Functional Observability, Structural Functional Observability and Optimal Sensor Placement

TL;DR

This paper introduces new characterizations for functional observability and structural functional observability, providing graph-theoretic conditions, NP-hardness results, and efficient heuristics for optimal sensor placement.

Contribution

It develops unified conditions for functional observability, redefines SFO from a generic perspective, and proposes polynomial-time heuristics with guarantees for sensor placement problems.

Findings

Modal functional observability conditions are established.

Sensor placement problems are proven NP-hard.

Greedy algorithms with provable guarantees are proposed.

Abstract

In this paper, new characterizations for functional observability, functional detectability, and structural functional observability (SFO) are developed, and based on them, the related optimal sensor placement problems are investigated. A novel concept of modal functional observability coinciding with the notion of modal observability is proposed. This notion introduces necessary and sufficient conditions for functional observability and detectability in a unified way without resorting to system observability decomposition, and facilitates the design of a functionally observable/detectable system. Afterwards, SFO is redefined rigorously from a generic perspective, contrarily to the definition of structural observability. A complete graph-theoretic characterization for SFO is proposed. Based on these results, the problems of selecting the minimal sensors from a prior set to achieve functional observability and SFO are shown to be NP-hard. Nevertheless, supermodular set functions are established, leading to greedy heuristics that can find approximation solutions to these problems with provable guarantees in polynomial time. A closed-form solution along with a constructive procedure is also given for the unconstrained case on systems with diagonalizable state matrices. Notably, our results also yield a polynomial-time verifiable case for structural target controllability, a problem that may be hard otherwise.