Abstract
We discuss the pseudo measurement method which is one of the main approaches to equality-constrained state estimation for a dynamic system. We demonstrate by the fundamental theory of Kalman filtering that reviewing the equality constraint as a pseudo measurement seems questionable. The main reason is that the additional pseudo measurement is actually a constant here which cannot help to estimate the state. More specifically, when the states in an unconstrained dynamic system model have already satisfied the equality constraint, the extra constraint is obviously not necessary. When the true equality-constrained states do not satisfy the unconstrained dynamic process equation, the effect of pseudo measurement is projecting the estimate which is not optimal onto the constraint set. However, since the performance of a projected estimate is also certainly influenced by its original estimate, we show through a numerical example that the pseudo measurement method is not always a good choice, especially when the process equation mismatch is large.
Highlights
Filtering problems often have certain inherent and known constraints in the physical dynamic systems [1], for example, target tracking [2, 3], robotics [4], multisensor data fusion [5, 6], vision-based systems [7], econometric modeling [8], biomedical systems [9] and others [10, 11]
By the sequential forms, they computed the equivalence of Kalman filtering with and without pseudo measurement and proved that, under certain conditions, the use of the pseudo measurement for filtering is redundant
Gupta [24] proved that, under certain conditions, the pseudo measurement method and the estimate projection method result in mathematically equivalent constrained estimate structures
Summary
Filtering problems often have certain inherent and known constraints in the physical dynamic systems [1], for example, target tracking [2, 3], robotics [4], multisensor data fusion [5, 6], vision-based systems [7], econometric modeling [8], biomedical systems [9] and others [10, 11]. For equality-constrained state estimation, numerous approaches have been developed, for example, the model reduction method [5, 6, 12], the pseudo measurement method [13,14,15,16,17,18], the estimate projection method [19, 20], the system projection method [21], the gain projection method [22], and some other methods [11, 23] Among these methods, the pseudo measurement method has been deeply investigated. By the sequential forms, they computed the equivalence of Kalman filtering with and without pseudo measurement and proved that, under certain conditions, the use of the pseudo measurement for filtering is redundant.
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