research article nonlinear robust observer-based fault...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 713560 7 pageshttpdxdoiorg1011552013713560

Research ArticleNonlinear Robust Observer-Based Fault Detection forNetworked Suspension Control System of Maglev Train

Yun Li Guang He and Jie Li

College of Mechatronics Engineering and Automation National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Guang He heguang410163com

Received 11 January 2013 Revised 22 April 2013 Accepted 26 April 2013

Academic Editor Engang Tian

Copyright copy 2013 Yun Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fault detection approach based on nonlinear robust observer is designed for the networked suspension control system of Maglevtrain with random induced time delay First considering random bounded time-delay and external disturbance the nonlinearmodel of the networked suspension control system is established Then a nonlinear robust observer is designed using the input ofthe suspension gap And the estimate error is proved to be bounded with arbitrary precision by adopting an appropriate parameterWhen sensor faults happen the residual between the real states and the observer outputs indicates which kind of sensor failuresoccurs Finally simulation results using the actual parameters of CMS-04maglev train indicate that the proposedmethod is effectivefor maglev train

1 Introduction

Maglev train is a new type of track transportation vehiclewith no need of traditional wheels which levitates thevehicle on the tracks using electromagnetic force With theadvantages of low noise no pollution small turning radiusand ease of maintenance Maglev train is considered to besuitable for urban transportation and has extensively beenstudied [1 2] The suspension control system is the pivotalcomponent of the electromagnetic suspension system whichis inherently instable and nonlinear and it is necessary todesign feedback controllers for adjusting the electromagneticforces between the electromagnets and the tracks to achievestable levitation Considering of the exiting complicatedelectromagnetic interference (EMI) in the suspension controlsystem the networked control system is adopted to avoidthe electromagnetic interference with wires and improve thereliability of data transmission [3 4] By doing this it can alsorealize data sharing and communication among suspensionmodules on the train and is helpful for the control of thesuspension modules

Thenetworked suspension control systemmainly consistsof three parts [5] the sensor group the controller and theelectromagnets acting as actuators In engineering practice

high security and reliability especially high reliability of thesuspension control system are required for the applicationof Maglev trains If any fault happens the suspension controlsystemmay turn to be unstable which endangers the vehiclersquosoperation It is proved by researches and statistics thatsensor faults eventually become the major cause for theinvalidation of suspension control system [6] Meanwhileconsidering the existing network-induced time delay in thedata transmission it is more complex and more difficult toseparate and recognize the faults of the sensors in networkedcontrol system than doing the job in other control systemTherefore it is of great theoretical and practical value todevelop the researches on the faults detection in networkedsuspension control system

For the present lots of the researches have been focusedon the modeling controllers design and stability of thenetworked control system [7ndash10] In recent years faultdetection and diagnosis on networked control systems hasalso been regarded as an important research direction forengineering requirements Wang et a1 [11] adopted referencemodel approaches to deal with the fault detection of the net-worked control systemswith randomand unknownnetwork-induced delay that might be larger than one sampling periodIn [12] Liu et al study fault detection of linear systems over

2 Mathematical Problems in Engineering

networks with bounded packet loss and a fault detectionfilter for switched system is designed to satisfy performancerequires Zhang et al [13] proposes the approach ofminimumerror entropy filter to indicate the existing fault in the net-worked control systems with random delays packet dropoutand noises by the residual generated by the filter In [14] T-S fuzzy model is adopted to deal with the fault detection ofthe networked control system with Markov transfer delaysBesides observer-based approaches have also been applied tofault detection of the networked control system As for a classof nonlinear networked control systems (NCSs) withMarkovtransfer delays a novel slidingmode observer approach [15] isproposed to solve the fault estimation problem He et al [16]study the problem of robust fault detection for a nonlinearnetworked control system model approximated by uncertainT-S fuzzy models Wherein the residual obtained from therobust fault observer can be sensitive to the fault but robustto exogenous disturbance In [17ndash19] for several networkedcontrol systems respective observer-based fault detectionapproaches have been developed Tian et al [20] concerns thefault tolerant control for discrete networked control systems(NCSs) with probabilistic sensor and actuator fault By usingLyapunov functional method and linear matrix inequalitytechnology sufficient conditions for the mean square stable(MSS) of the NCSs can be obtainedThe literature [21] studiesthe nonlinear networked control system with T-S fuzzymodel and the Lyapunov functional and the linear matrixinequality (LMI) are applied to develop two new stabilityconditions These conditions and an algorithm are used todesign a controller to achieve robust mean square stabilityof the system The literature [22] investigates robust andreliable 119867

infinfilter design for a class of nonlinear networked

control systems four new theorems are proved to cover theconditions for the robust mean square stability of the systemsunder study in terms of LMIs and a decoupling method forthe filter design is developed

The networked suspension control system is nonlinearand a precise mathematical model is always difficult to beobtained in a practical system for the existing model errortime delay and unknown disturbance Therefore designingthe fault detection methods with the property of robust andinsensitive to uncertain disturbance turns to be an importantresearch direction for the networked suspension controlsystem At present an observer-based approach is consideredto be one of the most promising fault detection methods thathave excellent application potential for nonlinear networkedcontrol system In [23] an adaptive nonlinear observerwhich can dominate the effects of unmodeled dynamicsindependently to prevent the state estimations fromdivergingand to get the precise estimations is designed However itis validated that the networked suspension control systemcannot satisfy the applied condition of this adaptive nonlin-ear observer By promoting the application domain of theobserver introduced by [23] we present a nonlinear robustfault observer to handle the fault detection of the networkedsuspension control systemAnovel nonlinear robust observerwith the compensator for the effects of network delay anduncertainty of the system is proposed by adaptive methodand the state estimate error is proved to be bounded and

with arbitrary precision by adopting an appropriate Lyapunovfunction When sensor faults happen the residual betweenthe real states and the observer outputs indicates the directionandmagnitude of the sensor fault Different from the existingmethod in dealing with the linear networked suspensioncontrol system the proposed method concerns the strongnonlinear characteristic in the system modeling and randominduced time delay

The overall purpose of this paper is to develop a nonlinearrobust fault observer for the fault detection of the networkedsuspension control systemThe rest of the paper is organizedas follows Section 2 introduces the networked suspensioncontrol system of Maglev train with CAN bus serving asthe information transmission network of the system andalso analyzes the possible sensor failures Section 3 designsa nonlinear robust fault observer and the estimate error isproved to be bounded by adopting an appropriate Lyapunovfunction And the residual between the real states and theobserver outputs indicates the sensor failures Simulationsin Section 4 using the actual parameters of CMS-04 Maglevtrain demonstrate the effectiveness of this method FinallySection 5 gives the conclusion of this paper

2 Mathematical Model of the NetworkedSuspension Control System

In a Maglev train the suspension control system is the keypart Its normal working is the premise that the train can runin safety The suspension system and motor propulsion sys-tem cause an adverse electromagnetic environment of controlfield of the Maglev train The analogue transmission methodapplied to the signal of the control system is easy to causeldquoglitchrdquo to signal conductors due to external disturbancewhich may cause a serious impact on the systemrsquos suspensioncontrol function Therefore a communication network is tobe introduced in the suspension control system to realize thedigital network transmission of signals which may ensurethat the signals can transmit effectively so as to avoid theimpact of electromagnetic interference

The networked sensors system design is adopted as thegeneral scheme in the networked suspension control systemNetwork nodes of the sensor collect the information ofsystem status with a constant frequency and transmit it afterdigitization of data to corresponding control nodes throughCANbus network As for the controllers working in an event-driven model once the data of sensors reach the node of acontroller through the network the node starts computationimmediately to get the control variables and send it to theelectromagnet in a shape of PWM wave to form a closedloopThe PMWwave itself is capable of resisting disturbanceand the control quantity needs no network transmission Andthe structure of Maglev network control system is shown inFigure 1

The single suspension node is the basic suspension unit ofMaglev train Therefore the analysis and simulations in thispaper are based on the single suspensionnode Figure 2 showsthe scheme of the networked suspension control system(single node) The whole system consists of the suspension

Mathematical Problems in Engineering 3

Electromagneticsuspension system Sensors

Network controller

CANnetwork

Other nodes

Other nodesPWM wave

Figure 1 The structure of networked suspension control system

network controller integrative sensors the network the wavechopper and the electromagnets Especially the networkcan link to other nodes and their corresponding uppersurveillance and control layers

A network structure is used for the Maglev controlsystem to strengthen the systemrsquos capacity restraining strongdisturbance and unavoidably brings some problems suchas delay of signal transmission and loss of network datapackages [23 24] Such problems may degrade the perfor-mance of the suspension control system and even make thesystem unstable under adverse circumstance Therefore itis necessary to consider the impact of the network on thecontrol system First we make assumptions as follows for thesystem

(1) The time-driven model is used for nodes of thenetwork sensors and the event-driven model for thecontrollers

(2) The network-induced time delay satisfies 0 le 120591 le 119879(3) There are three types of status informationmdashgaps

current and speedmdashtransmitted from the sensorsrsquonodes to the network

According to the data stream path in the CAN bus-controlled network the message delay time of the CAN busis

120591 = 119905sent + 119905wait + 119905rev (1)

where 119905sent 119905wait and 119905rev are transform processing delayCANbus access waiting delay and receiving processing delayrespectivelyThen 120591will be a parameter with a supremewhenthe CAN network is not busy This assumption is provedreasonable by engineering practice [3 25]

Denote downward as positive Let the electric resistanceof electromagnet be 119877 inductance 119871

119894 voltage of two ends

of electromagnet 119906 current 119894 mass of bogie and carriage119898 and acceleration of gravity 119892 Assume that the unknowndisturbance force applied on the system is 119889(119905) and definethat the status variable of the system is 119909 = (119909

1 1199092 1199093)1015840

where 1199091= 119911 is the suspension gap of the system 119909

2is the

systemrsquos vertical speed that can be obtained by integration ofacceleration transducer and 119909

3= 119894 is the current intensity

flowing through the electromagnet In accordance with theliterature [26] the dynamic equation of the suspensioncontrol system can be expressed as follows

= 119891 (119909) + 119861119906 (119909 119905 minus 120591) + 119889 (119905) (2)

Network Network

mg

PWM

CAN

CAN

Track

Chopper

Sens

or g

roup

nod

es

Gap

se

nsor

Acce

lera

tion

sens

orCu

rren

tse

nsor

Controllernode

Con

trolle

r

119894(119905)

119894(119905)

119865(119868 119905)

Φ119898

Φ119871 119885(119905)

Figure 2 The scheme of the networked suspension control system(single node)

where 119891(119909) = [1199092

minus1198961199092

31198981199092

1minus(119877119871)119909

3]119879 119861 = [0 0 1119871]

119879

is the nominal system model 119896 are parameters relevant tosuspension system structure and 119889 represents the impact ofuncertainty of parameter119896on the system

It is difficult to use the state equation (2) to analyze anddesign the control system So we change its form to someextent According to the working requirement the current119894eq of certain intensity flowing through the electromagnetgenerates the suspension force tomake the suspension systemmaintain stablity at the balance position 119911eq Accordingly thebalance point of the system is 119909 = (119911eq 0 119894eq) Using Taylorformula at the balance point the dynamic equation can beexpressed as follows

= 119860119909 + 119861119906 (119909 119905 minus 120591) + 119891hot (119909) + 119889 (119905) (3)

where 119860 = (120597119891120597119909)|119909=(119911eq 0119894eq)

= (

0 1 0

2119892119911eq 0 minus2119892119894eq0 0 minus119877119871

) and119891hot(119909) are the higher order terms by the systemrsquos Taylorexpansion

119891(119909) is bounded and hence the 119891hot(119909) is bounded119906(119909 119905 minus 120591) is the systemrsquos control output which makes thesystem (3) keep stable under the circumstance of certainnetwork delay and meet certain performance requirementBecause the higher order term 119891hot(119909)was not deleted in thelinearization process the dynamic equation (2) is equivalentto (3) Therefore the dynamic equation (2) is equivalent to(3)

Discrete the system (3) and we can get

119909 (119896 + 1) = 119860119909 (119896) + 1198610(120591119896) 119906 (119896)

+ 1198611(120591119896) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(4)


where

119909 (119896) = 119909 (119896119879) 119860 = 119890119860119879

119861119889= int

119879

0

119890119860119905

119889119905119861

1198610(120591119896) = int

119879minus120591119896

0

119890119860119905

119889119905119861 1198611(120591119896) = int

119879

119879minus120591119896

119890119860119905

119889119905119861

(5)

Denote 1198610(120591119896) and 119861

1(120591119896) as 119861

0 1198611 and we have the

expression

1198610= 1198610+ 119867119865 (120591

119896) 119864 119861

1= 1198611+ 119867119865 (120591

119896) 119864 (6)

where 1198610 1198611 119867 and 119864 are constant matrices 119865(120591

119896) is the

function of 120591119896 which has 119865

119879

(120591119896)119865(120591119896) le 119868 The proof and

expression of 1198610 1198611 119867 and 119864 can be found in the literature

[27] Hence the discrete model of networked Maglev controlsystem can be expressed as

119909 (119896 + 1) = 119860119909 (119896) + (1198610+ Δ119861) 119906 (119896)

+ (1198611minus Δ119861) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(7)

where Δ119861 = 119867119865(120591119896)119864 Let 119889

119891(119896) = Δ119861119906(119896) minus Δ119861119906(119896 minus 1) +

119861119889119891hot(119909) + 119861

119889119889(119896) and the system (7) can be rewritten as

119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1) + 119889

119891(119896) (8)

1199091(119896) is the output of the networked Maglev control system

so

119910 (119896) = 1199091(119896) = 119862119909 (119896) (9)

where 119862 = [0 0 1] Since the systemrsquos status in the actualsuspension system is boundedwe can always find the positiveconstants 120577 and 120601 to make 119889

119891(119896) satisfy the following

equation10038171003817100381710038171003817119889119891(119909 119905)

10038171003817100381710038171003817le 120577 119909 + 120601 (10)

That is 120577119909 + 120601 defines the range of uncertainty and outsidedisturbance of the system model

3 Fault Detection Approach Based on aNonlinear Robust Observer

The sensors group is made up of gap sensors current sensorsand acceleration sensors In practice it needs several gapsensors tomake theMaglev train get across the joints existingin the tracks By doing this the gap sensors realize theredundancy in hardware Therefore the faults diagnosis ofthe sensors in suspension control system is focused on thecurrent sensors and the acceleration sensors The aim offault detection is to recognize the direction and the extentof the sensor faults and produce an alarming signal Thismethod plays an important significance in improving themaintainability and reliability of the control system If highlyreliable gap signals can be used to estimate the correct currentand velocity signals a comparable big residual between the

estimating current and speed signals output and the physicalsensors will indicate that faults occur in the sensors System(3) is a typical nonlinear system therefore designing the faultdiagnosismethodswith the property of robust and insensitiveto uncertain disturbance turns to be an important researchdirection In this section a fault detection approach based ona nonlinear robust observer is proposed and designed

Lemma 1 (see [28]) For given Hurwitz matrix 1198600

isin 119877119899times119899

and constants V and 120576 gt 0 there exist positive definite andsymmetrical matrix P to ensure the validity of the followingRiccati equation

119860119879

0119875 + 119875119860

0+ V2

119875119875 + |V| 119868 lt 0 (11)

Assumption 2 For P in Lemma 1 there exist 119864 ℎ such that

119875119864 = 119862119879

ℎ (12)

where 119864 and ℎ are constant matrices with appropriatedimensions

Assumption 3 There exists a constant 120593 which ensures thevalidity of the following Riccati equation

1003817100381710038171003817120593119862119890 (119896)1003817100381710038171003817 ge 119890 (13)

Theorem4 Based on Assumption 2 one defines 119890(119896) = 119909(119896)minus

119909(119896) and then the estimation of the systemrsquos states from (15)of nonlinear observer is uniformly bounded If one choosesappropriate gains of120590 gt 0 the observation error of the observercan be made arbitrarily small

119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1)

+ 119864120573 + 119871 (119910 (119896) minus 119862119909 (119896))

(14)

where L is the feedback coefficient to make (AminusLC) be aHurwitz matrix and 120573 is a scalar self-adaptive parameter thatis expressed by the following equation

120573 (119896 + 1) = minus 120590120573 (119896) minus ℎ119879

119862119890 (119896) minus 120572 sgn (120573 (119896))

times (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+ 119909 (119896)2

)

(15)

Proof From (8) and (14) we can obtain

119890 (119896) = 119909 (119896) minus 119909 (119896) = (119860 minus 119871119862) 119890 (119896) minus 119889119891(119909 119896) + 119864120573 (119896)

(16)

Select the lyapunov function

119881 (119896) =1

2[119890119879

(119896) 119875119890 (119896) + 1205732

(119896)] (17)


The derivation ofV along the error equation (16) is written as

Δ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)

=1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

minus 119890119879

(119896) 119875119889119891(119909 119896)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

[(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 + 119875119890

times (120585 ||119909|| + 120601) + 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(18)

Based on 119909 le 119890 + 119909 we can obtain the following

Δ119881 (119896) le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+10038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817(120585 119909 (119896) + 120585 119890 (119896) + 120601)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+ 12058510038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817sdot 119890 (119896) + 120582max (119875) 120585 119890 119909 (119896)

+ 120582max (119875) 120601 119890 + 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)

+1

2119868 +

1

21205852

119875119875] 119890 (119896)

+ 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(19)

From Lemma 1 we can obtain

Δ119881 (119896) le 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(20)

For any positive constant 120572 120582max(119875)120601119890(119896) le

(12120572)(1205822

max(119875)1206012

+ 120572119890(119896)2

) and then we have

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(21)

Substituting (15) into (21) we obtain

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) minus 1205901205732

(119896) minus 120573 (119896) ℎ119879

119862119890 (119896)

minus 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

sdot 119909 (119896)2

)

(22)

From Assumption 2 we have

119889119881

119889119905le minus 120590120573

2

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1205852

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1206012

= minus 1205901205732

+ Ω

(23)

where Ω = (12120572|120573|)1205822

max(119875)1205852

+ (12120572|120573|)1205822

max(119875)1206012

Therefore119881(119890 120573) is decreasingmonotonously to a closed ballof 119861 = (119890 120573) isin 119877

119899

times 119877 119881(119890 120573) le 120590minus1

Ω so e and 120573 aregenerally and uniformly bounded If we choose that 120590 is smallenough the closed ball B will be small enough Thereby theerror of the observer could be arbitrarily small

If we let C = [1 0 0] in (9) namely the observed resultsof nonlinear observer only depending on the gap sensors itcan be seen that the observer can estimate correct systemstatus information Therefore if faults occur in the currentsensors or the acceleration sensors themeasured values of thestates of the observers and physical sensors would be notablydifferent Define the threshold 120575 If

|119890| ge 120575 (24)

it indicates that the fault has occurred and according to thedirection of e we can further diagnose which one has thefault It should be noted that due to the network delay theestimated values of general status and measured values ofthe physical sensors are not exactly the same so 120575 is slightlygreater than 0

4 Simulations and Results

With CMS-04 Maglev train developed by the National Uni-versity of Defense Technology as the object the simulationshave verified the effectiveness of the fault diagnose schemeTable 1 shows the actual parameters of the Maglev system

Let 119875 = (minus21667 minus18333 minus05000

minus18333 minus35000 minus11667


) 119871 = (10 1 0

0 20 0

minus1 minus2 48

) and120590 = 120 Assume the network delay range of the networksuspension system be 0 le 120591

119896le 119879 where 119879 = 1ms If

there are no faults occurring in the sensors the observeroutput diagram is shown in Figure 3 And the curves showlevitation gap the residual between the velocity obtainedfromobserver and computing from the accelerometer outputand the residual between the current obtained from observerand current sensors respectively FromFigure 3 the system is


0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)

Figure 3 Output of robust fault observer with disturbance and timedelay

0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)

Figure 4 Residual outputs when accelerometer failed

not sensitive to the disturbance caused by the induced delayof random network within certain range

Assume that the acceleration sensor is unexpectedlydamaged at 8 s by which a constant fault signal with theamplitude of 02 is superimposed on the normal signalFigure 4 shows the output of the robust fault observer

From Figure 4 it can be concluded that when theaccelerometer is abruptly faulted at 275 s the speed direction

0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)

Figure 5 Residual outputs when current sensors failed

Table 1 Parameters of suspension control system

Property Value119896 0000693119898 750 kg119892 981ms2

119911eq 002m119877 05Ω119871 06H

of residual output responds to the fault signal immediatelyMeanwhile the fault can be judged according to the phasestepWhen the accelerometer failed the systemhas no impacton the gap and current channel It shows a good directivityand can be judged that accelerometer faults have occurred

Similarly the system can detect the current faults in anexcellent way As shown in Figure 5 when a constant faultsignal of 8 A is superimposed on the current sensors it canbe analyzed that the faults come from the current sensorsfrom the residual signal Thereby the fault detection canbe completed The simulation results indicate that the faultdetection approach is effective

5 Conclusion

In engineering practice high security and reliability ofthe suspension control system are the most foundationalrequirements for the safety operation of Maglev train Inthis paper the sensors fault detection of the networkedsuspension control system with random bounded time delayis studied In order to realize the real statesrsquo estimationof the networked suspension control system with arbitrary


precision under the conditions of random bounded timedelay a nonlinear adaptive robust observer is presentedand well designed And the residual between the real statesand the observer outputs indicates which kind of sensorfailures occurs Finally simulations results demonstrate theeffectiveness of this proposed method

Acknowledgment

Thiswork is supported by theNational Natural Science Foun-dation of China under Grant no 11202230

References

[1] LG Yan ldquoProgress of themaglev transportation in chinardquo IEEETransactions on Applied Superconductivity vol 16 no 2 pp1138ndash1141 2006

[2] C Wensen ldquoMaglev technology development and automaticcontrolrdquo in Proceedings of the 22th Chinese Control Conferencevol 8 pp 27ndash30 Yichang China 2003

[3] L Yun and L Zhiqiang ldquoDesign and realization of suspensionsystem based on networked control systems for maglev trainrdquoJournal of System Simulation vol 21 no 14 pp 4420ndash44252009

[4] H Guang L Yun L Z Long and J I Zhide ldquoResearch on faulttolerant control technology based on networked control systemof Maglev trainrdquo in Proceddings of the International Conferenceon Intelligent SystemDesign andEngineeringApplication (ISDEArsquo10) 2010

[5] L Zhiqiang H Aming and C Chengkan ldquoSensitivity con-strained robust controller design of suspension controller forMaglev trainrdquo Journal of Control Theory and Applications vol21 no 3 pp 804ndash808 2004

[6] Z Zhizhou L Xiaolong and L Zhiqiang ldquoSensor fault toler-ance method for Maglev train based on state-observerrdquo ElectricDrive for Locomotives no 4 pp 39ndash42 2008

[7] J P Hespanha P Naghshtabrizia and X Yonggang ldquoSurvey ofrecent results in networked control systemsrdquo Proceedings of theIEEE vol 95 no 1 pp 138ndash162 2007

[8] L Zhang Y Shi T Chen et al ldquoA new method for stabilizationof networked control systems with random delaysrdquo IEEETransactions on Automatic Control vol 50 no 8 pp 1177ndash11812005

[9] X M Zhang G P Lu and Y E Zheng ldquoStabilization of net-worked stochastic time delay fuzzy systems with data dropoutrdquoIEEE Transactions on Fuzzy Systems vol 16 no 3 pp 798ndash8072008

[10] H G Zhang J Yang C Y Su et al ldquoT-S fuzzy model-based robust 119867

infindesign for networked control systems with

uncertaintiesrdquo IEEE Transactions on Industrial Informatics vol3 no 4 pp 289ndash301 2007

[11] Y Q Wang H Ye X S Ding and G Z Wang ldquoFault detectionof networked control systems based on optimal robust faultdetection filterrdquoActaAutomatica Sinica vol 34 no 12 pp 1534ndash1539 2008

[12] B Liu Y Xia Y Yang and M Fu ldquoRobust fault detection oflinear systems over networks with bounded packet lossrdquo Journalof the Franklin Institute vol 349 no 7 pp 2480ndash2499 2012

[13] J Zhang L Du M Ren and G Hou ldquoMinimum error entropyfilter for fault detection of networked control systemsrdquo Entropyno 14 pp 505ndash516 2012

[14] C Peng M R Fei and E Tian ldquoNetworked control for a classof T-S fuzzy systems with stochastic sensor faultsrdquo Fuzzy Setsand Systems vol 212 pp 62ndash77 2013

[15] B Jiang P Shi and Z Mao ldquoSliding mode observer-based faultestimation for nonlinear networked control systemsrdquo CircuitsSystems and Signal Processing vol 30 no 1 pp 1ndash16 2011

[16] H He X Dexia H Xiaodong Z Dengfeng and W ZhiquanldquoFault detection for a class of T-S Fuzzy model based nonlinearnetworked control systemrdquo Information and Control vol 38 no6 pp 703ndash710 2009

[17] M Y Zhong Y X Liu and C E Ma ldquoObserver-based faultdetection for networked control systems with random timedelaysrdquo in Proceedings of the 1st International Conference onInnovative Computing Information and Control pp 528ndash531Los Alamitos CA USA 2006

[18] M Y Zhong and Q L Hart ldquoFault detection filter design fora class of networked control systemsrdquo in Proceedings of the 6thWorld Congress on Intelligent Control and Automation pp 215ndash219 Piscataway NJ USA 2006

[19] Q Zong F Zeng W Liu Y Ji and Y Tao ldquoSliding modeobserver-based fault detection of distributed networked controlsystems with time delayrdquo Circuits Systems and Signal Process-ing vol 31 no 1 pp 203ndash222 2012

[20] E Tian C Peng and Z Gu ldquoFault tolerant control for discretenetworked control systems with random faultsrdquo InternationalJournal of Control Automation and Systems vol 10 no 2 pp444ndash448 2012

[21] E Tian D Yue T C Yang Z Gu and G Lu ldquoT-S fuzzy model-based robust stabilization for networked control systems withprobabilistic sensor and actuator failurerdquo IEEE Transactions onFuzzy Systems vol 19 no 3 pp 553ndash561 2011

[22] E Tian and D Yue ldquoReliable 119867infin

filter design for T-S fuzzymodel-based networked control systems with random sensorfailurerdquo International Journal of Robust and Nonlinear Controlvol 23 no 1 pp 15ndash32 2013

[23] Y Liu ldquoRobust adaptive observer for nonlinear systems withunmodeled dynamicsrdquoAutomatica vol 45 no 8 pp 1891ndash18952009

[24] Y Jose M Pau and M F Josep ldquoControl loop performanceanalysis over networked control systemsrdquo in IEEE 28th AnnualConference of the Industrial Electronics Society pp 2880ndash28852002

[25] W Jun L Xiaolong and L Zhiqiang ldquoThe analysis andmeasureof the real-time performance on CAN bus control networkrdquoIndustry Control Computer vol 17 no 10 pp 21ndash23 2004

[26] A Yetendje M M Seron J A De Dona and J J MartınezldquoSensor fault-tolerant control of a magnetic levitation systemrdquoInternational Journal of Robust and Nonlinear Control vol 20no 18 pp 2108ndash2121 2010

[27] L Xi-mai W Wan-yun W Ke-tai and Y Fei ldquoFault diagnosisof network control systems with uncertain time-delayrdquo SystemsEngineering and Electronics vol 30 no 4 pp 767ndash771 2008

[28] J Xiuqin Observer Design and PerFormance Analysis ForNonlinear Systems Shandong University Jinan China 2008

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


networks with bounded packet loss and a fault detectionfilter for switched system is designed to satisfy performancerequires Zhang et al [13] proposes the approach ofminimumerror entropy filter to indicate the existing fault in the net-worked control systems with random delays packet dropoutand noises by the residual generated by the filter In [14] T-S fuzzy model is adopted to deal with the fault detection ofthe networked control system with Markov transfer delaysBesides observer-based approaches have also been applied tofault detection of the networked control system As for a classof nonlinear networked control systems (NCSs) withMarkovtransfer delays a novel slidingmode observer approach [15] isproposed to solve the fault estimation problem He et al [16]study the problem of robust fault detection for a nonlinearnetworked control system model approximated by uncertainT-S fuzzy models Wherein the residual obtained from therobust fault observer can be sensitive to the fault but robustto exogenous disturbance In [17ndash19] for several networkedcontrol systems respective observer-based fault detectionapproaches have been developed Tian et al [20] concerns thefault tolerant control for discrete networked control systems(NCSs) with probabilistic sensor and actuator fault By usingLyapunov functional method and linear matrix inequalitytechnology sufficient conditions for the mean square stable(MSS) of the NCSs can be obtainedThe literature [21] studiesthe nonlinear networked control system with T-S fuzzymodel and the Lyapunov functional and the linear matrixinequality (LMI) are applied to develop two new stabilityconditions These conditions and an algorithm are used todesign a controller to achieve robust mean square stabilityof the system The literature [22] investigates robust andreliable 119867

infinfilter design for a class of nonlinear networked

control systems four new theorems are proved to cover theconditions for the robust mean square stability of the systemsunder study in terms of LMIs and a decoupling method forthe filter design is developed

The networked suspension control system is nonlinearand a precise mathematical model is always difficult to beobtained in a practical system for the existing model errortime delay and unknown disturbance Therefore designingthe fault detection methods with the property of robust andinsensitive to uncertain disturbance turns to be an importantresearch direction for the networked suspension controlsystem At present an observer-based approach is consideredto be one of the most promising fault detection methods thathave excellent application potential for nonlinear networkedcontrol system In [23] an adaptive nonlinear observerwhich can dominate the effects of unmodeled dynamicsindependently to prevent the state estimations fromdivergingand to get the precise estimations is designed However itis validated that the networked suspension control systemcannot satisfy the applied condition of this adaptive nonlin-ear observer By promoting the application domain of theobserver introduced by [23] we present a nonlinear robustfault observer to handle the fault detection of the networkedsuspension control systemAnovel nonlinear robust observerwith the compensator for the effects of network delay anduncertainty of the system is proposed by adaptive methodand the state estimate error is proved to be bounded and

with arbitrary precision by adopting an appropriate Lyapunovfunction When sensor faults happen the residual betweenthe real states and the observer outputs indicates the directionandmagnitude of the sensor fault Different from the existingmethod in dealing with the linear networked suspensioncontrol system the proposed method concerns the strongnonlinear characteristic in the system modeling and randominduced time delay

The overall purpose of this paper is to develop a nonlinearrobust fault observer for the fault detection of the networkedsuspension control systemThe rest of the paper is organizedas follows Section 2 introduces the networked suspensioncontrol system of Maglev train with CAN bus serving asthe information transmission network of the system andalso analyzes the possible sensor failures Section 3 designsa nonlinear robust fault observer and the estimate error isproved to be bounded by adopting an appropriate Lyapunovfunction And the residual between the real states and theobserver outputs indicates the sensor failures Simulationsin Section 4 using the actual parameters of CMS-04 Maglevtrain demonstrate the effectiveness of this method FinallySection 5 gives the conclusion of this paper

2 Mathematical Model of the NetworkedSuspension Control System

In a Maglev train the suspension control system is the keypart Its normal working is the premise that the train can runin safety The suspension system and motor propulsion sys-tem cause an adverse electromagnetic environment of controlfield of the Maglev train The analogue transmission methodapplied to the signal of the control system is easy to causeldquoglitchrdquo to signal conductors due to external disturbancewhich may cause a serious impact on the systemrsquos suspensioncontrol function Therefore a communication network is tobe introduced in the suspension control system to realize thedigital network transmission of signals which may ensurethat the signals can transmit effectively so as to avoid theimpact of electromagnetic interference

The networked sensors system design is adopted as thegeneral scheme in the networked suspension control systemNetwork nodes of the sensor collect the information ofsystem status with a constant frequency and transmit it afterdigitization of data to corresponding control nodes throughCANbus network As for the controllers working in an event-driven model once the data of sensors reach the node of acontroller through the network the node starts computationimmediately to get the control variables and send it to theelectromagnet in a shape of PWM wave to form a closedloopThe PMWwave itself is capable of resisting disturbanceand the control quantity needs no network transmission Andthe structure of Maglev network control system is shown inFigure 1

The single suspension node is the basic suspension unit ofMaglev train Therefore the analysis and simulations in thispaper are based on the single suspensionnode Figure 2 showsthe scheme of the networked suspension control system(single node) The whole system consists of the suspension



Network controller

CANnetwork

Other nodes

Other nodesPWM wave








120591 = 119905sent + 119905wait + 119905rev (1)





1 1199092 1199093)1015840


2is the




= 119891 (119909) + 119861119906 (119909 119905 minus 120591) + 119889 (119905) (2)

Network Network

mg

PWM

CAN

CAN

Track

Chopper

Sens

or g

roup

nod

es

Gap

se

nsor

Acce

lera

tion

sens

orCu

rren

tse

nsor

Controllernode

Con

trolle

r

119894(119905)

119894(119905)

119865(119868 119905)

Φ119898

Φ119871 119885(119905)


where 119891(119909) = [1199092

minus1198961199092

31198981199092

1minus(119877119871)119909

3]119879 119861 = [0 0 1119871]

119879



= 119860119909 + 119861119906 (119909 119905 minus 120591) + 119891hot (119909) + 119889 (119905) (3)

where 119860 = (120597119891120597119909)|119909=(119911eq 0119894eq)

= (

0 1 0

2119892119911eq 0 minus2119892119894eq0 0 minus119877119871




119909 (119896 + 1) = 119860119909 (119896) + 1198610(120591119896) 119906 (119896)

+ 1198611(120591119896) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(4)


where

119909 (119896) = 119909 (119896119879) 119860 = 119890119860119879

119861119889= int

119879

0

119890119860119905

119889119905119861

1198610(120591119896) = int

119879minus120591119896

0

119890119860119905

119889119905119861 1198611(120591119896) = int

119879

119879minus120591119896

119890119860119905

119889119905119861

(5)

Denote 1198610(120591119896) and 119861

1(120591119896) as 119861


expression

1198610= 1198610+ 119867119865 (120591

119896) 119864 119861

1= 1198611+ 119867119865 (120591

119896) 119864 (6)


119896) is the


119879

(120591119896)119865(120591119896) le 119868 The proof and



119909 (119896 + 1) = 119860119909 (119896) + (1198610+ Δ119861) 119906 (119896)

+ (1198611minus Δ119861) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(7)

where Δ119861 = 119867119865(120591119896)119864 Let 119889

119891(119896) = Δ119861119906(119896) minus Δ119861119906(119896 minus 1) +

119861119889119891hot(119909) + 119861


119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1) + 119889

119891(119896) (8)


so

119910 (119896) = 1199091(119896) = 119862119909 (119896) (9)



equation10038171003817100381710038171003817119889119891(119909 119905)

10038171003817100381710038171003817le 120577 119909 + 120601 (10)






isin 119877119899times119899


119860119879

0119875 + 119875119860

0+ V2

119875119875 + |V| 119868 lt 0 (11)


119875119864 = 119862119879

ℎ (12)



1003817100381710038171003817120593119862119890 (119896)1003817100381710038171003817 ge 119890 (13)



119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1)

+ 119864120573 + 119871 (119910 (119896) minus 119862119909 (119896))

(14)


120573 (119896 + 1) = minus 120590120573 (119896) minus ℎ119879

119862119890 (119896) minus 120572 sgn (120573 (119896))

times (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+ 119909 (119896)2

)

(15)



(16)


119881 (119896) =1

2[119890119879

(119896) 119875119890 (119896) + 1205732

(119896)] (17)



Δ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)

=1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

minus 119890119879

(119896) 119875119889119891(119909 119896)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

[(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 + 119875119890

times (120585 ||119909|| + 120601) + 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(18)


Δ119881 (119896) le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+10038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817(120585 119909 (119896) + 120585 119890 (119896) + 120601)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+ 12058510038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817sdot 119890 (119896) + 120582max (119875) 120585 119890 119909 (119896)

+ 120582max (119875) 120601 119890 + 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)

+1

2119868 +

1

21205852

119875119875] 119890 (119896)

+ 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(19)


Δ119881 (119896) le 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(20)


(12120572)(1205822

max(119875)1206012

+ 120572119890(119896)2

) and then we have

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(21)


Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) minus 1205901205732

(119896) minus 120573 (119896) ℎ119879

119862119890 (119896)

minus 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

sdot 119909 (119896)2

)

(22)


119889119881

119889119905le minus 120590120573

2

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1205852

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1206012

= minus 1205901205732

+ Ω

(23)

where Ω = (12120572|120573|)1205822

max(119875)1205852

+ (12120572|120573|)1205822

max(119875)1206012


119899

times 119877 119881(119890 120573) le 120590minus1



|119890| ge 120575 (24)







) 119871 = (10 1 0

0 20 0

minus1 minus2 48


119896le 119879 where 119879 = 1ms If



0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)


0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)





0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)




119911eq 002m119877 05Ω119871 06H



5 Conclusion




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Network controller

CANnetwork

Other nodes

Other nodesPWM wave








120591 = 119905sent + 119905wait + 119905rev (1)





1 1199092 1199093)1015840


2is the




= 119891 (119909) + 119861119906 (119909 119905 minus 120591) + 119889 (119905) (2)

Network Network

mg

PWM

CAN

CAN

Track

Chopper

Sens

or g

roup

nod

es

Gap

se

nsor

Acce

lera

tion

sens

orCu

rren

tse

nsor

Controllernode

Con

trolle

r

119894(119905)

119894(119905)

119865(119868 119905)

Φ119898

Φ119871 119885(119905)


where 119891(119909) = [1199092

minus1198961199092

31198981199092

1minus(119877119871)119909

3]119879 119861 = [0 0 1119871]

119879



= 119860119909 + 119861119906 (119909 119905 minus 120591) + 119891hot (119909) + 119889 (119905) (3)

where 119860 = (120597119891120597119909)|119909=(119911eq 0119894eq)

= (

0 1 0

2119892119911eq 0 minus2119892119894eq0 0 minus119877119871




119909 (119896 + 1) = 119860119909 (119896) + 1198610(120591119896) 119906 (119896)

+ 1198611(120591119896) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(4)


where

119909 (119896) = 119909 (119896119879) 119860 = 119890119860119879

119861119889= int

119879

0

119890119860119905

119889119905119861

1198610(120591119896) = int

119879minus120591119896

0

119890119860119905

119889119905119861 1198611(120591119896) = int

119879

119879minus120591119896

119890119860119905

119889119905119861

(5)

Denote 1198610(120591119896) and 119861

1(120591119896) as 119861


expression

1198610= 1198610+ 119867119865 (120591

119896) 119864 119861

1= 1198611+ 119867119865 (120591

119896) 119864 (6)


119896) is the


119879

(120591119896)119865(120591119896) le 119868 The proof and



119909 (119896 + 1) = 119860119909 (119896) + (1198610+ Δ119861) 119906 (119896)

+ (1198611minus Δ119861) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(7)

where Δ119861 = 119867119865(120591119896)119864 Let 119889

119891(119896) = Δ119861119906(119896) minus Δ119861119906(119896 minus 1) +

119861119889119891hot(119909) + 119861


119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1) + 119889

119891(119896) (8)


so

119910 (119896) = 1199091(119896) = 119862119909 (119896) (9)



equation10038171003817100381710038171003817119889119891(119909 119905)

10038171003817100381710038171003817le 120577 119909 + 120601 (10)






isin 119877119899times119899


119860119879

0119875 + 119875119860

0+ V2

119875119875 + |V| 119868 lt 0 (11)


119875119864 = 119862119879

ℎ (12)



1003817100381710038171003817120593119862119890 (119896)1003817100381710038171003817 ge 119890 (13)



119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1)

+ 119864120573 + 119871 (119910 (119896) minus 119862119909 (119896))

(14)


120573 (119896 + 1) = minus 120590120573 (119896) minus ℎ119879

119862119890 (119896) minus 120572 sgn (120573 (119896))

times (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+ 119909 (119896)2

)

(15)



(16)


119881 (119896) =1

2[119890119879

(119896) 119875119890 (119896) + 1205732

(119896)] (17)



Δ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)

=1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

minus 119890119879

(119896) 119875119889119891(119909 119896)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

[(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 + 119875119890

times (120585 ||119909|| + 120601) + 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(18)


Δ119881 (119896) le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+10038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817(120585 119909 (119896) + 120585 119890 (119896) + 120601)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+ 12058510038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817sdot 119890 (119896) + 120582max (119875) 120585 119890 119909 (119896)

+ 120582max (119875) 120601 119890 + 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)

+1

2119868 +

1

21205852

119875119875] 119890 (119896)

+ 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(19)


Δ119881 (119896) le 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(20)


(12120572)(1205822

max(119875)1206012

+ 120572119890(119896)2

) and then we have

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(21)


Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) minus 1205901205732

(119896) minus 120573 (119896) ℎ119879

119862119890 (119896)

minus 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

sdot 119909 (119896)2

)

(22)


119889119881

119889119905le minus 120590120573

2

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1205852

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1206012

= minus 1205901205732

+ Ω

(23)

where Ω = (12120572|120573|)1205822

max(119875)1205852

+ (12120572|120573|)1205822

max(119875)1206012


119899

times 119877 119881(119890 120573) le 120590minus1



|119890| ge 120575 (24)







) 119871 = (10 1 0

0 20 0

minus1 minus2 48


119896le 119879 where 119879 = 1ms If



0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)


0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)





0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)




119911eq 002m119877 05Ω119871 06H



5 Conclusion




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119909 (119896) = 119909 (119896119879) 119860 = 119890119860119879

119861119889= int

119879

0

119890119860119905

119889119905119861

1198610(120591119896) = int

119879minus120591119896

0

119890119860119905

119889119905119861 1198611(120591119896) = int

119879

119879minus120591119896

119890119860119905

119889119905119861

(5)

Denote 1198610(120591119896) and 119861

1(120591119896) as 119861


expression

1198610= 1198610+ 119867119865 (120591

119896) 119864 119861

1= 1198611+ 119867119865 (120591

119896) 119864 (6)


119896) is the


119879

(120591119896)119865(120591119896) le 119868 The proof and



119909 (119896 + 1) = 119860119909 (119896) + (1198610+ Δ119861) 119906 (119896)

+ (1198611minus Δ119861) 119906 (119896 minus 1) + 119861

119889119891hot (119909) + 119861

119889119889 (119896)

(7)

where Δ119861 = 119867119865(120591119896)119864 Let 119889

119891(119896) = Δ119861119906(119896) minus Δ119861119906(119896 minus 1) +

119861119889119891hot(119909) + 119861


119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1) + 119889

119891(119896) (8)


so

119910 (119896) = 1199091(119896) = 119862119909 (119896) (9)



equation10038171003817100381710038171003817119889119891(119909 119905)

10038171003817100381710038171003817le 120577 119909 + 120601 (10)






isin 119877119899times119899


119860119879

0119875 + 119875119860

0+ V2

119875119875 + |V| 119868 lt 0 (11)


119875119864 = 119862119879

ℎ (12)



1003817100381710038171003817120593119862119890 (119896)1003817100381710038171003817 ge 119890 (13)



119909 (119896 + 1) = 119860119909 (119896) + 1198610119906 (119896) + 119861

1119906 (119896 minus 1)

+ 119864120573 + 119871 (119910 (119896) minus 119862119909 (119896))

(14)


120573 (119896 + 1) = minus 120590120573 (119896) minus ℎ119879

119862119890 (119896) minus 120572 sgn (120573 (119896))

times (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+ 119909 (119896)2

)

(15)



(16)


119881 (119896) =1

2[119890119879

(119896) 119875119890 (119896) + 1205732

(119896)] (17)



Δ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)

=1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

minus 119890119879

(119896) 119875119889119891(119909 119896)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

[(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 + 119875119890

times (120585 ||119909|| + 120601) + 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(18)


Δ119881 (119896) le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+10038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817(120585 119909 (119896) + 120585 119890 (119896) + 120601)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+ 12058510038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817sdot 119890 (119896) + 120582max (119875) 120585 119890 119909 (119896)

+ 120582max (119875) 120601 119890 + 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)

+1

2119868 +

1

21205852

119875119875] 119890 (119896)

+ 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(19)


Δ119881 (119896) le 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(20)


(12120572)(1205822

max(119875)1206012

+ 120572119890(119896)2

) and then we have

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(21)


Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) minus 1205901205732

(119896) minus 120573 (119896) ℎ119879

119862119890 (119896)

minus 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

sdot 119909 (119896)2

)

(22)


119889119881

119889119905le minus 120590120573

2

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1205852

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1206012

= minus 1205901205732

+ Ω

(23)

where Ω = (12120572|120573|)1205822

max(119875)1205852

+ (12120572|120573|)1205822

max(119875)1206012


119899

times 119877 119881(119890 120573) le 120590minus1



|119890| ge 120575 (24)







) 119871 = (10 1 0

0 20 0

minus1 minus2 48


119896le 119879 where 119879 = 1ms If



0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)


0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)





0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)




119911eq 002m119877 05Ω119871 06H



5 Conclusion




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)

=1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

minus 119890119879

(119896) 119875119889119891(119909 119896)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

[(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 + 119875119890

times (120585 ||119909|| + 120601) + 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(18)


Δ119881 (119896) le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+10038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817(120585 119909 (119896) + 120585 119890 (119896) + 120601)

+ 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)] 119890 (119896)

+ 12058510038171003817100381710038171003817119890119879

(119896) 11987510038171003817100381710038171003817sdot 119890 (119896) + 120582max (119875) 120585 119890 119909 (119896)

+ 120582max (119875) 120601 119890 + 119890119879

(119896) 119875119864120573 + 120573 (119896) 120573 (119896 + 1)

le1

2119890119879

(119896) [(119860 minus 119871119862)119879

119875 + 119875 (119860 minus 119871119862)

+1

2119868 +

1

21205852

119875119875] 119890 (119896)

+ 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(19)


Δ119881 (119896) le 120582max (119875) 120585 119890 (119896) 119909 (119896) + 120582max (119875) 120601 119890 (119896)

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(20)


(12120572)(1205822

max(119875)1206012

+ 120572119890(119896)2

) and then we have

Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) + 120573 (119896) 120573 (119896 + 1)

(21)


Δ119881 (119896) le1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1205852

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

119909(119896)2

+1

21205721003816100381610038161003816120573 (119896)

1003816100381610038161003816

1205822

max (119875) 1206012

+ 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 119890(119896)2

+ 119890119879

(119896) 119875119864120573 (119896) minus 1205901205732

(119896) minus 120573 (119896) ℎ119879

119862119890 (119896)

minus 1205721003816100381610038161003816120573 (119896)

1003816100381610038161003816 (1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

+1003817100381710038171003817120593119862119890 (119896)

1003817100381710038171003817

2

sdot 119909 (119896)2

)

(22)


119889119881

119889119905le minus 120590120573

2

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1205852

+1

212057210038161003816100381610038161205731003816100381610038161003816

1205822

max (119875) 1206012

= minus 1205901205732

+ Ω

(23)

where Ω = (12120572|120573|)1205822

max(119875)1205852

+ (12120572|120573|)1205822

max(119875)1206012


119899

times 119877 119881(119890 120573) le 120590minus1



|119890| ge 120575 (24)







) 119871 = (10 1 0

0 20 0

minus1 minus2 48


119896le 119879 where 119879 = 1ms If



0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)


0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)





0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)




119911eq 002m119877 05Ω119871 06H



5 Conclusion




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 4 45 5

001

002

Time (s)

Gap

(m)

(a)

0 05 1 15 2 25 3 35 4 45 5

0

1

Velo

city

(ms

)

minus1

Time (s)

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)


0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)





0 05 1 15 2 25 3 35 4 45 5

001

002

Gap

(m)

Time (s)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Velo

city

(ms

)

0

1

minus1

(b)

0 05 1 15 2 25 3 35 4 45 5Time (s)

0

20

Curr

ent (

A)

minus20

(c)




119911eq 002m119877 05Ω119871 06H



5 Conclusion




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article nonlinear robust observer-based fault...

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