Estimação de estados de sistemas não lineares: abordagem robusta via métodos de otimização convexa

Abstract

While the state estimation for linear systems has well-established approaches in the literature, for nonlinear systems, several methodologies have been proposed to overcome the various difficulties encountered in obtaining a sub-optimal solution, according to some criterion. By considering a convex modeling of nonlinearities using polytopes, the Differential Mean Value Theorem (DMVT) or factorization techniques, it is possible to derive conditions based on convex optimization, through Linear Matrix Inequalities (LMIs), to achieve the project objectives. In this context, this work aims at developing new state estimation approaches for discrete-time or continuous-time nonlinear systems that employ LMI conditions for the synthesis of state estimators. More specifically, this work proposes three new methodologies based on the prediction and data assimilation steps of the classical Kalman filter. The first consists of a Guaranteed Covariance Kalman Filter (GCKF) approach, the second presents a nonlinear filter approach that guarantees robustness to unknown exogenous signals and, finally, the third solves a multiobjective optimization problem, which guarantees the minimization of the covariance matrix of the estimation error and robustness to noise, indicated in the first two approaches. Simulated examples are used to illustrate and validate the methodologies proposed in this work.