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Measurement partitioning and observational equivalence in state estimation

This letter studies measurement partitioning and equivalence in state estimation based on graph-theoretic principles. We show that a set of critical measurements (required to ensure LTI state-space observability) can be further partitioned into two types:~$α$ and~$β$. This partitioning is driven by different graphical (or algebraic) methods used to define the corresponding measurements. Subsequently, we describe observational equivalence, i.e. given an~$α$ (or~$β$) measurement, say~$y_i$, what is the set of measurements equivalent to~$y_i$, such that only one measurement in this set is required to ensure observability? Since~$α$ and~$β$ measurements are cast using different algebraic and graphical characteristics, their equivalence sets are also derived using different algebraic and graph-theoretic principles. We illustrate the related concepts on an appropriate system digraph.