research article robust fault detection for a class of...
TRANSCRIPT
Research ArticleRobust Fault Detection for a Class of Uncertain NonlinearSystems Based on Multiobjective Optimization
Bingyong Yan Huifeng Wang and Huazhong Wang
Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of EducationDepartment of Automation East China University of Science and Technology Shanghai 200237 China
Correspondence should be addressed to Huazhong Wang hzwangecusteducn
Received 30 August 2014 Accepted 12 October 2014
Academic Editor Wei Zhang
Copyright copy 2015 Bingyong Yan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A robust fault detection scheme for a class of nonlinear systems with uncertainty is proposed The proposed approach utilizesrobust control theory and parameter optimization algorithm to design the gainmatrix of fault tracking approximator (FTA) for faultdetectionThe gainmatrix of FTA is designed tominimize the effects of systemuncertainty on residual signals whilemaximizing theeffects of system faults on residual signalsThe design of the gainmatrix of FTA takes into account the robustness of residual signalsto systemuncertainty and sensitivity of residual signals to system faults simultaneously which leads to amultiobjective optimizationproblem Then the detectability of system faults is rigorously analyzed by investigating the threshold of residual signals Finallysimulation results are provided to show the validity and applicability of the proposed approach
1 Introduction
In recent years with the development of intelligent controltechnology and processing ability of microchips moderndynamic systems are becoming more and more complexFault detection and identification (FDI) can be deployed formonitoring and reacting to the faults occurring in thesemod-ern dynamic systems which has a broad range of applicationsincluding intelligent power grids underwater robot highvoltage direct current transmission lines and long transmis-sion lines in pneumatic chemical processes Effective FDIschemes can ensure safety and reliability of complex dynamicsystemsThemost fundamental problem for FDI is to developa fault diagnosis observer or fault diagnosis filter Due to thedevelopment of various nonlinear or linear states observersthe model-based analytical redundancy approaches receivedmore and more attention in the last two decades includingobserver based method parity space based method eigen-structure assignment based method and parameter identifi-cation based method and 119867
infinfilter based method [1ndash8]
Early fault diagnosis approaches often assumed the avail-ability of an accurate dynamic system model In practicehowever such an assumption can be invalid The mainreason is that unstructuredmodeling uncertainties are always
unavoidable when modeling system mathematical struc-tures Therefore there are a growing number of researchersfocusing their interests on FDI for nonlinear systems withuncertainty A novel integrated fault diagnosis and faulttolerant control algorithm for non-Gaussian singular time-delayed stochastic distribution control system was proposedbased on iterative learning observer The iterative learningobserver was developed to obtain estimation of systemfaults Recently in [9] the authors introduced a novel faultdetection and diagnosis for nonlinear non-Gaussian dynamicprocesses using kernel dynamic independent componentanalysis method Sensor fault diagnosis and identificationin nonlinear plants were discussed in [10] The faults underconsideration in [10] were assumed to be abruptly occur-ring calibration errors Thus an adaptive particle filter wasdeveloped to diagnose sensor faults and compensate fortheir effects Chen and Saif in [11] proposed an iterativelearning observer (ILO) based fault diagnosis approach forfault detection identification and accommodationThemaincharacteristic of the ILO was that its states were updated ordriven successively by the estimation errors of previous sys-tem outputs and control inputsThe observer gainmatrix andadaptive adjusting rule of the fault estimator are investigatedin detail
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 705725 9 pageshttpdxdoiorg1011552015705725
2 Mathematical Problems in Engineering
In our previous work [12] a robust fault tracking approx-imator (RFTA) based fault detection and identificationscheme for a class of nonlinear systems was developedand its stability properties were investigated as well Dueto the existence of unstructured modeling uncertaintiesthe convergence speed and fault tracking accuracy of theFTA will be influenced dramatically The objective of thiswork is to extend the previous research results to a class ofnonlinear systems with uncertainty and optimize the gainmatrix of FTA First of all we decompose the fault diagnosisproblem into a parameter optimization problem and a faultdetection problemThe parameter optimization problem canbe described as follows by using robust control technologyand parameter optimization algorithms the gain matrix ofFTA is designed tominimize the effects of system uncertaintyon residual signals while maximizing the effects of systemfaults on residual signals The design of the gain matrix ofFTA takes into account the robustness to system uncertaintyand sensitivity to system faults simultaneously which leadsto a multiobjective optimization problemThe fault detectionproblem can be described as follows we detect the systemfaults according to the relationship between the calculatedthreshold and residual signals The detectability of systemfaults is rigorously analyzed by investigating the thresholdof residual signals In the end an illustrative example isproposed to demonstrate the validity and applicability of theproposed approach
An outline of this paper is organized as follows InSection 2 we define a class of uncertain nonlinear systemsand present a multiobjective optimization problem Thedesign of gain matrix of FTA is investigated in Section 3The calculation of threshold for fault detection is designedin Section 4 In Section 5 simulation results are reportedillustrating the effectiveness of the proposed robust faultdetection scheme Some conclusion remarks are provided inSection 6
2 Problem Formulation
Consider a class of uncertain nonlinear systems described by
(119905) = 119860119909 (119905) + 119861119906 (119905) + 119892 (119909 (119905) 119905) + 119861119891119891 (119905) + 119861
119889119889 (119905)
(1)
119910 (119905) = 119862119909 (119905) + 119863119906 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905) (2)
where 119909(119905) isin 119877119899 is the system state vector 119906(119905) isin 119877
119901
is the control input vector 119910(119905) isin 119877119902 is the measurement
output vector 119889(119905) is the system uncertainty that belongs to119871119898
2[0 +infin] 119891(119905) isin 119877
119898 is the system fault to be detectedand 119892(sdot) is a known nonlinear function 119860 119861 119861
119891 119861119889 119862 119863
119863119891 and 119863
119889are known parameter matrix with appropriate
dimensions We take the following assumptions
Assumption 1 The observability matrix associated with thepair (119860 119862) is full rank
Assumption 2 The function 119889(119905) in (1) representing theunstructured modeling uncertainty is bounded by a knownconstant that is 119889(119905) le 119871
119908
Assumption 3 The nonlinear function 119892(sdot) satisfies Lipschitzcondition that is
1003817100381710038171003817119892 (119909 (119905) 119905) minus 119892 (119910 (119905) 119905)1003817100381710038171003817 le
1003817100381710038171003817120588 (119909 (119905) minus 119910 (119905))1003817100381710038171003817
forall119909 (119905) 119910 (119905) isin 119863
(3)
where 120588 is a known real constant Throughout the paper thenotation sdot will be used to denote the Euclidean norm of avector
To detect a fault the FTA is constructed as follows [12]
119909119896= 119860119909119896(119905) + 119892 (119909
119896(119905) 119905) + 119861119906
119896(119905) + 119861
119891119891119896(119905)
+ 119867 (119910 (119905) minus 119910 (119905))
(4)
119910119896(119905) = 119862119909
119896(119905) + 119863119906
119896(119905) (5)
119890119896(119905) = 119909
119896(119905) minus 119909
119896(119905) (6)
119903119896(119905) = 119862119890
119896(119905) (7)
119891119896+1
(119905) = 119891119896(119905) + Γ 119903
119896(119905) (8)
1003817100381710038171003817119910119896(119905) minus 119910119896(119905)
1003817100381710038171003817infinle 120574 119905 isin [119905
119886 119905119887] (9)
where 119909119896(119905) isin 119877
119899 119910119896(119905) isin 119877
119902 are estimated system state andoutput respectively 119896 is the iteration index and 120574 is the givenperformance index Γ is constant gainmatrix and its elementsare within the scope (0 1) 119891
119896(119905) is virtual fault which is an
estimate of119891(119905) and the value of119891(119905) is set to zero until a faultis detected 119890(119905) denotes the system state estimation error119890119896(119905) denotes the estimation error of 119890(119905) after 119896th iterative
operation 119903119896(119905) denotes the estimation error of 119903(119905) after 119896th
iterative operation119867 is the gain matrix to be optimized 119903(119905)is the so-called generated residual signal
The basic idea behind the FTA is to adjust the virtual fault119891119896(119905) within a specified time horizon by using iterative learn-
ing algorithm such that the virtual fault can approximate thesystem fault 119891(119905) as closely as possible Detailed descriptionabout the FTA can be found in [12]
From (4)we can clearly see that the value of gainmatrix119867
will have a great influence on the convergence speed of FTAAs a result the fault tracking accuracy will be influenced aswell In previous work [12] we have investigated the stabilityand fault tracking accuracy of the FTAon the assumption thatthe gainmatrix is a prespecified value However the design ofgain matrix 119867 still remains unresolved In this work we willutilize multiobjective parameter optimization algorithm androbust control theory to optimize the gain matrix 119867
3 Design of Gain Matrix
First we convert the parameter optimization problem into aperformance index optimization problem Then the designof gain matrix is given in terms of linear matrix inequality(LMI)
Mathematical Problems in Engineering 3
Define the state estimation error 119890(119905) = 119909(119905) minus 119909(119905) andthen it follows from (1)ndash(7) that [13ndash17]
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(10)
where gain matrix 119867 is to be designed such that the system(10) is asymptotically stable To optimize the gain matrix 119867we propose the following performance index [18]
119869 =
10038171003817100381710038171198791199031198891003817100381710038171003817infin
10038171003817100381710038171003817119879119903119891
10038171003817100381710038171003817minus
(11)
where 119879119903119889
denotes the transfer function from system uncer-tainty to residual signal and 119879
119903119891denotes the transfer function
from system faults to residual signal The objective of thisparameter optimization problem is to maximize the effects ofsystem faults on residual signals while minimizing the effectsof system uncertainty on residual signals Thus it leads tothe following optimization problem finding a gain matrix119867such that the system (10) remains asymptotically stable andthe performance index 119869 = 119879
119903119889infin
119879119903119891minusis minimized
Remark 4 The item 119879119903119889infinis used tomeasure the robustness
of residual signals to system uncertainty while the sensi-tivity of residual signals to system faults is measured by119879119903119891minus
= inf119891isin(0infin)
120590[119879119903119891(119895119908)] and 120590[119879
119903119891(119895119908)] denotes the
nonzero singular value of119879119903119891(119895119908)The optimization problem
described by (11) is actually a multiobjective optimizationproblem and it can be formulated as follows for a givenconstant 120574 gt 0 120573 gt 0 finding a gain matrix 119867 such thatthe system (10) remains asymptotically stable and satisfyingthe following inequality [18]
119903(119905)infin
le 120574 119889 (119905)infin
119903(119905)minus
gt 1205731003817100381710038171003817119891(119905)
1003817100381710038171003817minus (12)
with (10) note that the dynamics of the residual signaldepends not only on 119891(119905) and 119889(119905) but also on the nonlinearpart 119892(119909(119905) 119905) minus 119892(119909(119905) 119905) So the traditional fault detectionobserver design methods cannot be used here A novelmethod to design gain matrix 119867 meeting the performanceindex (12) is required In this paper we propose a method toresolve this problem in terms of LMIs
Lemma 5 (see [18 19]) Let 119860 and 119861 be real matrices withappropriate dimensions For any scalar 120576 gt 0 and vectors119909 119910 isin 119877
119899 then
2119909119879
119860119861119910 le 120576minus1
119909119879
119860119860119879
119909 + 120576119910119879
119861119879
119861119910 (13)
Theorem 6 Given a constant 120574 gt 0 in the condition of119891(119905) = 0 the system (10) is asymptotically stable and satisfies119903(119905)
infinle 120574119889(119905)
infin if there exists a positive symmetrical
matrix 119875 scalar 1205761gt 0 satisfying the following LMI
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0 (14)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(15)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119875119890 (119905) (16)
where 119875 is a positive symmetrical matrix In the fault-freecase when the system uncertainty 119889(119905) = 0 we have
(119905) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119875119890
= 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875] 119890 (119905)
+ 119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119875
(17)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 1205761(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 12057611205882
119890119879
(119905) 119890 (119905)
(18)
Thus
(119905) le 119890119879
(119905) lfloor119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875
+ 120576minus1
1119875119875119879
+ 12057611205882
119868rfloor 119890 (119905)
(19)
According to (14) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of no fault
When the system uncertainty 119889(119905) = 0 define
119867(119890 119889) = (119905) + 119903 (119905)infin
minus 1205742
119889 (119905)infin
(20)
So we have
119867(119890 119889) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905) + 119890119879
(119905) 119862119879
119862119890 (119905)
+ 119890119879
(119905) 119862119879
119863119889119889 (119905) + 119889
119879
(119905) 119863119879
119889119862119890 (119905)
+ 119889119879
(119905) 119863119879
119889119863119889119889 (119905) minus 120574
2
119889119879
(119905) 119889 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + (119861119889minus 119867119863
119889) 119889 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119889minus 119867119863
119889) 119889 (119905)]
119879
119875119890 (119905)
4 Mathematical Problems in Engineering
+ 119890119879
(119905) 119862119879
119862119890 (119905) + 119890119879
(119905) 119862119879
119863119889119889 (119905)
+ 119889119879
(119905) 119863119879
119889119862119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905)
le 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875 + 120576minus1
1119875119875119879
+119862119879
119862 + 12057611205882
119868] 119890 (119905) + 119890119879
(119905) [119862119879
119863119889] 119889 (119905)
+ 119889119879
(119905) [119863119879
119889119862] 119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905) + 119890119879
(119905) [119875 (119861119889minus 119867119863
119889)] 119889 (119905)
+ 119889119879
(119905) [(119861119889minus 119867119863
119889)119879
119875] 119890 (119905)
= [119890(119905)
119889(119905)]
119879
[
[
1198721
1198723
119872119879
31198722
]
]
[119890 (119905)
119889 (119905)]
(21)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 120576minus1
1119875119875119879
+ 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(22)
According to (14) and Schur theory we can obtain that
119867(119890 119889) = (119905) + 119903119879
(119905) 119903 (119905) minus 1205742
119889119879
(119905) 119889 (119905) lt 0 (23)
For any given time 119905 gt 0 integration of (23) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) lt 1205742
int
+infin
0
119889119879
(119905) 119889 (119905) (24)
Thus the inequality 119903(119905)infin
le 120574119889(119905)infin
holds This com-pletes the proof
Theorem 7 Given a constant 120573 gt 0 in the condition of 119889(119905) =
0 the system (10) is asymptotically stable and satisfies 119903(119905)minus
ge
120573119891(119905)minus if there exists a positive symmetrical matrix119876 scalar
1205781gt 0 satisfying the following LMI
[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (25)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(26)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119876119890 (119905) (27)
where119876 is a positive symmetrical matrix In the condition of119889(119905) = 0 when the system faults 119891(119905) = 0 we have
(119905) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119876119890
= 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876] 119890 (119905)
+ 119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119876
(28)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 1205781(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 12057811205882
119890119879
(119905) 119890 (119905)
(29)
Thus
(119905) le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876
+120578minus1
1119875119875119879
+ 12057811205882
119868] 119890 (119905)
(30)
According to (25) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of 119891(119905) = 0
When the system faults 119891(119905) = 0 define
119867(119890 119891) = (119905) + 1205732 1003817100381710038171003817119891 (119905)
1003817100381710038171003817infinminus 119903 (119905)
infin (31)
So we have
119867(119890 119891) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119891 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119891minus 119867119863
119891) 119891 (119905)]
119879
119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119889 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876 + 120578minus1
1119876119876119879
minus119862119879
119862 + 12057811205882
119868] 119890 (119905) minus 119890119879
(119905) [119862119879
119863119891] 119891 (119905)
Mathematical Problems in Engineering 5
minus 119891119879
(119905) [119863119879
119891119862] 119890 (119905) minus 119891
119879
(119905) 119863119879
119891119863119891119891 (119905)
+ 1205732
119891119879
(119905) 119891 (119905) minus 119890119879
(119905) [119876 (119861119889minus 119867119863
119889)] 119891 (119905)
minus 119891119879
(119905) [(119861119891minus 119867119863
119891)119879
119876] 119890 (119905)
= [119890(119905)
119891(119905)]
119879
[
[
1198731
1198733
119873119879
31198732
]
]
[119890 (119905)
119891 (119905)]
(32)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862
+ 120578minus1
1119876119876119879
+ 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(33)
According to Schur theory we can obtain that
119867(119890 119891) = (119905) + 1205732
119891119879
(119905) 119891 (119905) minus 119903119879
(119905) 119903 (119905) lt 0 (34)
For any given time 119905 gt 0 integration of (34) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) gt 1205732
int
+infin
0
119891119879
(119905) 119891 (119905) (35)
Thus the inequality 119903(119905)minus
gt 120573119891(119905)minusholdsThis completes
the proof
Theorem 8 Given constants 120574 gt 0 and 120573 gt 0 the system (10)is asymptotically stable and satisfies 119903(119905)
infinle 120574119889(119905)
infinand
119903(119905)minus
ge 120573119891(119905)minus if there exist matrices 119875 119876 and 119867 scalar
1205761gt 0 120578
1gt 0 satisfying matrix inequality
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (36)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(37)
Proof By combining Theorems 6 and 7 we have Theorem 8This completes the proof
Remark 9 Theorem 6 considers the robustness of residualsignals to system uncertainty and Theorem 7 considers thesensitivity of residual signals to system faults Theorem 8investigates the optimization of gain matrix 119867 by takinginto account the robustness of residual signals to systemuncertainty and sensitivity of residual signals to system faultssimultaneously If we iteratively useTheorem 8 we can get theoptimized solution of the performance indices 120574 120573 and gainmatrix 119867 by using LMI toolbox in MatLab
4 Fault Detection Threshold
In the above sections we have investigated the optimizationof the value of gain matrix 119867 in terms of LMIs In thissection we will investigate the calculation of threshold forfault detection
Theorem 10 Suppose Assumptions 1ndash3 hold Consider thedynamic system described by (1) (2) and the FTA described by(4)sim(9) let the initial state and output of the FTA be 119909
119896(0) =
119909(0) 119910119896(0) = 119910(0) (119896 = 1 2 ) respectively If the following
inequality holds1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(38)
then there is fault occurring in the system
Proof Without loss of generality we assume 119905 isin [119905119886 119905119887] and
119905119887minus 119905119886= 119875
Subtracting (4) from system equation (1) and subtracting(5) from system equation (2) result in the estimation errordynamics
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905)
minus 119892 (119909 (119905) 119905) minus 119861119891119891 (119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(39)
In the condition of 119891(119905) = 0 we have
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(40)
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
In our previous work [12] a robust fault tracking approx-imator (RFTA) based fault detection and identificationscheme for a class of nonlinear systems was developedand its stability properties were investigated as well Dueto the existence of unstructured modeling uncertaintiesthe convergence speed and fault tracking accuracy of theFTA will be influenced dramatically The objective of thiswork is to extend the previous research results to a class ofnonlinear systems with uncertainty and optimize the gainmatrix of FTA First of all we decompose the fault diagnosisproblem into a parameter optimization problem and a faultdetection problemThe parameter optimization problem canbe described as follows by using robust control technologyand parameter optimization algorithms the gain matrix ofFTA is designed tominimize the effects of system uncertaintyon residual signals while maximizing the effects of systemfaults on residual signals The design of the gain matrix ofFTA takes into account the robustness to system uncertaintyand sensitivity to system faults simultaneously which leadsto a multiobjective optimization problemThe fault detectionproblem can be described as follows we detect the systemfaults according to the relationship between the calculatedthreshold and residual signals The detectability of systemfaults is rigorously analyzed by investigating the thresholdof residual signals In the end an illustrative example isproposed to demonstrate the validity and applicability of theproposed approach
An outline of this paper is organized as follows InSection 2 we define a class of uncertain nonlinear systemsand present a multiobjective optimization problem Thedesign of gain matrix of FTA is investigated in Section 3The calculation of threshold for fault detection is designedin Section 4 In Section 5 simulation results are reportedillustrating the effectiveness of the proposed robust faultdetection scheme Some conclusion remarks are provided inSection 6
2 Problem Formulation
Consider a class of uncertain nonlinear systems described by
(119905) = 119860119909 (119905) + 119861119906 (119905) + 119892 (119909 (119905) 119905) + 119861119891119891 (119905) + 119861
119889119889 (119905)
(1)
119910 (119905) = 119862119909 (119905) + 119863119906 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905) (2)
where 119909(119905) isin 119877119899 is the system state vector 119906(119905) isin 119877
119901
is the control input vector 119910(119905) isin 119877119902 is the measurement
output vector 119889(119905) is the system uncertainty that belongs to119871119898
2[0 +infin] 119891(119905) isin 119877
119898 is the system fault to be detectedand 119892(sdot) is a known nonlinear function 119860 119861 119861
119891 119861119889 119862 119863
119863119891 and 119863
119889are known parameter matrix with appropriate
dimensions We take the following assumptions
Assumption 1 The observability matrix associated with thepair (119860 119862) is full rank
Assumption 2 The function 119889(119905) in (1) representing theunstructured modeling uncertainty is bounded by a knownconstant that is 119889(119905) le 119871
119908
Assumption 3 The nonlinear function 119892(sdot) satisfies Lipschitzcondition that is
1003817100381710038171003817119892 (119909 (119905) 119905) minus 119892 (119910 (119905) 119905)1003817100381710038171003817 le
1003817100381710038171003817120588 (119909 (119905) minus 119910 (119905))1003817100381710038171003817
forall119909 (119905) 119910 (119905) isin 119863
(3)
where 120588 is a known real constant Throughout the paper thenotation sdot will be used to denote the Euclidean norm of avector
To detect a fault the FTA is constructed as follows [12]
119909119896= 119860119909119896(119905) + 119892 (119909
119896(119905) 119905) + 119861119906
119896(119905) + 119861
119891119891119896(119905)
+ 119867 (119910 (119905) minus 119910 (119905))
(4)
119910119896(119905) = 119862119909
119896(119905) + 119863119906
119896(119905) (5)
119890119896(119905) = 119909
119896(119905) minus 119909
119896(119905) (6)
119903119896(119905) = 119862119890
119896(119905) (7)
119891119896+1
(119905) = 119891119896(119905) + Γ 119903
119896(119905) (8)
1003817100381710038171003817119910119896(119905) minus 119910119896(119905)
1003817100381710038171003817infinle 120574 119905 isin [119905
119886 119905119887] (9)
where 119909119896(119905) isin 119877
119899 119910119896(119905) isin 119877
119902 are estimated system state andoutput respectively 119896 is the iteration index and 120574 is the givenperformance index Γ is constant gainmatrix and its elementsare within the scope (0 1) 119891
119896(119905) is virtual fault which is an
estimate of119891(119905) and the value of119891(119905) is set to zero until a faultis detected 119890(119905) denotes the system state estimation error119890119896(119905) denotes the estimation error of 119890(119905) after 119896th iterative
operation 119903119896(119905) denotes the estimation error of 119903(119905) after 119896th
iterative operation119867 is the gain matrix to be optimized 119903(119905)is the so-called generated residual signal
The basic idea behind the FTA is to adjust the virtual fault119891119896(119905) within a specified time horizon by using iterative learn-
ing algorithm such that the virtual fault can approximate thesystem fault 119891(119905) as closely as possible Detailed descriptionabout the FTA can be found in [12]
From (4)we can clearly see that the value of gainmatrix119867
will have a great influence on the convergence speed of FTAAs a result the fault tracking accuracy will be influenced aswell In previous work [12] we have investigated the stabilityand fault tracking accuracy of the FTAon the assumption thatthe gainmatrix is a prespecified value However the design ofgain matrix 119867 still remains unresolved In this work we willutilize multiobjective parameter optimization algorithm androbust control theory to optimize the gain matrix 119867
3 Design of Gain Matrix
First we convert the parameter optimization problem into aperformance index optimization problem Then the designof gain matrix is given in terms of linear matrix inequality(LMI)
Mathematical Problems in Engineering 3
Define the state estimation error 119890(119905) = 119909(119905) minus 119909(119905) andthen it follows from (1)ndash(7) that [13ndash17]
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(10)
where gain matrix 119867 is to be designed such that the system(10) is asymptotically stable To optimize the gain matrix 119867we propose the following performance index [18]
119869 =
10038171003817100381710038171198791199031198891003817100381710038171003817infin
10038171003817100381710038171003817119879119903119891
10038171003817100381710038171003817minus
(11)
where 119879119903119889
denotes the transfer function from system uncer-tainty to residual signal and 119879
119903119891denotes the transfer function
from system faults to residual signal The objective of thisparameter optimization problem is to maximize the effects ofsystem faults on residual signals while minimizing the effectsof system uncertainty on residual signals Thus it leads tothe following optimization problem finding a gain matrix119867such that the system (10) remains asymptotically stable andthe performance index 119869 = 119879
119903119889infin
119879119903119891minusis minimized
Remark 4 The item 119879119903119889infinis used tomeasure the robustness
of residual signals to system uncertainty while the sensi-tivity of residual signals to system faults is measured by119879119903119891minus
= inf119891isin(0infin)
120590[119879119903119891(119895119908)] and 120590[119879
119903119891(119895119908)] denotes the
nonzero singular value of119879119903119891(119895119908)The optimization problem
described by (11) is actually a multiobjective optimizationproblem and it can be formulated as follows for a givenconstant 120574 gt 0 120573 gt 0 finding a gain matrix 119867 such thatthe system (10) remains asymptotically stable and satisfyingthe following inequality [18]
119903(119905)infin
le 120574 119889 (119905)infin
119903(119905)minus
gt 1205731003817100381710038171003817119891(119905)
1003817100381710038171003817minus (12)
with (10) note that the dynamics of the residual signaldepends not only on 119891(119905) and 119889(119905) but also on the nonlinearpart 119892(119909(119905) 119905) minus 119892(119909(119905) 119905) So the traditional fault detectionobserver design methods cannot be used here A novelmethod to design gain matrix 119867 meeting the performanceindex (12) is required In this paper we propose a method toresolve this problem in terms of LMIs
Lemma 5 (see [18 19]) Let 119860 and 119861 be real matrices withappropriate dimensions For any scalar 120576 gt 0 and vectors119909 119910 isin 119877
119899 then
2119909119879
119860119861119910 le 120576minus1
119909119879
119860119860119879
119909 + 120576119910119879
119861119879
119861119910 (13)
Theorem 6 Given a constant 120574 gt 0 in the condition of119891(119905) = 0 the system (10) is asymptotically stable and satisfies119903(119905)
infinle 120574119889(119905)
infin if there exists a positive symmetrical
matrix 119875 scalar 1205761gt 0 satisfying the following LMI
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0 (14)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(15)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119875119890 (119905) (16)
where 119875 is a positive symmetrical matrix In the fault-freecase when the system uncertainty 119889(119905) = 0 we have
(119905) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119875119890
= 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875] 119890 (119905)
+ 119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119875
(17)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 1205761(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 12057611205882
119890119879
(119905) 119890 (119905)
(18)
Thus
(119905) le 119890119879
(119905) lfloor119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875
+ 120576minus1
1119875119875119879
+ 12057611205882
119868rfloor 119890 (119905)
(19)
According to (14) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of no fault
When the system uncertainty 119889(119905) = 0 define
119867(119890 119889) = (119905) + 119903 (119905)infin
minus 1205742
119889 (119905)infin
(20)
So we have
119867(119890 119889) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905) + 119890119879
(119905) 119862119879
119862119890 (119905)
+ 119890119879
(119905) 119862119879
119863119889119889 (119905) + 119889
119879
(119905) 119863119879
119889119862119890 (119905)
+ 119889119879
(119905) 119863119879
119889119863119889119889 (119905) minus 120574
2
119889119879
(119905) 119889 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + (119861119889minus 119867119863
119889) 119889 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119889minus 119867119863
119889) 119889 (119905)]
119879
119875119890 (119905)
4 Mathematical Problems in Engineering
+ 119890119879
(119905) 119862119879
119862119890 (119905) + 119890119879
(119905) 119862119879
119863119889119889 (119905)
+ 119889119879
(119905) 119863119879
119889119862119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905)
le 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875 + 120576minus1
1119875119875119879
+119862119879
119862 + 12057611205882
119868] 119890 (119905) + 119890119879
(119905) [119862119879
119863119889] 119889 (119905)
+ 119889119879
(119905) [119863119879
119889119862] 119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905) + 119890119879
(119905) [119875 (119861119889minus 119867119863
119889)] 119889 (119905)
+ 119889119879
(119905) [(119861119889minus 119867119863
119889)119879
119875] 119890 (119905)
= [119890(119905)
119889(119905)]
119879
[
[
1198721
1198723
119872119879
31198722
]
]
[119890 (119905)
119889 (119905)]
(21)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 120576minus1
1119875119875119879
+ 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(22)
According to (14) and Schur theory we can obtain that
119867(119890 119889) = (119905) + 119903119879
(119905) 119903 (119905) minus 1205742
119889119879
(119905) 119889 (119905) lt 0 (23)
For any given time 119905 gt 0 integration of (23) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) lt 1205742
int
+infin
0
119889119879
(119905) 119889 (119905) (24)
Thus the inequality 119903(119905)infin
le 120574119889(119905)infin
holds This com-pletes the proof
Theorem 7 Given a constant 120573 gt 0 in the condition of 119889(119905) =
0 the system (10) is asymptotically stable and satisfies 119903(119905)minus
ge
120573119891(119905)minus if there exists a positive symmetrical matrix119876 scalar
1205781gt 0 satisfying the following LMI
[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (25)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(26)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119876119890 (119905) (27)
where119876 is a positive symmetrical matrix In the condition of119889(119905) = 0 when the system faults 119891(119905) = 0 we have
(119905) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119876119890
= 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876] 119890 (119905)
+ 119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119876
(28)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 1205781(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 12057811205882
119890119879
(119905) 119890 (119905)
(29)
Thus
(119905) le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876
+120578minus1
1119875119875119879
+ 12057811205882
119868] 119890 (119905)
(30)
According to (25) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of 119891(119905) = 0
When the system faults 119891(119905) = 0 define
119867(119890 119891) = (119905) + 1205732 1003817100381710038171003817119891 (119905)
1003817100381710038171003817infinminus 119903 (119905)
infin (31)
So we have
119867(119890 119891) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119891 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119891minus 119867119863
119891) 119891 (119905)]
119879
119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119889 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876 + 120578minus1
1119876119876119879
minus119862119879
119862 + 12057811205882
119868] 119890 (119905) minus 119890119879
(119905) [119862119879
119863119891] 119891 (119905)
Mathematical Problems in Engineering 5
minus 119891119879
(119905) [119863119879
119891119862] 119890 (119905) minus 119891
119879
(119905) 119863119879
119891119863119891119891 (119905)
+ 1205732
119891119879
(119905) 119891 (119905) minus 119890119879
(119905) [119876 (119861119889minus 119867119863
119889)] 119891 (119905)
minus 119891119879
(119905) [(119861119891minus 119867119863
119891)119879
119876] 119890 (119905)
= [119890(119905)
119891(119905)]
119879
[
[
1198731
1198733
119873119879
31198732
]
]
[119890 (119905)
119891 (119905)]
(32)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862
+ 120578minus1
1119876119876119879
+ 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(33)
According to Schur theory we can obtain that
119867(119890 119891) = (119905) + 1205732
119891119879
(119905) 119891 (119905) minus 119903119879
(119905) 119903 (119905) lt 0 (34)
For any given time 119905 gt 0 integration of (34) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) gt 1205732
int
+infin
0
119891119879
(119905) 119891 (119905) (35)
Thus the inequality 119903(119905)minus
gt 120573119891(119905)minusholdsThis completes
the proof
Theorem 8 Given constants 120574 gt 0 and 120573 gt 0 the system (10)is asymptotically stable and satisfies 119903(119905)
infinle 120574119889(119905)
infinand
119903(119905)minus
ge 120573119891(119905)minus if there exist matrices 119875 119876 and 119867 scalar
1205761gt 0 120578
1gt 0 satisfying matrix inequality
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (36)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(37)
Proof By combining Theorems 6 and 7 we have Theorem 8This completes the proof
Remark 9 Theorem 6 considers the robustness of residualsignals to system uncertainty and Theorem 7 considers thesensitivity of residual signals to system faults Theorem 8investigates the optimization of gain matrix 119867 by takinginto account the robustness of residual signals to systemuncertainty and sensitivity of residual signals to system faultssimultaneously If we iteratively useTheorem 8 we can get theoptimized solution of the performance indices 120574 120573 and gainmatrix 119867 by using LMI toolbox in MatLab
4 Fault Detection Threshold
In the above sections we have investigated the optimizationof the value of gain matrix 119867 in terms of LMIs In thissection we will investigate the calculation of threshold forfault detection
Theorem 10 Suppose Assumptions 1ndash3 hold Consider thedynamic system described by (1) (2) and the FTA described by(4)sim(9) let the initial state and output of the FTA be 119909
119896(0) =
119909(0) 119910119896(0) = 119910(0) (119896 = 1 2 ) respectively If the following
inequality holds1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(38)
then there is fault occurring in the system
Proof Without loss of generality we assume 119905 isin [119905119886 119905119887] and
119905119887minus 119905119886= 119875
Subtracting (4) from system equation (1) and subtracting(5) from system equation (2) result in the estimation errordynamics
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905)
minus 119892 (119909 (119905) 119905) minus 119861119891119891 (119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(39)
In the condition of 119891(119905) = 0 we have
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(40)
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Define the state estimation error 119890(119905) = 119909(119905) minus 119909(119905) andthen it follows from (1)ndash(7) that [13ndash17]
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(10)
where gain matrix 119867 is to be designed such that the system(10) is asymptotically stable To optimize the gain matrix 119867we propose the following performance index [18]
119869 =
10038171003817100381710038171198791199031198891003817100381710038171003817infin
10038171003817100381710038171003817119879119903119891
10038171003817100381710038171003817minus
(11)
where 119879119903119889
denotes the transfer function from system uncer-tainty to residual signal and 119879
119903119891denotes the transfer function
from system faults to residual signal The objective of thisparameter optimization problem is to maximize the effects ofsystem faults on residual signals while minimizing the effectsof system uncertainty on residual signals Thus it leads tothe following optimization problem finding a gain matrix119867such that the system (10) remains asymptotically stable andthe performance index 119869 = 119879
119903119889infin
119879119903119891minusis minimized
Remark 4 The item 119879119903119889infinis used tomeasure the robustness
of residual signals to system uncertainty while the sensi-tivity of residual signals to system faults is measured by119879119903119891minus
= inf119891isin(0infin)
120590[119879119903119891(119895119908)] and 120590[119879
119903119891(119895119908)] denotes the
nonzero singular value of119879119903119891(119895119908)The optimization problem
described by (11) is actually a multiobjective optimizationproblem and it can be formulated as follows for a givenconstant 120574 gt 0 120573 gt 0 finding a gain matrix 119867 such thatthe system (10) remains asymptotically stable and satisfyingthe following inequality [18]
119903(119905)infin
le 120574 119889 (119905)infin
119903(119905)minus
gt 1205731003817100381710038171003817119891(119905)
1003817100381710038171003817minus (12)
with (10) note that the dynamics of the residual signaldepends not only on 119891(119905) and 119889(119905) but also on the nonlinearpart 119892(119909(119905) 119905) minus 119892(119909(119905) 119905) So the traditional fault detectionobserver design methods cannot be used here A novelmethod to design gain matrix 119867 meeting the performanceindex (12) is required In this paper we propose a method toresolve this problem in terms of LMIs
Lemma 5 (see [18 19]) Let 119860 and 119861 be real matrices withappropriate dimensions For any scalar 120576 gt 0 and vectors119909 119910 isin 119877
119899 then
2119909119879
119860119861119910 le 120576minus1
119909119879
119860119860119879
119909 + 120576119910119879
119861119879
119861119910 (13)
Theorem 6 Given a constant 120574 gt 0 in the condition of119891(119905) = 0 the system (10) is asymptotically stable and satisfies119903(119905)
infinle 120574119889(119905)
infin if there exists a positive symmetrical
matrix 119875 scalar 1205761gt 0 satisfying the following LMI
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0 (14)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(15)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119875119890 (119905) (16)
where 119875 is a positive symmetrical matrix In the fault-freecase when the system uncertainty 119889(119905) = 0 we have
(119905) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119875119890
= 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875] 119890 (119905)
+ 119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119875
(17)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119875 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 1205761(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120576minus1
1119890119879
(119905) 119875119875119879
119890 (119905) + 12057611205882
119890119879
(119905) 119890 (119905)
(18)
Thus
(119905) le 119890119879
(119905) lfloor119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875
+ 120576minus1
1119875119875119879
+ 12057611205882
119868rfloor 119890 (119905)
(19)
According to (14) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of no fault
When the system uncertainty 119889(119905) = 0 define
119867(119890 119889) = (119905) + 119903 (119905)infin
minus 1205742
119889 (119905)infin
(20)
So we have
119867(119890 119889) = 119890119879
(119905) 119875 119890 (119905) + 119890119879
(119905) 119875119890 (119905) + 119890119879
(119905) 119862119879
119862119890 (119905)
+ 119890119879
(119905) 119862119879
119863119889119889 (119905) + 119889
119879
(119905) 119863119879
119889119862119890 (119905)
+ 119889119879
(119905) 119863119879
119889119863119889119889 (119905) minus 120574
2
119889119879
(119905) 119889 (119905)
= 119890119879
(119905) 119875 [(119860 minus 119867119862) 119890 (119905) + (119861119889minus 119867119863
119889) 119889 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119889minus 119867119863
119889) 119889 (119905)]
119879
119875119890 (119905)
4 Mathematical Problems in Engineering
+ 119890119879
(119905) 119862119879
119862119890 (119905) + 119890119879
(119905) 119862119879
119863119889119889 (119905)
+ 119889119879
(119905) 119863119879
119889119862119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905)
le 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875 + 120576minus1
1119875119875119879
+119862119879
119862 + 12057611205882
119868] 119890 (119905) + 119890119879
(119905) [119862119879
119863119889] 119889 (119905)
+ 119889119879
(119905) [119863119879
119889119862] 119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905) + 119890119879
(119905) [119875 (119861119889minus 119867119863
119889)] 119889 (119905)
+ 119889119879
(119905) [(119861119889minus 119867119863
119889)119879
119875] 119890 (119905)
= [119890(119905)
119889(119905)]
119879
[
[
1198721
1198723
119872119879
31198722
]
]
[119890 (119905)
119889 (119905)]
(21)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 120576minus1
1119875119875119879
+ 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(22)
According to (14) and Schur theory we can obtain that
119867(119890 119889) = (119905) + 119903119879
(119905) 119903 (119905) minus 1205742
119889119879
(119905) 119889 (119905) lt 0 (23)
For any given time 119905 gt 0 integration of (23) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) lt 1205742
int
+infin
0
119889119879
(119905) 119889 (119905) (24)
Thus the inequality 119903(119905)infin
le 120574119889(119905)infin
holds This com-pletes the proof
Theorem 7 Given a constant 120573 gt 0 in the condition of 119889(119905) =
0 the system (10) is asymptotically stable and satisfies 119903(119905)minus
ge
120573119891(119905)minus if there exists a positive symmetrical matrix119876 scalar
1205781gt 0 satisfying the following LMI
[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (25)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(26)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119876119890 (119905) (27)
where119876 is a positive symmetrical matrix In the condition of119889(119905) = 0 when the system faults 119891(119905) = 0 we have
(119905) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119876119890
= 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876] 119890 (119905)
+ 119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119876
(28)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 1205781(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 12057811205882
119890119879
(119905) 119890 (119905)
(29)
Thus
(119905) le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876
+120578minus1
1119875119875119879
+ 12057811205882
119868] 119890 (119905)
(30)
According to (25) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of 119891(119905) = 0
When the system faults 119891(119905) = 0 define
119867(119890 119891) = (119905) + 1205732 1003817100381710038171003817119891 (119905)
1003817100381710038171003817infinminus 119903 (119905)
infin (31)
So we have
119867(119890 119891) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119891 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119891minus 119867119863
119891) 119891 (119905)]
119879
119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119889 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876 + 120578minus1
1119876119876119879
minus119862119879
119862 + 12057811205882
119868] 119890 (119905) minus 119890119879
(119905) [119862119879
119863119891] 119891 (119905)
Mathematical Problems in Engineering 5
minus 119891119879
(119905) [119863119879
119891119862] 119890 (119905) minus 119891
119879
(119905) 119863119879
119891119863119891119891 (119905)
+ 1205732
119891119879
(119905) 119891 (119905) minus 119890119879
(119905) [119876 (119861119889minus 119867119863
119889)] 119891 (119905)
minus 119891119879
(119905) [(119861119891minus 119867119863
119891)119879
119876] 119890 (119905)
= [119890(119905)
119891(119905)]
119879
[
[
1198731
1198733
119873119879
31198732
]
]
[119890 (119905)
119891 (119905)]
(32)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862
+ 120578minus1
1119876119876119879
+ 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(33)
According to Schur theory we can obtain that
119867(119890 119891) = (119905) + 1205732
119891119879
(119905) 119891 (119905) minus 119903119879
(119905) 119903 (119905) lt 0 (34)
For any given time 119905 gt 0 integration of (34) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) gt 1205732
int
+infin
0
119891119879
(119905) 119891 (119905) (35)
Thus the inequality 119903(119905)minus
gt 120573119891(119905)minusholdsThis completes
the proof
Theorem 8 Given constants 120574 gt 0 and 120573 gt 0 the system (10)is asymptotically stable and satisfies 119903(119905)
infinle 120574119889(119905)
infinand
119903(119905)minus
ge 120573119891(119905)minus if there exist matrices 119875 119876 and 119867 scalar
1205761gt 0 120578
1gt 0 satisfying matrix inequality
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (36)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(37)
Proof By combining Theorems 6 and 7 we have Theorem 8This completes the proof
Remark 9 Theorem 6 considers the robustness of residualsignals to system uncertainty and Theorem 7 considers thesensitivity of residual signals to system faults Theorem 8investigates the optimization of gain matrix 119867 by takinginto account the robustness of residual signals to systemuncertainty and sensitivity of residual signals to system faultssimultaneously If we iteratively useTheorem 8 we can get theoptimized solution of the performance indices 120574 120573 and gainmatrix 119867 by using LMI toolbox in MatLab
4 Fault Detection Threshold
In the above sections we have investigated the optimizationof the value of gain matrix 119867 in terms of LMIs In thissection we will investigate the calculation of threshold forfault detection
Theorem 10 Suppose Assumptions 1ndash3 hold Consider thedynamic system described by (1) (2) and the FTA described by(4)sim(9) let the initial state and output of the FTA be 119909
119896(0) =
119909(0) 119910119896(0) = 119910(0) (119896 = 1 2 ) respectively If the following
inequality holds1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(38)
then there is fault occurring in the system
Proof Without loss of generality we assume 119905 isin [119905119886 119905119887] and
119905119887minus 119905119886= 119875
Subtracting (4) from system equation (1) and subtracting(5) from system equation (2) result in the estimation errordynamics
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905)
minus 119892 (119909 (119905) 119905) minus 119861119891119891 (119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(39)
In the condition of 119891(119905) = 0 we have
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(40)
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
+ 119890119879
(119905) 119862119879
119862119890 (119905) + 119890119879
(119905) 119862119879
119863119889119889 (119905)
+ 119889119879
(119905) 119863119879
119889119862119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905)
le 119890119879
(119905) [119875 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119875 + 120576minus1
1119875119875119879
+119862119879
119862 + 12057611205882
119868] 119890 (119905) + 119890119879
(119905) [119862119879
119863119889] 119889 (119905)
+ 119889119879
(119905) [119863119879
119889119862] 119890 (119905) + 119889
119879
(119905) 119863119879
119889119863119889119889 (119905)
minus 1205742
119889119879
(119905) 119889 (119905) + 119890119879
(119905) [119875 (119861119889minus 119867119863
119889)] 119889 (119905)
+ 119889119879
(119905) [(119861119889minus 119867119863
119889)119879
119875] 119890 (119905)
= [119890(119905)
119889(119905)]
119879
[
[
1198721
1198723
119872119879
31198722
]
]
[119890 (119905)
119889 (119905)]
(21)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 120576minus1
1119875119875119879
+ 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
(22)
According to (14) and Schur theory we can obtain that
119867(119890 119889) = (119905) + 119903119879
(119905) 119903 (119905) minus 1205742
119889119879
(119905) 119889 (119905) lt 0 (23)
For any given time 119905 gt 0 integration of (23) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) lt 1205742
int
+infin
0
119889119879
(119905) 119889 (119905) (24)
Thus the inequality 119903(119905)infin
le 120574119889(119905)infin
holds This com-pletes the proof
Theorem 7 Given a constant 120573 gt 0 in the condition of 119889(119905) =
0 the system (10) is asymptotically stable and satisfies 119903(119905)minus
ge
120573119891(119905)minus if there exists a positive symmetrical matrix119876 scalar
1205781gt 0 satisfying the following LMI
[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (25)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(26)
Proof We choose a Lyapunov function of the form
119881 (119905) = 119890119879
(119905) 119876119890 (119905) (27)
where119876 is a positive symmetrical matrix In the condition of119889(119905) = 0 when the system faults 119891(119905) = 0 we have
(119905) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119876119890
= 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876] 119890 (119905)
+ 119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]119879
119890119876
(28)
According to Lemma 5 and Assumption 3 we have
2119890119879
(119905) 119876 [119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 1205781(119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
119879
times (119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905))
le 120578minus1
1119890119879
(119905) 119876119876119879
119890 (119905) + 12057811205882
119890119879
(119905) 119890 (119905)
(29)
Thus
(119905) le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876
+120578minus1
1119875119875119879
+ 12057811205882
119868] 119890 (119905)
(30)
According to (25) we can obtain that (119905) le 0 So the system(10) is asymptotically stable in the condition of 119891(119905) = 0
When the system faults 119891(119905) = 0 define
119867(119890 119891) = (119905) + 1205732 1003817100381710038171003817119891 (119905)
1003817100381710038171003817infinminus 119903 (119905)
infin (31)
So we have
119867(119890 119891) = 119890119879
(119905) 119876 119890 (119905) + 119890119879
(119905) 119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119891 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
= 119890119879
(119905) 119876 [(119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)]
+ [(119860 minus 119867119862) 119890 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
+ (119861119891minus 119867119863
119891) 119891 (119905)]
119879
119876119890 (119905) minus 119890119879
(119905) 119862119879
119862119890 (119905)
minus 119890119879
(119905) 119862119879
119863119891119889 (119905) minus 119891
119879
(119905) 119863119879
119891119862119890 (119905)
minus 119891119879
(119905) 119863119879
119891119863119891119891 (119905) + 120573
2
119891119879
(119905) 119891 (119905)
le 119890119879
(119905) [119876 (119860 minus 119867119862) + (119860 minus 119867119862)119879
119876 + 120578minus1
1119876119876119879
minus119862119879
119862 + 12057811205882
119868] 119890 (119905) minus 119890119879
(119905) [119862119879
119863119891] 119891 (119905)
Mathematical Problems in Engineering 5
minus 119891119879
(119905) [119863119879
119891119862] 119890 (119905) minus 119891
119879
(119905) 119863119879
119891119863119891119891 (119905)
+ 1205732
119891119879
(119905) 119891 (119905) minus 119890119879
(119905) [119876 (119861119889minus 119867119863
119889)] 119891 (119905)
minus 119891119879
(119905) [(119861119891minus 119867119863
119891)119879
119876] 119890 (119905)
= [119890(119905)
119891(119905)]
119879
[
[
1198731
1198733
119873119879
31198732
]
]
[119890 (119905)
119891 (119905)]
(32)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862
+ 120578minus1
1119876119876119879
+ 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(33)
According to Schur theory we can obtain that
119867(119890 119891) = (119905) + 1205732
119891119879
(119905) 119891 (119905) minus 119903119879
(119905) 119903 (119905) lt 0 (34)
For any given time 119905 gt 0 integration of (34) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) gt 1205732
int
+infin
0
119891119879
(119905) 119891 (119905) (35)
Thus the inequality 119903(119905)minus
gt 120573119891(119905)minusholdsThis completes
the proof
Theorem 8 Given constants 120574 gt 0 and 120573 gt 0 the system (10)is asymptotically stable and satisfies 119903(119905)
infinle 120574119889(119905)
infinand
119903(119905)minus
ge 120573119891(119905)minus if there exist matrices 119875 119876 and 119867 scalar
1205761gt 0 120578
1gt 0 satisfying matrix inequality
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (36)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(37)
Proof By combining Theorems 6 and 7 we have Theorem 8This completes the proof
Remark 9 Theorem 6 considers the robustness of residualsignals to system uncertainty and Theorem 7 considers thesensitivity of residual signals to system faults Theorem 8investigates the optimization of gain matrix 119867 by takinginto account the robustness of residual signals to systemuncertainty and sensitivity of residual signals to system faultssimultaneously If we iteratively useTheorem 8 we can get theoptimized solution of the performance indices 120574 120573 and gainmatrix 119867 by using LMI toolbox in MatLab
4 Fault Detection Threshold
In the above sections we have investigated the optimizationof the value of gain matrix 119867 in terms of LMIs In thissection we will investigate the calculation of threshold forfault detection
Theorem 10 Suppose Assumptions 1ndash3 hold Consider thedynamic system described by (1) (2) and the FTA described by(4)sim(9) let the initial state and output of the FTA be 119909
119896(0) =
119909(0) 119910119896(0) = 119910(0) (119896 = 1 2 ) respectively If the following
inequality holds1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(38)
then there is fault occurring in the system
Proof Without loss of generality we assume 119905 isin [119905119886 119905119887] and
119905119887minus 119905119886= 119875
Subtracting (4) from system equation (1) and subtracting(5) from system equation (2) result in the estimation errordynamics
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905)
minus 119892 (119909 (119905) 119905) minus 119861119891119891 (119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(39)
In the condition of 119891(119905) = 0 we have
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(40)
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
minus 119891119879
(119905) [119863119879
119891119862] 119890 (119905) minus 119891
119879
(119905) 119863119879
119891119863119891119891 (119905)
+ 1205732
119891119879
(119905) 119891 (119905) minus 119890119879
(119905) [119876 (119861119889minus 119867119863
119889)] 119891 (119905)
minus 119891119879
(119905) [(119861119891minus 119867119863
119891)119879
119876] 119890 (119905)
= [119890(119905)
119891(119905)]
119879
[
[
1198731
1198733
119873119879
31198732
]
]
[119890 (119905)
119891 (119905)]
(32)
where
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862
+ 120578minus1
1119876119876119879
+ 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(33)
According to Schur theory we can obtain that
119867(119890 119891) = (119905) + 1205732
119891119879
(119905) 119891 (119905) minus 119903119879
(119905) 119903 (119905) lt 0 (34)
For any given time 119905 gt 0 integration of (34) from 0 to 119905 yields
int
+infin
0
119903119879
(119905) 119903 (119905) gt 1205732
int
+infin
0
119891119879
(119905) 119891 (119905) (35)
Thus the inequality 119903(119905)minus
gt 120573119891(119905)minusholdsThis completes
the proof
Theorem 8 Given constants 120574 gt 0 and 120573 gt 0 the system (10)is asymptotically stable and satisfies 119903(119905)
infinle 120574119889(119905)
infinand
119903(119905)minus
ge 120573119891(119905)minus if there exist matrices 119875 119876 and 119867 scalar
1205761gt 0 120578
1gt 0 satisfying matrix inequality
[[
[
1198721
1198723
119875
119872119879
31198722
0
119875119879
0 minus1205761119868
]]
]
lt 0[[
[
1198731
1198733
119876
119873119879
31198732
0
119876119879
0 minus1205781119868
]]
]
lt 0 (36)
where
1198721= 119875 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119875 + 119862119879
119862 + 12057611205882
119868
1198722= 119863119879
119889119863119889minus 1205742
119868
1198723= 119862119879
119863119889+ 119875 (119861
119889minus 119867119863
119889)
1198731= 119876 (119860 minus 119867119862) + (119860 minus 119867119862)
119879
119876 minus 119862119879
119862 + 12057811205882
119868
1198732= minus119863119879
119891119863119891+ 1205732
119868
1198733= minus119862119879
119863119891+ 119876 (119861
119891minus 119867119863
119891)
(37)
Proof By combining Theorems 6 and 7 we have Theorem 8This completes the proof
Remark 9 Theorem 6 considers the robustness of residualsignals to system uncertainty and Theorem 7 considers thesensitivity of residual signals to system faults Theorem 8investigates the optimization of gain matrix 119867 by takinginto account the robustness of residual signals to systemuncertainty and sensitivity of residual signals to system faultssimultaneously If we iteratively useTheorem 8 we can get theoptimized solution of the performance indices 120574 120573 and gainmatrix 119867 by using LMI toolbox in MatLab
4 Fault Detection Threshold
In the above sections we have investigated the optimizationof the value of gain matrix 119867 in terms of LMIs In thissection we will investigate the calculation of threshold forfault detection
Theorem 10 Suppose Assumptions 1ndash3 hold Consider thedynamic system described by (1) (2) and the FTA described by(4)sim(9) let the initial state and output of the FTA be 119909
119896(0) =
119909(0) 119910119896(0) = 119910(0) (119896 = 1 2 ) respectively If the following
inequality holds1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(38)
then there is fault occurring in the system
Proof Without loss of generality we assume 119905 isin [119905119886 119905119887] and
119905119887minus 119905119886= 119875
Subtracting (4) from system equation (1) and subtracting(5) from system equation (2) result in the estimation errordynamics
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905)
minus 119892 (119909 (119905) 119905) minus 119861119891119891 (119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(39)
In the condition of 119891(119905) = 0 we have
119890 (119905) = (119860 minus 119867119862) 119890 (119905) + (119861119891minus 119867119863
119891) 119891 (119905)
+ (119861119889minus 119867119863
119889) 119889 (119905) + 119892 (119909 (119905) 119905) minus 119892 (119909 (119905) 119905)
119903 (119905) = 119862119890 (119905) + 119863119891119891 (119905) + 119863
119889119889 (119905)
(40)
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
The solution of (40) is119890119896(119905) = Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(41)
119903119896(119905) = 119862Φ (119905 119905
119886) 119890119896(119905119886)
+ int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591)+(119861
119889minus119867119863119889) 119889 (120591)] 119889120591
(42)
where Φ(119905) = 119871minus1
[(119904119868 minus (119860 minus 119867119862))minus1
] 119871minus1 denotes inverseLaplacian transform
Due to 119909119896(0) = 119909(0) 119910
119896(0) = 119910(0) (119896 = 1 2 ) we have
119910119896+1
(119905119886) = 119862119909
119896+1(119905119886) = 119862119909
119896(119905119886) = 119910119896(119905119886) 119903
119896(119905119886) = 0
119890119896(119905119886) = 0
(43)
Substituting (43) into (42) yields
119903119896(119905) = int
119905
119905119886
119862Φ (119905 120591) [(119861119891minus 119867119863
119891) 119891 (120591) + 119892
119896(119909 (120591) 120591)
minus 119892119896(119909 (120591) 120591) + (119861
119889minus 119867119863
119889) 119889 (120591)] 119889120591
(44)
According to Assumptions 1ndash3 the time weighted norm of(44) is
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
10038171003817100381710038171003817119862Φ (119905 120591) (119861
119891minus 119867119863
119891) 119891 (120591)
10038171003817100381710038171003817119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(45)
In the condition of 119891(119905) = 0 we have
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
(46)
According to (41) we have
1003817100381710038171003817119890119896 (119905)1003817100381710038171003817 le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 Φ (119905 120591)1003817100381710038171003817119890119896 (120591)
1003817100381710038171003817 119889120591
+ 119871119908119875 sup119905120591isin[119905119886 119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
(47)
Using Gronwall-Bellman inequality (47) can be simplified as1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
(48)
where exp denotes exponential function Substituting (48)into (46) yields
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119892119896(119909 (120591) 120591) minus 119892
119896(119909 (120591) 120591))
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le int
119905
119905119886
120588 119862Φ (119905 120591) sdot1003817100381710038171003817119890119896 (119905)
1003817100381710038171003817 119889120591
+ int
119905
119905119886
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817 119889 (120591) 119889120591
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ (119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) (119905 minus 119905119886)
le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(49)
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
If faults occur in the system then1003817100381710038171003817119903119896 (119905)
1003817100381710038171003817 gt 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905119886 119905119887]
Φ (119905 120591) 119875
(50)If the following inequality holds
1003817100381710038171003817119903119896 (119905)1003817100381710038171003817 le 119871119908119875 sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817119862Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817
+ (( sup119905120591isin[119905
119886119905119887]
119862Φ (119905 120591) 119871119908119875
times sup119905120591isin[119905
119886119905119887]
1003817100381710038171003817Φ (119905 120591) (119861119889minus 119867119863
119889)1003817100381710038171003817)
times( sup119905120591isin[119905
119886119905119887]
Φ(119905 120591))
minus1
)
times exp 120588 sup119905120591isin[119905
119886119905119887]
Φ (119905 120591) 119875
(51)
then there are no faults occurring in the systemThis completes the proofIn the presence of unstructuredmodeling uncertainty we
have to determine the upper bounds of the residual signalsfor fault detection referred to as the threshold Theorem 10investigated the calculation of threshold for fault detectionOnce the residual signals exceed the threshold it indicatesthat system faults occur If the residual signal is below thethreshold there is no fault occurring Moreover we can usethe residual evaluation function 119903(119905)
2to detect system
faults [20 21]
5 Simulation Results
In this section we use the proposed method to detect faultsfor a class of uncertain nonlinear systems Let us consider theuncertain nonlinear system with parameters as follows
(119905) = [minus038 0
0 minus059] 119909 (119905) + [
1
1] 119906
+ [01 sin (119905)
01 sin (119905)] 119909 (119905)
+ [02
02] 119889 (119905) + [
01
01] 119891 (119905)
119910 (119905) = [1 1] 119909 (119905) + 01119891 (119905) + 01119889 (119905)
(52)
0 100 200 300 400 500
08
06
04
02
0
minus02
minus04
minus06
minus08
Time (s)
Noi
ses
Figure 1 White noise signal
0 100 200 300 400 500
12
1
08
06
04
02
0
minus02
minus04
Time (s)
Resid
ual s
igna
ls
Figure 2 Generated residual signal
Let 120574 = 03 let 120573 = 07 and let 119889(119905) be white noise withenergy 05 According to Theorem 8 the gain matrix of FTAcan be determined 119867 = [02071 01832]
119879 According toTheorem 10 the threshold for fault detection is 0426 Thewhite noise signal generated residual signal are shown inFigures 1 and 2 respectively
From Figure 2 we can clearly see that the residual signalis below the threshold at time 119905 lt 100 sTherefore there is nofault occurring at time 119905 lt 100 s At time 100 s lt 119905 lt 200 sthe residual signal exceeds the threshold it indicates thatsystem fault occurs Next we use residual evaluation function119903(119905)
2to detect system faults Figure 3 shows the evolution
of residual evaluation function 119903(119905)2 From Figure 3 we can
see that at time 100 s lt 119905 lt 200 s the residual evaluationfunction 119903(119905)
2jumps from 01 to 51 it indicates that system
fault occurs It can be seen from the simulation results thatthe proposed approach can improve not only the sensitivityof the FTA to system faults but also the robustness of the FTAto systems uncertainty
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
0 100 200 300 400 5000
2
4
6
8
10
12
Evol
utio
n of
resid
ual s
igna
ls
Time (s)
Figure 3 Evolution of residual evaluation function
6 Conclusions
This paper has proposed a fault detection scheme for a classof uncertain nonlinear systemsThemain contribution of thispaper is to utilize robust control theory and multiobjectiveoptimization algorithm to design the gainmatrix of FTAThegain matrix of FTA is designed to minimize the effects ofsystem uncertainty on residual signals while maximizing theeffects of system faults on residual signals The calculationof the gain matrix is given in terms of LMIs formulationsThe selection of threshold for fault detection is rigorouslyinvestigated as well In the end an illustrative example hasdemonstrated the validity and applicability of the proposedapproach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Natural Science Founda-tion of China (nos 51407078 61201124) and Special Fund ofEast China University of Science and Technology for BasicScientific Research (WJ1313004-1WH1114027 H200-4-13192and WH1414022)
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyti-cal and knowledge-based redundancymdasha survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] R Isermann ldquoProcess fault detection based on modeling andestimation methods a surveyrdquo Automatica vol 20 no 4 pp387ndash404 1984
[3] K Zhang B Jiang and V Cocquempot ldquoFast adaptive faultestimation and accommodation for nonlinear time-varying
delay systemsrdquo Asian Journal of Control vol 11 no 6 pp 643ndash652 2009
[4] X Zhang T Parisini and M M Polycarpou ldquoSensor bias faultisolation in a class of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 370ndash376 2005
[5] A T Vemuri ldquoSensor bias fault diagnosis in a class of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 46 no6 pp 949ndash954 2001
[6] W Zhang X-S Cai and Z-Z Han ldquoRobust stability criteriafor systems with interval time-varying delay and nonlinear per-turbationsrdquo Journal of Computational and AppliedMathematicsvol 234 no 1 pp 174ndash180 2010
[7] L Guo and HWang ldquoFault detection and diagnosis for generalstochastic systems using B-spline expansions and nonlinearfiltersrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 52 no 8 pp 1644ndash1652 2005
[8] X ZhangMM Polycarpou and T Parisini ldquoFault diagnosis ofa class of nonlinear uncertain systems with Lipschitz nonlinear-ities using adaptive estimationrdquo Automatica vol 46 no 2 pp290ndash299 2010
[9] J Fan and Y Wang ldquoFault detection and diagnosis of non-linear non-Gaussian dynamic processes using kernel dynamicindependent component analysisrdquo Information Sciences vol259 pp 369ndash379 2014
[10] P Tadic and Z Durovic ldquoParticle filtering for sensor faultdiagnosis and identification in nonlinear plantsrdquo Journal ofProcess Control vol 24 no 4 pp 401ndash409 2014
[11] W Chen and M Saif ldquoAn iterative learning observer forfault detection and accommodation in nonlinear time-delaysystemsrdquo International Journal of Robust and Nonlinear Controlvol 16 no 1 pp 1ndash19 2006
[12] B Yan Z Tian and S Shi ldquoA novel distributed approach torobust fault detection and identificationrdquo International Journalof Electrical Power amp Energy Systems vol 30 no 5 pp 343ndash3602008
[13] W Zhang H Su H Wang and Z Han ldquoFull-order andreduced-order observers for one-sided Lipschitz nonlinearsystems using Riccati equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 12 pp 4968ndash49772012
[14] W Zhang H-S Su Y Liang and Z-Z Han ldquoNon-linearobserver design for one-sided Lipschitz systems an linearmatrix inequality approachrdquo IET Control Theory and Applica-tions vol 6 no 9 pp 1297ndash1303 2012
[15] W Zhang H Su F Zhu and D Yue ldquoA note on observers fordiscrete-time lipschitz nonlinear systemsrdquo IEEETransactions onCircuits and Systems II Express Briefs vol 59 no 2 pp 123ndash1272012
[16] X ZhangMM Polycarpou andT Parisini ldquoA robust detectionand isolation scheme for abrupt and incipient faults in nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 4pp 576ndash593 2002
[17] Y Bingyong T Zuohua and L Dongmei ldquoFault diagnosis fora class of nonlinear stochastic time-delay systemsrdquo Control andInstruments in Chemical Industry vol 34 no 1 pp 12ndash15 2007
[18] M Zhong S X Ding J Lam and H Wang ldquoAn LMI approachto design robust fault detection filter for uncertain LTI systemsrdquoAutomatica vol 39 no 3 pp 543ndash550 2003
[19] S Xie S Tian and Y Fu ldquoA new algorithm of iterative learningcontrol with forgetting factorsrdquo inProceedings of the 8thControlAutomation Robotics and Vision Conference (ICARCV rsquo04) vol1 pp 625ndash630 2004
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
[20] H Yang and M Saif ldquoState observation failure detection andisolation (FDI) in bilinear systemsrdquo International Journal ofControl vol 67 no 6 pp 901ndash920 1997
[21] D Yu and D N Shields ldquoA bilinear fault detection filterrdquoInternational Journal of Control vol 68 no 3 pp 417ndash430 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of