Exponential Convergent Projection-Based Estimator onto a Closed Convex Set

States are typically unavailable in practical control system applications. The best approximation of the true states of the observable system is generated by state estimators operating in two steps, including measurement and prediction. Moreover, constrained problems, whether they may describe physical limitations, may form a closed convex set. In such cases, it is possible that the predicted states fall outside of the admissible closed convex set. Therefore, it is necessary to guarantee constraint satisfaction. We need to ensure that the estimator meets the constraints. To this end, it is required to design a projection-based state estimator. Then, implausible states will be projected back onto the desired closed convex set. In this study, we present a general strategy for projection onto multiple closed convex sets with arbitrary boundaries, utilizing Dykstra’s algorithm.