On the use of the time-integrals of the output in observer design for non-linear autonomous system

The goal of this note is to show that the observation scheme recently proposed in L. Menini, C. Possieri, and A. Tornambe, "Observers for linear systems by the time-integrals and moving average of the output" for linear systems, achieving good performance in the presence of a high-frequency measurement disturbance, can be adapted to be used for non-linear autonomous systems. The design is based on the use of the time-integrals of the measured output and requires the availability of a state observer able to provide a rough estimate of the state; the proposed scheme can be then understood as an observer post-processing algorithm. For the implementation of the proposed scheme, it is needed to express the time-integrals of the output as functions of the state of the system. Such functions are called directional anti-derivatives; in the note, conditions to guarantee their existence and methods for their computation, in closed form or by approximation, are resumed. The performance of the proposed observation scheme is illustrated through an example.