Adaptive observers for time-delay nonlinear systems in triangular form

A simple nonlinear observer with a dynamic gain is proposed for a class of bounded-state nonlinear systems subject to state delay. By saturating the states of the observer nonlinearities with either symmetric or non-symmetric saturation functions, we show that the observer exists, whatever the delay is. Furthermore, it will be highlighted that the observer design is free from any preliminary analysis of the time-delay system such as estimating the Lipschitz constants of nonlinearities. The proposed design encompasses a wide class of nonlinear and time-delay systems written in triangular form and generalizes previous results on delayless nonlinear systems.