  In this paper, we propose a novel nonlinear observer based on neural
networks, called neural observer, for observation tasks of linear
time-invariant (LTI) systems and uncertain nonlinear systems. In particular,
the neural observer designed for uncertain systems is inspired by the active
disturbance rejection control, which can measure the uncertainty in real-time.
The stability analysis (e.g., exponential convergence rate) of LTI and
uncertain nonlinear systems (involving neural observers) are presented and
guaranteed, where it is shown that the observation problems can be solved only
using the linear matrix inequalities (LMIs). Also, it is revealed that the
observability and controllability of the system matrices are required to
demonstrate the existence of solutions of LMIs. Finally, the effectiveness of
neural observers is verified on three simulation cases, including the X-29A
aircraft model, the nonlinear pendulum, and the four-wheel steering vehicle.
