Lp String Stability of Cascaded Systems: Application to Vehicle Platooning

Ploeg, Jeroen; van de Wouw, Nathan; Nijmeijer, Henk · 2014 · OpenAlex-citations

DOI: 10.1109/tcst.2013.2258346

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Summary

This paper addresses the ambiguity surrounding the definition of "string stability" in the context of vehicle platooning, a technique used to increase road throughput by reducing inter-vehicle distances via Cooperative Adaptive Cruise Control (CACC). Existing literature offers conflicting interpretations, ranging from Lyapunov stability focusing on initial conditions to performance-oriented criteria for linear systems that ignore external disturbances. To resolve this, the authors propose a rigorous, novel definition of $L_p$ string stability for nonlinear cascaded systems based on input-output properties. This definition accounts for both external disturbances (such as lead vehicle velocity variations) and initial condition perturbations, providing a unified framework that generalizes existing linear stability conditions. The study derives a dynamic model for a homogeneous vehicle platoon, incorporating spacing policies and driveline dynamics. The authors formalize $L_p$ string stability by requiring that the $L_p$ norm of the output signal for any vehicle in the string remains bounded by a class $\mathcal{K}$ function of the external input and initial state norms, regardless of the string length. For linear systems, they prove that this definition is equivalent to standard $H_\infty$ norm conditions for $L_2$ stability and $L_1$ norm conditions for $L_\infty$ stability. Specifically, they establish that a linear unidirectionally interconnected system is strictly $L_2$ string stable if and only if the $H_\infty$ norm of the transfer function from the preceding vehicle’s output to the current vehicle’s output is less than or equal to one. The theoretical findings are validated through experiments using a platoon of six passenger vehicles equipped with CACC. The results demonstrate that controller design based on the proposed criteria is not only theoretically sound but also practically feasible, ensuring disturbance attenuation along the vehicle string. The paper concludes that the proposed definition provides a rigorous basis for analyzing both homogeneous and heterogeneous platoons, bridging the gap between stability theory and performance-oriented controller design. This work clarifies the distinction between stability and performance interpretations, offering a scalable and robust framework for the design of automated vehicle-following systems.

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