Nonlinear set‐membership state estimation based on the Koopman operator

Abstract

SummaryIn this study, the Koopman operator is applied to solve the challenging nonlinear set‐membership (SM) state estimation problem. The basic idea is to lift the nonlinear system into a linear one with a higher dimension, and then linear SM estimation methods can be adopted. The Koopman operator is an infinite‐dimensional linear decomposition of nonlinear dynamics. This linearized system can fully describe the state evolution of the nonlinear dynamics with input, output, and noises. To apply SM state estimation algorithms, a finite‐dimensional approximation is obtained by using the data‐driven method. Meanwhile, the statistical properties of the approximation errors are also utilized to reduce the conservativeness of the filter. The probability distributions of approximation errors are described by sample particles. Coupled with the unknown but bounded noises, the linearized system is associated with two types of uncertainties. To this end, the merging SM and stochastic strategy is adopted to construct a class of confidence state set whose level approaches 1. After that, the optimal filter gain is given to minimize the size of the confidence state set. Finally, three numerical examples are given to demonstrate the effectiveness of the proposed method.