Asymptotic behavior of a fault diagnosis performance measure for linear systems

doi:10.1016/j.automatica.2019.04.041

Automatica

Volume 106, August 2019, Pages 143-149

https://doi.org/10.1016/j.automatica.2019.04.041 Get rights and content

Abstract

Fault detection and fault isolation performance of a model based diagnosis system mainly depends on the level of model uncertainty and the time allowed for detection. The longer time for detection that can be accepted, the more certain detection can be achieved and the main objective of this paper is to show how the window length relates to a diagnosis performance measure. A key result is an explicit expression for asymptotic performance with respect to window length and it is shown that there exists a linear asymptote as the window length tends to infinity. The gradient of the asymptote is a system property that can be used in the evaluation of diagnosis performance when designing a system. A key property of the approach is that the model of the system is analyzed directly, which makes the approach independent of detection filter design.

Introduction

A quantitative fault diagnosability performance measure for fault detection and isolation of time-discrete linear descriptor models with Gaussian distributed noise has been proposed in Eriksson, Frisk, and Krysander (2013) and Harrou, Fillatre, and Nikiforov (2014). The measure, called distinguishability in Eriksson et al. (2013), quantifies diagnosability performance given a model and takes process noise, measurement noise, and fault time profiles into consideration and gives an upper limit of achievable fault-to-noise ratio of any linear residual generator generated from that model.

The asymptotic behavior of the distinguishability measure, when the number of samples goes to infinity, gives useful information about the fault detection and isolation properties of the system. In this paper, an expression of the distinguishability asymptote, as function of window length, is derived. It is shown that there exists a linear asymptote as the window length tends to infinity and it is also shown how to compute the asymptote.

Section snippets

Problem formulation

Before stating the problem formulation, a brief summary of the definition of the distinguishability measure in Eriksson et al. (2013) is presented. Consider a time-discrete linear model in the form $x_{t + 1} = A x_{t} + B^{u} u_{t} + B^{f} f_{t} + v_{t},$ $y_{t} = C x_{t} + D^{u} u_{t} + D^{f} f_{t} + ε_{t},$ where $x \in R^{n}$ are state variables, $y \in R^{n_{y}}$ are measured signals, $u \in R^{n_{u}}$ are input signals, $f \in R^{n_{f}}$ are modeled faults, $v_{t}$ and $ε_{t}$ are i.i.d. Gaussian random vectors with zero mean. Without loss of generality, identity covariance matrices are assumed. Furthermore, it

Asymptotic performance of distinguishability

In this section the main results of the paper are presented. The proofs are given in Section 6. It is assumed in the asymptotic analysis that the fault is constant and it is sufficient to consider the constant fault time profile $θ = {(1, 1, \dots, 1)}^{T}$ , since distinguishability for an arbitrary fault size $λ$ is given by $D^{i} (λ θ, N) = λ^{2} D^{i} (θ, N)$ or $D^{i, j} (λ θ, N) = λ^{2} D^{i, j} (θ, N)$ . The main result for asymptotic behavior of the detectability performance measure is the following.

Theorem 1

There exist constants $k^{i}$ , $m^{i}$ , and $0 < ϱ < 1$

The least squares problem

In the forthcoming sections, the asymptotic behavior of least squares problem in the form (6) will be studied, and now a procedure to transform the least squares problem (5) into the form (6) will be presented. The procedure is divided into three cases, which together cover all possible combinations.

Case 1: $B_{f}^{j} = 0$ and $D_{f}^{j} \neq 0$ . After some row operations on the measurement equations, it can be assumed that the fault is affecting a single sensor, and computing $D^{i, j}$ , is reduced to computing $D^{i}$ for the

Asymptotics of the minimizing sequence

In this section, the two point boundary value problem (14a), (14b), and (14c), will be studied and asymptotic results will be derived for the solution $x_{k}$ , i.e., the minimizing sequence of the least squares problem (13). This is the first step in the derivation of the asymptotics of the distinguishability measure, i.e., the minimal value of the objective function in (13). The main result of this section is the following.

Theorem 3

If the system (1) is observable, then there exists a unique solution $x_{t}$ , $t = 1$

Proofs of the theorems in Section 3

The basic idea in the first proof is to substitute the asymptotic approximation of the solution of the least squares problem, given by Theorem 3, into the corresponding objective function to obtain the asymptotic results for the distinguishability measure. The details are given in the following proof of Theorem 1:

Proof

Using the inequality $| {‖ x ‖}^{2} - {‖ y ‖}^{2} | \leq (‖ x ‖ + ‖ y ‖) ‖ x - y ‖$ together with inequality (24), $r_{t} = x_{t} - s_{t}$ and $k = \frac{1}{2} ({‖ x^{†} - A x^{†} - b ‖}^{2} + {‖ C x^{†} + d ‖}^{2})$ the relation $\frac{1}{2} \sum_{t = 1}^{N} {‖ x_{t + 1} - A x_{t} - b ‖}^{2} + {‖ C x_{t} + d ‖}^{2} = \frac{1}{2} \sum_{t = 1}^{N} ({‖ s_{t + 1}^{(1)} - A s_{t}^{(1)} - b ‖}^{2}$

Conclusions

In many fault diagnosis applications, the negative impact of sensor noise and model uncertainties on fault detection and isolation performance cannot be neglected. The analysis of the asymptote of the distinguishability measure gives useful information about fault detection and isolation properties for a given model. The gradient of the asymptote is relevant since it gives information about how much fault detection performance will improve for each new sample of sensor data which is useful for,

Jan Åslund was born in Boden, Sweden in 1971. He received a Ph.D. in Mathematics in 2002 from Linköping University, Sweden, and currently he has a position as Associate Professor at the Department of Electrical Engineering at Linköping University. His research interests include model based diagnosis and optimal control.

References (6)

ErikssonDaniel et al.
A method for quantitative fault diagnosability analysis of stochastic linear descriptor models
Automatica
(2013)
BassevilleMichèle et al.
Detection of abrupt changes: theory and application, vol. 104
(1993)
HarrouFouzi et al.
Anomaly detection/detectability for a linear model with a bounded nuisance parameter
Annual Reviews in Control
(2014)

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^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Angelo Alessandri under the direction of Editor Thomas Parisini.

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