Similar to observability, detectability indicates the ability of a dynamic system to use its output signals from sensors and the model to “observe” the behaviour of the state that can fully capture the characteristic of the system. Thus it plays an important role in many engineering applications in order to design a feedback control law using the measured output signals. For a time-invariant system, the well-defined detectability also indicates its uniform attractivity under mild assumptions. However, it is usually much harder to define such detectability for complex time-varying dynamic systems such as switching systems and hybrid systems. The main objective of this tutorial is to present a few different definitions of well-defined detectability for different types of systems; to reveal the link among detectability, the persistent excitation (PE) and uniform attractivity; and to demonstrate the power of detectability in analysing stability properties for switched nonlinear time-varying systems to researchers in the area of systems and control who may not be familiar with the concept. This tutorial will
revisit the concept of detectability and its link with uniform attractivity for linear- time-invariant systems;demonstrate the difficulty of the well-defined detectability for time-varying systems and revisit various detectability definitions of different dynamic systems in literature;introduce the concept of Signal Set, which can serve as a unified framework to characterize a large class of dynamic systems;introduce a well-defined detectability, weak detectability (WD) in the Signal Set framework;reveal the link among WD, PE condition and uniform attractivity;provide sufficient conditions to check WD;demonstrate the power of WD in stability analysis of switched nonlinear time-varying systems including arbitrarily switching cases and constrained switching cases with new stability results;consider a few examples from the literature to illustrate the strength of the proposed framework.