Optimal fusion β-quaternion filtering from delayed observations and packet losses with compensation strategy under properness

This study addresses the problem of centralized fusion filtering for β-quaternion signals obtained from multisensor observations. It is assumed that observations from each sensor may be subject to random updates or delays, and may also experience packet losses. In such cases, a compensation strategy is employed, where the missing observation is replaced by its predictor. Furthermore, the correlation between the additive noise of the signal and the observations is taken into account. Under first-order properness conditions, a recursive algorithm is developed to compute the optimal filter. A key feature of the proposed algorithm is its ability to reduce the dimensionality of operations in each iteration by a quarter of that required by the optimal β-quaternion processing algorithm, which relies on augmented processes. This reduction significantly enhances computational efficiency while maintaining identical estimators under first-order properness conditions. A numerical example is presented to illustrate, among other aspects, the substantial reduction in computational time achieved by the proposed algorithm, as well as its superior estimation accuracy compared to conventional quaternion processing methods.