# Detectability and Observer Design for Switched Differential Algebraic Equations

## TL;DR

This paper introduces the concept of interval-wise detectability for switched differential-algebraic equations and applies it to design observers that ensure asymptotic state estimation accuracy.

## Contribution

It proposes a new detectability notion for switched DAEs and demonstrates its use in synthesizing observers with guaranteed convergence.

## Key findings

- Interval-wise detectability implies asymptotic stability of zero-output trajectories.
- Observer design based on state decomposition ensures error convergence.
- Conditions for stability and convergence are established.

## Abstract

This paper studies detectability for switched linear differential-algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption.