On the relationship between parity space and H2 approaches to fault detection

doi:10.1016/j.sysconle.2005.05.006

Systems & Control Letters

Volume 55, Issue 2, February 2006, Pages 94-100

https://doi.org/10.1016/j.sysconle.2005.05.006 Get rights and content

Abstract

Parity space approach and $H_{2}$ approach are two important fault detection approaches. This paper studies the relationship between these two approaches, which reveals frequency domain characteristics of the optimal solution of the parity space approach on the one side and provides a numerical solution of the $H_{2}$ -optimal design of residual generators on the other side.

Introduction

Parity space approach and $H_{2}$ 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 $H_{2}$ -optimal solution of residual generator design. It is proven that the optimal parity vector approximates the $H_{2}$ -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 $H_{2}$ approach in the context of discrete-time systems. The relationship between the parity space approach and the $H_{2}$ 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 by $x (k + 1) = Ax (k) + Bu (k) + E_{d} d (k) + E_{f} f (k),$ $y (k) = Cx (k) + Du (k) + F_{d} d (k) + F_{f} f (k),$ where $x \in R^{n}, u \in R^{k_{u}}, y \in R^{m}, d \in R^{k_{d}}, f \in R^{k_{f}}$ denote the vector of states, control inputs, measurement outputs, unknown disturbances and faults to be detected, respectively. $A, B, C, D, E_{d}, E_{f}, F_{d}$ and $F_{f}$ are known matrices of appropriate dimensions. It is assumed that $(C, A)$ is observable.

A parity relation based residual generator can be constructed as [1], [10], [12]

Optimal solution of the $H_{2}$ approach

The $H_{2}$ 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), use $G_{u} (z) = C (zI - A)^{- 1} B + D, G_{d} (z) = C (zI - A)^{- 1} E_{d} + F_{d}, G_{f} (z) = C (zI - A)^{- 1} E_{f} + F_{f}$ to denote the transfer function matrices from $u, d$ and f to y, respectively. It is well-known that all linear time-invariant residual generators can be expressed by [9] $r (z) = R (z) ({\hat{M}}_{u} (z) y (z) - {\hat{N}}_{u} (z) u (z)),$ where $R (z) \in {RH}_{\infty}$ 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 $H_{2}$ approach.

Suppose that ${g_{d} (0), g_{d} (1), \dots}$ is the impulse response of system(1)–(2) to the unknown disturbances. Apparently, $g_{d} (0) = F_{d}, g_{d} (1) = {CE}_{d}, \dots, g_{d} (s) = {CA}^{s - 1} E_{d}, \dots$ The matrix $H_{d, s}$ can then be expressed in terms of the impulse response as follows $H_{d, s} = [\begin{matrix} g_{d} (0) & O & \dots & O \\ g_{d} (1) & g_{d} (0) & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & O \\ g_{d} (s) & \dots & g_{d} (1) & g_{d} (0) \end{matrix}] .$ Partition the parity vector $v_{s}$ as $v_{s} = [\begin{matrix} v_{s, 0} & v_{s, 1} & \dots & v_{s, s} \end{matrix}],$

Numerical example

Given a discrete-time system modelled by (1)–(2), where $A = [\begin{matrix} 1 & - 1.30 \\ 0.25 & - 0.25 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}], C = [\begin{matrix} 0 & 1 \end{matrix}],$ $E_{d} = [\begin{matrix} 0.4 \\ 0.5 \end{matrix}], E_{f} = [\begin{matrix} 0.6 \\ 0.1 \end{matrix}], D = F_{d} = F_{f} = 0 .$ As system (38) is stable, matrix L in (14) can be selected to be zero matrix and thus ${\hat{M}}_{u} (z)$ is an identity matrix. To solve the generalized eigenvalue–eigenvector problem (19) to get $ω_{0}$ that achieves $σ_{\min} (ω_{0}) = \inf_{ω} σ_{\min} (ω)$ , note that $σ_{\min} (ω) = \frac{0.41 - 0.4 \cos ω}{0.0125 + 0.01 \cos ω} .$ Therefore, the optimal performance index of the $H_{2}$ approach is $J_{opt} = 0.4444$ and the selective frequency is $ω_{0} = 0$ .

Fig. 1

Conclusion

The relationship between the parity space approach and the $H_{2}$ 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 $H_{2}$ approach, and the frequency response of the optimal parity vector also converges to the optimal post-filter $R_{opt} (z)$ in the $H_{2}$ 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.

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On the relationship between parity space and H2 approaches to fault detection

Abstract

Introduction

Section snippets

Brief review of the parity space approach

Optimal solution of the H2 approach

Relationship between two approaches

Numerical example

Conclusion

Acknowledgements

Automatica

Automatica

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Analytical redundancy and the design of robust failure detection systems

IEEE Trans. Automat. Control

F-8 DFBW sensor failure identification using analytic redundancy

IEEE Trans. Automat. Control

A characterization of parity space and its application to robust fault detection

IEEE Trans. Automat. Control

On the relationship between parity space and $H_{2}$ approaches to fault detection

Optimal solution of the $H_{2}$ approach