Brief paperAsymptotic behavior of a fault diagnosis performance measure for linear systems☆
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 where are state variables, are measured signals, are input signals, are modeled faults, and 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 , since distinguishability for an arbitrary fault size is given by or . The main result for asymptotic behavior of the detectability performance measure is the following.
Theorem 1 There exist constants , , and
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: and . After some row operations on the measurement equations, it can be assumed that the fault is affecting a single sensor, and computing , is reduced to computing 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 , 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 ,
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 together with inequality (24), and the relation
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)
- et al.
A method for quantitative fault diagnosability analysis of stochastic linear descriptor models
Automatica
(2013) - et al.
Detection of abrupt changes: theory and application, vol. 104
(1993) - et al.
Anomaly detection/detectability for a linear model with a bounded nuisance parameter
Annual Reviews in Control
(2014)
Cited by (5)
Fault Diagnosability Evaluation for Markov Jump Systems With Multiple Time Delays
2022, IEEE Transactions on Systems, Man, and Cybernetics: SystemsA Fault Diagnosability Evaluation Method for Dynamic Systems Without Distribution Knowledge
2022, IEEE Transactions on CyberneticsFault detection optimization for controllable dynamic systems
2019, Informatsionno-Upravliaiushchie Sistemy
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.
Erik Frisk was born in Stockholm, Sweden in 1971 and currently he has a position as Professor at the Department of Electrical Engineering at Linköping University, Sweden. Current research interests include fault diagnosis, prognostics, and autonomous functions using data driven and model based techniques.
Daniel Jung was born in Linköping, Sweden in 1984. He received a Ph.D. degree in 2015 from Linköping University, Sweden. During 2017 he was a Research Associate at the Center for Automotive Research at The Ohio State University, Columbus, OH, USA. Since 2018, he is an Assistant Professor at Linköping University. His current research interests include theory and applications of model based and data driven diagnosis, and optimal control of hybrid electric vehicles.
- ☆
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.