Research Article Robust FDI for a Class of Nonlinear Networked Systems …downloads.hindawi.com/journals/mpe/2014/473901.pdf · 2019-07-31 · is paper considers the robust fault

Research ArticleRobust FDI for a Class of Nonlinear NetworkedSystems with ROQs

An-quan Sun Lin Zhang Wen-feng Wang Jun Wang Tian-lin Niu and Jia-li Zhang

Air and Missile Defense College Air Force Engineering University Xirsquoan 710051 China

Correspondence should be addressed to Wen-feng Wang wwf981612163com

Received 9 September 2014 Accepted 3 November 2014 Published 20 November 2014

Academic Editor Zidong Wang

Copyright copy 2014 An-quan Sun 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

This paper considers the robust fault detection and isolation (FDI) problem for a class of nonlinear networked systems (NSs)with randomly occurring quantisations (ROQs) After vector augmentation Lyapunov function is introduced to ensure theasymptotically mean-square stability of fault detection system By transforming the quantisation effects into sector-boundedparameter uncertainties sufficient conditions ensuring the existence of fault detection filter are proposed which can reduce thedifference between output residuals and fault signals as small as possible under119867

infinframework Finally an example linearized from

a vehicle system is introduced to show the efficiency of the proposed fault detection filter

1 Introduction

With the wide use of computer network and the decreaseof networked techniquersquos cost structures of control systemsare changed The wires of traditional control systems werereplaced by computer network reserved for special or publicuse thus information can be transmitted via controllerssensors actuators and other system units [1] This type ofcontrol systems is used or will be used not only in large-scaledistributed systems (such as large-scale industrial controlsystems) but also in centralized small-scale local area systems(such as space vehicles airliners ships high-performancemotors etc) [2 3] For this type of control systems infor-mation like detection control coordinate instruction andso forth is transmitted via public network While functionslike estimation control diagnosis and so forth can be con-ducted among different network nodes distributively Controlsystems with at least one loop or several loops to make upclosed loop by networks are called networked systems (NSs)Compared to traditional control systems NSs have less wireswhich make them convenient to install and service In thissituation NSs can reduce systemsrsquo weight and size and theyare also convenient to extend As well as the advantagesbrought by NSs through sharing resources on network newchallenges and opportunities were brought to its systems

and control theories such as data transmission delay datadropout and immanent quantisation problemDuring recentdecades many famous journals about automatic control havepublished special issues on NSs [4ndash7] Specifically in 2003report on the development of automatic control made byAstrom and some famous scholars attracted unprecedentedattention to NSs also plenty of research results published[8] However compared to the comprehensive researches onNSsrsquo stability [9ndash11] control [12 13] and filtration [14ndash16]researches on NSsrsquo fault detection seem to be less

Fault detection is a technique to determine whether somecharacteristics or parameters of the system arouse largerdeviation which is unacceptable It is a key technique whichcan improve systemrsquos security and reliability and developedrapidly in the past thirty years Fault detection belongs tothe fault determination part in fault diagnosis procedure Atypical fault diagnosis procedure includes residual genera-tion residual evaluation separation strategy fault estimateand performance evaluation Firstly generate residual signalswhen fault occurs and the signals are sensitive to the fault androbust to unknown input and system parameters uncertainty[16 17] The most ideal condition to reach the above demandis to make residuals decoupled with unknown inputs relatedonly to faults But the decoupling conditions are usuallysevere and most systems are unable to reach In the latest

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 473901 8 pageshttpdxdoiorg1011552014473901

2 Mathematical Problems in Engineering

years a generally adopted approach is transforming the resid-ual generation problem into optimization problem Design afault detection filter (residual generator) which makes thetransfer functionrsquos 119867

infinbound norm from disturbance to

residual as small as possible but the transfer functionrsquos 119867infin

bound norm from (or other bound norms) fault to residualas large as possible Some existing researches demonstratethat there exists a unified solving method among differentoptimal indices [18 19] After the residuals are generatedselect a suitable residual evaluation function and a thresholdvalue Residual evaluation functions and threshold values canbe different in form but usually we take residual evaluationfunction over than threshold value as the judgment of faultalarm Here is the description of fault detection problem thisis what we care about in this paper After fault detection isdone faults are located with suitable separation strategy levelof the faults is estimated by kinds of approaches superiorityof the diagnosis algorithm is evaluated This is the completefault diagnosis procedure

Most of the existing fault detection theories are set upbased on traditional control systems whose data transmis-sions are without delay or with ideal data dropout amongsystem components The appearance of NSs brings newchallenges to traditional fault detection theories and out ofthis the main problem is that the signals transmitted vianetwork became imperfect including delay data dropoutquantisation effect and other signal changes not expectedduring the design Compared to traditional control systemsthe difficulty of NSsrsquo fault detection mainly depends on theimperfection of observation signals By now most of theexisting research results of NSsrsquo diagnosis detection resultsfocus on dealingwith network induced delay and few of themfocus on data dropout multipacket transmission or signalquantisation [20ndash23] In [24ndash27] the authors studied systemrsquosstability and control problem with information coder In[28] Fu and Xie discussed the logarithmic quantiser anddesigned the corresponding state feedback controller Yang etal in [29] designed a networked filter based on logarithmicquantiser In [30] Li et al studied NSsrsquo fault diagnosisproblem whose residual errors have quantisation errors andthen designed fault diagnosis filters respectively based onstatic and dynamic coders But so far to the best of theauthorsrsquo knowledge there are very few related results in theliterature that focus on quantisation problems for NSs withrandomly occurring quantisations (ROQs) This inspires ourcurrent research

In this paper the robust fault detection and isolation(FDI) problem is investigated for a class of nonlinearNSswithROQs A time-variant system with unknown disturbancesand nonlinear terms is studiedThe parameters of the systemare time-variant and satisfy norm-bounded condition andall the quantisers are assumed to be in the logarithmic typewith different quantisation lawsOn the other hand Bernoullidistributed stochastic sequence is used to decide which quan-tiser is to be used at the moment After vector augmentationthe fault detection problem is transformed into robust 119867

infin

filter problem design a robust fault detection filter for allallowable time-variant parameters external disturbance andpossible quantisations reduce the difference between fault

detection filterrsquos output and fault signal as small as possibleunder119867

infinframework The Lyapunov function is introduced

to ensure the asymptotically mean-square stability of faultdetection system and satisfies the corresponding119867

infindistur-

bance restraint conditions A sufficient condition is presentedto ensure the existence of robust fault detection filter inthe form of Linear Matrix Inequality (LMI) as well as thecalculation of fault detection filter parameters An examplelinearized from a vehicle system is used to show the efficiencyof the proposed method

Notations R119899 is the 119899-dimensional Euclidean space 1198972[0infin)

is the space of all square-summable vector functions over[0infin) 120575(119909 119910) is the Kronecker delta function for twointegers 119909 and 119910 if 119909 = 119910 120575(119909 119910) = 1 otherwise 120575(119909 119910) = 0Prob120591

119896= 119894means the probability of stochastic event 120591

119896= 119894

119868 stands for unit matrixE119909 is themathematical expectationof a stochastic variable 119909 |119909

119896| is the Euclidean norm vector of

119909119896 119909 = (suminfin

119896=0|119909119896|2

)12 is the 119897

2norm of 119909

119896 In symmetric

matrices ldquolowastrdquo represents a term that is induced by symmetryand diagsdot sdot sdot stands for a block-diagonal matrix

2 Problem Formulation

Consider a class of nonlinear NSs with ROQs

119909119896+1= 119860119896119909119896+ 119864119896119892 (119909119896) + 119861119896119908119896+119872119896119891119896

119910119896= 119862119909119896+ 119863119908

119896

(1)

where 119909119896isin R119899 is the system state vector 119908

119896isin 1198972[0infin) sub

R119899119908 is the external disturbance 119891119896isin R119899119891 is the fault signal

119860119896 119864119896 119861119896 and 119872

119896are time-variant matrices with suitable

dimensions and 119860119896= 119860 + Δ119860

119896 119864119896= 119864 + Δ119864

119896 119861119896= 119861 +

Δ119861119896 and119872

119896= 119872 + Δ119872

119896 where 119860 119864 119861 and119872 are known

constant matrices Δ119860119896 Δ119864119896 Δ119861119896 and Δ119872

119896are unknown

matrices with sector-bounded conditions given by

[Δ119860119896Δ119864119896Δ119861119896Δ119872119896] = 119873119865

119896[1198671119867211986731198674] (2)

where 119873 1198671 1198672 1198673 and 119867

4are known matrices 119865

119896

satisfies 119865119896119865119879

119896le 119868 119910

119896is the measurement signal under

ideal transmission conditions 119862 and 119863 are known constantmatrices119892(119909119896) is nonlinear vector function with sector restricted

conditions described as [31]

[119892 (119909119896) minus 1198791119909119896]119879

[119892 (119909119896) minus 1198792119909119896] le 0 forall119909

119896isin R119899119909 (3)

where1198791and1198792are known constantmatrices and119879 = 119879

1minus1198792

is symmetric and positive definite matrixConsidering the quantisation issue of NCs in this paper

we model the observing function as

119910119896=

119873

sum

119894=1

120575 (120591119896 119894) 119892119894(119910119896) (4)

where 120591119896is a Bernoulli distributed stochastic variable with

a known distribution law Prob120591119896= 119894 = E120575(120591

119896 119894) = 119901

119894

sum119873

119894=1119901119894le 1 119910

119896is the measurement signal with quantisation

effect

Mathematical Problems in Engineering 3

For each 119894 (1 le 119894 le 119873) 119892119894(sdot) is a time-invariant

logarithmic quantiser which can be described by

119892119894(119910119896) = [119892

(1)

119894(119910(1)

119896) 119892(2)

119894(119910(2)

119896) sdot sdot sdot 119892

(119899119910)

119894(119910(119899119910)

119896)]

119879

(5)

where 119892(119895)119894(sdot) (1 le 119895 le 119899

119910) is a scalar quantisation function

with the quantisation levels described by

U(119895)

119894= plusmn119906

(119895)

119894119898| 119906(119895)

119894119898= (120588(119895)

119894)119898

119906(119895)

1198940 119898 = 0 plusmn1 plusmn2 cup 0

0 lt 120588(119895)

119894lt 1 119906

(119895)

1198940gt 0

(6)

Each of the quantisation levels corresponds to a seg-ment the data in one segment has the same quantisationoutput after being quantized The logarithmic quantisers aredescribed as

119892(119895)

119894(119910(119895)

119896) =

119906(119895)

119894119898

1

1 + 120575(119895)

119894

119906(119895)

119894119898lt119910(119895)

119896le

1

1 minus 120575(119895)

119894

119906(119895)

119894119898

0 119910(119895)

119896= 0

minus119892119895(minus119910(119895)

119896) 119910

(119895)

119896lt 0

(7)

where 120575(119895)119894

= (1 minus 120588(119895)

119894)(1 + 120588

(119895)

119894) Then we can obtain

120575(119895)

119894(119910(119895)

119896) = (1+Δ

(119895)

119894119896)119910(119895)

119896 where |Δ(119895)

119894119896| le 120575(119895)

119894 By definingΔ

119894119896=

diagΔ(1)119894119896 Δ

(119899119910)

119894119896 Δ119894= diag120575(1)

119894 120575

(119899119910)

119894 and 119865

119894119896=

Δ119894119896Δminus1

119894 we can obtain an unknown real-valued time-variant

matrix satisfying 119865119894119896119865119879

119894119896le 119868 Furthermore the measurements

with quantisation effects are given by

119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119894119896Δ119894) 119910119896 (8)

Without loss of generality the subscript 119894 of 119865119894119896can be

ignored after vector augmentation and rewrite (8) as

119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119910119896

=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119862119909119896+

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894)119863119908119896

(9)

Consider a full-order filter described as119909119896+1= 119866119909119896+ 119870119910119896

119903119896= 119871119909119896

(10)

where 119909119896is the state vector of the filter 119903

119896is output residual

119866 119870 and 119871 are the parameters to be determined

By defining 120578119896= [119909119879

119896119909119879

119896]119879 V119896= [119908119879

119896119908119879

119896]119879 and 119903

119896= 119903119896minus

119891119896 the augmented model is given by

120578119896+1= A119896120578119896+

119873

sum

119894=1

[120575 (120591119896 119894) minus 119901

119894]A0120578119896+E119896119892 (119885120578

119896) +B

119896V119896

119903119896= 119903119896minus 119891119896= C120578119896+DV119896

(11)where

A119896=[[

[

119860 0

119873

sum

119894=1

119901119894119870(119868 + 119865

119896Δ119894) 119862 119866

]]

]

+ [119873

0]119865119896[11986710] = A +N119865

119896H1

E119896= [119864

0] + [

119873

0]1198651198961198672= E +N119865

1198961198672

B119896=[[

[

119861 119872

119873

sum

119894=1

120575 (120591119896 119894) 119870 (119868 + 119865

119896Δ119894)119863 0

]]

]

+ [119873

0]119865119896[11986731198674] =B +N119865

119896H3

A0= [

0 0

119870 (119868 + 119865119896Δ119894) 119862 0

]

C = [0 119871] D = [0 minus119868] 119885 = [119868 0]

(12)

After the above augmentation the fault detection filterproblem can be formulated as design robust119867

infinfilter

The design of fault detection filter can be converted intodetermining a series of filter parameters119866119870 and 119871 tomakethe system satisfy two conditions

(1) under zero input condition system (11) is asymptoti-cally mean-square stable

(2) under zero initial condition for all possible nonzeroinputs V

119896 the output 119903

119896of (11) satisfies

E 1199032

lt 1205742

]2 (13)

where 120574 gt 0 is a prescribed119867infin

disturbance scalar

3 Main Results

31 Robust Fault Detection Filter Analysis

Theorem 1 For a given scalar 120574 gt 0 if there exist scalar 120575 andsymmetric and positive definite matrix 119875 = 119875119879 gt 0making thefollowing LMI holds

Φ =

[[[[

[

A119879119896119875A119896+

119873

sum

119894=1

119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1+C119879C A119879

119896119875E119896minus 120575T2

A119879119896119875B119896+C119879D

lowast E119879119896119875E119896minus 120575119868 E119879

119896119875B119896

lowast lowast B119879119896119875B119896minus 1205742

119868 +D119879D

]]]]

]

lt 0 (14)


whereT1= (TT1T2+ TT2T1)2 and T

2= minus(TT

1+ TT2)2 then

system (11) satisfies the aforementioned two conditions

Proof Consider the following Lyapunov function

119881119896= 120578119879

119896119875120578119896 (15)

ConsiderEsum119873119894=1[120575(120591119896 119894)minus119901

119894]2

= sum119873

119894=1119901119894(1minus119901119894) calculate

the difference of 119881119896under V

119896= 0 and its mathematical

expectation is

E Δ119881119896 = 120578119879

119896+1119875120578119896+1minus 120578119879

119896119875120578119896 (16)

where 120578119896+1= A119896120578119896+sum119873

119894=1119901119894(1minus119901119894)A0120578119896+E119896119892(119885120578119896) Formula

(3) can be rewritten as

[119909119896

119892(119909119896)]

119879

[T1

T2

T1198792119868]

119879

[119909119896

119892 (119909119896)] le 0 (17)

By defining 120585119896= [120578119879

119896119892119879

(119885120578119896)]119879 we can obtain

E Δ119881119896 le 120585119879

119896[Φ11Φ12

lowast Φ22

] 120585119896 (18)

where Φ11= A119879119896119875A119896+ sum119873

119894=1119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1

Φ12= A119879119896119875E119896minus 120575T2 and Φ

22= E119879119896119875E119896minus 120575119868

It can be inferred from LMI (14) that

[Φ11Φ12

lowast Φ22

] + [C119879C 0

0 0] lt 0 (19)

furthermore we can obtain EΔ119881119896 lt 0 From the above

proof it can be determined that the fault detection generalsystem (11) is asymptotically mean-square stable [32]

Let us consider the119867infin

disturbance restraint conditionsFor optional V

119896= 0 it can be inferred from (14) and (18) that

EΔ119881119896 + E119903119879

119896119903119896 minus 1205742EV119879119896V119896 = E120585119879

119896Φ120585119896 lt 0 where 120585

119896=

[120578119879

119896119892119879

(119885120578119896) V119879119896]119879 Then sum up the inequality from 0 toinfin

to obtaininfin

sum

119896=0

E 10038161003816100381610038161199031198961003816100381610038161003816

2

lt 1205742

infin

sum

119896=0

E 1003816100381610038161003816V1198961003816100381610038161003816

2

minus E 119881infin + E Δ119881

0 (20)

Since the Lyapunov function is positive-definite it is easy tosee that (13) holds under the initial conditionThis concludesthe proof

32 Robust Fault Detection Filter Design Before we discussthe robust fault detection filter design problem the followingtwo lemmas are useful in deriving the filter design results

Lemma 2 (see [33 34] (Schur complement)) Symmetricalmatrix

119878 = [1198781111987812

lowast 11987822

] (21)

is negative-definite when and only when

119887119900119905ℎ 11987811lt 0 119886119899119889 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (22)

or

119887119900119905ℎ 11987822lt 0 119886119899119889 119878

11minus 119878119879

12119878minus1

2211987812lt 0 (23)

Lemma 3 (see [30]) If119872 = 119872119879119867 and 119864 are real matrices

and 119865 satisfies 119865119896119865119879

119896le 119868 then

119872+119867119865119864 + 119864119879

119865119879

119867119879

lt 0 (24)

holds when and only when there exists a constant 120576 gt 0making the following LMI holds

119872+1

120576119867119867119879

+ 120576119864119879

119864 lt 0 (25)

or

[

[

119872 119867 120576119864119879

lowast minus120576119868 0

lowast lowast minus120576119868

]

]

lt 0 (26)

Theorem4 For a given scalar 120574 gt 0 and the nonlinear discretetime-variant system characterized by (1)sim(3) there exists arobust fault detection filter in the form of (10) which makes thesystem asymptotically mean-square stable And the sufficientcondition satisfying the119867

infindisturbance restraint conditions is

that there exist real matrices 119877 = 119877119879 gt 0 119878 = 119878119879 gt 0 1198801 1198802

1198803and real scalar 120575 120576 gt 0 to make the following LMI holds

[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575T2

0 0 119860119879

119878 Ψ17

0 Ψ19119880119879

30 120576119867

119879

1

lowast Ψ22minus120575T2

0 0 119860119879

119878 Ψ27

0 Ψ29

0 0 120576119867119879

1

lowast lowast minus120575119868 0 0 119864119879

119878 119864119879

119877 0 0 0 0 120576119867119879

2

lowast lowast lowast minus1205742

119868 0 119861119879

119878 Ψ47

0 0 0 0 120576119867119879

3

lowast lowast lowast lowast minus1205742

119868 119872119879

119878 119872119879

119877 0 0 minus119868 0 120576119867119879

4

lowast lowast lowast lowast lowast minus119878 minus119878 0 0 0 119873 0

lowast lowast lowast lowast lowast lowast minus119877 0 0 0 119873 0

lowast lowast lowast lowast lowast lowast lowast minus119878 minus119878 0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast minus119877 0 0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119868 0 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus120576119868 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus120576119868

]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (27)


where Ψ11= Ψ12= minus120575T

1minus 119878 Ψ

17= 119860119879

119877 + sum119873

119894=1119901119894(119868 +

119865119896Δ119894)119862119879

119880119879

2+ 119880119879

1 Ψ19= Ψ29= 120590(119868 + 119865

119896Δ119894)119862119879

119880119879

2 Ψ22=

minus120575T1minus119877Ψ

27= 119860119879

119877+sum119873

119894=1119901119894(119868+119865119896Δ119894)119862119879

119880119879

2Ψ47= 119861119879

119877+

sum119873

119894=1119901119894(119868+119865119896Δ119894)119863119879

119880119879

2 and 120590 = sum119873

119894=1radic119901119894(1 minus 119901

119894) If LMI (27)

holds then the robust fault detection filter parameters satisfyingthe above conditions can be calculated by

119866 = 119883minus1

121198801119878minus1

119884minus119879

12 119870 = 119883

minus1

121198802 119871 = 119880

3119878minus1

119884minus119879

12

(28)

and 11988312

and 11988412

here can be determined by 119868 minus 119877119878minus1 =11988312119884119879

12lt 0

Proof It can be inferred from Lemmas 2 and 3 that the LMI(14) equals

[[[[[[[[[[[[[[

[

minus119875 minus 120575T1minus120575T2

0 A119875 120590A1198790119875 C119879 0 120576H119879

1

lowast minus120575119868 0 E119875 0 0 0 120576119867119879

2


119868 B119875 0 D119879 0 120576H1198793

lowast lowast lowast minus119875 0 0 N 0

lowast lowast lowast lowast minus119875 0 0 0

lowast lowast lowast lowast lowast minus119868 0 0

lowast lowast lowast lowast lowast lowast minus120576119868 0

lowast lowast lowast lowast lowast lowast lowast minus120576119868

]]]]]]]]]]]]]]

]

lt 0 (29)

Define 119875 and 119875minus1 as the following blocks

119875 = [119877 119883

12

119883119879

1211988322

] 119875minus1

= [119878minus1

11988412

119884119879

1211988422

] (30)

where the dimension of a block is as the same asA

By defining

1198691= [119878minus1

119868

119884119879

120] 119869

2= [119868 119877

0 119883119879

12

] (31)

it can be see that 1198751198691= 1198692 11986911987911198751198691= 119869119879

11198692

By conducting contragradient transformation to thematrix of (29) with diag119869119879

1 119868 119868 119869

119879

1 119869119879

1 119868 119868 119868 we can obtain

[[[[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575119878minus119879T2

0 0 119878minus119879

119860119879

Ψ17

0 Ψ1911988412119871119879

0 120576119878minus119879

119867119879

1

lowast Ψ22

minus120575T2

0 0 119860119879

Ψ27

0 Ψ29

0 0 120576119867119879

1


119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119872119879

119877 0 0 minus119868 0 120576119867119879

4

lowast lowast lowast lowast lowast minus119878minus119879

minus119868 0 0 0 119878minus119879

119873 0


lowast lowast lowast lowast lowast lowast lowast minus119878minus119879

minus119868 0 0 0





]]]]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (32)

where Ψ11= minus120575119878

minus119879T1119878minus1

minus 119878minus119879 Ψ12= minus120575119878

minus119879T1minus 119868

Ψ17= 119878minus119879

119860119879

119877 + 119878minus119879

sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119862119879

119870119879

119883119879

12 Ψ19=

120590119878minus119879

(119868 + 119865119896Δ119894)119862119879

119870119879

119883119879

12 Ψ22= minus120575T

1minus 119877 Ψ

27= 119860119879

119877 +

sum119873

119894=1119901119894(119868 +119865119896Δ119894)119862119879

119870119879

119883119879

12Ψ29= 120590(119868+119865

119896Δ119894)119862119879

119870119879

119883119879

12 and

Ψ47= 119861119879

119877 + sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119863119879

119870119879

119883119879

12

By introducing newmatrices1198801= 11988312119866119884119879

121198781198802=11988312119870

and 1198803= 119871119884119879

12119878 and conducting contragradient transforma-

tion to thematrix of (32) with diag119878 119868 119868 119868 119868 119878 119868 119878 119868 119868 119868 119868we can obtain LMI (27) It is to say that if the LMI (27) holdsthe fault detection augmentation system (11) is ensured to


be asymptotically mean-square stable and satisfies the 119867infin

disturbance restraint conditions (13)On the other hand if LMI (27) holds then

[minus119878 minus119878

minus119878 minus119877] lt 0 (33)

and it can be inferred that 119878 minus 119877 lt 0 119878 gt 0 Since119875119875minus1

= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879

12lt 0 By matrix singular

value decomposing two nonsingular square matrices 11988312

11988412

satisfying conditions can always be obtained thus thefilter parameters can be calculated by (28)This concludes theproof

33 Residual Evaluation Strategy After the fault detectionfilter is designed the residual evaluation function 119869

119896and

threshold value 119869th are introduced to evaluate NSs (whetherthere are faults in NSs)

119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)

The following inequalities are used to determine whetherthe fault occurs

119869119896gt 119869th 997904rArr Fault 997904rArr ALARM

119869119896le 119869th 997904rArr No Fault

(35)

4 Simulation Results

In this section in order to show the effectiveness of the pro-posed method a discrete-time linear system linearized froma vehicle system is employed with the following parameters

119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)

The nonlinear term 119892(119909119896) is defined as [119892

1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times

sin(1199092119896) Then we can obtain that

1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)

The parameters of logarithmic quantisers are set as 119906(1)10=

119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t

Quantizer 1Quantizer 2Quantizer 3

minus20 minus15 minus10 minus5minus25

minus20

minus15

minus10

minus5

Figure 1 Quantisation law of three quantisers

0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal

Figure 2 Quantiser selection indicator

120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)

3= 04 as shown in Figure 1The distribution law of

the quantiser selection variable 120591119896is set as Prob120591

119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say

that at a certain time the probability of quantiser 1 is 04the probability of quantiser 2 is 035 and the probability ofquantiser 3 is 025 which are illustrated in Figure 2

This paper aims to determine the parameters of the 119867infin

robust fault detection filter by using Theorem 4 With thehelp of mincx in Matlab [35] we can calculate the minimumvalue of 120574 as 120574lowast = radic1205742opt = 33321 where 120574

2

opt is the optimalsolution of the corresponding convex optimisation problems


0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk

Figure 3 The time-domain simulations of 119908119896and 119865

119896

The parameters of the 119867infin

robust fault detection filter arecalculated as

119866 = [

[

minus01985 minus02718 minus0856

00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)

The unknown input 119908119896is defined as 05 times exp(minus11989610) times

(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly

distributed in [minus1 1] The system parameter 119865119896is taken as

sin(1198962) as illustrated in Figure 3The fault signal 119891119896is taken

as 05 when 119896 isin [200 300] otherwise 119891119896= 0

As shown in Figure 4 (a) represents systemrsquos fault signal(b) represents the square of residuals 119903

119896 (c) represents the

output of residuals evaluation function 119869119896 By fixing the form

of 119908119896 we obtain 119869th = (1500)sum

500

119905=1119869500= 15997 times 10

minus4

after 500 times of simulation without fault As demonstratedin Figure 4(c) residuals evaluation function 119869

260= 15975 times

10minus4

lt 119869th lt 119869261 = 16462 times 10minus4 then it can be seen that the

fault is detected in the 61st step after it occurred

5 Conclusion

In this paper the analysis and design problem of an 119867infin

robust fault detection filter for a class of discrete-time NSswith ROQs have been investigated Because the bandwidthof the communication channel is limited signal must bequantised before it is transmitted via network ROQs areintroduced in the form of randomly occurring logarithmicquantisers which are transformed into randomly occurringuncertainties of system parameters By vector augmentingthe original fault detection problem is transformed intorobust119867

infinfilter problem Then a sufficient condition is pre-

sented to ensure the existence of robust fault detection filterwhich guarantees the asymptotically mean-square stability offault detection system and satisfies a certain119867

infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)

Figure 4 The time-domain simulations of 119891119896 1199032119896and 119869119896

restraint condition The simulation results demonstrate thatthis proposed approach is efficient for NSs with ROQs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W Zhang M S Branicky and S M Phillips ldquoStability ofnetworked control systemsrdquo IEEE Control Systems Magazinevol 21 no 1 pp 84ndash99 2001

[2] M-Y Chow and Y Tipsuwan ldquoNetwork-based control systemsa tutorialrdquo in Proceedings of the 27th Annual Conference of theIEEE Industrial Electronics Society (IECON rsquo01) pp 1593ndash1602Denver Colo USA December 2001

[3] J P Hespanha P Naghshtabrizi and Y Xu ldquoA survey of recentresults in networked control systemsrdquo Proceedings of the IEEEvol 95 no 1 pp 138ndash172 2007

[4] L G Bushnell ldquoSpecial issue on networks and controlrdquo IEEEControl Systems Magazine vol 21 no 1 pp 132ndash143 2001

[5] P Antsaklis and J Baillieul ldquoSpecial issue on networked controlsystemsrdquo IEEE Transactions on Automatic Control vol 49 no4 pp 35ndash46 2004

[6] F-Y Wang D Liu S X Yang and L Li ldquoNetworking sensingand control for networked control systems architectures algo-rithms and applicationsrdquo IEEE Transactions on Systems Manand Cybernetics Part C Applications and Reviews vol 37 no 2pp 157ndash159 2007

[7] T J Koo and S Sastry ldquoSpecial issue on networked embeddedhybrid control systemsrdquo Asian Journal of Control vol 10 no 1pp 103ndash115 2008


[8] R M Murray K M Astrom S P Boyd R W Brockett andG Stein ldquoFuture directions in control in an information-richworldrdquo IEEE Control Systems vol 23 no 2 pp 20ndash33 2003

[9] D Liberzon ldquoOn stabilization of linear systems with limitedinformationrdquo IEEE Transactions on Automatic Control vol 48no 2 pp 304ndash307 2003

[10] S Wlmadssia K Saadaoui and M Benrejeb ldquoNew delay-dependent stability conditions for linear systems with delayrdquoSystem Science and Control Engineering vol 1 no 1 pp 2ndash112013

[11] L Zhang Y Shi T Chen and B Huang ldquoA new method forstabilization of networked control systemswith randomdelaysrdquoIEEE Transactions on Automatic Control vol 50 no 8 pp 1177ndash1181 2005

[12] D Yue Q-L Han and J Lam ldquoNetwork-based robust 119867infin

control of systems with uncertaintyrdquo Automatica vol 41 no 6pp 999ndash1007 2005

[13] F Yang Z Wang Y S Hung and M Gani ldquo119867infin

control fornetworked systems with random communication delaysrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 511ndash5182006

[14] H Ahmada and T Namerikawa ldquoExtended Kalman filter-based mobile robot localization with intermittent measure-mentsrdquo System Science and Control Engineering vol 1 no 1 pp113ndash126 2013

[15] M Sahebsara T Chen and S L Shah ldquoOptimal1198672filtering in

networked control systems withmultiple packet dropoutrdquo IEEETransactions on Automatic Control vol 52 no 8 pp 1508ndash15132007

[16] H J Gao Y Zhao J Lam et al ldquo119867yen fuzzy filtering of nonlinearsystemswith intermittentmeasurementsrdquo IEEETransactions onSystems Fuzzy Systems vol 17 no 2 pp 291ndash300 2009

[17] J Chen and R J Patton Robust Model-Based Fault Diagnosis forDynamic Systems Kluwer Academic Boston Mass USA 1999

[18] X S Ding T Jeinsch M P Frank et al ldquoA unified approachto the optimization of fault detection systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 14 no 7pp 725ndash745 2000

[19] L Qin X He and D H Zhou ldquoA survey of fault diagnosisfor swarm systemsrdquo Systems Science amp Control Engineering AnOpen Access Journal vol 2 no 1 pp 13ndash23 2014

[20] H Fang H Ye and M Zhong ldquoFault diagnosis of networkedcontrol systemsrdquo Annual Reviews in Control vol 31 no 1 pp55ndash68 2007

[21] Y-Q Wang H Ye and G-Z Wang ldquoRecent developmentof fault detection techniques for networked control systemsrdquoControl Theory and Applications vol 26 no 4 pp 400ndash4092009 (Chinese)

[22] X He Z Wang and D H Zhou ldquoRobust fault detectionfor networked systems with communication delay and datamissingrdquo Automatica vol 45 no 11 pp 2634ndash2639 2009

[23] W S Wong and R W Brockett ldquoSystems with finite commu-nication bandwidth constraints II Stabilization with limitedinformation feedbackrdquo IEEE Transactions on Automatic Con-trol vol 44 no 5 pp 1049ndash1053 1999

[24] X He Z Wang X Wang and D H Zhou ldquoNetworked strongtracking filteringwithmultiple packet dropouts algorithms andapplicationsrdquo IEEE Transactions on Industrial Electronics vol61 no 3 pp 1454ndash1463 2014

[25] R W Brockett and D Liberzon ldquoQuantized feedback stabiliza-tion of linear systemsrdquo IEEE Transactions on Automatic Controlvol 45 no 7 pp 1279ndash1289 2000

[26] S Tatikonda and S Mitter ldquoControl under communicationconstraintsrdquo IEEE Transactions on Automatic Control vol 49no 7 pp 1056ndash1068 2004

[27] X He Z Wang Y Liu and D H Zhou ldquoLeast-squaresfault detection and diagnosis for networked sensing systemsusing a direct state estimation approachrdquo IEEE Transactions onIndustrial Informatics vol 9 no 3 pp 1670ndash1679 2013

[28] M Fu and L Xie ldquoThe sector bound approach to quantizedfeedback controlrdquo IEEE Transactions on Automatic Control vol50 no 11 pp 1698ndash1711 2005

[29] Y H Yang W Wang and F W Yang ldquoQuantified Hinfinfilter for

discrete-timeMIMO systemsrdquoControl andDecision vol 24 no6 pp 932ndash936 2009 (Chinese)

[30] W Li P Zhang S X Ding and O Bredtmann ldquoFault detectionover noisy wireless channelsrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 5050ndash5055New Orleans La USA December 2007

[31] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005

[32] X Mao Differential Equations and Applications HorwoodPublishing Chichester UK 1997

[33] S Boyd L El Ghaoui E Feron and V Balakrishnan Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1994

[34] K M Grigoriadis and J T Watson Jr ldquoReduced-order119867infinand

1198712-119871infin

filtering via linear matrix inequalitiesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 33 no 4 pp1326ndash1338 1997

[35] P Gahinet A Nemirovski J A Laub et al LMI ControlToolbox-for Use with Matlab The Math Works Natick MassUSA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


years a generally adopted approach is transforming the resid-ual generation problem into optimization problem Design afault detection filter (residual generator) which makes thetransfer functionrsquos 119867

infinbound norm from disturbance to

residual as small as possible but the transfer functionrsquos 119867infin

bound norm from (or other bound norms) fault to residualas large as possible Some existing researches demonstratethat there exists a unified solving method among differentoptimal indices [18 19] After the residuals are generatedselect a suitable residual evaluation function and a thresholdvalue Residual evaluation functions and threshold values canbe different in form but usually we take residual evaluationfunction over than threshold value as the judgment of faultalarm Here is the description of fault detection problem thisis what we care about in this paper After fault detection isdone faults are located with suitable separation strategy levelof the faults is estimated by kinds of approaches superiorityof the diagnosis algorithm is evaluated This is the completefault diagnosis procedure

Most of the existing fault detection theories are set upbased on traditional control systems whose data transmis-sions are without delay or with ideal data dropout amongsystem components The appearance of NSs brings newchallenges to traditional fault detection theories and out ofthis the main problem is that the signals transmitted vianetwork became imperfect including delay data dropoutquantisation effect and other signal changes not expectedduring the design Compared to traditional control systemsthe difficulty of NSsrsquo fault detection mainly depends on theimperfection of observation signals By now most of theexisting research results of NSsrsquo diagnosis detection resultsfocus on dealingwith network induced delay and few of themfocus on data dropout multipacket transmission or signalquantisation [20ndash23] In [24ndash27] the authors studied systemrsquosstability and control problem with information coder In[28] Fu and Xie discussed the logarithmic quantiser anddesigned the corresponding state feedback controller Yang etal in [29] designed a networked filter based on logarithmicquantiser In [30] Li et al studied NSsrsquo fault diagnosisproblem whose residual errors have quantisation errors andthen designed fault diagnosis filters respectively based onstatic and dynamic coders But so far to the best of theauthorsrsquo knowledge there are very few related results in theliterature that focus on quantisation problems for NSs withrandomly occurring quantisations (ROQs) This inspires ourcurrent research

In this paper the robust fault detection and isolation(FDI) problem is investigated for a class of nonlinearNSswithROQs A time-variant system with unknown disturbancesand nonlinear terms is studiedThe parameters of the systemare time-variant and satisfy norm-bounded condition andall the quantisers are assumed to be in the logarithmic typewith different quantisation lawsOn the other hand Bernoullidistributed stochastic sequence is used to decide which quan-tiser is to be used at the moment After vector augmentationthe fault detection problem is transformed into robust 119867

infin

filter problem design a robust fault detection filter for allallowable time-variant parameters external disturbance andpossible quantisations reduce the difference between fault

detection filterrsquos output and fault signal as small as possibleunder119867

infinframework The Lyapunov function is introduced

to ensure the asymptotically mean-square stability of faultdetection system and satisfies the corresponding119867

infindistur-

bance restraint conditions A sufficient condition is presentedto ensure the existence of robust fault detection filter inthe form of Linear Matrix Inequality (LMI) as well as thecalculation of fault detection filter parameters An examplelinearized from a vehicle system is used to show the efficiencyof the proposed method

Notations R119899 is the 119899-dimensional Euclidean space 1198972[0infin)

is the space of all square-summable vector functions over[0infin) 120575(119909 119910) is the Kronecker delta function for twointegers 119909 and 119910 if 119909 = 119910 120575(119909 119910) = 1 otherwise 120575(119909 119910) = 0Prob120591

119896= 119894means the probability of stochastic event 120591

119896= 119894

119868 stands for unit matrixE119909 is themathematical expectationof a stochastic variable 119909 |119909

119896| is the Euclidean norm vector of

119909119896 119909 = (suminfin

119896=0|119909119896|2

)12 is the 119897

2norm of 119909

119896 In symmetric

matrices ldquolowastrdquo represents a term that is induced by symmetryand diagsdot sdot sdot stands for a block-diagonal matrix

2 Problem Formulation

Consider a class of nonlinear NSs with ROQs

119909119896+1= 119860119896119909119896+ 119864119896119892 (119909119896) + 119861119896119908119896+119872119896119891119896

119910119896= 119862119909119896+ 119863119908

119896

(1)

where 119909119896isin R119899 is the system state vector 119908

119896isin 1198972[0infin) sub

R119899119908 is the external disturbance 119891119896isin R119899119891 is the fault signal

119860119896 119864119896 119861119896 and 119872

119896are time-variant matrices with suitable

dimensions and 119860119896= 119860 + Δ119860

119896 119864119896= 119864 + Δ119864

119896 119861119896= 119861 +

Δ119861119896 and119872

119896= 119872 + Δ119872

119896 where 119860 119864 119861 and119872 are known

constant matrices Δ119860119896 Δ119864119896 Δ119861119896 and Δ119872

119896are unknown

matrices with sector-bounded conditions given by

[Δ119860119896Δ119864119896Δ119861119896Δ119872119896] = 119873119865

119896[1198671119867211986731198674] (2)

where 119873 1198671 1198672 1198673 and 119867

4are known matrices 119865

119896

satisfies 119865119896119865119879

119896le 119868 119910

119896is the measurement signal under

ideal transmission conditions 119862 and 119863 are known constantmatrices119892(119909119896) is nonlinear vector function with sector restricted

conditions described as [31]

[119892 (119909119896) minus 1198791119909119896]119879

[119892 (119909119896) minus 1198792119909119896] le 0 forall119909

119896isin R119899119909 (3)

where1198791and1198792are known constantmatrices and119879 = 119879

1minus1198792

is symmetric and positive definite matrixConsidering the quantisation issue of NCs in this paper

we model the observing function as

119910119896=

119873

sum

119894=1

120575 (120591119896 119894) 119892119894(119910119896) (4)

where 120591119896is a Bernoulli distributed stochastic variable with

a known distribution law Prob120591119896= 119894 = E120575(120591

119896 119894) = 119901

119894

sum119873

119894=1119901119894le 1 119910

119896is the measurement signal with quantisation

effect




119892119894(119910119896) = [119892

(1)

119894(119910(1)

119896) 119892(2)

119894(119910(2)


(119899119910)

119894(119910(119899119910)

119896)]

119879

(5)

where 119892(119895)119894(sdot) (1 le 119895 le 119899



U(119895)

119894= plusmn119906

(119895)

119894119898| 119906(119895)

119894119898= (120588(119895)

119894)119898

119906(119895)


0 lt 120588(119895)

119894lt 1 119906

(119895)

1198940gt 0

(6)


119892(119895)

119894(119910(119895)

119896) =

119906(119895)

119894119898

1

1 + 120575(119895)

119894

119906(119895)

119894119898lt119910(119895)

119896le

1

1 minus 120575(119895)

119894

119906(119895)

119894119898

0 119910(119895)

119896= 0

minus119892119895(minus119910(119895)

119896) 119910

(119895)

119896lt 0

(7)

where 120575(119895)119894

= (1 minus 120588(119895)

119894)(1 + 120588

(119895)


120575(119895)

119894(119910(119895)

119896) = (1+Δ

(119895)

119894119896)119910(119895)

119896 where |Δ(119895)

119894119896| le 120575(119895)


119894119896=

diagΔ(1)119894119896 Δ

(119899119910)

119894119896 Δ119894= diag120575(1)

119894 120575

(119899119910)

119894 and 119865

119894119896=

Δ119894119896Δminus1





119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119894119896Δ119894) 119910119896 (8)



119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119910119896

=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119862119909119896+

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894)119863119908119896

(9)


119903119896= 119871119909119896

(10)




By defining 120578119896= [119909119879

119896119909119879

119896]119879 V119896= [119908119879

119896119908119879

119896]119879 and 119903

119896= 119903119896minus


120578119896+1= A119896120578119896+

119873

sum

119894=1

[120575 (120591119896 119894) minus 119901

119894]A0120578119896+E119896119892 (119885120578

119896) +B

119896V119896

119903119896= 119903119896minus 119891119896= C120578119896+DV119896

(11)where

A119896=[[

[

119860 0

119873

sum

119894=1

119901119894119870(119868 + 119865

119896Δ119894) 119862 119866

]]

]

+ [119873

0]119865119896[11986710] = A +N119865

119896H1

E119896= [119864

0] + [

119873

0]1198651198961198672= E +N119865

1198961198672

B119896=[[

[

119861 119872

119873

sum

119894=1

120575 (120591119896 119894) 119870 (119868 + 119865

119896Δ119894)119863 0

]]

]

+ [119873

0]119865119896[11986731198674] =B +N119865

119896H3

A0= [

0 0

119870 (119868 + 119865119896Δ119894) 119862 0

]

C = [0 119871] D = [0 minus119868] 119885 = [119868 0]

(12)


infinfilter




119896 the output 119903


E 1199032

lt 1205742

]2 (13)


disturbance scalar

3 Main Results



Φ =

[[[[

[

A119879119896119875A119896+

119873

sum

119894=1

119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1+C119879C A119879

119896119875E119896minus 120575T2

A119879119896119875B119896+C119879D

lowast E119879119896119875E119896minus 120575119868 E119879

119896119875B119896


119868 +D119879D

]]]]

]

lt 0 (14)



2= minus(TT

1+ TT2)2 then



119881119896= 120578119879

119896119875120578119896 (15)


119894]2

= sum119873

119894=1119901119894(1minus119901119894) calculate



expectation is

E Δ119881119896 = 120578119879

119896+1119875120578119896+1minus 120578119879

119896119875120578119896 (16)

where 120578119896+1= A119896120578119896+sum119873

119894=1119901119894(1minus119901119894)A0120578119896+E119896119892(119885120578119896) Formula


[119909119896

119892(119909119896)]

119879

[T1

T2

T1198792119868]

119879

[119909119896

119892 (119909119896)] le 0 (17)

By defining 120585119896= [120578119879

119896119892119879

(119885120578119896)]119879 we can obtain

E Δ119881119896 le 120585119879

119896[Φ11Φ12

lowast Φ22

] 120585119896 (18)

where Φ11= A119879119896119875A119896+ sum119873

119894=1119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1

Φ12= A119879119896119875E119896minus 120575T2 and Φ

22= E119879119896119875E119896minus 120575119868


[Φ11Φ12

lowast Φ22

] + [C119879C 0

0 0] lt 0 (19)






EΔ119881119896 + E119903119879

119896119903119896 minus 1205742EV119879119896V119896 = E120585119879

119896Φ120585119896 lt 0 where 120585

119896=

[120578119879

119896119892119879


to obtaininfin

sum

119896=0

E 10038161003816100381610038161199031198961003816100381610038161003816

2

lt 1205742

infin

sum

119896=0

E 1003816100381610038161003816V1198961003816100381610038161003816

2


0 (20)




119878 = [1198781111987812

lowast 11987822

] (21)


119887119900119905ℎ 11987811lt 0 119886119899119889 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (22)

or

119887119900119905ℎ 11987822lt 0 119886119899119889 119878

11minus 119878119879

12119878minus1

2211987812lt 0 (23)


and 119865 satisfies 119865119896119865119879

119896le 119868 then

119872+119867119865119864 + 119864119879

119865119879

119867119879

lt 0 (24)


119872+1

120576119867119867119879

+ 120576119864119879

119864 lt 0 (25)

or

[

[

119872 119867 120576119864119879



]

]

lt 0 (26)





[[[[[[[[[[[[[[[[[[[[[[

[


0 0 119860119879

119878 Ψ17

0 Ψ19119880119879

30 120576119867

119879

1


0 0 119860119879

119878 Ψ27

0 Ψ29

0 0 120576119867119879

1


119878 119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

119878 Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119878 119872119879

119877 0 0 minus119868 0 120576119867119879

4








]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (27)



1minus 119878 Ψ

17= 119860119879

119877 + sum119873

119894=1119901119894(119868 +

119865119896Δ119894)119862119879

119880119879

2+ 119880119879

1 Ψ19= Ψ29= 120590(119868 + 119865

119896Δ119894)119862119879

119880119879

2 Ψ22=


27= 119860119879

119877+sum119873

119894=1119901119894(119868+119865119896Δ119894)119862119879

119880119879

2Ψ47= 119861119879

119877+

sum119873

119894=1119901119894(119868+119865119896Δ119894)119863119879

119880119879

2 and 120590 = sum119873

119894=1radic119901119894(1 minus 119901

119894) If LMI (27)


119866 = 119883minus1

121198801119878minus1

119884minus119879

12 119870 = 119883

minus1

121198802 119871 = 119880

3119878minus1

119884minus119879

12

(28)

and 11988312

and 11988412


12lt 0


[[[[[[[[[[[[[[

[


0 A119875 120590A1198790119875 C119879 0 120576H119879

1

lowast minus120575119868 0 E119875 0 0 0 120576119867119879

2


119868 B119875 0 D119879 0 120576H1198793






]]]]]]]]]]]]]]

]

lt 0 (29)


119875 = [119877 119883

12

119883119879

1211988322

] 119875minus1

= [119878minus1

11988412

119884119879

1211988422

] (30)


By defining

1198691= [119878minus1

119868

119884119879

120] 119869

2= [119868 119877

0 119883119879

12

] (31)

it can be see that 1198751198691= 1198692 11986911987911198751198691= 119869119879

11198692


1 119868 119868 119869

119879

1 119869119879

1 119868 119868 119868 we can obtain

[[[[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575119878minus119879T2

0 0 119878minus119879

119860119879

Ψ17

0 Ψ1911988412119871119879

0 120576119878minus119879

119867119879

1

lowast Ψ22

minus120575T2

0 0 119860119879

Ψ27

0 Ψ29

0 0 120576119867119879

1


119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119872119879

119877 0 0 minus119868 0 120576119867119879

4


minus119868 0 0 0 119878minus119879

119873 0



minus119868 0 0 0





]]]]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (32)

where Ψ11= minus120575119878




Ψ17= 119878minus119879

119860119879

119877 + 119878minus119879

sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119862119879

119870119879

119883119879

12 Ψ19=

120590119878minus119879

(119868 + 119865119896Δ119894)119862119879

119870119879

119883119879

12 Ψ22= minus120575T

1minus 119877 Ψ

27= 119860119879

119877 +

sum119873

119894=1119901119894(119868 +119865119896Δ119894)119862119879

119870119879

119883119879

12Ψ29= 120590(119868+119865

119896Δ119894)119862119879

119870119879

119883119879

12 and

Ψ47= 119861119879

119877 + sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119863119879

119870119879

119883119879

12


121198781198802=11988312119870

and 1198803= 119871119884119879







minus119878 minus119877] lt 0 (33)


= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879



11988412



119896and


119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)




(35)



119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)


1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times


1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)


119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t



minus20

minus15

minus10

minus5


0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal


120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)



119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say




2



0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119892119894(119910119896) = [119892

(1)

119894(119910(1)

119896) 119892(2)

119894(119910(2)


(119899119910)

119894(119910(119899119910)

119896)]

119879

(5)

where 119892(119895)119894(sdot) (1 le 119895 le 119899



U(119895)

119894= plusmn119906

(119895)

119894119898| 119906(119895)

119894119898= (120588(119895)

119894)119898

119906(119895)


0 lt 120588(119895)

119894lt 1 119906

(119895)

1198940gt 0

(6)


119892(119895)

119894(119910(119895)

119896) =

119906(119895)

119894119898

1

1 + 120575(119895)

119894

119906(119895)

119894119898lt119910(119895)

119896le

1

1 minus 120575(119895)

119894

119906(119895)

119894119898

0 119910(119895)

119896= 0

minus119892119895(minus119910(119895)

119896) 119910

(119895)

119896lt 0

(7)

where 120575(119895)119894

= (1 minus 120588(119895)

119894)(1 + 120588

(119895)


120575(119895)

119894(119910(119895)

119896) = (1+Δ

(119895)

119894119896)119910(119895)

119896 where |Δ(119895)

119894119896| le 120575(119895)


119894119896=

diagΔ(1)119894119896 Δ

(119899119910)

119894119896 Δ119894= diag120575(1)

119894 120575

(119899119910)

119894 and 119865

119894119896=

Δ119894119896Δminus1





119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119894119896Δ119894) 119910119896 (8)



119910119896=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119910119896

=

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894) 119862119909119896+

119873

sum

119894=1

120575 (120591119896 119894) (119868 + 119865

119896Δ119894)119863119908119896

(9)


119903119896= 119871119909119896

(10)




By defining 120578119896= [119909119879

119896119909119879

119896]119879 V119896= [119908119879

119896119908119879

119896]119879 and 119903

119896= 119903119896minus


120578119896+1= A119896120578119896+

119873

sum

119894=1

[120575 (120591119896 119894) minus 119901

119894]A0120578119896+E119896119892 (119885120578

119896) +B

119896V119896

119903119896= 119903119896minus 119891119896= C120578119896+DV119896

(11)where

A119896=[[

[

119860 0

119873

sum

119894=1

119901119894119870(119868 + 119865

119896Δ119894) 119862 119866

]]

]

+ [119873

0]119865119896[11986710] = A +N119865

119896H1

E119896= [119864

0] + [

119873

0]1198651198961198672= E +N119865

1198961198672

B119896=[[

[

119861 119872

119873

sum

119894=1

120575 (120591119896 119894) 119870 (119868 + 119865

119896Δ119894)119863 0

]]

]

+ [119873

0]119865119896[11986731198674] =B +N119865

119896H3

A0= [

0 0

119870 (119868 + 119865119896Δ119894) 119862 0

]

C = [0 119871] D = [0 minus119868] 119885 = [119868 0]

(12)


infinfilter




119896 the output 119903


E 1199032

lt 1205742

]2 (13)


disturbance scalar

3 Main Results



Φ =

[[[[

[

A119879119896119875A119896+

119873

sum

119894=1

119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1+C119879C A119879

119896119875E119896minus 120575T2

A119879119896119875B119896+C119879D

lowast E119879119896119875E119896minus 120575119868 E119879

119896119875B119896


119868 +D119879D

]]]]

]

lt 0 (14)



2= minus(TT

1+ TT2)2 then



119881119896= 120578119879

119896119875120578119896 (15)


119894]2

= sum119873

119894=1119901119894(1minus119901119894) calculate



expectation is

E Δ119881119896 = 120578119879

119896+1119875120578119896+1minus 120578119879

119896119875120578119896 (16)

where 120578119896+1= A119896120578119896+sum119873

119894=1119901119894(1minus119901119894)A0120578119896+E119896119892(119885120578119896) Formula


[119909119896

119892(119909119896)]

119879

[T1

T2

T1198792119868]

119879

[119909119896

119892 (119909119896)] le 0 (17)

By defining 120585119896= [120578119879

119896119892119879

(119885120578119896)]119879 we can obtain

E Δ119881119896 le 120585119879

119896[Φ11Φ12

lowast Φ22

] 120585119896 (18)

where Φ11= A119879119896119875A119896+ sum119873

119894=1119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1

Φ12= A119879119896119875E119896minus 120575T2 and Φ

22= E119879119896119875E119896minus 120575119868


[Φ11Φ12

lowast Φ22

] + [C119879C 0

0 0] lt 0 (19)






EΔ119881119896 + E119903119879

119896119903119896 minus 1205742EV119879119896V119896 = E120585119879

119896Φ120585119896 lt 0 where 120585

119896=

[120578119879

119896119892119879


to obtaininfin

sum

119896=0

E 10038161003816100381610038161199031198961003816100381610038161003816

2

lt 1205742

infin

sum

119896=0

E 1003816100381610038161003816V1198961003816100381610038161003816

2


0 (20)




119878 = [1198781111987812

lowast 11987822

] (21)


119887119900119905ℎ 11987811lt 0 119886119899119889 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (22)

or

119887119900119905ℎ 11987822lt 0 119886119899119889 119878

11minus 119878119879

12119878minus1

2211987812lt 0 (23)


and 119865 satisfies 119865119896119865119879

119896le 119868 then

119872+119867119865119864 + 119864119879

119865119879

119867119879

lt 0 (24)


119872+1

120576119867119867119879

+ 120576119864119879

119864 lt 0 (25)

or

[

[

119872 119867 120576119864119879



]

]

lt 0 (26)





[[[[[[[[[[[[[[[[[[[[[[

[


0 0 119860119879

119878 Ψ17

0 Ψ19119880119879

30 120576119867

119879

1


0 0 119860119879

119878 Ψ27

0 Ψ29

0 0 120576119867119879

1


119878 119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

119878 Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119878 119872119879

119877 0 0 minus119868 0 120576119867119879

4








]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (27)



1minus 119878 Ψ

17= 119860119879

119877 + sum119873

119894=1119901119894(119868 +

119865119896Δ119894)119862119879

119880119879

2+ 119880119879

1 Ψ19= Ψ29= 120590(119868 + 119865

119896Δ119894)119862119879

119880119879

2 Ψ22=


27= 119860119879

119877+sum119873

119894=1119901119894(119868+119865119896Δ119894)119862119879

119880119879

2Ψ47= 119861119879

119877+

sum119873

119894=1119901119894(119868+119865119896Δ119894)119863119879

119880119879

2 and 120590 = sum119873

119894=1radic119901119894(1 minus 119901

119894) If LMI (27)


119866 = 119883minus1

121198801119878minus1

119884minus119879

12 119870 = 119883

minus1

121198802 119871 = 119880

3119878minus1

119884minus119879

12

(28)

and 11988312

and 11988412


12lt 0


[[[[[[[[[[[[[[

[


0 A119875 120590A1198790119875 C119879 0 120576H119879

1

lowast minus120575119868 0 E119875 0 0 0 120576119867119879

2


119868 B119875 0 D119879 0 120576H1198793






]]]]]]]]]]]]]]

]

lt 0 (29)


119875 = [119877 119883

12

119883119879

1211988322

] 119875minus1

= [119878minus1

11988412

119884119879

1211988422

] (30)


By defining

1198691= [119878minus1

119868

119884119879

120] 119869

2= [119868 119877

0 119883119879

12

] (31)

it can be see that 1198751198691= 1198692 11986911987911198751198691= 119869119879

11198692


1 119868 119868 119869

119879

1 119869119879

1 119868 119868 119868 we can obtain

[[[[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575119878minus119879T2

0 0 119878minus119879

119860119879

Ψ17

0 Ψ1911988412119871119879

0 120576119878minus119879

119867119879

1

lowast Ψ22

minus120575T2

0 0 119860119879

Ψ27

0 Ψ29

0 0 120576119867119879

1


119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119872119879

119877 0 0 minus119868 0 120576119867119879

4


minus119868 0 0 0 119878minus119879

119873 0



minus119868 0 0 0





]]]]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (32)

where Ψ11= minus120575119878




Ψ17= 119878minus119879

119860119879

119877 + 119878minus119879

sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119862119879

119870119879

119883119879

12 Ψ19=

120590119878minus119879

(119868 + 119865119896Δ119894)119862119879

119870119879

119883119879

12 Ψ22= minus120575T

1minus 119877 Ψ

27= 119860119879

119877 +

sum119873

119894=1119901119894(119868 +119865119896Δ119894)119862119879

119870119879

119883119879

12Ψ29= 120590(119868+119865

119896Δ119894)119862119879

119870119879

119883119879

12 and

Ψ47= 119861119879

119877 + sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119863119879

119870119879

119883119879

12


121198781198802=11988312119870

and 1198803= 119871119884119879







minus119878 minus119877] lt 0 (33)


= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879



11988412



119896and


119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)




(35)



119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)


1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times


1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)


119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t



minus20

minus15

minus10

minus5


0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal


120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)



119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say




2



0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= minus(TT

1+ TT2)2 then



119881119896= 120578119879

119896119875120578119896 (15)


119894]2

= sum119873

119894=1119901119894(1minus119901119894) calculate



expectation is

E Δ119881119896 = 120578119879

119896+1119875120578119896+1minus 120578119879

119896119875120578119896 (16)

where 120578119896+1= A119896120578119896+sum119873

119894=1119901119894(1minus119901119894)A0120578119896+E119896119892(119885120578119896) Formula


[119909119896

119892(119909119896)]

119879

[T1

T2

T1198792119868]

119879

[119909119896

119892 (119909119896)] le 0 (17)

By defining 120585119896= [120578119879

119896119892119879

(119885120578119896)]119879 we can obtain

E Δ119881119896 le 120585119879

119896[Φ11Φ12

lowast Φ22

] 120585119896 (18)

where Φ11= A119879119896119875A119896+ sum119873

119894=1119901119894(1 minus 119901

119894)A1198790119875A0minus 119875 minus 120575T

1

Φ12= A119879119896119875E119896minus 120575T2 and Φ

22= E119879119896119875E119896minus 120575119868


[Φ11Φ12

lowast Φ22

] + [C119879C 0

0 0] lt 0 (19)






EΔ119881119896 + E119903119879

119896119903119896 minus 1205742EV119879119896V119896 = E120585119879

119896Φ120585119896 lt 0 where 120585

119896=

[120578119879

119896119892119879


to obtaininfin

sum

119896=0

E 10038161003816100381610038161199031198961003816100381610038161003816

2

lt 1205742

infin

sum

119896=0

E 1003816100381610038161003816V1198961003816100381610038161003816

2


0 (20)




119878 = [1198781111987812

lowast 11987822

] (21)


119887119900119905ℎ 11987811lt 0 119886119899119889 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (22)

or

119887119900119905ℎ 11987822lt 0 119886119899119889 119878

11minus 119878119879

12119878minus1

2211987812lt 0 (23)


and 119865 satisfies 119865119896119865119879

119896le 119868 then

119872+119867119865119864 + 119864119879

119865119879

119867119879

lt 0 (24)


119872+1

120576119867119867119879

+ 120576119864119879

119864 lt 0 (25)

or

[

[

119872 119867 120576119864119879



]

]

lt 0 (26)





[[[[[[[[[[[[[[[[[[[[[[

[


0 0 119860119879

119878 Ψ17

0 Ψ19119880119879

30 120576119867

119879

1


0 0 119860119879

119878 Ψ27

0 Ψ29

0 0 120576119867119879

1


119878 119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

119878 Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119878 119872119879

119877 0 0 minus119868 0 120576119867119879

4








]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (27)



1minus 119878 Ψ

17= 119860119879

119877 + sum119873

119894=1119901119894(119868 +

119865119896Δ119894)119862119879

119880119879

2+ 119880119879

1 Ψ19= Ψ29= 120590(119868 + 119865

119896Δ119894)119862119879

119880119879

2 Ψ22=


27= 119860119879

119877+sum119873

119894=1119901119894(119868+119865119896Δ119894)119862119879

119880119879

2Ψ47= 119861119879

119877+

sum119873

119894=1119901119894(119868+119865119896Δ119894)119863119879

119880119879

2 and 120590 = sum119873

119894=1radic119901119894(1 minus 119901

119894) If LMI (27)


119866 = 119883minus1

121198801119878minus1

119884minus119879

12 119870 = 119883

minus1

121198802 119871 = 119880

3119878minus1

119884minus119879

12

(28)

and 11988312

and 11988412


12lt 0


[[[[[[[[[[[[[[

[


0 A119875 120590A1198790119875 C119879 0 120576H119879

1

lowast minus120575119868 0 E119875 0 0 0 120576119867119879

2


119868 B119875 0 D119879 0 120576H1198793






]]]]]]]]]]]]]]

]

lt 0 (29)


119875 = [119877 119883

12

119883119879

1211988322

] 119875minus1

= [119878minus1

11988412

119884119879

1211988422

] (30)


By defining

1198691= [119878minus1

119868

119884119879

120] 119869

2= [119868 119877

0 119883119879

12

] (31)

it can be see that 1198751198691= 1198692 11986911987911198751198691= 119869119879

11198692


1 119868 119868 119869

119879

1 119869119879

1 119868 119868 119868 we can obtain

[[[[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575119878minus119879T2

0 0 119878minus119879

119860119879

Ψ17

0 Ψ1911988412119871119879

0 120576119878minus119879

119867119879

1

lowast Ψ22

minus120575T2

0 0 119860119879

Ψ27

0 Ψ29

0 0 120576119867119879

1


119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119872119879

119877 0 0 minus119868 0 120576119867119879

4


minus119868 0 0 0 119878minus119879

119873 0



minus119868 0 0 0





]]]]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (32)

where Ψ11= minus120575119878




Ψ17= 119878minus119879

119860119879

119877 + 119878minus119879

sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119862119879

119870119879

119883119879

12 Ψ19=

120590119878minus119879

(119868 + 119865119896Δ119894)119862119879

119870119879

119883119879

12 Ψ22= minus120575T

1minus 119877 Ψ

27= 119860119879

119877 +

sum119873

119894=1119901119894(119868 +119865119896Δ119894)119862119879

119870119879

119883119879

12Ψ29= 120590(119868+119865

119896Δ119894)119862119879

119870119879

119883119879

12 and

Ψ47= 119861119879

119877 + sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119863119879

119870119879

119883119879

12


121198781198802=11988312119870

and 1198803= 119871119884119879







minus119878 minus119877] lt 0 (33)


= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879



11988412



119896and


119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)




(35)



119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)


1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times


1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)


119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t



minus20

minus15

minus10

minus5


0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal


120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)



119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say




2



0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1minus 119878 Ψ

17= 119860119879

119877 + sum119873

119894=1119901119894(119868 +

119865119896Δ119894)119862119879

119880119879

2+ 119880119879

1 Ψ19= Ψ29= 120590(119868 + 119865

119896Δ119894)119862119879

119880119879

2 Ψ22=


27= 119860119879

119877+sum119873

119894=1119901119894(119868+119865119896Δ119894)119862119879

119880119879

2Ψ47= 119861119879

119877+

sum119873

119894=1119901119894(119868+119865119896Δ119894)119863119879

119880119879

2 and 120590 = sum119873

119894=1radic119901119894(1 minus 119901

119894) If LMI (27)


119866 = 119883minus1

121198801119878minus1

119884minus119879

12 119870 = 119883

minus1

121198802 119871 = 119880

3119878minus1

119884minus119879

12

(28)

and 11988312

and 11988412


12lt 0


[[[[[[[[[[[[[[

[


0 A119875 120590A1198790119875 C119879 0 120576H119879

1

lowast minus120575119868 0 E119875 0 0 0 120576119867119879

2


119868 B119875 0 D119879 0 120576H1198793






]]]]]]]]]]]]]]

]

lt 0 (29)


119875 = [119877 119883

12

119883119879

1211988322

] 119875minus1

= [119878minus1

11988412

119884119879

1211988422

] (30)


By defining

1198691= [119878minus1

119868

119884119879

120] 119869

2= [119868 119877

0 119883119879

12

] (31)

it can be see that 1198751198691= 1198692 11986911987911198751198691= 119869119879

11198692


1 119868 119868 119869

119879

1 119869119879

1 119868 119868 119868 we can obtain

[[[[[[[[[[[[[[[[[[[[[[[[[

[

Ψ11Ψ12minus120575119878minus119879T2

0 0 119878minus119879

119860119879

Ψ17

0 Ψ1911988412119871119879

0 120576119878minus119879

119867119879

1

lowast Ψ22

minus120575T2

0 0 119860119879

Ψ27

0 Ψ29

0 0 120576119867119879

1


119864119879

119877 0 0 0 0 120576119867119879

2


119868 0 119861119879

Ψ47

0 0 0 0 120576119867119879

3


119868 119872119879

119872119879

119877 0 0 minus119868 0 120576119867119879

4


minus119868 0 0 0 119878minus119879

119873 0



minus119868 0 0 0





]]]]]]]]]]]]]]]]]]]]]]]]]

]

lt 0 (32)

where Ψ11= minus120575119878




Ψ17= 119878minus119879

119860119879

119877 + 119878minus119879

sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119862119879

119870119879

119883119879

12 Ψ19=

120590119878minus119879

(119868 + 119865119896Δ119894)119862119879

119870119879

119883119879

12 Ψ22= minus120575T

1minus 119877 Ψ

27= 119860119879

119877 +

sum119873

119894=1119901119894(119868 +119865119896Δ119894)119862119879

119870119879

119883119879

12Ψ29= 120590(119868+119865

119896Δ119894)119862119879

119870119879

119883119879

12 and

Ψ47= 119861119879

119877 + sum119873

119894=1119901119894(119868 + 119865

119896Δ119894)119863119879

119870119879

119883119879

12


121198781198802=11988312119870

and 1198803= 119871119884119879







minus119878 minus119877] lt 0 (33)


= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879



11988412



119896and


119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)




(35)



119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)


1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times


1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)


119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t



minus20

minus15

minus10

minus5


0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal


120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)



119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say




2



0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus119878 minus119877] lt 0 (33)


= 119868 then 119868 minus 119877119878minus1 = 11988312119884119879



11988412



119896and


119869119896=

119896

sum

119904=0

119903119879

119903119903119904

12

119869th = sup119908isin1198972119891=0

E 119869119896

(34)




(35)



119860 = [

[

minus01 01 04

03 02 minus01

02 02 03

]

]

119864 = [

[

02 01 07

01 02 04

03 08 02

]

]

119861 = [

[

04

10825

40687

]

]

119872 = [

[

02

04

05

]

]

119873 = [

[

03

01

03

]

]

1198671= [02 01 05] 119867

2= [03 01 02]

1198673= [035] 119867

4= [02]

119862 = [02 01 05] 119863 = [03]

(36)


1(1199091119896) 0 119892

2(1199092119896)]119879

where 1198921(1199091119896) = 02119909

1119896times sin(119909

1119896) and 119892

2(1199092119896) = minus01119909

2119896times


1198791= [

[

025 0 0

0 0 0

0 0 0

]

]

1198792= [

[

0 0 0

0 0 0

0 0 015

]

]

(37)


119906(2)

10= 119906(3)

10= 119906(1)

20= 119906(2)

20= 119906(3)

20= 119906(1)

30= 119906(2)

30= 119906(3)

30= 1

0 5 10 15 20

0

5

10

15

20

25

Quantizer input

Qua

ntiz

er o

utpu

t



minus20

minus15

minus10

minus5


0 50 100 150 200 250 300 350 400 450 50005

1

15

2

25

3

35

Time (min)

Qua

ntiz

ater

sele

ctio

n sig

nal


120588(1)

1= 120588(2)

1= 120588(3)

1= 06 120588(1)

2= 120588(2)

2= 120588(3)

2= 05 and 120588(1)

3=

120588(2)

3= 120588(3)



119896= 1 = 05

Prob120591119896= 2 = 035 and Prob120591

119896= 3 = 015 It is to say




2



0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 500

0

05

Time k

Time k0 50 100 150 200 250 300 350 400 450 500

005

1

minus05

wk

minus1

minus05

Fk


119896



119866 = [

[


00005 03711 07378

minus00113 01834 06084

]

]

119870 = [

[

0044

minus01722

minus0031

]

]

119871 = [minus0029 minus00192 00945]

(38)


(2119899119896minus 1) for 119896 = 0 1 500 where 119899

119896is uniformly








500

119905=1119869500= 15997 times 10

minus4


260= 15975 times

10minus4



5 Conclusion





infindisturbance

0 50 100 150 200 250 300 350 400 450 500

0020406

minus02

fk

Time k

(a)

0 50 100 150 200 250 300 350 400 450 5000

05

1

r2 k

times10minus5

Time k

(b)

0 50 100 150 200 250 300 350 400 450 500024

J k

times10minus4

Time k

(c)





References








































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































1198712-119871infin










Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Robust FDI for a Class of Nonlinear Networked Systems …downloads.hindawi.com/journals/mpe/2014/473901.pdf · 2019-07-31 · is paper considers the robust fault