Minimum Energy Estimation of Nonlinear Dynamical Systems in Continuous and Discrete Time

Abstract

In many applications, accurately estimating a system's state is crucial for effective control. This thesis gives an in-depth introduction to the minimum energy estimator (MEE), an optimal estimator that is based on the minimum energy cost function. Although less widely used than the extended Kalman filter or the unscented Kalman filter, the MEE offers a rigorous optimality guarantee. Using tools from optimal control theory - such as the Pontryagin maximum principle and the Hamilton-Jacobi-Bellman (HJB) equation - we derive both continuous and discrete time formulations of the MEE. Conditions for the stability and converge of the MEE are reviewed. We also demonstrate that the MEE coincides with the Kalman filter in the case of linear dynamics. The derivation leads to an estimator equation for the MEE that requires solving an HJB equation, prompting the introduction of various numerical methods to address this challenge. To further explore the relationship between continuous and discrete time filtering, simulations are conducted to compare the accuracy of the discrete time Kalman filter and a "hybrid" Kalman filter, where the continuous time filter is discretized in the update step.