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Graph-Theoretic Analysis of Residual Generation Under Computational Constraints

Jan Åslund

Abstract

A unified structural framework is presented for model-based fault diagnosis that explicitly incorporates both fault locations and constraints imposed by the residual generation methodology. Building on the concepts of proper and minimal structurally overdetermined (PSO/MSO) sets and Test Equation Supports (TES/MTES), the framework introduces testable PSO sets, Residual Generation (RG) sets, irreducible fault signatures (IFS), and Irreducible RG (IRG) sets to characterize which submodels are suitable for residual generation under given computational restrictions. An operator $M^*$ is defined to extract, from any model, the largest testable PSO subset consistent with a specified residual generation method. Using this operator, an algorithm is developed to compute all RG sets, and it is shown that irreducible fault signature sets form the join-irreducible elements of a join-semilattice of sets and fully capture the multiple-fault isolability properties in the method-constrained setting. The approach is exemplified on a semi-explicit linear DAE model, where low structural differential index can be used to define $M^*$. The results demonstrate that the proposed framework generalizes MTES-based analysis to residual generation scenarios with explicit computational limitations.

Graph-Theoretic Analysis of Residual Generation Under Computational Constraints

Abstract

A unified structural framework is presented for model-based fault diagnosis that explicitly incorporates both fault locations and constraints imposed by the residual generation methodology. Building on the concepts of proper and minimal structurally overdetermined (PSO/MSO) sets and Test Equation Supports (TES/MTES), the framework introduces testable PSO sets, Residual Generation (RG) sets, irreducible fault signatures (IFS), and Irreducible RG (IRG) sets to characterize which submodels are suitable for residual generation under given computational restrictions. An operator is defined to extract, from any model, the largest testable PSO subset consistent with a specified residual generation method. Using this operator, an algorithm is developed to compute all RG sets, and it is shown that irreducible fault signature sets form the join-irreducible elements of a join-semilattice of sets and fully capture the multiple-fault isolability properties in the method-constrained setting. The approach is exemplified on a semi-explicit linear DAE model, where low structural differential index can be used to define . The results demonstrate that the proposed framework generalizes MTES-based analysis to residual generation scenarios with explicit computational limitations.

Paper Structure

This paper contains 10 sections, 3 theorems, 21 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Basic definitions
  4. Restrictions on the residual generation methodology
  5. Residual generation
  6. Fault signature
  7. Fault detectability and fault isolability
  8. Algorithm
  9. Example: Differential algebraic equations
  10. Conclusions

Key Result

Lemma 1

Given a fault signature $F$, there exists a unique testable PSO set $M$ such that $F(M) = F$ and $M' \subset M$ for all testable PSO sets $M'$ satisfying $F(M') = F$.

Figures (1)

  • Figure 1: Illustration of the bi-adjacency matrix in a Dulmage--Mendelsohn decomposition, where all edges lie within the shaded area

Theorems & Definitions (16)

  • Definition 1: PSO set, MSO set, and redundancy $\varphi$
  • Definition 2: TES and MTES
  • Definition 3: Structurally testable
  • Definition 4: The operator $M^*$
  • Definition 5: Fault signature
  • Lemma 1
  • proof
  • Definition 6: RG set
  • Definition 7: Structurally detectable
  • Definition 8: Structurally isolable
  • ...and 6 more