On the equivalence relations of detectability and PE conditions

This note investigates equivalent relations of detectability and persistency 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 nonlinear-version 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. Moreover, by combining a newly developed stability criterion with the proposed results, an interesting example is given to illustrate how the classic PE conditions for linear systems can be extended and used in nonlinear systems. From this example, it can also be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the PE conditions based on our approach.