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On stochastic string stability with applications to platooning over additive noise channels

Francisco J. Vargas, Marco A. Gordon, Andrés A. Peters, Alejandro I. Maass

TL;DR

This work defines stochastic string stability for vehicular platoons under additive white noise channels by introducing $\mathcal{L}_p$-mean $\mathcal{L}_q$-variance string stability and mean-square string stability, and applies these notions to a predecessor-following topology with AWN channels. The authors derive necessary and sufficient conditions, showing that stability of the mean and bounded stationary variance are governed by a size-invariant bound on the closed-loop transfer function $||T(z)||_{\infty}$ and related spectral factors; notably, when $\rho(A)<1$ the mean error converges to zero and the stationary variance remains bounded as the platoon grows, with asymptotic variance given by $P^{\infty}_{\zeta_N} = (\|S(z)/M(z)\|_2^2 -1) P_d$ as $N \to \infty$. A key finding is that the AWN-channel conditions closely align with the ideal-communication case, offering practical design criteria for robust platooning under stochastic disturbances. Numerical examples validate the theory, illustrating stable versus unstable string behavior and the impact of initial conditions on transient statistics, while confirming that the framework generalizes deterministic results to stochastic communication models.

Abstract

This paper addresses the string stabilization of vehicular platooning when stochastic phenomena are inherent in inter-vehicle communication. To achieve this, we first provide two definitions to analytically assess the string stability in stochastic scenarios, considering the mean and variance of tracking errors as the platoon size grows. Subsequently, we analytically derive necessary and sufficient conditions to achieve this notion of string stability in predecessor-following linear platoons that communicate through additive white noise channels. We conclude that the condition ensuring string stability with ideal communication is essentially the same that achieves stochastic string stability when additive noise channels are in place and guarantees that the tracking error means and variances converge.

On stochastic string stability with applications to platooning over additive noise channels

TL;DR

This work defines stochastic string stability for vehicular platoons under additive white noise channels by introducing -mean -variance string stability and mean-square string stability, and applies these notions to a predecessor-following topology with AWN channels. The authors derive necessary and sufficient conditions, showing that stability of the mean and bounded stationary variance are governed by a size-invariant bound on the closed-loop transfer function and related spectral factors; notably, when the mean error converges to zero and the stationary variance remains bounded as the platoon grows, with asymptotic variance given by as . A key finding is that the AWN-channel conditions closely align with the ideal-communication case, offering practical design criteria for robust platooning under stochastic disturbances. Numerical examples validate the theory, illustrating stable versus unstable string behavior and the impact of initial conditions on transient statistics, while confirming that the framework generalizes deterministic results to stochastic communication models.

Abstract

This paper addresses the string stabilization of vehicular platooning when stochastic phenomena are inherent in inter-vehicle communication. To achieve this, we first provide two definitions to analytically assess the string stability in stochastic scenarios, considering the mean and variance of tracking errors as the platoon size grows. Subsequently, we analytically derive necessary and sufficient conditions to achieve this notion of string stability in predecessor-following linear platoons that communicate through additive white noise channels. We conclude that the condition ensuring string stability with ideal communication is essentially the same that achieves stochastic string stability when additive noise channels are in place and guarantees that the tracking error means and variances converge.
Paper Structure (14 sections, 4 theorems, 48 equations, 6 figures)

This paper contains 14 sections, 4 theorems, 48 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Platoon over additive noise channels
  4. String stability for stochastic systems
  5. String stability analysis for platoon under additive noise channels
  6. Analysis for $k \rightarrow \infty$
  7. Analysis for $N \rightarrow \infty$
  8. String stability conditions
  9. Numerical example
  10. Conclusions and future work
  11. Proof of Lemma \ref{['lem:mean_square_all']}
  12. Proof of Theorem \ref{['lem:zdom_scalability_all']}
  13. Proof of Corollary \ref{['cor:stationary_all']}
  14. Proof of Theorem \ref{['teo:Lpmean_Lqvariance_all']}

Key Result

Lemma 3

Consider the platoon described by eq:ss_sys_all. Then, the mean of the state and the tracking error are given by The corresponding covariance matrices are given by where $P_{\bm{\xi}}(0) = P_{\bm{\xi}_0}$ and $\Upsilon = \mathbf{B}_{a} P_{\mathbf{d}} {\mathbf{B}_{a}}^\top + \mathbf{B}_{b} P_{\mathbf{d}} {\mathbf{B}_{b}}^\top + \mathbf{A}\mathbf{B}_{a} P_{\mathbf{d}} {\mathbf{B}_{b}}^\top + \math

Figures (6)

  • Figure 1: Platoon configuration
  • Figure 2: Feedback control loop for the $i$-th agent
  • Figure 3: Concatenation of the platoon with channel noise on each vehicle.
  • Figure 4: Platoon behavior with zero initial conditions.
  • Figure 5: Platoon behavior with initial conditions different from zero.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 3
  • Theorem 4
  • Corollary 5
  • Theorem 6