Measurement partitioning and observational equivalence in state estimation

This paper studies observability of linear systems from both algebraic and graph-theoretic arguments, and further draws a parallel between the two. We show that a set of critical measurements (for state-space observability) can be 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 yi, what is the set of measurements equivalent to yi, such that only one measurement in this set is required? 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. The need to make such equivalence arises in areas, e.g., meter placement for power systems, where relevant lines of study include: (a) to guarantee state-space observability with as few sensors as possible; and, (b) to find candidate replacement measurements when a sensor incurs a fault. We illustrate the related concepts on a simple, yet insightful, system digraph.