Contraction analysis of time-varying DAE systems via auxiliary ODE systems

TL;DR

This paper investigates the contraction properties of time-varying DAE systems by embedding them into auxiliary ODE systems, establishing conditions for contraction and stability analysis, and designing observers.

Contribution

It introduces a novel approach to analyze DAE contraction via auxiliary ODE systems and provides new stability criteria and observer design methods.

Findings

Contraction of DAE systems is equivalent to UGES of their variational systems.

A sufficient condition for DAE contraction using matrix measures is derived.

The methods are applied to stability analysis and observer design for DAE and ODE systems.

Abstract

This paper studies the contraction property of time-varying differential-algebraic equation (DAE) systems by embedding them to higher-dimension ordinary differential equation (ODE) systems. The first result pertains to the equivalence of the contraction of a DAE system and the uniform global exponential stability (UGES) of its variational DAE system. Such equivalence inherits the well-known property of contracting ODE systems on a specific manifold. Subsequently, we construct an auxiliary ODE system from a DAE system whose trajectories encapsulate those of the corresponding variational DAE system. Using the auxiliary ODE system, a sufficient condition for contraction of the time-varying DAE system is established by using matrix measure which allows us to estimate an lower bound on the parameters of the auxiliary system. Finally, we apply the results to analyze the stability of time-invariant DAE systems, and to design observers for time-varying ODE systems.