Elsevier

Systems & Control Letters

Volume 82, August 2015, Pages 115-120
Systems & Control Letters

H/H fault detection observer design for two-dimensional Roesser systems

https://doi.org/10.1016/j.sysconle.2015.04.005Get rights and content

Abstract

This paper addresses the problem of H/H fault detection for two-dimensional (2-D) systems with disturbances in Roesser model. A fault detection observer is designed to satisfy a finite frequency H index and an H index simultaneously. The finite frequency H index is used to increase the fault sensitivity in low frequency domain while the H index is used to attenuate the effects of disturbances in full frequency domain. By the generalized Kalman–Yakubovich–Popov lemma and some useful lemmas, sufficient design conditions are obtained. An example is given to illustrate the effectiveness of the proposed method.

Introduction

In the past several decades, a large attention has been paid to two-dimensional (2-D) systems  [1], [2], [3], [4]. The interest in 2-D systems has mainly been motivated by a wide variety of applications, arising in theory and in practice, such as image processing, multidimensional signal processing, multidimensional digital filtering, thermal process, repetitive process, etc. Originated from Roesser  [1], Fornasini and Marchesini [2], 2-D state-space models have attracted a lot of interest due to its advantage of providing a simple and intuitive research method for 2-D signals and systems. Based on state-space models, several properties have been investigated, such as stability [5], [6], controllability and observability [7], [8], etc. The problems of feedback control and filtering for 2-D systems have been extensively studied, e.g.,  [4], [9], [10], [11], [12], [13], [14], [15], [16], [17].

Due to the increasing demand for reliability and safety in industrial processes, the fault detection problem has drawn great attention  [18], [19]. The main challenge here is to distinguish faults from other disturbances. Many approaches have been proposed to solve this problem. Among them, the H/H approach is an effective one [18], [19], [20], [21], [22], [23], [24], [25]. It is worth noticing that most of the approaches in  [18], [19], [20], [21], [22], [23], [24], [25] consider faults in full frequency domain. In practice, however, faults usually emerge in low frequency domain, e.g., for an incipient signal, the fault information is contained within a low frequency band as the fault development is slow  [18], and the actuator stuck failures that occur in flight control systems just belong to low frequency domain  [26]. This motivates the fault detection observer design for one-dimensional (1-D) linear systems in low frequency domain  [26], [27], [28].

To date, however, few works have been reported on fault detection for 2-D systems  [29], [30], [31]. In  [29], a complete theory of dead-beat observer-based fault detectors and isolators has been developed. The method in  [29] has been extended to deal with the fault detection and isolation problem for 2-D systems in a quite general setting in  [30]. The generalized H2 fault detection for 2-D discrete-time Markovian jump systems has been investigated in  [31], where LMI-type sufficient conditions for the existence of a desired fault detection filter has been derived. It is noted that the fault detection and isolation or fault detection problems in  [29], [30], [31] are considered in full frequency domain. Motivated by [26], [27], [28], in this paper, we propose a finite frequency fault detection method for 2-D systems in Roesser model. A fault detection observer is designed which satisfies the finite frequency H and H indices to increase the fault sensitivity in finite frequency domain and attenuate the effects of disturbances in full frequency domain, respectively. The proposed fault detection observer can distinguish faults in finite frequency domain from other disturbances effectively. By the newly developed generalized Kalman–Yakubovich–Popov lemma  [32] and some useful lemmas, sufficient design conditions are obtained. The effectiveness of the proposed method is illustrated by an example.

The rest of the paper is organized as follows. Section  2 gives the problem statement and preliminaries. The main results of the paper are presented in Section  3, where the fault detection observer design conditions are developed in detail. Section  4 gives an example to illustrate the effectiveness of the proposed method. Finally, conclusions are given in Section  5.

Notations. We use standard notations throughout this paper. For a matrix M, M, M denote its conjugate transpose and orthogonal complement, respectively. M>0(M<0) means that M is positive definite (negative definite). The symbol will be used in some matrix expressions to induce a symmetric structure. The Hermitian part of a square matrix M is denoted by He(M)M+M. σmin(G), σmax(G) denote the minimum singular value and the maximum singular value of the transfer matrix G, respectively.

Section snippets

Problem formulation

Consider the following 2-D discrete-time system described by the Roesser model [xh(i+1,j)xv(i,j+1)]=A[xh(i,j)xv(i,j)]+Buu(i,j)+Bff(i,j)+Bdd(i,j)y(i,j)=C[xh(i,j)xv(i,j)]+Duu(i,j)+Dff(i,j)+Ddd(i,j) where i,j indicate the horizontal (or spatial) and vertical (or time) indices, xh(i,j)Rnh is the state evolving horizontally (or spatially), xv(i,j)Rnv is the state evolving vertically (or in time), u(i,j)Rnu is the control input, y(i,j)Rny is the measured output, d(i,j)Rnd is the disturbance

Main results

In this section, a fault detection observer design method satisfying condition (i)–(iii) will be given.

Example

Heat exchangers play an important role in gas turbine engines including preheating fuel to increase the engine efficiency [36], [37], [33]. As pointed out in  [37], [33], a heat exchanger is usually subject to two different types of faults, namely fouling and leakage. In this section, the proposed fault detection method is applied to a 2-D Roesser model which represents an approximation model of a heat exchanger.

The mathematical model of a typical heat exchanger is governed by the following

Conclusions

This paper has investigated the problem of H/H fault detection observer design for 2-D Roesser systems. A fault detection observer has been designed to satisfy two performance indices: the finite frequency H index is used to increase the fault sensitivity in low frequency domain, and H index is used to attenuate the effects of disturbances in full frequency domain. By the generalized Kalman–Yakubovich–Popov lemma and some useful lemmas, sufficient design conditions have been obtained. The

Acknowledgments

This work was supported in part by National Natural Science Foundation of China under Grants 61104013, 61473032, and 61473034, Program for New Century Excellent Talents in University under Grant NCET-13-0662, Beijing Natural Science Foundation under Grant 4153061, and the Fundamental Research Funds for the Central Universities under Grant FRF-TP-14-010C1.

References (37)

  • R.P. Roesser

    A discrete state-space model for linear image processing

    IEEE Trans. Automat. Control

    (1975)
  • E. Fornasini et al.

    State-space realization theory of two-dimensional filters

    IEEE Trans. Automat. Control

    (1976)
  • T. Kaczorek

    Two-Dimensional Linear Systems

    (1985)
  • C. Du et al.

    Control and Filtering of Two-Dimensional Systems

    (2002)
  • T. Hinamoto

    Stability of 2-D discrete systems described by the Fornasini–Marchesini second model

    IEEE Trans. Circuits Syst. I, Fund. Theory Appl.

    (1997)
  • T. Bose et al.

    Stability of the 2-D Givone–Roesser model with periodic coefficients

    IEEE Trans. Circuits Syst. I, Reg. Papers

    (2007)
  • R. Eising

    Controllability and observability of 2-D systems

    IEEE Trans. Automat. Control

    (1979)
  • E. Rogers et al.

    Stability and controllability of a class of 2-D linear systems with dynamic boundary conditions

    IEEE Trans. Circuits Syst. I

    (2002)
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