Zonotopic set-membership state estimation for nonlinear systems based on the deep Koopman operator

In this study, a novel nonlinear zonotopic set-membership estimation (SME) approach is proposed based on the Koopman operator and the deep neural networks (DNNs). The core concept involves transforming the original nonlinear system into a higher-dimensional linear system, enabling the development of linear SME algorithms. To address noise in real-world data, we propose a three-step offline training strategy for computing the linear approximation of nonlinear dynamics. Building on the deep-Koopman-based linearized system, we construct a reduced-order filter structure that directly estimates the original states and then computes the lifted states for improved observability. We derive two optimal gains based on different size criteria and present a data-driven nonlinear mapping approach that ensures a closed-form solution for the algorithm. Additionally, we introduce a local model update method that refines estimation accuracy using the set of state estimates obtained from SME. The effectiveness and superiority of our proposed methods is demonstrated through two illustrative examples.