Tracking Differentiators for Both the Real Time Signals and the Time Delayed Signals

The differential tracking for a given signal is a well-known and challenging problem in control theory and practice. In this note, we introduce a novel method to design the differentiators. The proposed differentiator, in contrast to existing differentiators that take the form of a dynamical system, is represented through convolutions. This idea is primarily inspired by the mollifier technique, which is well-known in the theory of partial differential equations. Both the weighted moving average technique and the mollifier technique are used in the differentiator's design. By the proper choice of the kernel function, we can obtain the derivative of the given signal by integrating rather than differentiating the signal itself directly. As a result, the proposed tracking differentiators can be robust to the high-frequency signals. Although our approach is simple, it is very effective, both for the real time signals and the time delayed signals.