The further study of sliding mode observers

In this report, our new research results on the error sign propagation, the finite error convergence and the learning mechanism of the sliding mode observer systems, developed by Slotine et al. in 1980s, for a class of high-order nonlinear systems are presented. It will be shown that, as the switching component of the output tracking error, between the output estimate and the measurable system output, drives the output tracking error dynamics (leading subsystem) to converge to zero in a finite time, this switching component is also used as the driving force of the other subsystems (followers) in the observer error dynamics, to force them to learn the dynamic behaviours of the leading subsystem, for achieving the finite error convergence. Our new findings are that (i) on the sliding surfaces, the signs of the error derivatives of the followers are the same as the one of the leading subsystem; (ii) The error derivatives of all followers can be treated as the modulations of the error derivatives of leading subsystem by the bounded positive functions. Based on these observations and considerations, the error sign propagation rule on the sliding surfaces is formulated, the supervised learning mechanism embedded in the sliding mode observer systems is explored in detail, and the finite error convergence of the sliding mode observer systems is also proved.