Controle extremal para sistemas descritos por equações diferenciais parciais

Abstract

This dissertation addresses the design and analysis of fast Newton-based extremum seeking feedback for scalar and vectorial static maps in cascade with partial differential equation (PDE) dynamics in its actuation path. Although more general classes of PDEbased systems could be envisaged, we concentrated our efforts in handling diffusion PDEs. The proposed adaptive control scheme for real-time optimization follows two basic steps: first, it cancels out the effects of the actuation dynamics in the dither signals, and second, it applies a boundary control for the diffusion process via backstepping transformation. In particular, the diffusion compensator employs perturbation-based (averaging-based) estimates for gradient and Hessian's inverse of the nonlinear-scalar static map to be optimized. The complete stability analysis of the closed-loop system is carried out using Lyapunov's method and applying averaging for infinite-dimensional systems in order to capture the infinite-dimensional state of the actuator model. Local exponential convergence to a small neighborhood of the unknown extremum is guaranteed and verified by means of a numerical example.