Passivity-based PI control of first-order systems with I/O communication delays: A complete sigma-stability analysis

TL;DR

This paper analyzes the stability of first-order passive systems with PI control over delayed communication channels, providing a complete stability map and explicit PI tuning guidelines based on sigma-stability and spectral methods.

Contribution

It introduces a sigma-stability framework for passivity-based PI control with delays, offering explicit stability regions and maximum decay rate analysis.

Findings

Complete sigma-stability regions for PI control with delays

Explicit PI tuning formulas based on decay rate and system parameters

Spectral methods applied to neutral system stability analysis

Abstract

The PI control of first-order linear passive systems through a delayed communication channel is revisited in light of the relative stability concept called sigma-stability. Treating the delayed communication channel as a transport PDE, the passivity of the overall control-loop is guaranteed, resulting in a closed-loop system of neutral nature. Spectral methods are then applied to the system to obtain a complete stability map. In particular, we perform the D-subdivision method to declare the exact sigma-stability regions in the space of PI parameters. This framework is then utilized to analytically determine the maximum achievable exponential decay rate of the system while achieving the PI tuning as an explicit function of the decay rate and the system parameters.