Detectability and Observer Design for Switched Differential Algebraic Equations
For control theorists working with switched DAE systems, this work provides a theoretical framework for observer design, though it is incremental as it builds on prior interval-wise observability concepts.
The paper addresses detectability for switched linear differential-algebraic equations and proposes interval-wise detectability, which ensures asymptotic stability of zero-output-constrained state trajectories. This enables the design of observers that generate asymptotically converging state estimates.
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.