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

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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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

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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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