This paper presents new methods for set-valued state estimation of discrete-time nonlinear systems whose trajectories are known to satisfy nonlinear equality constraints, called invariants (e.g., conservation laws). Set-valued estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. Most existing methods employ a standard prediction-update framework with set-based prediction and update steps based on various set representations and techniques. However, achieving accurate enclosures for nonlinear systems remains a significant challenge. This paper presents new methods based on constrained zonotopes that improve the standard prediction-update framework for systems with invariants by adding a consistency step. This new step uses invariants to reduce conservatism and is enabled by new algorithms for refining constrained zonotopes based on nonlinear constraints. This paper also presents significant improvements to existing prediction and update steps for constrained zonotopes. Specifically, new update algorithms are developed that allow nonlinear measurement equations for the first time, and existing prediction methods based on conservative approximation techniques are modified to allow a more flexible choice of the approximation point, which can lead to tighter enclosures. Numerical results demonstrate that the resulting methods can provide significantly tighter enclosures than existing zonotope-based methods while maintaining comparable efficiency.
