On the relationship between parity space and approaches to fault detection
Introduction
Parity space approach and approach are two commonly used approaches for designing robust fault detection systems [1], [9], [14]. The former is initially proposed by [3], [4] and has been extensively studied since then [2], [5], [6], [7], [10], [12], [13], [15]. The latter is proposed by [8].
In this paper, some insight will be shed on the relationship between these two approaches, which simultaneously enhances our understanding about the optimal solution of the parity space approach and provides us a numerical way to calculate the -optimal solution of residual generator design. It is proven that the optimal parity vector approximates the -optimal residual generator and thus it is a bandpass filter whose bandwidth will become narrower as the order of the parity relation increases.
The paper is organized as follows. First, the parity space approach is briefly reviewed in Section 2. Then, Section 3 gives the optimal solution of the approach in the context of discrete-time systems. The relationship between the parity space approach and the approach is studied in Section 4. Finally, the results are illustrated by an example in Section 5.
Section snippets
Brief review of the parity space approach
In this contribution, we consider linear discrete time-invariant systems described bywhere denote the vector of states, control inputs, measurement outputs, unknown disturbances and faults to be detected, respectively. and are known matrices of appropriate dimensions. It is assumed that is observable.
A parity relation based residual generator can be constructed as [1], [10], [12]
Optimal solution of the approach
The approach is originally proposed in [8] in the context of linear continuous-time systems. In this section, a discrete-time version of this approach will be presented.
Given system (1)–(2), useto denote the transfer function matrices from and f to y, respectively. It is well-known that all linear time-invariant residual generators can be expressed by [9]where is called post-filter and
Relationship between two approaches
In this section, we present the main result of this paper, the discussion on the relationship between the optimal solutions of the parity space approach and the approach.
Suppose that is the impulse response of system(1)–(2) to the unknown disturbances. Apparently,The matrix can then be expressed in terms of the impulse response as follows Partition the parity vector as
Numerical example
Given a discrete-time system modelled by (1)–(2), where As system (38) is stable, matrix L in (14) can be selected to be zero matrix and thus is an identity matrix. To solve the generalized eigenvalue–eigenvector problem (19) to get that achieves , note that Therefore, the optimal performance index of the approach is and the selective frequency is .
Fig. 1
Conclusion
The relationship between the parity space approach and the approach to fault detection of linear discrete time-invariant systems has been discussed in this paper. It is shown that with the increase of the order of the parity relation s, the optimal performance index of the parity space approach converges to that of the approach, and the frequency response of the optimal parity vector also converges to the optimal post-filter in the approach. This result not only leads to a
Acknowledgements
The authors would like to thank the anonymous reviewer for the insightful comments. This work was in part supported by the DAAD, the National Natural Science Foundation of China and the National Education Ministry of China.
References (17)
- et al.
A new structural framework for parity equation-based failure detection and isolation
Automatica
(1990) - et al.
Optimally robust redundancy relations for failure detection in uncertain system
Automatica
(1986) - et al.
Robust Model-based Fault Diagnosis for Dynamic Systems
(1999) - et al.
Analytical redundancy and the design of robust failure detection systems
IEEE Trans. Automat. Control
(1984) - et al.
F-8 DFBW sensor failure identification using analytic redundancy
IEEE Trans. Automat. Control
(1977) - M. Desai, A. Ray, A fault detection and isolation methodology. Proceedings of the 20th IEEE CDC, 1981, pp....
- et al.
A characterization of parity space and its application to robust fault detection
IEEE Trans. Automat. Control
(1999) - S.X. Ding, E.L. Ding, T. Jeinsch, An approach to analysis and design of observer and parity relation based FDI systems,...
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