ANALYSIS OF THE DMPC ALGORITHM

The stability analysis uses the sum of local costs as a Lyapunov function and derives explicit weight conditions sufficient for asymptotic stability.

  • The sum of local cost functions serves as a Lyapunov candidate for proving asymptotic stability.Source: §IV · p. 6
  • Stability is guaranteed through explicit sufficient conditions on the weights in the local cost functions.Source: §IV · p. 6

A. Terminal constraint analysis

Under a leader-rooted spanning tree, terminal states converge to the desired state; unidirectional topology strengthens this result to convergence within at most N steps.

A. Terminal constraint analysis

B. Analysis of local cost function

The local cost function supports recursive feasibility and decreases monotonically under the stated conditions, providing a Lyapunov function for asymptotic stability.

B. Analysis of local cost function

  • Lemma 5 establishes recursive feasibility by showing a feasible control remains available at future times.The result uses a feasible shifted control construction for each local optimization problem.Source: B. Analysis of local cost function · p. 8
  • Under the theorem’s conditions, each local cost function satisfies a bound on its one-step change.The bound is derived using the terminal constraint, index changes, and the triangle inequality.Source: B. Analysis of local cost function · p. 8
  • The local cost function decreases monotonically and therefore serves as a proper Lyapunov function for asymptotic stability.This establishes stability at the individual local-cost level, although selecting parameters directly from the bound is not intuitive.Source: B. Analysis of local cost function · pp. 8–9

C. Sum of local cost functions

The analysis aggregates all local cost functions into a single Lyapunov candidate, yielding a system-level decrease condition for the platoon.

C. Sum of local cost functions

  • The sum of all local cost functions is defined as the platoon-level Lyapunov candidate.This construction considers the collective behavior of all vehicles rather than an individual local problem.Source: C. Sum of local cost functions · p. 9
  • Theorem 4 derives an upper bound on the decline of the summed local cost function.The proof combines the local-cost result with a reorganization of the follower-index sets.Source: C. Sum of local cost functions · p. 9
  • Using the summed cost simplifies controller design because designers can seek a sufficient condition directly from the aggregate decrease bound.The paper identifies this as more intuitive than adjusting parameters through the individual local-cost inequality.Source: C. Sum of local cost functions · p. 9

D. Sufficient condition of DMPC stability

The paper derives an explicit distributed condition on cost-function weights that guarantees asymptotic stability for heterogeneous platoons with unidirectional topologies.

D. Sufficient condition of DMPC stability

  • Theorem 5 states that satisfying the explicit sufficient condition makes the DMPC platoon asymptotically stable.The proof establishes strict monotonic decrease of the aggregate cost function.Source: D. Sufficient condition of DMPC… · p. 9
  • Only the weights penalizing errors between predicted and assumed trajectories need adjustment to guarantee asymptotic stability.The condition is distributed, so followers can select penalty weights without centralized information.Source: D. Sufficient condition of DMPC… · p. 9
  • Stability requires downstream nodes not to rely heavily on a node unless that node is sufficiently consistent with its assumed trajectory.This interpretation follows from the information-sharing sets used in the sufficient condition.Source: D. Sufficient condition of DMPC… · p. 9
  • Theorem 5 extends the admissible communication structures from special cases to arbitrary unidirectional topologies satisfying the paper’s assumption.The broader class includes the topology examples illustrated in Fig. 1.Source: D. Sufficient condition of DMPC… · p. 10

V. SIMULATION RESULTS

Simulations evaluate an eight-vehicle heterogeneous platoon across four unidirectional communication topologies. The DMPC remains stable in every tested topology, with spacing errors below 1 m.

V. SIMULATION RESULTS

  • Eight heterogeneous vehicles—one leader and seven followers—are simulated under four communication topologies.The topologies are PF, PLF, TPF, and TPLF.Source: §V · p. 10
  • Spacing errors remain below 1 m for all four communication topologies.The initial spacing and velocity errors are zero in the reported scenario.Source: §V · p. 10
  • The DMPC is stable for all tested topologies, consistent with Theorem 5.The simulation also reports no collisions during the transient process.Source: §V · p. 10

VI. CONCLUSIONS

The paper develops a DMPC algorithm for heterogeneous vehicle platoons with nonlinear dynamics, unidirectional topologies, and unknown desired set points. It establishes asymptotic stability under an explicit weight condition, while identifying broader topologies, disturbances, uncertainty, packet drops, and delays as future challenges.

VI. CONCLUSIONS

  • The proposed DMPC handles nonlinear vehicle dynamics and unidirectional topologies without requiring every vehicle to know the desired set point beforehand.Each vehicle solves a local optimal control problem using neighboring information and sends its assumed output trajectory to neighbors.Source: §VI · p. 10
  • A neighboring-average terminal constraint makes predictive-horizon terminal states converge to the desired state in finite time for unidirectional topologies containing a spanning tree.The constraint is included in each vehicle’s local optimal control problem.Source: §VI · pp. 10–11
  • The sum of local cost functions serves as a Lyapunov function, yielding asymptotic stability under an explicit sufficient condition on the cost weights.The stability guarantee is derived for the proposed distributed control framework.Source: §VI · p. 11
  • The stability condition is currently limited to the considered topology class, motivating future work on more general topologies.The paper also identifies disturbances, dynamic uncertainty, packet drops, and communication delays as open issues.Source: §VI · p. 11

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