On the equivalence relations of detectability and PE conditions with applications to stability analysis of time-varying systems

The paper investigates equivalent relations of detectability and persistency of excitation (PE) for general nonlinear time-varying systems. Three new definitions relating to the detectability are proposed and discussed. As a preliminary result, we show that these detectability conditions are all equivalent under a mild assumption. Furthermore, two PE conditions that are a nonlinear-version extension of the PE proposed in present literature are defined. Very interestingly, it can be shown that these PE conditions are both equivalent to the proposed detectability conditions under the same assumption. To the best of our knowledge, this is a first result about the connection between detectability and PE in the area of nonlinear time-varying systems. Furthermore, we revisit a class of nonlinear time-varying interconnected systems modified from adaptive control systems. By combining a newly developed stability criterion with the proposed results, the globally uniformly asymptotic stability of the origin can be guaranteed by verifying a PE condition. From these applications, it can be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the PE conditions based on our approach.