State observation for a class of uncertain nonlinear systems via deterministic learning

In this paper, based on the recently proposed theory of deterministic learning, a new neural observer is proposed for a class of uncertain nonlinear systems undergoing periodic or recurrent motions. Firstly, it is shown that the state observation error converges to a small neighborhood of zero exponentially in finite time, and a partial persistent excitation (PE) condition is satisfied. Secondly, by imposing an auxiliary filter and constructing a new Lyapunov function, it is obtained that the neural weight estimation error also converges to a small neighborhood of zero. Thus, the system uncertain dynamics along the periodic trajectory can be locally-accurately identified by the radial basis function(RBF) neural networks(NNs) and then stored in a constant RBF NNs. Finally, a constant NNs observer is also implemented, with guaranteed stability and good performance. The proposed observer scheme does not require high-gain design, and the embedded localized NNs can learn (identify) the nonlinear uncertain dynamics along the estimated periodic or recurrent system trajectory.