Modeling, analysis and design of linear systems with switching delays

TL;DR

This paper addresses the modeling, stability analysis, and control design for discrete-time linear systems with switching delays, providing algorithms for stability assessment and insights into control strategies under uncertain delays.

Contribution

It models systems with switching delays as regular switching linear systems and offers algorithms for stability analysis and controllability, highlighting their structural properties.

Findings

Provided an exponential-time algorithm for robust stability analysis.

Showed stability analysis is NP-hard in general.

Developed an algorithm to determine minimal delay knowledge for controllability.

Abstract

We consider the modeling, stability analysis and controller design problems for discrete-time LTI systems with state feedback, when the actuation signal is subject to switching propagation delays, due to e.g. the routing in a multi-hop communication network. We show how to model these systems as regular switching linear systems and, as a corollary, we provide an (exponential-time) algorithm for robust stability analysis. We also show that the general stability analysis problem is NP-hard in general. Even though the systems studied here are inherently switching systems, we show that their particular structure allows for analytical understanding of the dynamics, and even efficient algorithms for some problems: for instance, we give an algorithm that computes in a finite number of steps the minimal look-ahead knowledge of the delays necessary to achieve controllability. We finally show that when the switching signal cannot be measured it can be necessary to use nonlinear controllers for stabilizing a linear plant.