Observer design for a class of nonlinear systems under a persistent excitation

The problem of state reconstruction from input and output measurements for nonlinear time delay systems remain open in many cases. In this paper we propose an adaptive observer to solve this problem for a class of unknown variable time-delay nonlinear systems where the state matrix depends on the input persistency excitation. To achieve this we combine the use of a Kalman-like observer with a suitable choice for the Lyapunov-Krasovskii functional. This is done under a sufficient number of hypothesis to guarantee the convergence of the observer inside a sphere depending of the delay upper bound. The proposed strategy is tested in simulation by considering a mixed piece-wise and sinusoid time delay function and its efficiency when the problem of persistency excitation occurs.