Towards a comprehensive impossibility result for string stability

TL;DR

This paper establishes a broad impossibility result showing that achieving string stability in large chains of subsystems is fundamentally impossible under various controller types and definitions, including nonlinear and time-varying controllers.

Contribution

It extends existing impossibility results from linear time-invariant systems to general homogeneous controllers, covering nonlinear, time-varying, and locally communicating controllers.

Findings

Impossibility holds for multiple norm-based definitions including $L_2$ and BIBO criteria.

Results apply to general discrete-time controllers, broadening applicability.

String stability cannot be achieved independently of chain size under the considered conditions.

Abstract

We provide a comprehensive impossibility result towards achieving string stability, i.e. keeping local relative errors in check with local controllers independently of the size of a chain of subsystems. We significantly extend existing results, from the LTI setting to any homogeneous controllers that can be nonlinear, time-varying, and locally communicating. We prove this impossibility for a set of definitions with various norm choices, including the $L_2$-type which is more standard in the literature and a BIBO type criterion. All results hold for a general discrete-time controller which should cover most applications.