This paper deals with the Robust Fault Detection (RFD) problem with the aid of the $H∞$ norm and $H−$ index optimization techniques and the LMI approach. First, a necessary and sufficient condition is proposed for the design of RFD observers in the nominal case. Then, the RFD problem for systems with structured uncertainties in the system matrices is considered. Approaches are proposed to design robust fault detection observers to enhance the fault detection and to attenuate the effects due to unknown inputs and uncertainties. Furthermore, the design of the threshold of fault detection is investigated. We also consider the fault sensitivity over finite frequency range in which case no constraint is required on $D$ being of full column rank for a system $(A,B,C,D)$. Numerical examples are employed to demonstrate the effectiveness of the proposed methods.

1.
Ding
,
X.
,
Jeinsch
,
T.
,
Frank
,
P. M.
, and
Ding
,
E. L.
, 2000, “
A Unified Approach to the Optimization of Fault Detection Systems
,”
Int. J. Adapt. Control Signal Process.
0890-6327,
14
(
7
), pp.
725
745
.
2.
Frank
,
P. M.
, and
Ding
,
X.
, 1997, “
Survey or Robust Residual Generation and Evaluation Methods in Observer-Based Fault Detection Systems
,”
J. Process Control
0959-1524,
7
(
6
), pp.
403
424
.
3.
Hou
,
M.
, and
Patton
,
R. J.
, 1996, “
An LMI Approach to H−∕H∞ Fault Detection Observers
,”
Proc. of the UKACC Int. Conf. on Contr.: CONTROL’96
,
IEE/University of Exter
, UK, pp.
305
310
.
4.
Wang
,
H. B.
,
Lam
,
J.
,
Ding
,
X.
, and
Zhong
,
M.
, 2002, “
An LMI Approach for Fault Detection Performance Enhancement
,” The 2002 International Conference on Control and Automation, Xiamen, China, pp.
1238
1242
.
5.
Wang
,
H. B.
,
Lam
,
J.
,
Ding
,
X.
, and
Zhong
,
M.
, 2005, “
Iterative LMI Algorithms for Fault Detection With Unknown Inputs
,”
Proc. Inst. Mech. Eng. Part I—J. Syst. Control Eng.
,
219
(
2
), pp.
161
172
.
6.
Zhong
,
M.
,
Ding
,
X.
,
Lam
,
J.
, and
Wang
,
H. B.
, 2003, “
An LMI Approach to Design Robust Fault Detection Filter for Uncertain LTI Systems
,”
Automatica
0005-1098,
39
(
3
), pp.
543
550
.
7.
Patton
,
R. J.
, and
Hou
,
M.
, 1997, “
H∞ Estimation and Robust Fault Detection
,”
Proc. of the 1997 European Control Conference
, Brussels, Belgium.
8.
Rank
,
M. L.
, and
Niemann
,
H.
, 1999, “
Norm Based Design of Fault Detectors
,”
Int. J. Control
0020-7179,
72
(
9
), pp.
773
783
.
9.
Liu
,
J.
,
Wang
,
J. L.
, and
Yang
,
G. H.
, 2002, “
Worst-Case Fault Detection Observer Design: An LMI Approach
,” The 2002 International Conference on Control and Automation, Xiamen, China, pp.
1243
1247
.
10.
Liu
,
J.
,
Wang
,
J. L.
, and
Yang
,
G. H.
, 2003, “
An LMI Approach to Worst Case Analysis for Fault Detection Observers
,”
Proc. 2003 American Control Conference
2985
2990
.
11.
Liu
,
J.
,
Wang
,
J. L.
, and
Yang
,
G. H.
, 2005, “
An LMI Approach to Minimum Sensitivity Analysis With Application to Fault Detection
,”
Automatica
0005-1098,
41
(
11
), pp.
1995
2004
.
12.
Wang
,
J. L.
,
Yang
,
G. H.
, and
Liu
,
J.
, 2007, “
An LMI Approach to H− Index and Mixed H−∕H∞ Fault Detection Observer Design
,” Automatica,
43
, to appear.
13.
Duan
,
G.
,
Howe
,
D.
, and
Patton
,
R. J.
, 2002, “
Robust Fault Detection in Descriptor Linear Systems via Generalized Unknown Input Observers
,”
Int. J. Syst. Sci.
0020-7721,
33
(
5
), pp.
369
377
.
14.
Zhou
,
K.
,
Doyle
,
J. C.
, and
Glover
,
K.
, 1996,
Robust and Optimal Control
,
Prentice-Hall
, Englewood Cliffs, NewJersey.
15.
Li
,
X.
, and
de Souza
,
C. E.
, 1997, “
Delay-Dependent Robust Stability and Stabilization of Uncertain Linear Delay Systems: A Linear Matrix Inequality Approach
,”
IEEE Trans. Autom. Control
0018-9286,
42
(
8
), pp.
1144
1148
.
16.
Xu
,
S.
,
Lam
,
J.
, and
Yang
,
C.
, 2001, “
H∞ and Positive Real Control for Linear Neutral Delay Systems
,”
IEEE Trans. Autom. Control
0018-9286,
46
(
8
), pp.
1321
1326
.
17.
Wang
,
H. B.
,
Wang
,
J. L.
,
Liu
,
J.
, and
Lam
,
J.
, 2003, “
Iterative LMI Approach for Robust Fault Detection Observer Design
,”
Proceedings of the 42th IEEE Conference on Decision and Control
, pp.
1974
1979
.