State estimation with point and set measurements

Numerous state estimation problems (e.g., under linear or nonlinear inequality constraints, with quantized measurements) can be formulated as those with point and set measurements. Inspired by the estimation with quantized measurements developed by Curry, under a Gaussian assumption, the minimum mean-squared error (MMSE) filtering with point measurements and set measurements of any shape is proposed by discretizing continuous set measurements. Possible ways to relax the Gaussian assumption and to discretize the involved Gaussian and truncated Gaussian distributions are discussed. Through an inequality constrained state estimation example, it is shown that under a certain condition, the update by inequality constraints as set measurements is redundant, otherwise the update is necessary and helpful. Supporting numerical examples are provided.