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Functional Observability, Structural Functional Observability and Optimal Sensor Placement

Yuan Zhang, Tyrone Fernando, Mohamed Darouach

TL;DR

This paper advances functional observability by introducing modal functional observability and deriving necessary and sufficient conditions that operate mode-by-mode in the eigenstructure. It rigorously redefines structural functional observability (SFO), contrasting it with classical structural observability and presenting a complete graph-theoretic characterization via maximum independent walks. It proves that minimal sensor-placement problems for functional observability and SFO are NP-hard, yet provides supermodular-optimization-based greedy guarantees and a closed-form solution for diagonalizable systems using real Jordan forms. The results offer polynomial-time verifiable SFO criteria and practical sensor-design guidance, with implications for structural target controllability and scalable analysis in large-scale networks.

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.

Functional Observability, Structural Functional Observability and Optimal Sensor Placement

TL;DR

This paper advances functional observability by introducing modal functional observability and deriving necessary and sufficient conditions that operate mode-by-mode in the eigenstructure. It rigorously redefines structural functional observability (SFO), contrasting it with classical structural observability and presenting a complete graph-theoretic characterization via maximum independent walks. It proves that minimal sensor-placement problems for functional observability and SFO are NP-hard, yet provides supermodular-optimization-based greedy guarantees and a closed-form solution for diagonalizable systems using real Jordan forms. The results offer polynomial-time verifiable SFO criteria and practical sensor-design guidance, with implications for structural target controllability and scalable analysis in large-scale networks.

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.
Paper Structure (18 sections, 23 theorems, 45 equations, 2 figures)

This paper contains 18 sections, 23 theorems, 45 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notations
  4. Graph theory
  5. Structured system theory
  6. Submodular set function
  7. Functional observability and detectability
  8. Modal functional observability and new criteria for functional observability/detectability
  9. Redefined structural functional observability
  10. Criteria for SFO
  11. Linking and independent walk family
  12. Characterizations of structural functional observability
  13. Implication to structural target controllability
  14. Minimal sensor placement for functional observability
  15. Computational complexity
  16. ...and 3 more sections

Key Result

Lemma 1

genericRamos2022AnOO The pair $(\bar{A}, \bar{C})$ is structurally observable, if and only if the whole state vertex set $X$ is covered by a cactus configuration in ${\cal G}(\bar{A}, \bar{C})$.

Figures (2)

  • Figure 1: An example of ${\cal G}(\bar{A}, \bar{C})$ and its associated ${\cal D}(\bar{A}, \bar{C})$, where $(\bar{A}, \bar{C})$ is given in Example \ref{['counter-example-2']}.
  • Figure 2: Illustration of the construction from a set cover problem to an instance of ${\cal P}_2$. Filled nodes represent functional states. Since each state has a dedicated sensor, only ${\cal G}(\bar{A})$ is present.

Theorems & Definitions (48)

  • Definition 1
  • Lemma 1
  • Definition 2: Functional observability, fernando2010functional2
  • Definition 3: Functional detectability, Darouach2023FunctionalDA
  • Definition 4: Modal functional observability
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • Corollary 1
  • ...and 38 more