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Output behavior equivalence and simultaneous subspace identification of systems and faults

Gabriel de Albuquerque Gleizer

TL;DR

This paper tackles the problem of identifying an LTI system in the presence of additive unknown faults while reconstructing the fault signal, without assuming a fault model or nominal data. It combines a subspace identification approach (PI-MOESP) to consistently estimate the nominal matrices $A,B,C,D$ under mild fault-signal conditions with a novel notion of output behavioral equivalence to characterize when different fault representations yield the same outputs. An exact fault-identification method is developed using left-invertibility and a structured factorization to recover fault matrices $F,G$ up to a right-multiplication, along with a procedure to recover the initial state and fault trajectory. The paper also discusses a rank-minimization alternative and demonstrates the methods on a numerical example and Monte-Carlo studies, highlighting both potential and limitations, particularly in the presence of transmission zeros and noise. Overall, the framework enables fault-aware system identification in operation, supporting applications such as digital twins and online fault reconstruction.

Abstract

We address the problem of identifying a system subject to additive faults, while simultaneously reconstructing the fault signal via subspace methods. We do not require nominal data for the identification, neither do we impose any assumption on the class of faults, e.g., sensor or actuator faults. We show that, under mild assumptions on the fault signal, standard PI-MOESP can recover the system matrices associated to the input-output subsystem. Then we introduce the concept of output behavior equivalence, which characterizes systems with the same output behavior set, and present a method to establish this equivalence from system matrices. Finally, we show how to estimate from data the complete set of fault matrices for which there exist a fault signal with minimal dimension that explains the data.

Output behavior equivalence and simultaneous subspace identification of systems and faults

TL;DR

This paper tackles the problem of identifying an LTI system in the presence of additive unknown faults while reconstructing the fault signal, without assuming a fault model or nominal data. It combines a subspace identification approach (PI-MOESP) to consistently estimate the nominal matrices under mild fault-signal conditions with a novel notion of output behavioral equivalence to characterize when different fault representations yield the same outputs. An exact fault-identification method is developed using left-invertibility and a structured factorization to recover fault matrices up to a right-multiplication, along with a procedure to recover the initial state and fault trajectory. The paper also discusses a rank-minimization alternative and demonstrates the methods on a numerical example and Monte-Carlo studies, highlighting both potential and limitations, particularly in the presence of transmission zeros and noise. Overall, the framework enables fault-aware system identification in operation, supporting applications such as digital twins and online fault reconstruction.

Abstract

We address the problem of identifying a system subject to additive faults, while simultaneously reconstructing the fault signal via subspace methods. We do not require nominal data for the identification, neither do we impose any assumption on the class of faults, e.g., sensor or actuator faults. We show that, under mild assumptions on the fault signal, standard PI-MOESP can recover the system matrices associated to the input-output subsystem. Then we introduce the concept of output behavior equivalence, which characterizes systems with the same output behavior set, and present a method to establish this equivalence from system matrices. Finally, we show how to estimate from data the complete set of fault matrices for which there exist a fault signal with minimal dimension that explains the data.
Paper Structure (11 sections, 8 theorems, 27 equations, 2 figures)

This paper contains 11 sections, 8 theorems, 27 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM FORMULATION
  3. Mathematical notation
  4. System model and problem statement
  5. IDENTIFICATION OF INPUT-OUTPUT SUBSYSTEM
  6. FAULT SYSTEM IDENTIFICATION
  7. Rank minimization approach
  8. A notion of output behavioral equivalence
  9. Exact approach for fault system identification
  10. NUMERICAL EXAMPLE
  11. CONCLUSION

Key Result

Proposition 1

Let Assumptions ass:ergodic_input--ass:noise hold. Then, given data sequences $\bm{y}(k),$$\bm{u}(k),$ the PI-MOESP method as described in verhaegen1993subspace,verhaegen2007filtering provides a consistent estimate of the matrices $\bm{A},\bm{B},\bm{C},\bm{D}$.

Figures (2)

  • Figure 1: Singular values of Hankel residual matrices $\bm{R}_5$ and $\bm{R}_{6}$ when using identified system to generate the residual. Clearly, only the seventh singular value of $\bm{R}_5$ is negligible compared to that of $\bm{R}_{6}$, indicating that ${n_{\mathrm{v}}}=1$.
  • Figure 2: Grassmanian error between real and recovered fault matrices in the Monte-Carlo simulation.

Theorems & Definitions (20)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Definition 1: Output behavior equivalence
  • Lemma 1
  • proof
  • Theorem 1
  • Definition 2: Left-invertibility kirtikar2011delay
  • Theorem 2: Adapted from sanjeevini2020counting
  • ...and 10 more