A Geometric Approach to Fault Detection and Isolation of Two-Dimensional (2D) Systems

TL;DR

This paper introduces a geometric fault detection and isolation method for 2D systems modeled by FMII, extending invariant subspace concepts from 1D to 2D, with algorithms for finite-step computation and practical validation.

Contribution

It generalizes invariant subspace theory to 2D systems using infinite-dimensional representations, enabling finite-step algorithms for FDI and providing conditions for solvability.

Findings

Developed algorithms for finite-step computation of invariant subspaces in 2D systems.

Provided necessary and sufficient conditions for FDI problem solvability.

Validated the approach through numerical simulations on a heat exchanger PDE model.

Abstract

In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time two-dimensional (2D) systems that are represented by the Fornasini-Marchesini model II (FMII). This is accomplished by generalizing the basic invariant subspaces including unobservable, conditioned invariant and unobservability subspaces of 1D systems to 2D models. These extensions have been achieved and facilitated by representing a 2D model as an infinite dimensional (Inf-D) system on a Banach vector space, and by particularly constructing algorithms that compute these subspaces in a \emph{finite and known} number of steps. By utilizing the introduced subspaces the FDI problem is formulated and necessary and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the FDI problem for 2D systems using both deadbeat and LMI filters are also developed. Moreover, the capabilities and advantages of our proposed approach are demonstrated by performing an analytical comparison with the currently available 2D geometric methods in the literature. Finally, numerical simulations corresponding to an approximation of a hyperbolic partial differential equation (PDE) system of a heat exchanger, that is mathematically represented as a 2D model, have also been provided.