Design of parameterized state observers and controllers for a class of nonlinear continuous-time systems

The design of observers and controllers for a class of continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities is addressed. Observers and controllers that depend on a linear gain and a parameterized function implemented by a feedforward neural network are considered. The gain and the weights of the neural network are optimized in such way to ensure the convergence of the estimation error for the observer and the stability of the closed-loop system for the controller, respectively. This is achieved by constraining the derivative of a quadratic Lyapunov function to be negative definite on a grid of points, penalizing the constraints that are not satisfied. It is shown that suitable sampling techniques such as low-discrepancy sequences, commonly employed in quasi-Monte Carlo methods for high-dimensional integration allow one to reduce the computational burden required to optimize the network parameters. Simulations results are presented to illustrate the effectiveness of the method.