Design of observers for continuous-time nonlinear systems using neural networks

Observers design is addressed for a class of continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities. A full-order state estimator is considered that depends on an innovation function made up of two terms: a linear gain and a feedforward neural network that provides a nonlinear contribution. The gain and the weights of the neural network are chosen in such way to ensure the convergence of the estimation error. Such a goal is achieved by constraining the derivative of a Lyapunov function to be negative definite on a sampling grid of points. Under assumptions on the smoothness of the Lyapunov function and of the distribution of the sampling points, the negative definiteness of the derivative of the Lyapunov function is obtained by minimizing a cost function that penalizes the constraints that are not satisfied. Suitable sampling techniques allow to reduce the computational burden required by the network's weights optimization. Simulations results are presented to illustrate the effectiveness of the proposed method.