Research Article Dynamic Output Feedback Based Active...

Research ArticleDynamic Output Feedback Based ActiveDecentralized Fault-Tolerant Control for ReconfigurableManipulator with Concurrent Failures

Yuanchun Li1 Fan Zhou1 and Bo Zhao12

1Department of Control Engineering Changchun University of Technology Changchun 130012 China2State Key Laboratory of Management and Control for Complex Systems Institute of AutomationChinese Academy of Sciences Beijing 100190 China

Correspondence should be addressed to Bo Zhao zhaob09mailsjlueducn

Received 10 July 2014 Revised 27 November 2014 Accepted 29 November 2014

Academic Editor Gerhard-WilhelmWeber

Copyright copy 2015 Yuanchun Li 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

The goal of this paper is to describe an active decentralized fault-tolerant control (ADFTC) strategy based on dynamic outputfeedback for reconfigurable manipulators with concurrent actuator and sensor failures Consider each joint module of thereconfigurable manipulator as a subsystem and treat the fault as the unknown input of the subsystem Firstly by virtue oflinear matrix inequality (LMI) technique the decentralized proportional-integral observer (DPIO) is designed to estimate andcompensate the sensor fault online hereafter the compensated systemmodel could be derivedThen the actuator fault is estimatedsimilarly by another DPIO using LMI as well and the sufficient condition of the existence of 119867

infinfault-tolerant controller in the

dynamic output feedback is presented for the compensated system model Furthermore the dynamic output feedback controlleris presented based on the estimation of actuator fault to realize active fault-tolerant control Finally two 3-DOF reconfigurablemanipulators with different configurations are employed to verify the effectiveness of the proposed scheme in simulationThemainadvantages of the proposed scheme lie in that it can handle the concurrent faults act on the actuator and sensor on the same jointmodule as well as there is no requirement of fault detection and isolation process moreover it is more feasible to the modularityof the reconfigurable manipulator

1 Introduction

The rapid development of robotics leads the reconfigurablemanipulators to be variously applied to the potential unstruc-tured environments especially in the fields where humancannot intervene directly such as the space station nuclearpower plant and battle field However once the faultappeared in the system it might deteriorate the performanceor cause the loss of the system functionality even stabil-ity As a result there is an increasing demand for safetyreliability and performance of reconfigurable manipulatorsystems Therefore it is an urgent requirement to designcontrol systems which can tolerate the occurrence of failuresduring the operation in order to guarantee the stability andfunctionality and maintain the acceptable performance aswell

Generally speaking two strategies namely the passivefault-tolerant control (PFTC) and the active fault-tolerantcontrol (AFTC) were carried out to achieve the aim at FTC[1] For the PFTC the control structure and parameters havebeen redesigned to go against the occurring of failures Thismeans that the FTC was fixed to tolerate a certain set of faultswithout any change in the controller Du et al [2] obtained thefault information by estimating the outputs of the actuatorsand then compared them with the corresponding prescribedcontrol inputs hereafter the FTC was developed by choosinga safe-park point Jiang et al [3] presented the sliding modeFTC method in the view of the nonlinear flexible spacecraftflywheel failure while in fact it was difficult to obtain theminimum value of the spacecraft flywheel fault Brambillaet al [4] adopted an optimal second-order sliding modecontrol method to design observer-control law by using

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 528086 14 pageshttpdxdoiorg1011552015528086

2 Mathematical Problems in Engineering

the unknown input observer and generalized observer toanalyze residuals but this method can only detect a singlecomponent failure In [5] a decentralized tuning PID outputfeedback controller was utilized to ensure the stability of largeflexible space structures (LFSS) suffered sensor and actuatorfailures Moreover a common solution in the PFTC whensome severe failures are taken into account does not alwaysexist In addition it usually presented a low performanceeven though it exists On the other hand the AFTC maychange the control structure andor parameters to overcomethe bad effect on the whole control systems aroused by thefault Even when necessary it needs to introduce a detectionand estimation module to detect and estimate when thefault occurs Hereafter a supervisory controller should bereconstructed based on the estimated information in the caseof the occurrence of severe faults [6 7] such that it canguarantee the faulty systemrsquos stability and provide acceptablecontrol performance In [8ndash12] only single fault is handledwith AFTC but the concurrent failures on actuator andsensor always occur in actual fact In this regard Rotondo etal [13] used virtual actuator and sensor to correct the actualactuator and sensor faults which achieved the objective ofFTC based on the dynamical controller reconfiguration Samiand Patton [14] proposed a new architecture based on acombination of actuator and sensor Takagi-Sugeno (T-S)proportional state estimators augmented with proportionaland integral feedback (PPI) fault estimators together witha T-S dynamic output feedback control for time-varyingreference tracking

In recent literatures some effort has been made forthe reconfigurable manipulators in fault Yuan et al [15]introduced an energy efficiency monitor approach to detectthe fault where the operation failure was reflected by theefficiency decline of mechanical systemThemeasurement ofeach joint torque is used not only to control the running statebut also to reflect the output capacity The method based ontorque measurement is independent of the whole dynamicmodel of robot systems Ahmad et al [16] presented a dis-tributed fault detection method which can gain the forecasterror through comparing the joint torque signal and torqueestimation being filtered Zhao and Li [17] were concernedwith the active fault-tolerant control problem for reconfig-urablemanipulator actuator based on local joint informationThis scheme processed a simple control structure as well asthe fault could be isolated and tolerated in subsystem and itcan be easily applied to different configurations without anyparameters modification However only a single fault in theactuator or sensor was taken into account to be handled inthe aforementioned methods which limited the availabilityin practice

There are severalmethods successfully used in controllingreconfigurable manipulator In centralized control approachLi et al [18] utilized the elastic parameters of joint modulewhich were identified by fuzzy logic to build finite elementmodel of reconfigurable robot then based on the BP neuralnetwork and genetic algorithm the vibration control methodwas proposed based on the finite element model Sun et al[19] divided 4-DOF module into posture coupled subsys-tems and position feedback subsystem and simultaneously

decomposedworkspace into the above two subspaces to solvethe inverse position problem through forecasting methodBiglarbegian et al [20] presented Type-2 TSK fuzzy logiccontrol method aimed at the reconfigurable manipulatorswith uncertain dynamic parameters This control structurehad a complex and fixed control structure and lackedflexibility thus it was difficult to be implemented to thereconfigurable manipulator when its configuration changedThe other one was distributed control method Muller etal [21] simplified the hardware of control system to ensurethe flexibility of system reconstruction and coordinated tooperate all modular robots through independent centralcontrol Zhu and Lamarche [22] described the system as aset of subsystems through virtual decomposition and thenused the exchange information amongmodules to design thesubsystemrsquos controller The distributed control method canreduce computational complexity and has a more harmo-nious and flexible structure compared to those in centralizedcontrol it makes the system compatibility not only betterbut also more suitable to the concept of modularizationThisdistributed control could conduct more thorough coordinatecontrol however the time delay in communication can resultin imprecise control performance To reduce the difficultyin controller design the decentralized control strategy isdeveloped in a large-scale system In fact the main propertyof the reconfigurable manipulator system lies in differentconfigurations and different degree of freedom Therefore itis more suitable to take a jointmodule as a subsystem and thedecentralized control method can satisfy its main propertyKirchoff and Melek [23] designed a PID robust controllerbased on independent joint information for industrial robotLi [24] introduced a dispersion saturated type of robustcontrol method only considering the single joint dynamicsafter the system was decoupled and treated the influence ofother jointsrsquo dynamics as external disturbanceThe controllerdesign in decentralized control approach utilizes only localinformation thus it is more suitable for the system with anuncertain degree of freedom and different configurations

This paper tries to address an ADFTC for reconfigurablemanipulator with concurrent failures This idea focuses onthe observer design for isolating and estimating the actuatorand sensor faults for the purpose of fault compensation Itdecomposes the entire system into a set of interconnectedsubsystems for developing decentralized control architectureADPIO is designed through using LMI technique to estimateand compensate the sensor fault online and the compen-sated system model is derived Similarly another DPIO isestablished with the sufficient condition of the existence of119867infinfault-tolerant controller and presented in the presence of

the dynamic output feedback Simultaneously the ADFTC isrealized by the estimation of the faults based on the dynamicoutput feedback Finally simulation results show the stabilityand accuracy in the tracking system with simultaneouslyacting actuators and sensors faults

The main advantages of the proposed approach lie inthe following (i) Only local information is used to designthe ADFTC for reconfigurable manipulator with the conceptof decentralized control which can tolerate the concurrentfaults acting in actuator and sensor in an independent joint

Mathematical Problems in Engineering 3

module (ii) LMI technique is used in the design procedure ofDPIOs and dynamic output feedback controller simplifyingthe control structure and making the proof process of systemstability easier on the condition of ensuring the systemstability (iii) There is no requirement of FDI unit here soit saves the reconfiguration time which is necessary in theconventional AFTC (iv) Compared to the existing resultsthe dynamic output feedback is utilized as the state feedbackin the proposed scheme meanwhile it could balance thecontradictions between the irreplaceable state feedback andthe difficulty in physical realization

This paper is presented in the following order Section 2describes nonlinear interconnected subsystem dynamicmodel of the reconfigurable manipulator including thesystems with fault or without fault Section 3 enters into adescription of the observers followed by two subsectionswhich illustrate the stability and performance designconditions for (i) sensor fault estimate observer and (ii)the actuator fault estimate observer In Section 4 thedynamic output feedback controller is designed and itillustrates the stability and performance design conditionsIn Section 5 the effectiveness of the proposed ADFTCmethod is verified by the simulation results of two 3-DOFreconfigurable manipulators with different configurationsSome conclusions are drawn in Section 6

2 Problem Description

For the development of decentralized control consider theentire reconfigurable manipulator with 119899-DOF as a set ofnonlinear interconnected subsystems which are composedof a general joint module And the subsystem 119868 in thereconfigurable manipulator system can be presented by thefollowing state equation [25]

119878119894

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) 119906119894+ ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894

(1)

where 119909119894= [1199091198941 1199091198942]119879

= [119902119894 119902119894]119879

(119894 = 1 2 119899) is the statevector of the subsystem 119878

119894and 119910

119894is the output of the sub-

system 119878119894 The matrices

119860119894= [

0 1

0 0] 119861

119894= [

0

1] 119862

119894= [

1 0

0 1]

119891119894(119902119894 119902119894) = 119872

minus1

119894(119902119894) [minus119862119894(119902119894 119902119894) 119902119894minus 119866119894(119902119894)]

119892119894(119902119894) = 119872

minus1

119894(119902119894)

ℎ119894(119902 119902 119902) = minus119872

minus1

119894(119902119894) 119885119894(119902 119902 119902)

(2)

For the subsystem suffering actuator and sensor failuresconcurrently the faulty dynamic model can be expressed as

119878119894119891

119894= 119860119894119909119894+ 119861119894[119892119894(119902119894) (119906119894+ 119891119894119886)

+119891119894(119902119894 119902119894) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119891119894119904

(3)

here 119891119894119886(119902119894 119902119894 119894) denotes the actuator fault function 119863

119894=

[1 0

0 1] is the sensor fault distribution matrix and 119891

119894119904=

[11989111989411990411198911198941199042]119879 satisfies 119891

119894119904 le 120588119894(119905) where 120588

119894(119905) is a continuous

positive-definite functionThe control objective is to design an active decentralized

fault-tolerant controller in order to guarantee the wholeclosed-loop system stability in the case of the system sufferingconcurrent actuator and sensor faults In other words theproposed fault-tolerant control scheme should make theoutputs of the entire system follow the desired trajectorieseven though concurrent faults occur

3 Decentralized Proportional-IntegralObserver Design

31 Sensor Fault Observer Design In this subsection adecentralized proportional-integral observer is designed forthe faulty dynamic model (3) in order to estimate the sensorfault

Assumption 1 The desired trajectories 119902119889119894 119902119889

119894 and 119902

119889

119894are

bounded

Assumption 2 The subsystem actuator fault function119891119894119886(119902119894 119902119894 119894) and the sensor fault function 119891

119894119904are bounded as

119891119894119886 le 119891lowast

119894119886and 119891

1198941199041 le 119891lowast

1198941199041

Introduce a first order filter as [26]

119894= minus119860

119894119886119911119894+ 119861119894119886119910119894 (4)

Next putting the output of (3) into (4) we have

119894= minus119860

119894119886119911119894+ 119861119894119886119862119894119909119894+ 119861119894119886119863119894119891119894119904 (5)

where 119911119894= [11991111989411199111198942]119879 119910119894is the joint position sensor and

velocity sensor output signal and minus119860119894119886isin 1198772times2 is a Hurwitz

matrix with det(119861119894119886) = 0

Combining the states of (3) and (5) it is gained that

119878119894119891

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+ℎ119894(119902 119902 119902)] + 119863

119894119891119894119904

119910119894= 119862119894119909119894

(6)

where 119909119894= [119909119894

119911119894] 119860119894= [119860119894 0

119861119894119886119862119894 minus119860119894119886

] 119861119894= [119861119894

0] 119862119894= [0 119868] and

119863119894= [0

119861119894119886119863119894]

Now utilize the RBF neural networks to approximatethe unknown term 119891

119894(119902119894 119902119894) and uncertainty term 119892

119894(119902119894) as

follows

119891119894(119902119894 119902119894119882119894119891) = 119882

119879

119894119891Φ119894119891(119902119894 119902119894) + 120576119894119891

119892119894(119902119894119882119894119892) = 119882

119879

119894119892Φ119894119892(119902119894) + 120576119894119892

(7)

where 119882119894119891

and 119882119894119892

are the ideal neural network weightsrespectively Φ(sdot) is the neural network basis function and120576119894119891

and 120576119894119892

are the neural network approximation errorsrespectively


Define 119894119891

and 119894119892

as the estimations of 119882119894119891

and119882119894119892 119891119894(119902119894 119902119894 119894119891) and 119892

119894(119902119894 119894119892) expressed as (8) are the

estimations of 119891119894(119902119894 119902119894119882119894119891) and 119892

119894(119902119894119882119894119892) respectively

119891119894(119902119894 119902119894 119879

119894119891) =

119879

119894119891Φ119894119891(119902119894 119902119894)

119892119894(119902119894 119879

119894119892) =

119879

119894119892Φ119894119892(119902119894)

(8)

where the adjustable parameters are updated by the followingadaptive laws

119882119894119891= 120578119894119891119890119879

119894119875119894119861119894Φ119894119891(119902119894 119902119894)

119882119894119892= 120578119894119892119890119879

119894119875119894119861119894Φ119894119892(119902119894) 119906119894

(9)

where 120578119894119891and 120578119894119892are positive constants

Note that a challenge in implementing the decentralizedcontrol is to compensate the coupling torque caused bythe interconnected joint modules In such a scenario thefollowing assumption is presented

Assumption 3 The interconnection term ℎ119894(119902 119902 119902) is

bounded by [25]

1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le

119899

sum

119895=1

119889119894119895119864119895 (10)

with 119889119894119895ge 0 and 119864

119895= 1 + 119890

119879

119894119875119894119861119894 + 119890

119879

1198941198751198941198611198942

Similarly another RBF neural network term 119894(119890119879

119894119875119894119861119894

119894119901) is introduced to compensate the effect of interconnec-

tion term and defined as follows

V119894(119905) = minus sgn (119890119879

119894119875119894119861119894) 119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) (11)

Similarly another RBF neural network expressed as (12)is proposed to achieve this goal

119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) =

119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817) (12)

where 119882119894119901

is the ideal neural network weight and Φ119894119901(sdot)

is the neural network basis function 119894119901

and Φ119894119901

are theestimations of 119882

119894119901and Φ

119894119901 respectively

119894119901= 119894119901minus 119882119894119901

and Φ119894119901= Φ119894119901minus Φ119894119901are relative estimation errors And

119894119901

can be updated by

119882119894119901= 120578119894119901

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817Φ119894119901 (13)

also 120578119894119901is a positive constant

Finally define approximation error

1205961198941= 119882119879

119894119891Φ119894119891(119902119894 119902119894) + 119882

119879

119894119892Φ119894119892(119902119894) 119906119894+ 120576119894119891+ 120576119894119892119906119894

1205961198942= 119877119894

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817minus 119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817)

120596119894=100381610038161003816100381612059611989411003816100381610038161003816 +100381610038161003816100381612059611989421003816100381610038161003816

(14)

where 119877119894(119890119879

119894119875119894119861119894) = 119899max

119894119895119889119894119895119864119894

Next the decentralized proportional-integral observer(DPIO) as (15) is proposed to simultaneously estimate thesystem states and sensor fault [27]

119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + V119894

minus120573119894sgn (119890119879

119894119875119894119861119894)]

+119863119894119891119894119904+ 119870119894119901(119910119894minus 119894)

119894= 119862119894119894

119891119894119904= 119870119894V119862119894 119890119894119909 + 119870119894119897119862119894119890119894119909

(15)

where 120573119894sgn(119890119879119894119875119894119861119894) is a robust term which is utilized to go

against the effects of neural network approximation error onthe observer

Now define the state estimation errors as 119890119894119909= 119909119894minus 119894and

sensor fault estimation errors as 119890119891119894119904= 119891119894119904minus 119891119894119904 where

119894is

the estimation of the state vector 119909119894and 119891

119894119904is the estimation

of the sensor fault 119891119894119904

Combining (6) and (15) the error dynamics are as follows

119890119894119909= 119894minus119909119894= (119860119894minus 119870119901119904119862119894) 119890119894+ 119863119894119890119891119894119904+ 119861119894119898119904

119890119891119894119904= 119891119894119897minus 119891119894119897

= 119891119894119897minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894119909

+ 119870119894V119862119894119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119904

(16)

where

119898119904= [(119891119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119894119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(17)

Note that the augmented estimator will then be of thefollowing form

119890119894119904= 119860119894119904119890119894119904+ 119873119894119904119911119894119904 (18)

where 119890119894119904= [119890119894119909

119890119891119894119904] and

119860119894119904= [

119860119894minus 119870119901119904119862119894

119863119894

119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894

]

119873119894119904= [

119861119894

0

119870119894V119862119894119861119894 119868

] 119911119894119904= [

119898119904

119891119894119904

]

(19)

Lemma4 (see [28]) In the given system the eigenvalues of thesystem are located in a LMI region in the complex plane definedby 119863(119902 119903) which is defined by merging different eigenvaluesconstraints to produce a119863(119902 119903)LMI region inwhich 119902 and 119903 arethe radius and center of the disc region If there exist symmetric


positive-definite matrices 119875 and 119876 and matrices 119870119894119901 119870119894119897 and

119870119894V as well as the corresponding LMI such that

[[[[[[[

[

11986011

11986012

119875119861119894

0

119860119879

1211986022

119870119894V119862119894119861119894 119868

119875119861119894119870119894V119862119894119861119894 120574119868

10

0 119868 0 1205741198681

]]]]]]]

]

lt 0 (20)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119863

119894+ [119876 (119870

119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119863119894 + (119876119870119894V119862119894119863119894)119879

+ 119868119904

(21)

hold the system is stable and the 119867infin

performance is guaran-teed with an attenuation level 120574

Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870

119894119901 119870119894119897 and 119870

119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867

infinperformance indicator as follows

10038171003817100381710038171198901198941199041003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941199041003817100381710038171003817

2

+ 119881 (0) (22)

where 1198901198941199042

= int1199051

0

(119890119894119904

119879

119890119894119904)119889119905 119911

1198941199042

= int1199051

0

(119911119894119904

119879

119911119894119904)119889119905 and 119868

119890 119868119904

and 1198681are unit matrices

Proof Choosing the Lyapunov candidate as 119881 = 119890119879

1198941199041198751119890119894119904

combine (16) along with the time derivative of 119881 is given by

= 119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904) 119890119894119904+ 2119890119879

1198941199041198751119873119894119904119911119894119904 (23)

Consider the following index

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904) 119889119905 (24)

Thus

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904+ 119868) 119890

119894119904

+119890119879

1198941199041198751119873119894119911119894119904minus 120574119911119879

119894119904119911119894119904] 119889119905 + 119881 (0)

= int

1199051

0

([

119890119894119904

119911119894119904

]

119879

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894

119873119879

1198941198751

minus1205741198681

][

119890119894119904

119911119894119904

])119889119905

+ 119881 (0)

(25)

Considering that the estimation error is bounded definethe following inequality

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894119904

119873119879

1198941199041198751

minus1205741198681

] lt 0 (26)

Next define symmetric positive-definite matrix 1198751=

[119875 0

0 119868] then

1198751119873119894119904= [

119875119861119894

0

119870119894V119862119894119861119894 119868

]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(27)

where 11986011 11986012 and 119860

22are defined in (21) then

int

1199051

0

(119890119879

119894119904119890119894119904) 119889119905 le int

1199051

0

120574119911119879

119894119904119911119894119904119889119905 + 119881 (0) (28)

Inequality (22) can be obtained Therefore the observersatisfies the119867

infinperformance indicator and this completes the

proof of Theorem 5

32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to

119878119894119891

119894= 119860119894119909119894+ 119861119894

sdot [119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119890119891119894119904

(29)

Equivalently design another DPIO as (30) to simultane-ously estimate the system states and actuator fault

119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+V119894minus 120573119894sgn (119890119879

119894119875119894119861119894)]

+119870119894119901(119910119894minus 119894+ 119863119894119890119891119894119904)

119894= 119862119894119894+ 119863119894119890119891119894119904

119891119894119886= 119870119894119897119862119894119890119894119909+ 119870119894V119862119894119890119894119909

(30)


By using (29) and (30) the error dynamics can be changedas follows

119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)

Therefore the augmented estimator will then be of thefollowing form

119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894

minus119870119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119901119886119862119894) 119870119894V119862119894119861119894119892119894

]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)

Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870

119875119886 119870119894119886 and 119870V119886 as well as

matrix LMI such that

[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894

lowast lowast 1205741198681

0 0

lowast lowast lowast 1205741198681

0

lowast lowast lowast lowast 1205741198681

]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)

hold then system (31) is robust asymptotically stable andsatisfies the119867

infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices

Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted

4 Active Decentralized Fault-TolerantController Design

In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently

Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows

119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)

Aggregation of (29) and (38) gives the following system

119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)

Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7

Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860

119894119888 119861119894119888 119862119894119888 and 119863

119894119888as well as matrix LMI such

that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0

lowast lowast 120574119868 0 0

lowast lowast lowast 120574119868 0


]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)

hold then the closed-loop system (39) is robust asymptoticallystable and satisfies the119867


10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904


Proof Choosing the candidate function 119881 = 1199091198791198941198971198751119909119894119897 along

with the trajectories of (39) is then given by

= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)

Considering that estimation error is bounded define thefollowing inequality

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)

Inequality (43) can be obtained Therefore the systemsatisfies the119867


proof of Theorem 7

5 Simulation Results

To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results

The initial position and velocity are set as 1199021(0) = 119902

2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902

3(0) = 0 respectively

By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are

1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB

Configuration bConfiguration a

Figure 1 Configurations of 3-DOF reconfigurable manipulators

1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)

The sensor fault observer gains are calculated as

1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)

Actuator faultEstimated fault

minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)

Sensor faultEstimated fault

minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)

Desired trajectoryWithout sensor faults compensation trajectory

minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)

Actuator faultEstimatede fault

(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)

Sensor faultEstimatede fault

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)

Desired trajectoryWithout sensor fault compensation trajectory

minus08

minus06

minus04

minus02

(f)

Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)

Desired trajectoryActual trajectory



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02

Figure 3 The tolerant tracking performance of configuration 119886

The actuator fault observer gains are calculated as

1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)

Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578

119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance

indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added

to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

Desired trajectoryWithout sensor fault compensatsion trajectory

minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)

Without sensor fault compensation trajectoryDesired trajectory

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02


The fault functions are as follows

1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)

As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant

tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint

To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887


1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)


The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification

6 Conclusion

The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867

infinperformance indicatorsThe effectiveness

of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter

Conflict of Interests

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

Acknowledgment

Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH

References

[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005

[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012

[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010

[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008

[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012

[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006

[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin

fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012

[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012

[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012

[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012

[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013

[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012

[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014

[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013

[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011

[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013

[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014

[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004

[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010

[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011

[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011

[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007


[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007

[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007

[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013

[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994


[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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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


the unknown input observer and generalized observer toanalyze residuals but this method can only detect a singlecomponent failure In [5] a decentralized tuning PID outputfeedback controller was utilized to ensure the stability of largeflexible space structures (LFSS) suffered sensor and actuatorfailures Moreover a common solution in the PFTC whensome severe failures are taken into account does not alwaysexist In addition it usually presented a low performanceeven though it exists On the other hand the AFTC maychange the control structure andor parameters to overcomethe bad effect on the whole control systems aroused by thefault Even when necessary it needs to introduce a detectionand estimation module to detect and estimate when thefault occurs Hereafter a supervisory controller should bereconstructed based on the estimated information in the caseof the occurrence of severe faults [6 7] such that it canguarantee the faulty systemrsquos stability and provide acceptablecontrol performance In [8ndash12] only single fault is handledwith AFTC but the concurrent failures on actuator andsensor always occur in actual fact In this regard Rotondo etal [13] used virtual actuator and sensor to correct the actualactuator and sensor faults which achieved the objective ofFTC based on the dynamical controller reconfiguration Samiand Patton [14] proposed a new architecture based on acombination of actuator and sensor Takagi-Sugeno (T-S)proportional state estimators augmented with proportionaland integral feedback (PPI) fault estimators together witha T-S dynamic output feedback control for time-varyingreference tracking

In recent literatures some effort has been made forthe reconfigurable manipulators in fault Yuan et al [15]introduced an energy efficiency monitor approach to detectthe fault where the operation failure was reflected by theefficiency decline of mechanical systemThemeasurement ofeach joint torque is used not only to control the running statebut also to reflect the output capacity The method based ontorque measurement is independent of the whole dynamicmodel of robot systems Ahmad et al [16] presented a dis-tributed fault detection method which can gain the forecasterror through comparing the joint torque signal and torqueestimation being filtered Zhao and Li [17] were concernedwith the active fault-tolerant control problem for reconfig-urablemanipulator actuator based on local joint informationThis scheme processed a simple control structure as well asthe fault could be isolated and tolerated in subsystem and itcan be easily applied to different configurations without anyparameters modification However only a single fault in theactuator or sensor was taken into account to be handled inthe aforementioned methods which limited the availabilityin practice

There are severalmethods successfully used in controllingreconfigurable manipulator In centralized control approachLi et al [18] utilized the elastic parameters of joint modulewhich were identified by fuzzy logic to build finite elementmodel of reconfigurable robot then based on the BP neuralnetwork and genetic algorithm the vibration control methodwas proposed based on the finite element model Sun et al[19] divided 4-DOF module into posture coupled subsys-tems and position feedback subsystem and simultaneously

decomposedworkspace into the above two subspaces to solvethe inverse position problem through forecasting methodBiglarbegian et al [20] presented Type-2 TSK fuzzy logiccontrol method aimed at the reconfigurable manipulatorswith uncertain dynamic parameters This control structurehad a complex and fixed control structure and lackedflexibility thus it was difficult to be implemented to thereconfigurable manipulator when its configuration changedThe other one was distributed control method Muller etal [21] simplified the hardware of control system to ensurethe flexibility of system reconstruction and coordinated tooperate all modular robots through independent centralcontrol Zhu and Lamarche [22] described the system as aset of subsystems through virtual decomposition and thenused the exchange information amongmodules to design thesubsystemrsquos controller The distributed control method canreduce computational complexity and has a more harmo-nious and flexible structure compared to those in centralizedcontrol it makes the system compatibility not only betterbut also more suitable to the concept of modularizationThisdistributed control could conduct more thorough coordinatecontrol however the time delay in communication can resultin imprecise control performance To reduce the difficultyin controller design the decentralized control strategy isdeveloped in a large-scale system In fact the main propertyof the reconfigurable manipulator system lies in differentconfigurations and different degree of freedom Therefore itis more suitable to take a jointmodule as a subsystem and thedecentralized control method can satisfy its main propertyKirchoff and Melek [23] designed a PID robust controllerbased on independent joint information for industrial robotLi [24] introduced a dispersion saturated type of robustcontrol method only considering the single joint dynamicsafter the system was decoupled and treated the influence ofother jointsrsquo dynamics as external disturbanceThe controllerdesign in decentralized control approach utilizes only localinformation thus it is more suitable for the system with anuncertain degree of freedom and different configurations

This paper tries to address an ADFTC for reconfigurablemanipulator with concurrent failures This idea focuses onthe observer design for isolating and estimating the actuatorand sensor faults for the purpose of fault compensation Itdecomposes the entire system into a set of interconnectedsubsystems for developing decentralized control architectureADPIO is designed through using LMI technique to estimateand compensate the sensor fault online and the compen-sated system model is derived Similarly another DPIO isestablished with the sufficient condition of the existence of119867infinfault-tolerant controller and presented in the presence of

the dynamic output feedback Simultaneously the ADFTC isrealized by the estimation of the faults based on the dynamicoutput feedback Finally simulation results show the stabilityand accuracy in the tracking system with simultaneouslyacting actuators and sensors faults

The main advantages of the proposed approach lie inthe following (i) Only local information is used to designthe ADFTC for reconfigurable manipulator with the conceptof decentralized control which can tolerate the concurrentfaults acting in actuator and sensor in an independent joint






119878119894

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) 119906119894+ ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894

(1)

where 119909119894= [1199091198941 1199091198942]119879

= [119902119894 119902119894]119879


119894and 119910



119860119894= [

0 1

0 0] 119861

119894= [

0

1] 119862

119894= [

1 0

0 1]

119891119894(119902119894 119902119894) = 119872

minus1

119894(119902119894) [minus119862119894(119902119894 119902119894) 119902119894minus 119866119894(119902119894)]

119892119894(119902119894) = 119872

minus1

119894(119902119894)

ℎ119894(119902 119902 119902) = minus119872

minus1

119894(119902119894) 119885119894(119902 119902 119902)

(2)


119878119894119891

119894= 119860119894119909119894+ 119861119894[119892119894(119902119894) (119906119894+ 119891119894119886)

+119891119894(119902119894 119902119894) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119891119894119904

(3)


119894=

[1 0


119894119904=

[11989111989411990411198911198941199042]119879 satisfies 119891

119894119904 le 120588119894(119905) where 120588







119894 and 119902

119889

119894are

bounded



119891119894119886 le 119891lowast

119894119886and 119891

1198941199041 le 119891lowast

1198941199041


119894= minus119860

119894119886119911119894+ 119861119894119886119910119894 (4)


119894= minus119860

119894119886119911119894+ 119861119894119886119862119894119909119894+ 119861119894119886119863119894119891119894119904 (5)



matrix with det(119861119894119886) = 0


119878119894119891

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+ℎ119894(119902 119902 119902)] + 119863

119894119891119894119904

119910119894= 119862119894119909119894

(6)

where 119909119894= [119909119894

119911119894] 119860119894= [119860119894 0

119861119894119886119862119894 minus119860119894119886

] 119861119894= [119861119894

0] 119862119894= [0 119868] and

119863119894= [0

119861119894119886119863119894]



119894(119902119894) as

follows

119891119894(119902119894 119902119894119882119894119891) = 119882

119879

119894119891Φ119894119891(119902119894 119902119894) + 120576119894119891

119892119894(119902119894119882119894119892) = 119882

119879

119894119892Φ119894119892(119902119894) + 120576119894119892

(7)

where 119882119894119891

and 119882119894119892


and 120576119894119892



Define 119894119891

and 119894119892


and119882119894119892 119891119894(119902119894 119902119894 119894119891) and 119892



119894(119902119894119882119894119892) respectively

119891119894(119902119894 119902119894 119879

119894119891) =

119879

119894119891Φ119894119891(119902119894 119902119894)

119892119894(119902119894 119879

119894119892) =

119879

119894119892Φ119894119892(119902119894)

(8)


119882119894119891= 120578119894119891119890119879

119894119875119894119861119894Φ119894119891(119902119894 119902119894)

119882119894119892= 120578119894119892119890119879

119894119875119894119861119894Φ119894119892(119902119894) 119906119894

(9)




bounded by [25]

1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le

119899

sum

119895=1

119889119894119895119864119895 (10)

with 119889119894119895ge 0 and 119864

119895= 1 + 119890

119879

119894119875119894119861119894 + 119890

119879

1198941198751198941198611198942


119894119875119894119861119894



V119894(119905) = minus sgn (119890119879

119894119875119894119861119894) 119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) (11)


119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) =

119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817) (12)

where 119882119894119901



and Φ119894119901


119894119901and Φ


119894119901= 119894119901minus 119882119894119901


119894119901

can be updated by

119882119894119901= 120578119894119901

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817Φ119894119901 (13)



1205961198941= 119882119879

119894119891Φ119894119891(119902119894 119902119894) + 119882

119879

119894119892Φ119894119892(119902119894) 119906119894+ 120576119894119891+ 120576119894119892119906119894

1205961198942= 119877119894

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817minus 119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817)

120596119894=100381610038161003816100381612059611989411003816100381610038161003816 +100381610038161003816100381612059611989421003816100381610038161003816

(14)

where 119877119894(119890119879

119894119875119894119861119894) = 119899max

119894119895119889119894119895119864119894


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + V119894

minus120573119894sgn (119890119879

119894119875119894119861119894)]

+119863119894119891119894119904+ 119870119894119901(119910119894minus 119894)

119894= 119862119894119894

119891119894119904= 119870119894V119862119894 119890119894119909 + 119870119894119897119862119894119890119894119909

(15)





119894is





119890119894119909= 119894minus119909119894= (119860119894minus 119870119901119904119862119894) 119890119894+ 119863119894119890119891119894119904+ 119861119894119898119904

119890119891119894119904= 119891119894119897minus 119891119894119897

= 119891119894119897minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894119909

+ 119870119894V119862119894119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119904

(16)

where

119898119904= [(119891119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119894119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(17)


119890119894119904= 119860119894119904119890119894119904+ 119873119894119904119911119894119904 (18)

where 119890119894119904= [119890119894119909

119890119891119894119904] and

119860119894119904= [

119860119894minus 119870119901119904119862119894

119863119894

119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894

]

119873119894119904= [

119861119894

0

119870119894V119862119894119861119894 119868

] 119911119894119904= [

119898119904

119891119894119904

]

(19)





[[[[[[[

[

11986011

11986012

119875119861119894

0

119860119879

1211986022

119870119894V119862119894119861119894 119868

119875119861119894119870119894V119862119894119861119894 120574119868

10

0 119868 0 1205741198681

]]]]]]]

]

lt 0 (20)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119863

119894+ [119876 (119870

119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119863119894 + (119876119870119894V119862119894119863119894)119879

+ 119868119904

(21)




119894119901 119870119894119897 and 119870



10038171003817100381710038171198901198941199041003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941199041003817100381710038171003817

2

+ 119881 (0) (22)

where 1198901198941199042

= int1199051

0

(119890119894119904

119879

119890119894119904)119889119905 119911

1198941199042

= int1199051

0

(119911119894119904

119879

119911119894119904)119889119905 and 119868

119890 119868119904



1198941199041198751119890119894119904


= 119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904) 119890119894119904+ 2119890119879

1198941199041198751119873119894119904119911119894119904 (23)


119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904) 119889119905 (24)

Thus

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904+ 119868) 119890

119894119904

+119890119879

1198941199041198751119873119894119911119894119904minus 120574119911119879

119894119904119911119894119904] 119889119905 + 119881 (0)

= int

1199051

0

([

119890119894119904

119911119894119904

]

119879

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894

119873119879

1198941198751

minus1205741198681

][

119890119894119904

119911119894119904

])119889119905

+ 119881 (0)

(25)


[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894119904

119873119879

1198941199041198751

minus1205741198681

] lt 0 (26)


[119875 0

0 119868] then

1198751119873119894119904= [

119875119861119894

0

119870119894V119862119894119861119894 119868

]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(27)

where 11986011 11986012 and 119860


int

1199051

0

(119890119879

119894119904119890119894119904) 119889119905 le int

1199051

0

120574119911119879

119894119904119911119894119904119889119905 + 119881 (0) (28)



proof of Theorem 5


119878119894119891

119894= 119860119894119909119894+ 119861119894

sdot [119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119890119891119894119904

(29)


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+V119894minus 120573119894sgn (119890119879

119894119875119894119861119894)]

+119870119894119901(119910119894minus 119894+ 119863119894119890119891119894119904)

119894= 119862119894119894+ 119863119894119890119891119894119904

119891119894119886= 119870119894119897119862119894119890119894119909+ 119870119894V119862119894119890119894119909

(30)



119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)


119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894


]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)


119875119886 119870119894119886 and 119870V119886 as well as


[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894


0 0


0


]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)



2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices





119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)


119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)



119894119888 119861119894119888 119862119894119888 and 119863


that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0




]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119878119894

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) 119906119894+ ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894

(1)

where 119909119894= [1199091198941 1199091198942]119879

= [119902119894 119902119894]119879


119894and 119910



119860119894= [

0 1

0 0] 119861

119894= [

0

1] 119862

119894= [

1 0

0 1]

119891119894(119902119894 119902119894) = 119872

minus1

119894(119902119894) [minus119862119894(119902119894 119902119894) 119902119894minus 119866119894(119902119894)]

119892119894(119902119894) = 119872

minus1

119894(119902119894)

ℎ119894(119902 119902 119902) = minus119872

minus1

119894(119902119894) 119885119894(119902 119902 119902)

(2)


119878119894119891

119894= 119860119894119909119894+ 119861119894[119892119894(119902119894) (119906119894+ 119891119894119886)

+119891119894(119902119894 119902119894) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119891119894119904

(3)


119894=

[1 0


119894119904=

[11989111989411990411198911198941199042]119879 satisfies 119891

119894119904 le 120588119894(119905) where 120588







119894 and 119902

119889

119894are

bounded



119891119894119886 le 119891lowast

119894119886and 119891

1198941199041 le 119891lowast

1198941199041


119894= minus119860

119894119886119911119894+ 119861119894119886119910119894 (4)


119894= minus119860

119894119886119911119894+ 119861119894119886119862119894119909119894+ 119861119894119886119863119894119891119894119904 (5)



matrix with det(119861119894119886) = 0


119878119894119891

119894= 119860119894119909119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+ℎ119894(119902 119902 119902)] + 119863

119894119891119894119904

119910119894= 119862119894119909119894

(6)

where 119909119894= [119909119894

119911119894] 119860119894= [119860119894 0

119861119894119886119862119894 minus119860119894119886

] 119861119894= [119861119894

0] 119862119894= [0 119868] and

119863119894= [0

119861119894119886119863119894]



119894(119902119894) as

follows

119891119894(119902119894 119902119894119882119894119891) = 119882

119879

119894119891Φ119894119891(119902119894 119902119894) + 120576119894119891

119892119894(119902119894119882119894119892) = 119882

119879

119894119892Φ119894119892(119902119894) + 120576119894119892

(7)

where 119882119894119891

and 119882119894119892


and 120576119894119892



Define 119894119891

and 119894119892


and119882119894119892 119891119894(119902119894 119902119894 119894119891) and 119892



119894(119902119894119882119894119892) respectively

119891119894(119902119894 119902119894 119879

119894119891) =

119879

119894119891Φ119894119891(119902119894 119902119894)

119892119894(119902119894 119879

119894119892) =

119879

119894119892Φ119894119892(119902119894)

(8)


119882119894119891= 120578119894119891119890119879

119894119875119894119861119894Φ119894119891(119902119894 119902119894)

119882119894119892= 120578119894119892119890119879

119894119875119894119861119894Φ119894119892(119902119894) 119906119894

(9)




bounded by [25]

1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le

119899

sum

119895=1

119889119894119895119864119895 (10)

with 119889119894119895ge 0 and 119864

119895= 1 + 119890

119879

119894119875119894119861119894 + 119890

119879

1198941198751198941198611198942


119894119875119894119861119894



V119894(119905) = minus sgn (119890119879

119894119875119894119861119894) 119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) (11)


119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) =

119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817) (12)

where 119882119894119901



and Φ119894119901


119894119901and Φ


119894119901= 119894119901minus 119882119894119901


119894119901

can be updated by

119882119894119901= 120578119894119901

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817Φ119894119901 (13)



1205961198941= 119882119879

119894119891Φ119894119891(119902119894 119902119894) + 119882

119879

119894119892Φ119894119892(119902119894) 119906119894+ 120576119894119891+ 120576119894119892119906119894

1205961198942= 119877119894

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817minus 119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817)

120596119894=100381610038161003816100381612059611989411003816100381610038161003816 +100381610038161003816100381612059611989421003816100381610038161003816

(14)

where 119877119894(119890119879

119894119875119894119861119894) = 119899max

119894119895119889119894119895119864119894


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + V119894

minus120573119894sgn (119890119879

119894119875119894119861119894)]

+119863119894119891119894119904+ 119870119894119901(119910119894minus 119894)

119894= 119862119894119894

119891119894119904= 119870119894V119862119894 119890119894119909 + 119870119894119897119862119894119890119894119909

(15)





119894is





119890119894119909= 119894minus119909119894= (119860119894minus 119870119901119904119862119894) 119890119894+ 119863119894119890119891119894119904+ 119861119894119898119904

119890119891119894119904= 119891119894119897minus 119891119894119897

= 119891119894119897minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894119909

+ 119870119894V119862119894119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119904

(16)

where

119898119904= [(119891119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119894119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(17)


119890119894119904= 119860119894119904119890119894119904+ 119873119894119904119911119894119904 (18)

where 119890119894119904= [119890119894119909

119890119891119894119904] and

119860119894119904= [

119860119894minus 119870119901119904119862119894

119863119894

119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894

]

119873119894119904= [

119861119894

0

119870119894V119862119894119861119894 119868

] 119911119894119904= [

119898119904

119891119894119904

]

(19)





[[[[[[[

[

11986011

11986012

119875119861119894

0

119860119879

1211986022

119870119894V119862119894119861119894 119868

119875119861119894119870119894V119862119894119861119894 120574119868

10

0 119868 0 1205741198681

]]]]]]]

]

lt 0 (20)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119863

119894+ [119876 (119870

119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119863119894 + (119876119870119894V119862119894119863119894)119879

+ 119868119904

(21)




119894119901 119870119894119897 and 119870



10038171003817100381710038171198901198941199041003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941199041003817100381710038171003817

2

+ 119881 (0) (22)

where 1198901198941199042

= int1199051

0

(119890119894119904

119879

119890119894119904)119889119905 119911

1198941199042

= int1199051

0

(119911119894119904

119879

119911119894119904)119889119905 and 119868

119890 119868119904



1198941199041198751119890119894119904


= 119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904) 119890119894119904+ 2119890119879

1198941199041198751119873119894119904119911119894119904 (23)


119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904) 119889119905 (24)

Thus

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904+ 119868) 119890

119894119904

+119890119879

1198941199041198751119873119894119911119894119904minus 120574119911119879

119894119904119911119894119904] 119889119905 + 119881 (0)

= int

1199051

0

([

119890119894119904

119911119894119904

]

119879

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894

119873119879

1198941198751

minus1205741198681

][

119890119894119904

119911119894119904

])119889119905

+ 119881 (0)

(25)


[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894119904

119873119879

1198941199041198751

minus1205741198681

] lt 0 (26)


[119875 0

0 119868] then

1198751119873119894119904= [

119875119861119894

0

119870119894V119862119894119861119894 119868

]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(27)

where 11986011 11986012 and 119860


int

1199051

0

(119890119879

119894119904119890119894119904) 119889119905 le int

1199051

0

120574119911119879

119894119904119911119894119904119889119905 + 119881 (0) (28)



proof of Theorem 5


119878119894119891

119894= 119860119894119909119894+ 119861119894

sdot [119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119890119891119894119904

(29)


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+V119894minus 120573119894sgn (119890119879

119894119875119894119861119894)]

+119870119894119901(119910119894minus 119894+ 119863119894119890119891119894119904)

119894= 119862119894119894+ 119863119894119890119891119894119904

119891119894119886= 119870119894119897119862119894119890119894119909+ 119870119894V119862119894119890119894119909

(30)



119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)


119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894


]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)


119875119886 119870119894119886 and 119870V119886 as well as


[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894


0 0


0


]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)



2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices





119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)


119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)



119894119888 119861119894119888 119862119894119888 and 119863


that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0




]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Define 119894119891

and 119894119892


and119882119894119892 119891119894(119902119894 119902119894 119894119891) and 119892



119894(119902119894119882119894119892) respectively

119891119894(119902119894 119902119894 119879

119894119891) =

119879

119894119891Φ119894119891(119902119894 119902119894)

119892119894(119902119894 119879

119894119892) =

119879

119894119892Φ119894119892(119902119894)

(8)


119882119894119891= 120578119894119891119890119879

119894119875119894119861119894Φ119894119891(119902119894 119902119894)

119882119894119892= 120578119894119892119890119879

119894119875119894119861119894Φ119894119892(119902119894) 119906119894

(9)




bounded by [25]

1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le

119899

sum

119895=1

119889119894119895119864119895 (10)

with 119889119894119895ge 0 and 119864

119895= 1 + 119890

119879

119894119875119894119861119894 + 119890

119879

1198941198751198941198611198942


119894119875119894119861119894



V119894(119905) = minus sgn (119890119879

119894119875119894119861119894) 119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) (11)


119894(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817 119894119901) =

119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817) (12)

where 119882119894119901



and Φ119894119901


119894119901and Φ


119894119901= 119894119901minus 119882119894119901


119894119901

can be updated by

119882119894119901= 120578119894119901

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817Φ119894119901 (13)



1205961198941= 119882119879

119894119891Φ119894119891(119902119894 119902119894) + 119882

119879

119894119892Φ119894119892(119902119894) 119906119894+ 120576119894119891+ 120576119894119892119906119894

1205961198942= 119877119894

10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817minus 119879

119894119901Φ119894119901(10038171003817100381710038171003817119890119879

119894119875119894119861119894

10038171003817100381710038171003817)

120596119894=100381610038161003816100381612059611989411003816100381610038161003816 +100381610038161003816100381612059611989421003816100381610038161003816

(14)

where 119877119894(119890119879

119894119875119894119861119894) = 119899max

119894119895119889119894119895119864119894


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + V119894

minus120573119894sgn (119890119879

119894119875119894119861119894)]

+119863119894119891119894119904+ 119870119894119901(119910119894minus 119894)

119894= 119862119894119894

119891119894119904= 119870119894V119862119894 119890119894119909 + 119870119894119897119862119894119890119894119909

(15)





119894is





119890119894119909= 119894minus119909119894= (119860119894minus 119870119901119904119862119894) 119890119894+ 119863119894119890119891119894119904+ 119861119894119898119904

119890119891119894119904= 119891119894119897minus 119891119894119897

= 119891119894119897minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894119909

+ 119870119894V119862119894119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119904

(16)

where

119898119904= [(119891119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119894119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(17)


119890119894119904= 119860119894119904119890119894119904+ 119873119894119904119911119894119904 (18)

where 119890119894119904= [119890119894119909

119890119891119894119904] and

119860119894119904= [

119860119894minus 119870119901119904119862119894

119863119894

119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894

]

119873119894119904= [

119861119894

0

119870119894V119862119894119861119894 119868

] 119911119894119904= [

119898119904

119891119894119904

]

(19)





[[[[[[[

[

11986011

11986012

119875119861119894

0

119860119879

1211986022

119870119894V119862119894119861119894 119868

119875119861119894119870119894V119862119894119861119894 120574119868

10

0 119868 0 1205741198681

]]]]]]]

]

lt 0 (20)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119863

119894+ [119876 (119870

119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119863119894 + (119876119870119894V119862119894119863119894)119879

+ 119868119904

(21)




119894119901 119870119894119897 and 119870



10038171003817100381710038171198901198941199041003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941199041003817100381710038171003817

2

+ 119881 (0) (22)

where 1198901198941199042

= int1199051

0

(119890119894119904

119879

119890119894119904)119889119905 119911

1198941199042

= int1199051

0

(119911119894119904

119879

119911119894119904)119889119905 and 119868

119890 119868119904



1198941199041198751119890119894119904


= 119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904) 119890119894119904+ 2119890119879

1198941199041198751119873119894119904119911119894119904 (23)


119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904) 119889119905 (24)

Thus

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904+ 119868) 119890

119894119904

+119890119879

1198941199041198751119873119894119911119894119904minus 120574119911119879

119894119904119911119894119904] 119889119905 + 119881 (0)

= int

1199051

0

([

119890119894119904

119911119894119904

]

119879

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894

119873119879

1198941198751

minus1205741198681

][

119890119894119904

119911119894119904

])119889119905

+ 119881 (0)

(25)


[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894119904

119873119879

1198941199041198751

minus1205741198681

] lt 0 (26)


[119875 0

0 119868] then

1198751119873119894119904= [

119875119861119894

0

119870119894V119862119894119861119894 119868

]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(27)

where 11986011 11986012 and 119860


int

1199051

0

(119890119879

119894119904119890119894119904) 119889119905 le int

1199051

0

120574119911119879

119894119904119911119894119904119889119905 + 119881 (0) (28)



proof of Theorem 5


119878119894119891

119894= 119860119894119909119894+ 119861119894

sdot [119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119890119891119894119904

(29)


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+V119894minus 120573119894sgn (119890119879

119894119875119894119861119894)]

+119870119894119901(119910119894minus 119894+ 119863119894119890119891119894119904)

119894= 119862119894119894+ 119863119894119890119891119894119904

119891119894119886= 119870119894119897119862119894119890119894119909+ 119870119894V119862119894119890119894119909

(30)



119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)


119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894


]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)


119875119886 119870119894119886 and 119870V119886 as well as


[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894


0 0


0


]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)



2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices





119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)


119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)



119894119888 119861119894119888 119862119894119888 and 119863


that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0




]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[[[[[[[

[

11986011

11986012

119875119861119894

0

119860119879

1211986022

119870119894V119862119894119861119894 119868

119875119861119894119870119894V119862119894119861119894 120574119868

10

0 119868 0 1205741198681

]]]]]]]

]

lt 0 (20)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119863

119894+ [119876 (119870

119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119863119894 + (119876119870119894V119862119894119863119894)119879

+ 119868119904

(21)




119894119901 119870119894119897 and 119870



10038171003817100381710038171198901198941199041003817100381710038171003817

2

le 12057410038171003817100381710038171199111198941199041003817100381710038171003817

2

+ 119881 (0) (22)

where 1198901198941199042

= int1199051

0

(119890119894119904

119879

119890119894119904)119889119905 119911

1198941199042

= int1199051

0

(119911119894119904

119879

119911119894119904)119889119905 and 119868

119890 119868119904



1198941199041198751119890119894119904


= 119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904) 119890119894119904+ 2119890119879

1198941199041198751119873119894119904119911119894119904 (23)


119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904) 119889119905 (24)

Thus

119869 = int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119890119879

119894119904119890119894119904minus 120574119911119879

119894119904119911119894119904+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119890119879

119894119904(119860119879

1198941199041198751+ 1198751119860119894119904+ 119868) 119890

119894119904

+119890119879

1198941199041198751119873119894119911119894119904minus 120574119911119879

119894119904119911119894119904] 119889119905 + 119881 (0)

= int

1199051

0

([

119890119894119904

119911119894119904

]

119879

[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894

119873119879

1198941198751

minus1205741198681

][

119890119894119904

119911119894119904

])119889119905

+ 119881 (0)

(25)


[

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 119875

1119873119894119904

119873119879

1198941199041198751

minus1205741198681

] lt 0 (26)


[119875 0

0 119868] then

1198751119873119894119904= [

119875119861119894

0

119870119894V119862119894119861119894 119868

]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(27)

where 11986011 11986012 and 119860


int

1199051

0

(119890119879

119894119904119890119894119904) 119889119905 le int

1199051

0

120574119911119879

119894119904119911119894119904119889119905 + 119881 (0) (28)



proof of Theorem 5


119878119894119891

119894= 119860119894119909119894+ 119861119894

sdot [119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886) + ℎ119894(119902 119902 119902)]

119910119894= 119862119894119909119894+ 119863119894119890119891119894119904

(29)


119878119894119891

119909119894= 119860119894119894+ 119861119894[119891119894(119902119894 119902119894) + 119892119894(119902119894) (119906119894+ 119891119894119886)

+V119894minus 120573119894sgn (119890119879

119894119875119894119861119894)]

+119870119894119901(119910119894minus 119894+ 119863119894119890119891119894119904)

119894= 119862119894119894+ 119863119894119890119891119894119904

119891119894119886= 119870119894119897119862119894119890119894119909+ 119870119894V119862119894119890119894119909

(30)



119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)


119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894


]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)


119875119886 119870119894119886 and 119870V119886 as well as


[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894


0 0


0


]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)



2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices





119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)


119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)



119894119888 119861119894119888 119862119894119888 and 119863


that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0




]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

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t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

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t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890119894= 119894minus119909119894

= (119860119894minus 119870119894119901119862119894) 119890119894+ 119870119894119901119863119894119890119891119894119904

+ 119861119894119898119886+ 119861119894119892119894119890119891119894119886

119890119891119894119886= 119891119894119886minus119891119894119886

= 119891119894119886minus [119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894)] 119890119894

+ 119870119894V119862119894119870119894119901119863119894119890119891119894119904 + 119870119894V119862119894119861119894119898119886

+ 119870119894V119862119894119861119894119892119894119890119891119894119886

(31)

where

119898119886= [(119891

119894minus 119891119894) + (119892

119894minus 119892119894) (119906119894+ 119891119886)

minusV119894+ ℎ119894+ 120573119894sgn (119890119879

119894119875119894119861119894)]

(32)


119890119894119886= 119860119894119886119890119894119886+ 119873119894119886119911119894119886+ 119864119894119886119861119894119898119886 (33)

where 119890119894119886= [119890119894

119890119891119894119886] and

119860119894119886= [

119860119894minus 119870119894119901119862119894

119861119894119892119894


]

119873119894119886= [

119870119894119901119863119894

0

119870119894V119862119894119870119894119901119863119894 119868

]

119911119894119886= [

119890119891119894119904

119891119894119886

] 119864119894119886= [

119868

119870119894V119862119894]

(34)


119875119886 119870119894119886 and 119870V119886 as well as


[[[[[[[[

[

1198601111986012

119875119870119894119901119863119894

0 119875

lowast 11986022119870119894V119862119894119870119894119901119863119894 119868 119870

119894V119862119894


0 0


0


]]]]]]]]

]

lt 0 (35)

where

11986011= 119875119860119894+ 119860119879

119894119875 minus 119875119870

119894119901119862119894minus (119875119870

119894119901119862)119879

+ 119868119890

11986012= 119875119861119894119892119894+ [119876 (minus119870

119894119897119862119894minus 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894))]

119879

11986022= 119876119870

119894V119862119894119861119894119892119894 + (119876119870119894V119862119894119861119894119892119894)119879

+ 119868119904

(36)



2

le 12057410038171003817100381710038171199111198941198861003817100381710038171003817

2

+ 119881 (0) (37)

where 1198901198941198862

= int1199051

0

(119890119894119886

119879

119890119894119886)119889119905 119911

1198941198862

= int1199051

0

(119911119894119886

119879

119911119894119886)119889119905 and 119868

119890

119868119904 and 119868

1are unit matrices





119894119888= 119860119894119888119909119894119888+ 119861119894119888119910119894

119906119894= 119862119894119888119909119894119888+ 119863119894119888119910119894minus 119891119894119886

(38)


119894119897= 119860119894119897119909119894119897+ 119864119894119897119889119894119897+119882119894119897119861119894

119910119894= 119862119894119897119909119894119897+ 119863119894119897119889119894119897

(39)

where

119909119894119897= [

119909119894

119909119894119888

] 119860119894119897= [

119860119894+ 119861119894119892119894119863119894119888119862119894119861119894119892119894119862119894119888

119861119894119888119862119894

119860119894119888

]

119889119894119897= [

119890119891119894119886

119890119891119894119904

] 119882119894119897= [

119891119894+ ℎ119894

0]

119864119894119897= [

119861119894119892119894119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

119862119894119897= [1198621198940] 119863

119894119897= [0 119863

119894]

(40)



119894119888 119861119894119888 119862119894119888 and 119863


that

[[[[[[[[

[

1198601111986012119875119861119894119892119894119875 (119861119894119892119894119863119894119888119863119894) 119875 (119891

119894+ ℎ119894)

119860119879

1211986022

0 119861119894119888119863119894

0




]]]]]]]]

]

lt 0 (41)


where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

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or fa

ult r

econ

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(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

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5

Time (s)

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t 2 se

nsor

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t rec

onstr

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minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

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minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

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lt re

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(d)

0 1 2 3 4 5 6 7 8 9 10

0

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Time (s)

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minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

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04

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Time (s)

Join

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0 1 2 3 4 5 6 7 8 9 10

0

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08

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Time (s)

Join

t 1 p

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on (r

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0 1 2 3 4 5 6 7 8 9 10

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05

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Time (s)

Join

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on (r

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0 1 2 3 4 5 6 7 8 9 10

0

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06

Time (s)

Join

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minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

11986011= 119875119860119894+ 119860119879

119894119875 + 119875119861

119894119892119894119863119894119888119862119894

+ (119875119861119894119892119894119863119894119888119862119894)119879

+ 119868119890

11986012= 119875119861119894119892119894119862119894119888+ (119876119861

119894119888119862119894)119879

11986022= 119876119860

119894119888+ (119876119860

119894119888)119879

+ 119868119904

(42)



10038171003817100381710038171199091198941198971003817100381710038171003817

2

le 12057410038171003817100381710038171198891198941003817100381710038171003817

2

+ 119881 (0) (43)

where 1199091198941198972

= int1199051

0

(119909119894119897

119879

119909119894119897)119889119905 119889

1198942

= int1199051

0

(119889119894

119879

119889119894)119889119905 and 119868

119890 119868119904




= 119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897) 119909119894119897+ 119909119879

1198941198971198751119864119894119897119889119894

+ 119889119879

119894119864119879

1198941198971198751119909119894119897+ 2119909119879

1198941198971198751119882119894119897119861119894

(44)

Given the index

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119897119889119894119897) 119889119905 (45)

thus

119869 = int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus int

1199051

0

119889119905

= int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 minus 119881 (119905

1) + 119881 (0)

le int

1199051

0

(119909119879

119894119897119909119894119897minus 120574119889119879

119894119889119894119897+ ) 119889119905 + 119881 (0)

= int

1199051

0

[119909119879

119894119897(119860119879

1198941198971198751+ 1198751119860119894119897+ 119868) 119909

119894119897

+ 119909119879

1198941198971198751119864119894119897119889119894+ 119889119879

119894119864119879

1198941198971198751119909119894119897

+ 2119909119879

1198941198971198751119882119894119897119861119894minus 120574119889119879

119894119889119894] 119889119905 + 119881 (0)

= int

1199051

0

(

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

119879

[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

sdot

[[[

[

119909119894119897

119889119894

119861119894

]]]

]

)119889119905 + 119881 (0)

(46)


[[[

[

119860119879

1198941198971198751+ 1198751119860119894119897+ 119868 119875

11198641198941198971198751119882119894119897

1198751119864119879

119894119897minus1205741198681

0

1198751119882119879

1198941198970 minus120574119868

1

]]]

]

lt 0 (47)


[119875 0

0 119868] then

1198751119864119894119897= [

119875119861119894119892119894119875119861119894119892119894119863119894119888119863119894

0 119861119894119888119863119894

]

1198751119882119894119897= [

119875 (119891119894+ ℎ119894)

0]

119860119879

1198941199041198751+ 1198751119860119894119904+ 119868 = [

1198601111986012

119860119879

1211986022

]

(48)

where 11986011 11986012 and 119860


int

1199051

0

(119909119879

119894119897119909119894119897) 119889119905 le int

1199051

0

120574119889119879

119894119889119894119889119905 + 119881 (0) (49)



proof of Theorem 7




2(0) =

1199023(0) = 1 and 119902

1(0) = 119902

2(0) = 119902



1198601119888=

[[[[[

[

17261 29781 44521 63562

11781 21535 33562 53070

minus10682 minus15344 45082 51247

minus09344 minus14606 31247 41075

]]]]]

]

1198602119888=

[[[[[

[

10261 28531 40951 59062

11141 20031 31639 50250

minus10009 minus13614 41163 49428

minus08314 minus12171 27319 40014

]]]]]

]


I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

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minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

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onstr

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minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

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1

15

Time (s)

Join

t 2 p

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minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

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20

Time (s)

Join

t 3 ac

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0 1 2 3 4 5 6 7 8 9 10

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Time (s)

Join

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t rec

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minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

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on (r

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minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

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on (r

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0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

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0 1 2 3 4 5 6 7 8 9 10

0

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04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

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(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

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minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










I[1]

I[2]

I[2]

I[2]

I[2]

I[2]

O[2]

O[2]

O[2]O[2]

O[2]O[2]

L[0]

L[0]

L[1]L[0]

L[0]

L[1]

L[3]

L[3]

T

T

BB



1198603119888=

[[[[[

[

09841 20781 40194 58376

10781 19535 30062 49030

minus09718 minus10981 35462 50007

minus09004 minus09261 29654 99612

]]]]]

]

1198611119888=

[[[[[

[

215639 404866 404866 208926

29584 minus228672 270025 441431

249533 57852 minus82747 193143

minus59514 132063 388838 348115

]]]]]

]

1198612119888=

[[[[[

[

211543 400039 398753 200034

27635 minus220753 265025 439471

246494 50735 minus80415 190001

minus56401 129083 381307 340671

]]]]]

]

1198613119888=

[[[[[

[

209680 401496 394986 205639

26835 minus220681 264517 431931

242583 51327 minus80017 189153

minus53454 128963 380818 339911

]]]]]

]

1198621119888= [07261 minus10665 minus09665 03214]

1198622119888= [03274 minus07765 minus05665 02341]

1198623119888= [00661 minus01735 minus00913 00648]

1198631119888= [26557 minus19799 minus20486 51928]

1198632119888= [19471 minus10006 minus20936 49718]

1198633119888= [09557 minus08799 minus10486 31928]

(50)


1198701119901=

[[[[[[

[

minus142064 99445 85239 113651

05600 minus124741 106921 142561

74114 51880 44468 minus59291

36582 25608 21949 minus29266

]]]]]]

]

1198702119901=

[[[[[[

[

minus122064 86255 79152 108921

04725 minus110031 98513 129151

64164 51880 43518 minus4831

26532 19375 21949 minus20216

]]]]]]

]

1198703119901=

[[[[[[

[

minus122257 86415 79514 103621

04651 minus124741 106921 142561

69416 49480 39723 minus53611

22418 15308 20009 minus25136

]]]]]]

]

1198701V = [105289 51970 132071 65190]

1198702V = [104079 41870 129731 60520]

1198703V = [99539 40940 128070 59510]

1198701119897= [75289 31970 82071 55190]

1198702119897= [65636 30070 79872 50079]

1198703119897= [65258 29963 72547 49130]

(51)


0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

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econ

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minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

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02

Time (s)

Join

t 2 se

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minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

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Time (s)

Join

t 2 p

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minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

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20

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0 1 2 3 4 5 6 7 8 9 10

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Time (s)

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minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

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minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

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on (r

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0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

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minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus8

minus6

minus4

minus2

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus02

(b)

0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus15

minus1

minus05

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

Time (s)

Join

t 3 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus08

minus06

minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 p

ositi

on (r

ad)




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 2 p

ositi

on (r

ad)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 3 p

ositi

on (r

ad)

minus1

minus05

minus08

minus06

minus04

minus02

minus06

minus04

minus02



1198701119901=

[[[[[[

[

271343 minus224461 157691 08612

40364 01558 177803 11852

141557 117100 82266 minus129699

69872 57800 40606 minus64019

]]]]]]

]

1198702119901=

[[[[[[

[

258643 minus204813 125191 07132

40374 01098 137013 10352

13977 117100 80136 minus119819

58924 51320 39836 minus53419

]]]]]]

]

1198703119901=

[[[[[[

[

221463 minus204001 117251 0702112

40364 09958 129803 09562

138937 117100 82266 minus129699

58924 50972 39001 minus52919

]]]]]]

]

1198701V = [65289 31970 52071 65190]

1198702V = [60963 30023 51325 62351]

1198703V = [59235 28971 50701 61180]

1198701119897= [50231 35576 75657 64152]

1198702119897= [49421 30893 71097 62321]

1198703119897= [48511 29416 65725 60757]

(52)


119894119891= 0002

120578119894119892= 0002 and 120578

119894119901= 500 and the 119867

infinperformance




0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 1 ac

tuat

or fa

ult r

econ

struc

tion

(rad

)


minus1

minus05

(a)

0 1 2 3 4 5 6 7 8 9 10

0

5

Time (s)

Join

t 2 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus5

(b)

0 1 2 3 4 5 6 7 8 9 10

002040608

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


minus1

minus08

minus06

minus04

minus02

(c)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (s)

Join

t 3 ac

uato

r fau

lt re

cons

truc

tion

(rad

)


(d)

0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

04

Time (s)

Join

t 3 se

nsor

faul

t rec

onstr

uctio

n (r

ad)


minus04

minus03

minus02

minus01

(e)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus04

minus02

(f)



0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

Join

t 1 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

Time (s)

Join

t 2 p

ositi

on (r

ad)


0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

Time (s)

Join

t 3 p

ositi

on (r

ad)


minus06

minus04

minus02

minus1

minus05

minus04

minus02



1198911119886=

0 119905 le 3

5 sin (1199021) 1199021 119905 gt 3

1198912119904=

0 119905 le 5

minus03 cos (1199022) 119905 gt 5

1198913119886=

0 119905 le 3

15 119905 gt 3

1198913119904=

0 119905 le 6

minus031199022 119905 gt 6

(53)





1198911119886=

0 119905 le 3

2 sin (1199021) 119905 gt 3

1198912119904=

0 119905 le 5

3 sin (119905) 119905 gt 5

1198913119886=

0 119905 le 6

20 119905 gt 6

1198913119904=

0 119905 le 4

minus051199022 119905 gt 4

(54)



6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Conclusion






Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Dynamic Output Feedback Based Active...

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