Set-Membership Estimation for Fault Diagnosis of Nonlinear Systems

TL;DR

The methodology advances fault diagnosis by continuously evaluating an estimate of the fault parameter and a feasible parameter set where the true fault parameter belongs, and adaptive regularization of the parameter estimates is introduced to enhance the estimation process when the input-output data are sparse or non-informative, enhancing fault identifiability.

Abstract

This paper introduces a Fault Diagnosis (Detection, Isolation, and Estimation) method using Set-Membership Estimation (SME) designed for a class of nonlinear systems that are linear to the fault parameters. The methodology advances fault diagnosis by continuously evaluating an estimate of the fault parameter and a feasible parameter set where the true fault parameter belongs. Unlike previous SME approaches, in this work, we address nonlinear systems subjected to both input and output uncertainties by utilizing inclusion functions and interval arithmetic. Additionally, we present an approach to outer-approximate the polytopic description of the feasible parameter set by effectively balancing approximation accuracy with computational efficiency resulting in improved fault detectability. Lastly, we introduce adaptive regularization of the parameter estimates to enhance the estimation process when the input-output data are sparse or non-informative, enhancing fault identifiability. We demonstrate the effectiveness of this method in simulations involving an Autonomous Surface Vehicle in both a path-following and a realistic collision avoidance scenario, underscoring its potential to enhance safety and reliability in critical applications.

Figures (8)

• Figure 1: Overview of the different steps in SME in a 2-D example: First, the UPS, $\mathrm{\bm{\Delta}}_{k}$, is computed at each time step $k$, based on the latest input-output measurements. The FPS, denoted as $\mathrm{\bm{\Theta}}_k$, is recursively computed based on the intersection of the existing FPS, $\mathrm{\bm{\Theta}}_{k-1}$, and the current update from $\mathrm{\bm{\Delta}}_{k}$. The FPS is then outer-approximated by a simpler polytope, denoted as $\overline{\mathrm{\bm{\Theta}}}_{k}$. Lastly, an estimate, denoted as $\bm{\hat{\theta}}_{k} \in \overline{\mathrm{\bm{\Theta}}}_{k}$, is computed.
• Figure 2: On the right is our formulation of the UPS that additionally accounts for measurement noise and thus all possible parameter realizations are guaranteed to lie inside the UPS. In contrast, if the noise is not accounted for, several realizations will be outside the UPS.
• Figure 3: The steps to compute the FPS. This starts with computing the predefined directions offline (Step 1). Then the process continues with computing the new FPS from the current UPS (Step 2), computing the vertices of the new FPS (Step 3), and lastly outer-approximating the new FPS based on the predefined directions (Step 4). The process starts again from Step 2, starting from the new outer approximation. Different choices of predefined directions will result in a different outer approximation of the FPS (top and bottom rows).
• Figure 4: Schematic representation of fault detection using the inverse test on the FPS. In this example, the UPS arises from a measurement corrupted by a fault. In the left figure, the outer approximation of the FPS is more conservative, preventing fault detection, as $\mathrm{\overline{\boldsymbol{\Theta}}}_{k-1} \cap \boldsymbol{\Delta}_{k} \neq \emptyset$. In the right figure, due to a tighter outer approximation of the FPS, the same measurement results in $\mathrm{\overline{\boldsymbol{\Theta}}}_{k-1} \cap \boldsymbol{\Delta}_{k} = \emptyset$, successfully revealing the fault.
• Figure 5: Evolution of the FPS in healthy conditions. In blue, the FPS considers measurement noise, converging towards the "healthy" region. In contrast, the orange FPS, which neglects noise, becomes infeasible multiple times and fails to converge uniformly.