Teoria da dissipatividade aplicada à estabilização assintótica de sistemas racionais com incertezas

Abstract

The present work aims to implement a robust asymptotic stabilization strategy for nonlinear systems, restricted to polynomial or rational nonlinearities. The stabilization technique uses the dissipation theory together with the Finsler’s Lemma and the definition of annihilators to develop conditions expressed in the form of linear matrix inequalities (LMIs). These LMIs are used to determine a static feedback gain that ensures the asymptotic stability of the nonlinear system with uncertainties in a polytopic region around the equilibrium point. In addition, is also made an estimate of the region in which any initial conditions within it result in trajectories that asymptotically converge to the equilibrium point without leaving that region, called the system attraction domain. It is important to highlight that, using the same strategy, it is possible to design both state feedback gains and output feedback gains. The strategy was applied in a microelectromechanical system (MEMS), which presents polynomial nonlinearities in its state equation, a static output feedback was designed to ensure the local stability of the system. The application was also carried out in a theoretical system with polynomial nonlinearities and a static output feedback was designed to ensure the local stability of the system. Lastly, the application was carried out in a system of an inverted pendulum, in which the nonlinearity in the differential equation that represents the system are transformed into rational nonlinearities through a change of variable, a static state feedback was designed to guarantee the local stability of the system. The simulation results of the applications show the efficiency of the strategy.