PaperlayerView exact revision on arXiv

Source-linked AI summary

Platooning of Connected Vehicles with Undirected Topologies: Robustness Analysis and Distributed H-infinity Controller Synthesis

Yang Zheng, Shengbo Eben Li, Keqiang Li, Wei Ren

arXiv:1611.01412v2math.OC

TL;DR

Large-scale connected-vehicle platoons with undirected communication topologies require robustness analysis and distributed H-infinity controller synthesis under disturbances. The paper decouples collective dynamics through spectral decomposition and shows that robustness degrades with platoon size, while leader information and mini-platoon coordination strongly influence scalable control.

  • Problem

    The paper addresses robustness analysis and distributed H-infinity controller synthesis for large-scale homogeneous platoons with undirected topologies and external disturbances.

  • Method

    The paper uses graph-theoretic unified models and spectral decomposition of L+P to decouple platoon dynamics into single-vehicle-sized subsystems for distributed controller synthesis.

  • Results

    Robustness γ-gain grows at least as O(N) with a fixed number of leader-pinned followers and deteriorates to O(N^2) for bidirectional topologies.

  • Takeaways & Limitations

    Leader information is more important than follower-to-follower communication, and coordinating multiple mini-platoons is identified as a reasonable large-scale control architecture.

  • Takeaways & Limitations

    The analysis assumes perfect communication, whereas practical channels may involve time delays or data losses, especially for long-range leader-data transmission.

Abstract

from arXiv · show

This paper considers the robustness analysis and distributed $\mathcal{H}_{\infty}$ (H-infinity) controller synthesis for a platoon of connected vehicles with undirected topologies. We first formulate a unified model to describe the collective behavior of homogeneous platoons with external disturbances using graph theory. By exploiting the spectral decomposition of a symmetric matrix, the collective dynamics of a platoon is equivalently decomposed into a set of subsystems sharing the same size with one single vehicle. Then, we provide an explicit scaling trend of robustness measure $γ$-gain, and introduce a scalable multi-step procedure to synthesize a distributed $\mathcal{H}_{\infty}$ controller for large-scale platoons. It is shown that communication topology, especially the leader's information, exerts great influence on both robustness performance and controller synthesis. Further, an intuitive optimization problem is formulated to optimize an undirected topology for a platoon system, and the upper and lower bounds of the objective are explicitly analyzed, which hints us that coordination of multiple mini-platoons is one reasonable architecture to control large-scale platoons. Numerical simulations are conducted to illustrate our findings.

I. INTRODUCTION · II. PROBLEM STATEMENT · A. Vehicle Longitudinal Dynamics

The paper studies robustness and distributed H-infinity controller synthesis for homogeneous connected-vehicle platoons with undirected topologies and external disturbances. It models collective behavior using algebraic graph theory, derives robustness scaling results, and motivates topology and mini-platoon design for large systems.

  • I. INTRODUCTION: The paper addresses performance limitations studied through measures including string stability and stability margin, while prior work links scaling behavior to topology and control design.String stability concerns attenuation of spacing errors, whereas stability margin characterizes convergence speed of initial errors,.
  • I. INTRODUCTION: The paper targets robustness analysis and distributed H-infinity controller synthesis for large-scale homogeneous platoons with undirected topologies and external disturbances.The approach derives a unified platoon model in both time and frequency domains using algebraic graph theory.
  • I. INTRODUCTION: The proposed distributed H-infinity design provides guaranteed performance while accounting for limited communication resources.The paper further identifies coordination of multiple mini-platoons as a reasonable architecture for controlling large-scale platoons.
  • I. INTRODUCTION: The robustness measure γ-gain is lower bounded by the minimal eigenvalue of a topology-associated matrix and grows at least as O(N) when the number of leader-pinned followers is fixed.For bidirectional topologies, the scaling deteriorates to O(N^2).
  • II. PROBLEM STATEMENT: Platooning seeks to make connected vehicles travel at the same speed while maintaining a rigid formation geometry with desired inter-vehicle spacing.The considered homogeneous platoon contains one leader and N followers, for a total of N + 1 vehicles.
  • II. PROBLEM STATEMENT: The platoon is organized through four interacting components: vehicle dynamics, communication topology, distributed controller, and formation geometry.Topology specifies information exchange, the controller uses locally available information, and formation geometry defines desired inter-spacing.
  • A. Vehicle Longitudinal Dynamics: Each vehicle’s longitudinal dynamics are inherently nonlinear because of engine, braking, aerodynamic, rolling-resistance, and gravitational effects, so theoretical analysis uses a simplified linear model.The model takes position, velocity, and acceleration as states; the control input represents desired acceleration, includes powertrain delay, and accounts for external disturbance.

B. Model of Communication Topology · C. Design of Linear Distributed Controller

The paper models follower communication with directed-graph tools while analyzing undirected follower topologies augmented by leader pinning, requiring global leader reachability for internal stability. It then formulates constant-spacing platoon control using identical local linear state feedback based only on neighboring and leader information.

  • B. Model of Communication Topology: The follower network is represented by a directed graph with adjacency matrix A, no self-loops, degree matrix D, and Laplacian L = D − A.An edge (j, i) indicates that node i receives communication from node j.
  • B. Model of Communication Topology: Leader-to-follower communication is encoded by an augmented graph and diagonal pinning matrix P, where p_i = 1 when follower i directly receives the leader’s information.The pinning matrix distinguishes direct leader access from information propagated through follower connections.
  • B. Model of Communication Topology: For undirected follower communication, the augmented graph must contain a leader-rooted spanning tree, ensuring every follower receives leader information directly or indirectly and enabling internal stability.The number of followers connected directly to the leader is not restricted.
  • B. Model of Communication Topology: For any undirected topology, λ_min(L) = 0 with eigenvector 1_N; under the spanning-tree assumption this eigenvalue is simple and every eigenvalue of L + P is positive.These spectral properties characterize the connectivity and leader-pinning conditions used in the analysis.
  • C. Design of Linear Distributed Controller: The control objective is asymptotic tracking of the leader’s constant-speed trajectory while maintaining the desired inter-vehicle spacing under a constant-spacing policy, d_i,i−1 = d_0.The leader is assumed to satisfy a_0 = 0 and p_0 = v_0t.
  • C. Design of Linear Distributed Controller: Each follower’s local controller uses only information from its neighboring followers and directly connected leader nodes, collected in I_i = N_i ∪ P_i.The controller is designed as an identical linear state-feedback law for all nodes.
  • C. Design of Linear Distributed Controller: The distributed feedback law uses local position, velocity, and acceleration gains k_p, k_v, and k_a, together with a coupling-strength parameter c.The gain vector is k = [k_p, k_v, k_a]^T.
  • C. Design of Linear Distributed Controller: Tracking errors are defined relative to the leader’s position, velocity, and acceleration, yielding the compact local error state x̂_i = [p̂_i, v̂_i, â_i]^T.The position error also subtracts the desired offset d_i,0, while velocity and acceleration errors subtract v_0 and a_0, respectively.

D. Formulation of Closed-loop Platoon Dynamics

The section formulates homogeneous platoon dynamics in unified state-space and output form, incorporating distributed feedback, communication topology, and external disturbances. Position-tracking errors define the platoon output used to characterize robustness across topologies.

  • Closed-loop model: The collective follower state and input stack individual vehicle states and controls into platoon-wide vectors X and U using Kronecker-product notation.This construction enables a unified representation for all following vehicles.
  • Closed-loop model: The closed-loop dynamics combine N-vehicle local dynamics, distributed feedback through L + P, and external disturbances W in a single state equation.The disturbance input matrix is B = I_N ⊗ B_2.
  • Output and transfer representation: Position-tracking errors are defined as the platoon output Y = C X, with C = I_N ⊗ C_1 and C_1 = [1, 0, 0].Assuming zero initial tracking errors, the formulation also defines the transfer function from disturbances W to tracking outputs Y.
  • Topology-dependent robustness: The unified time- and frequency-domain models describe platoon dynamics under varied communication topologies, whose robustness depends on both topology and distributed feedback gains.Topology can fundamentally limit stability margin, string stability, and coherence behavior.

A. Robustness Measure: γ-gain

This section defines γ-gain as an energy-amplification measure from finite-energy disturbances to platoon position-tracking errors. It interprets γ-gain through H-infinity transfer-function norms and explains why robustness analysis is difficult for general directed topologies.

  • Definition and interpretation: γ-gain measures the amplification of finite-energy disturbances acting on all vehicles into their position-tracking errors.The disturbances are assumed to have bounded L2 norm.
  • Definition and interpretation: Physically, γ-gain quantifies a platoon’s sensitivity or attenuation of external-disturbance energy.
  • Definition and interpretation: γ-gain differs from standard string stability, which concerns attenuation of spacing errors along the vehicle string.
  • H-infinity computation: γ-gain can be computed from the H-infinity norms of corresponding transfer functions using their maximum singular values.
  • Analytical challenge: Robustness analysis is difficult for general directed topologies because the transfer function requires an inverse involving the communication-topology factor L + P.The feasibility of analysis depends on whether the relevant H-infinity norms are explicitly computable.

B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies

For undirected platoon topologies, spectral decomposition links the robustness γ-gain to the minimum eigenvalue of L + P. Consequently, with a fixed number of leader-pinned followers, γ-gain increases at least as O(N), highlighting the importance of leader information and topology selection.

  • B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies: Theorem 1 establishes that γ-gain is lower bounded by the minimum eigenvalue λmin of L + P for any stabilizing feedback gains.This result connects robustness directly to the communication topology and motivates selecting topologies with larger λmin.
  • B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies: Spectral decomposition of the symmetric matrix L + P decomposes the platoon transfer function into single-node subsystems whose dynamics are modified by the topology eigenvalues.The proof reduces the H-infinity analysis of the collective system to the H-infinity norms of scalar transfer functions associated with individual eigenvalues.
  • B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies: Increasing communication range is proposed as a practical way to enlarge λmin and improve robustness performance in large-scale platoons.Topology selection is therefore tied to maximizing the minimum eigenvalue of L + P.
  • B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies: With the number of leader-pinned followers fixed, γ-gain increases at least as O(N) as the platoon size grows, regardless of stabilizing feedback-gain design.The bound follows from the relationship between λmin and the number of followers pinned to the leader.
  • B. Scaling Trend of γ-gain for Large-scale Platoons with Undirected Topologies: Leader information has greater influence on robustness scaling than follower-to-follower information, and bidirectional topology provides a tighter lower bound as a special undirected case.The paper identifies bidirectional topology as the case with one leader-pinned follower and notes that its bound is consistent with, but tighter than, the general undirected-topology result.

IV. DISTRIBUTED H∞CONTROLLER SYNTHESIS OF VEHICLE PLATOONS

This section formulates distributed H∞ controller synthesis as selecting feedback gains and coupling strength to achieve a desired γ-gain, then solves it through decoupling and a scalable multi-step procedure. It also discusses communication-topology selection based on the analytical results.

  • Problem formulation: The distributed H∞ control problem seeks feedback gains k = [kp, kv, ka]T and coupling strength c such that γ = ∥G(s)∥H∞ < γd.The objective is to guarantee a prescribed γd-gain for the platoon.
  • Decoupling technique: A decoupling technique converts platoon-level synthesis into independent H∞ control problems for systems with the same dimension as a single vehicle.This decomposition provides the basis for scalable controller synthesis.
  • Synthesis procedure: A multi-step procedure solves the vehicle-platoon H∞ control problem while preserving the favorable decoupling property.The section also discusses communication-topology selection using the resulting analytical insights.

A. Decoupling of Platoon Dynamics

Spectral decomposition of L + P converts the 3N-dimensional platoon dynamics into N single-vehicle-sized subsystems, reducing controller-synthesis complexity. Platoon stability and H-infinity robustness are equivalent to those properties holding for every decoupled subsystem, whose dynamics depend on L + P eigenvalues.

  • Decoupling of Platoon Dynamics: Spectral decomposition reduces the 3N-dimensional collective model to N subsystems sharing a single vehicle’s dimension, enabling scalable distributed H-infinity controller synthesis.The original closed-loop matrix is computationally demanding for large N, whereas the decoupled formulation provides significant computational benefits.
  • Decoupling of Platoon Dynamics: The platoon is asymptotically stable with robustness measure γ < γd if and only if all N decoupled subsystems are stable and have H-infinity norms below γd.This equivalence permits robustness analysis of the full platoon through individual subsystem conditions.
  • Decoupling of Platoon Dynamics: Communication topology influences the decoupled dynamics through the eigenvalues of L + P, while distributed H-infinity control decomposes into N single-vehicle-sized subsystem control problems.Thus, topology-dependent collective behavior can be addressed through lower-dimensional controller-synthesis problems.

B. Synthesis of Distributed H∞Controller

The section establishes an LMI-based distributed H∞ synthesis method whose feasibility is independent of platoon size and communication topology. It guarantees any prescribed H∞ performance in principle, but potentially requires impractically large coupling strength and controller gains.

  • B. Synthesis of Distributed H∞Controller: Lemma 5 gives a necessary-and-sufficient matrix condition for a controller achieving ||G(s)||H∞ < γd.The condition requires Q ≻ 0 and α > 0 satisfying the stated LMI.
  • B. Synthesis of Distributed H∞Controller: The synthesis solves LMI (26) for feedback gains and adjusts coupling strength c to satisfy (27), using an LMI dimension independent of platoon size N.Because the LMI dimension matches one vehicle’s dynamics, the procedure substantially reduces computational complexity for large-scale platoons.
  • B. Synthesis of Distributed H∞Controller: Controller synthesis is decoupled from communication-topology design: topology does not affect LMI feasibility and enters only through condition (27).This permits controller synthesis and topology selection to be performed in separate design stages.
  • B. Synthesis of Distributed H∞Controller: Any prescribed H∞ performance can require very large coupling strength c, producing high-gain controllers that may be impractical because actuators saturate.The guarantee is therefore theoretically broad but constrained by real implementation limits.

C. Selection of Communication Topology

Communication topology affects platoon robustness and controller synthesis through the minimum eigenvalue λmin of L + P. Larger λmin improves robustness and reduces feedback gains, while leader pinning maximizes λmin but may motivate mini-platoons under communication imperfections.

  • Selection principle: The minimum eigenvalue λmin of L + P governs both robustness scaling and distributed H-infinity controller synthesis.A larger λmin improves robustness for a fixed controller and reduces feedback gains for a fixed H-infinity performance.
  • Topology optimization: With only communication-link count constrained, the star topology maximizes λmin using the minimum number of links under Assumption 1.This topology has P = I and L = 0, but the conclusion considers only global consensus and ignores local safety.
  • Topology optimization: The proposed topology optimization seeks a large λmin under limited communication resources, but its exact formulation and solutions remain future work.The design must also account for local safety, including collision avoidance between consecutive vehicles.
  • Eigenvalue bounds: The bound 0 < λmin ≤ 1 is tight, with λmin = 1 if and only if every follower is pinned to the leader.Thus, leader information determines the maximum achievable λmin regardless of follower-to-follower connections.
  • Practical architecture: Because perfect communication is unrealistic for large platoons, dividing vehicles into mini-platoons with each first node pinned to the leader offers a balanced architecture.This addresses potential time-delays or data-losses in long-range, high-volume leader-data transmission.

V. NUMERICAL SIMULATIONS · A. Scaling Trend of γ-gain

Numerical simulations validate the predicted robustness scaling under undirected topologies and demonstrate distributed H-infinity control computation. Increasing communication range improves robustness slightly but does not remove its growth with platoon size, even under full follower connectivity.

  • V. NUMERICAL SIMULATIONS: The numerical experiments verify γ-gain scaling under different undirected topologies and demonstrate computation of a distributed H-infinity controller.The broader simulation program also includes realistic nonlinear vehicle dynamics.
  • V. NUMERICAL SIMULATIONS: Realistic nonlinear vehicle-dynamics simulations, using models from,, are conducted to demonstrate effectiveness in real environments.Additional spacing-error simulations consider h-neighbor topologies and multiple mini-platoons for total platoon size N = 10.
  • A. Scaling Trend of γ-gain: The simulations use inertial delay τ = 0.5 s and distributed gains kp = 1, kv = 2, and ka = 0.5, which stabilize the considered platoons.The gains follow stability-region discussions for linear platoons in,.
  • A. Scaling Trend of γ-gain: At least O(N^2) growth of γ-gain with platoon size is observed for BD topology, while Corollary 1’s lower bound is mathematically correct but not tight.Fig. 6 varies N from 10 to 100 and compares the computed H-infinity norm with the corresponding transfer-function lower bound.
  • A. Scaling Trend of γ-gain: The experiments vary platoon size N from 10 to 100 and numerically calculate the H-infinity norm and corresponding transfer-function lower bounds.These calculations are used to illustrate the scaling trend of robustness performance.
  • A. Scaling Trend of γ-gain: At least O(N) growth persists as reliable communication range h increases, although larger h slightly improves robustness performance.This behavior confirms the scaling predicted by Corollary 2.
  • A. Scaling Trend of γ-gain: Even with h = 100 and full follower connectivity, γ-gain continues increasing with platoon size, consistent with the robustness analysis in Section III.With h = 100, every follower has access to all other followers’ information.

B. Calculation of Distributed H∞Controller

Theorem 2 provides a scalable multi-step procedure for synthesizing distributed H∞ controllers by solving an LMI and selecting topology-dependent coupling strengths. For four 11-vehicle platoon topologies, the resulting controllers stabilize spacing errors and achieve error amplifications well below the desired performance γd = 1.

  • Controller synthesis: Theorem 2 synthesizes a distributed H∞ controller by solving LMI (26) for γd = 1, using YALMIP and SeDuMi with inertial delay τ = 0.5 s.The resulting feasible solution determines the feedback gain matrix.
  • Topology-dependent design: For four communication topologies in an 11-vehicle platoon, coupling strength c is selected according to (27), including configurations with multiple mini-platoons.The considered mini-platoon cases include two groups of size 5 and three groups of sizes 3, 4, and 3.
  • Topology-dependent design: Theorem 2 yields high-gain controllers for γd = 1 because the minimum eigenvalue λmin is small.This condition drives the required coupling and controller gains upward.
  • Numerical validation: The four platoons are stable under the synthesized H∞ controller, with error amplifications γ1 = 0.0226, γ2 = 0.0234, γ3 = 0.0166, and γ4 = 0.0187.These values are below the desired performance γd = 1 and validate Theorem 2 for the tested constant-speed and disturbance scenario.

C. Simulations with realistic vehicle dynamics · VI. CONCLUSION

The nonlinear vehicle simulations show that the distributed controller stabilizes platoons under realistic vehicle dynamics and varied undirected communication topologies. The conclusion summarizes a graph-theoretic robustness and controller-synthesis framework while identifying imperfect communication and nonidentical gains as future directions.

  • C. Simulations with realistic vehicle dynamics: The simulations implement upper-level desired-acceleration control together with lower-level throttle or braking commands generated through an inverse vehicle model.The lower-level controller tracks the desired acceleration trajectory, while the vehicle powertrain is modeled as a first-order inertial transfer function.
  • C. Simulations with realistic vehicle dynamics: The realistic-vehicle simulation uses randomly selected passenger-vehicle parameters and a leader trajectory that accelerates from 20 m/s to 30 m/s before maintaining 30 m/s.The platoon contains one leader and ten followers, with desired spacing 25 m and initially zero spacing and velocity errors.
  • C. Simulations with realistic vehicle dynamics: The distributed controller (6) stabilizes an 11-vehicle platoon with nonlinear vehicle dynamics under different communication topologies.The dynamics include road friction, approximated engine dynamics, and aerodynamic drag forces; Fig. 9 reports spacing errors for the tested topologies.
  • VI. CONCLUSION: The paper develops unified time- and frequency-domain models and decouples collective platoon dynamics into subsystems using the spectral decomposition of L+P.This framework underpins the robustness analysis and distributed H-infinity controller synthesis for homogeneous platoons with external disturbances.
  • VI. CONCLUSION: The conclusion identifies imperfect communication, including time delay and packet loss, as a future robustness-analysis direction.These effects are not addressed in the present study.
  • VI. CONCLUSION: The study restricts attention to identical feedback gains (6), leaving the robustness benefits of nonidentical controllers for future investigation.The conclusion notes that asymmetric controllers in BD topologies have shown certain robustness improvements in recent work.
Loading 1611.01412v2…