Adaptive consensus for high-order unknown nonlinear multi-agent systems with unknown control directions and switching topologies

Information Sciences 0 0 0 (2018) 1–14

ContentslistsavailableatScienceDirect

Information

Sciences

journalhomepage:www.elsevier.com/locate/ins

Adaptive

consensus

for

high-order

unknown

nonlinear

multi-agent

systems

with

unknown

control

directions

and

switching

topologies

Mohammad

Hadi

Rezaei

a,b

_,

_Meisam

_Kabiri

a,b

_,

_Mohammad

_Bagher

_Menhaj

a,b,∗

a Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

b The Center of Excellence on Control and Robotics, Department of Electrical Engineering, Amirkabir University of Technology (Tehran

Polytechnic), Tehran, Iran

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 7 January 2018 Revised 4 April 2018 Accepted 29 April 2018 Available online xxx Keywords: Consensus

High-order nonlinear multi-agent systems Unknown control directions

Input saturation Switching topologies Unknown dynamics

a

b

s

t

r

a

c

t

Inthis paper, we provide acomprehensive assessment of the consensusof high-order nonlinear multi-agentsystems with input saturationand time-varying disturbance un-derswitchingtopologies.Thecontroldirectionsandmodelparametersofagentsare sup-posedtobeunknown.Ourapproachisbasedontransformingtheproblemofconsensus foranetworkthatconsistsofhigh-ordernonlinearagentstothatofperturbedfirst-order multi-agentsystems.Theunknownpartofdynamicsiscancelledusingradialbasisneural networks.Nussbaumgainsand auxiliarysystemsarerespectivelyemployedtoovercome theunknown input directionand the saturation.Adaptivesliding mode controlis used tocompensateforthe time-varyingdisturbance andtheimperfectapproximationofthe developedneuralnetworkaswell.ThroughLyapunovanalysis,itisshownthatthe over-allclosed-loopsystemmaintainsasymptoticstability.Finally,ourapproachisappliedtoa groupofmultiplesingle-linkflexiblejointmanipulatorstohighlightbetteritsmerit.

© 2018PublishedbyElsevierInc.

1. Introduction

Advancesin networkedcyber-physicalsystems andembeddedsystems technologyhave createdan increasing interest amongthecontrolcommunitytostudymulti-agentsystems(MASs).ThecontroltechniquesdevelopedsofarforMASs en-ableustoapplyresilient,cheap,andflexiblemethodologiestodiversecooperativetasksinmanydomainsincluding main-tenance,surveillance,reconnaissance,searchandrescuemission,cooperativeconstruction,andmanipulation[6,18,28,31,47]. ConsensusisafundamentalcooperativetaskinMASswherealltheagentsinateamaresupposedtoagreeonacertain valueofinterestwhile eachagentupdatesits statesmerelyonthe basisofitsown statesandthelocalinformationfrom itsneighbors.Consensushasapplicationsina varietyofdomains,includingcooperationofnetworksensors [35],decision making[26],motioncoordinationofunmannedaerialvehicles(UAVs)[23,40] andautonomousunderwatervehicles(AUVs)

[2],attitudesynchronizationinspacecraft[48],flockingcontrol[33],andloadsharinginmicrogrids[12].

Early studies of consensus mainly focused on the cooperation of agents with first- and second-order dynamics

[20,27,29,34].In[34],the problemofaverageconsensus forfirst-order integratorsunder switchingtopologiesand

identi-cal time delayswas fullydeveloped. The authors in [20] fully discussed the consensus problemof second-order system ∗ _{Corresponding author.}

E-mail addresses: hadi.rezaei@aut.ac.ir (M.H. Rezaei), mk13@aut.ac.ir (M. Kabiri), menhaj@aut.ac.ir (M.B. Menhaj). https://doi.org/10.1016/j.ins.2018.04.089

0020-0255/© 2018 Published by Elsevier Inc.

multi-2 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14

modelsunderjointly-connectedswitchingtopologiesbasedonaspacedecompositiontechnique.In[29],necessaryand suf-ficientconditionswerederivedtoguaranteeconsensusinanetworkofsecond-ordersystemsoverdirectedtopologywitha uniformconstantdelay.

However, first-and second-orderkinematics fail tomodel manypractical systemsdescribedby high-order differential equations.Hence,severalstudiesconcentratedontheconsensusproblemforlinearhigh-orderdynamics[4,11,39,41]. Tech-niques suchasfeedbacklinearizationcanbeused toconvertnonlinear systemstolinearones. Theperfectcancellationof the nonlinearities requireshaving an exactmodel forthe systemwhich isnot feasible inreality.Therefore, applyingthe resultsoflinearMASstounknownnonlinearMASsisnotstraightforward.However,despiteitsgreatimportance,thereare rather few studies dedicated to the consensus of high-order nonlinear systems [19,21,37,38,45,46]. For example, the au-thorsin[37] developeda consensusframework fora networkconsistingofuncertainhigh-ordernonlinearsystemsunder jointly-connected switchingtopologies.Theworkof[21],investigatesconsensus problemforhigh-order stochastic nonlin-ear systems underfixed topology. In[19], theleader-followingconsensus fornonlinear homogenousMASs withnetwork induceddelaysisstudied.

Cooperativecontrolofhigh-ordernonlinearMASscanbeevenmorechallenginginthepresenceofinputsaturation.This practical concernresultsfromthephysical constraintsofactuators.Thisissuehasbeen studiedin[7,36,43,44].The study of [36] wasdedicated to the semi-global bipartite consensus of generallinear MASs with switching topologies. In [43], leader-followingoutputconsensusoflineardiscrete-timeMASssubjecttoactuatorsaturationandexternaldisturbanceswas examined. Aleader-followerframework forconsensus ofagroup oflinearMASswithinputsaturation wasestablishedin

[44].

Alltheaforementionedstudiessharedtheassumptionthatthecontroldirectionsareknown.Nevertheless,insome prac-tical situations, the controlling effect is not accessible. This issue is already addressed fora singlesystem by using the Nussbaum-typefunction initially introducedin[32].Tackling thisissueforcontrolofMASsis challengingdueto thefact thateachagentNussbaumgainparametermaymoveinadifferentdirectionwhichimpedesusingtheusualmethodof con-tradictionintheestablishmentofthestabilityoftheoverallsystem[13].Recentlysomestudieshaveappearedtoovercome thischallenge.Theauthorsin[3] investigatedadaptiveconsensusproblemoffirst-orderandsecond-orderlinearly parame-terizedsystemswherethecontroldirectionsareassumedtohaveknownlowerandupperbounds.Theproblemofadaptive output regulationinthepresence ofunknown identicalcontroldirectionwasaddressedin [13,30] inwhich theneed for priorknowledgeofthelowerandupperboundswasremoved.Researchersin[3,13,30] investigatedthespecialcasewhere allthecontroldirectionsofthesubsystemsareidentical.Althoughthisconditionwasrelaxedin[1],itstillreliesonknowing someofthecontroldirections.

Theabove-mentionedfactsmotivatedustoaddresstheproblemofconsensusforhigh-orderunknownnonlinearsystems withinputsaturation,time-varyingdisturbance,andunknowncontroldirectionunderswitchinginteractiontopologies.We developanapproachthatconvertstheproblemintotheconsensusoffirst-orderMASswithboundedperturbationtermsand thenemploysaproperlydevelopedstabilizingcontroller.Radialbasisfunctionneuralnetworksareusedtoapproximatethe unknown nonlinear part of the systemdynamics, while their update rules are derived based on the Lyapunov analysis. In orderto dealwiththe unknown control direction, theNussbaum-type functionis utilized.An auxiliary systemisalso implemented foreachagentto compensate fortheinput saturation whiletheeffect ofthetime-varyingdisturbance and theapproximationerroroftheneuralnetworkiscounteractedbyanadaptiveslidingmodecontrolscheme.

Comparedtothemostrelevantstudy[37],ourapproachtakesintoaccounttheeffectsofbothunknown control direc-tionandinputsaturation.Besides,intheworkof[37],thecontrolparametersweredeterminedinsuchawaythatalinear matrixinequality (LMI),whichisdependent onthenumberofagents,shouldhold.Hence,growing thenumberofagents inthenetworkwouldincreasethecomputationalburden.Tothebestofourknowledge,nostudieshaveyetconsideredthe problemathandinthepresenceoftheallaforementionedpracticalissues.Forexample,in[21,37,38,45],theunknown con-troldirectionwasnotaddressed.Existingapproachesthatcopewiththisdifficulty[1,3,13,30],notonlyneglectedtheeffects ofactuatorsaturationandswitchingtopologiesbutalsorequiredtheinputdirectionstosatisfysomelimitingconditions.As acaseinpoint,in[3],itwassupposedthat theupperandlower boundsofcontrol effectsareknown.In[3,13,30],itwas assumedthat allcontrol directionshavean identicalsignandin[1] somepartofthe controlcoefficientswasconsidered tobeknown.InTable1,thedifferencesofourmethodwiththosepresentedinother relevantstudiesarehighlighted.The contributionsofthispapercanbesummarizedasfollows:

• Consensus for a class of general complex high-order nonlinear MASs with unknown nonlinearity and disturbance is studied.

• Theconsensus isachievedasymptotically inthepresenceofjointlyconnectedtopology forundirectedgraphs and uni-formlyjointlyquasi-stronglyconnectedandbalancedtopologyfordirectedgraphs.

• TheimpactoftheactuatorsaturationinMASsiseffectivelyhandled.

• Theproposedapproachiscapableofcompletelytacklingunknowncontroleffectswhereitallowscontroleffectstohave non-identicalsignsandtheneedfortheknowledgeoftheboundariesofthecontrolcoefficientsisremoved.

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 3

Table 1

Comparisons between our approach and those proposed in the other existing relevant studies in the literature

2. Preliminaries

Inthissection,weintroducethenotationsandthegraphtheoryterminologywhichwillbeusedthroughoutthepaper. 2.1. Notations

Throughoutthispaper, symbolsRandR+ _{denotethe set ofreals andpositive reals,respectively. sgn(.)expresses the}

signfunction.1_N isacolumnvectorofNelementsallequaltoone.0_N denotesacolumnvectorofNwithzeroentries.L_∞ presentsthespaceofboundedsignals.x(t)_∈L1meansthat0∞

|

x

(

t

)

|

dt<∞.

2.2.Graphtheory

The interaction among N agents is represented by an undirected or directed graph G

(

V,E,A

)

with a node set V=

{

1_,...,n

}

_,edgeset_E∈

{

_{V× V}

}

_,andadjacencymatrixA₌[a_{i j} ]_∈Rn×n_{.Theadjacencymatrixisdefinedsuchthatthe}

diago-nalentriesareequaltozero(a_ii =0)andtheoff-diagonalentriesarea_ij >0if

(

i,j

)

∈E anda_{i j} =0,otherwise.Forundirected graphs,wealsohavea_ij =a_ji .Thesetofneighborsofnodev_i isNi =

{

v

j ∈V

|

(

v

j ,

v

i

)

∈E

}

.TheLaplacianmatrixL=



li j



is definedas

l_{i j} =



_N

j =1, j =i ai j i=j

−ai j i=j. (1)

The in-degreeand out-degreeof node v_i are defined asd_in

(

v

i

)

=Nj=1ai j and dout

(

v

i

)

=Nj=1aji , respectively. Agraph isbalanced if andonly ifd_in

(

v

i

)

=dout

(

v

i

)

,

∀

v

i ∈V. It isclear that an undirectedgraph is balanced.

(

1/√N

)

1N andwT 1

(

(

1/√N

)

1_N wT ₁=1)aretherightandlefteigenvectorsassociatedwiththezeroeigenvalueofL,respectively.w₁T is

(

1/√N

)

1T _N if the digraph D is balanced. A class of piecewise right-continuous switching function

σ

(

t

)

(

inshort

σ

)

:[0,∞]→P=

{

1_,2_,...,n_v

}

isusedtodescribetheswitchingtopologieswheren_v isthetotalnumberofallpossiblecommunicationgraphs. WeuseGσ(t )_{todenotethecommunicationgraphattimet.Anundirectedgraphissaidtobeconnectedifeverytwodistinct}

nodescanbeconnectedbyapathwhereapathisasequenceofadjacentedgesoftheform

(

v

i 1,

v

i 2

)

,

(

v

i 2,

v

i 3

)

,...,

(

v

i j−1,

v

i j

)

inwhichi_j ∈V.Adirectedgraphissaidtocontainadirectedspanningtreeifithasatleastonenodefromwhichthereexist directedpathstoallothernodes.TheunionofacollectionofgraphsG1,...,Gm withthenodesetV isagraphdenotedby

G1−mwiththesamenodesetwhoseedgesetistheunionoftheedgesetsofallgraphs.Acollectionofswitchingundirected

ordirectedgraphsiscalledjointlyconnectedoruniformlyjointlyquasi-stronglyconnected[5] if,respectively,theunionof thegraphs isconnected orhas adirected spanningtree.We assume that there exists an infinitesequence ofnonempty, boundedandcontiguoustime intervals [tr ,tr+1],r=0,1,.... witht0=0andtr+1− tr ≤ T1 forsome constant T1>0 such

4 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14

wheret_{r =}t_{r 0,}tr+1=tr mr andtr j+1− tr j≥ T2,0≤ j<mr − 1forsomeintegermr andgivenconstantT2.Thecommunication

topologyissupposedtobefixedoneachtimesubintervalandswitchesattr j.

3. Problemstatement

ConsiderateamofNhigh-ordernonlinearagentswhichthedynamicsoftheithoneisasfollows:

x(_i n )=f_i

(

Xi

)

+bi ui +di

(

t

)

(2)

where _{Xi =}[x_{i ,}x˙_{i ,...,}x(_i n −1)]T_∈Rn_,u

i ∈Rare systemstates and the control input, respectively. xi (h )∈Ris the hth-order stateoftheithagentandx_i denotestheposition. f_i

(

Xi

)

∈Risan unknownnonlinearfunction.bi isanunknownnonzero constant gain withan unknown sign. d_i (t) isan external boundeddisturbance.In the sequel,we make thefollowing as-sumptions:

Assumption1. u_i isgainedbypassingthedesignedinputv_i throughanon-symmetricsaturationconstraintdefinedas

ui =

⎧

⎨

⎩

u_maxi if

v

i >umaxi

v

i if u_min i≤

v

i≤ umaxi u_min i if

v

i <umini (3)

whereumax_i andu_min

i areknownboundsofsaturationnonlinearity.

Assumption2. |d_i |≤ Di whereDi isanunknownconstant.

Inordertotackletheunknowncontroldirection,theNussbaum-typefunctiontechniqueisexploitedinthispaper._Nm

(

ξ

)

isaNussbaumfunctionsatisfiesthetwo-sidedproperties[22]

lim θ→±∞sup 1

θ

_θ 0 Nm

(

ξ

)

d

ξ

=∞ lim θ→±∞inf 1

θ

_θ 0 Nm

(

ξ

)

d

ξ

=−∞. (4)

Inourframework,weutilize_Nm

(

ξ

)

₌

ξ

2_cos



₍

π

_/2

₎

ξ



_{asanexampleofanevenNussbaum-typefunction.}

Inthispaper,weaimatdesigningaprotocolforagroupofagentswithdynamicsgivenin(2) suchthatallagentsreach consensusonthepositionwhilethehigher-orderstatesconvergetozero.

Lemma1. Letusconsiderthefirst-orderintegralMAS.Thedynamicsofeachagentis

˙ zi =−

j∈Ni(G σ (t))

ai j

(

t

)(

zi − zj

)

+ wi

wherez_i ∈Risthestateoftheithagentandw_i ∈Riscontinuousfunctionon[0,∞)exceptforatmostasetwithmeasurezero. For

∀

zi(0)and

∀

wisatisfyingwi∈L1,thenzi∈L∞andlimt→∞



zi

(

t

)

− zj

(

t

)



=0ifGσ(t )_{isjointlyconnectedoruniformlyjointly} quasi-stronglyconnectedforrespectivelytheundirectedordirectedgraphs[5 ,43] .

4. Consensusforhigh-ordernonlinearMASs

Consensusforhigh-order nonlinearMASsisdiscussedinthissection andthemainresultispresentedintheformofa theorem.

Letusdefinethevariables_i fortheithagentas: si =

_γ

1

n



x(_i n −1)+

γ

2x_i (n −2)+...+

γ

n −1x˙i +

γ

n xi



(5)

wherethepolynomialsn −1+

γ

2s(n −2)+...+

γ

n −1s+

γ

n hasrootsintheopen-lefthalfplane. Todealwiththeeffectofthesaturationconstraint,theauxiliarysystemisdefinedas:

˙

τ

i =−

βτ

i +ki

_γ

(

t

)

n sgn

(

ei

)

|

ui

|

(6)

where

β

_∈R+_,e_{i =}s˜_{i − zi ,}withs˜_{i =}s_{i −}

τ

_{i ,}and

u_{i =}u_{i −}

v

_i ,i.e.,

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 5 where

α

_{i 1∈R}+andanauxiliarystatez_i isdefinedas

˙

z_i =− j∈Ni(G σ (t))

a_{i j}

(

t

)(

s˜_i − ˜s_j

)

(9)

withtheinitialconditionz_i

(

0

)

=x_i

(

0

)

.Based onuniversal approximationtheorem[14,24],anycontinuousfunction canbe estimatedwitharbitrarilysmallerror.Havingthispropertyinmindandexploitingthefactthat f_i

(

Xi

)

+



i iscontinuous, wecansay

fi

(

Xi

)

+



i =Wi T

ψ

i

(

i

)

+



i

(

i

)

(10)

where



_i =n −1

i =1

γ

i +1xi(n −i)+

βγ

n

τ

i ,

i =[Xi T ,



i ]T ,



i isaboundedapproximation error,i.e.,|



i |≤

ε

i where

ε

i isan un-knownconstant,W_i istheboundedidealweightvectorwithl_i neurons,and

ψ

_i =[

ψ

_i 1,...,

ψ

il i]

T _isdefinedby

ψ

i j

(

i

)

=exp



−

(

i − ¯

i j

)

T

(

i − ¯

i j

)

ν

2 i j



forj=1,...,l_i inwhich

¯_{i j} and

ν

_ij denotethecenterofthereceptivefieldandthewidthoftheGaussianfunction, respec-tively.Sincetheidealweightsareunknown,wecanapproximate f_i

(

Xi

)

+



i by

ˆ

f_i

(

Xi

)

+



ˆi =Wˆi T

ψ

i

(

i

)

. (11)

Exploitingthe above structure,ourproblemis convertedto theconsensus ofa networkof first-ordersystems(9) witha boundedperturbationandthestabilizationproblemofe_i .Theideabehindsucha structureistofirsttheerrorvariablee_i foreachagentisdriventozero.Tothisend,weusetheauxiliarysystem(6) tocounteractthesaturationeffect,andneural network to approximate the unknown part of each agentdynamics. The Nussbaum gain parameter is also exploited to handletheunknown signofthecontrol direction.Todealwiththetime-varyingdisturbanceandtheapproximationerror ofthe developedneural network, we use adaptivesliding mode control.Since we have noknowledge about thebounds onthe disturbanceandapproximation error,we make useofadaptive gainstoestimate thesebounds. After the variable e_i goes to zero,we can show that the state of the auxiliary system

τ

_i is driven to zeroforeach agent whileconsensus on a commonvalue for the variables z_i is achieved, i.e., z_i →N

i=1zi

(

0

)

/N. This impliesthat s˜i =si andhence si →zi → N

i =1zi

(

0

)

/N. Afterward,weprovethatchoosingtheinitialconditionsaszi

(

0

)

=xi

(

0

)

fortheauxiliarysystem(9),onecan havex_i →N i xi

(

0

)

/N.Nowweputourmainresultinthefollowingtheorem.

Theorem1. ConsidertheMASthatisdefinedby(2) .Providedthattime-varyingtopologyforgraphGσ(t )_{isjointlyconnectedor} uniformlyjointlyquasi-strongly connectedfortheundirected orbalanceddirectedgraphs,respectively,theagentsreachaverage consensusunderthefollowingprotocol:

v

i =Nm

(

ξ

i

)



Wˆi

(

t

)

T

ψ

i

(

i

)

+



i



(12) where ˙

ξ

i =

_γ

1 n



_ˆ Wi

(

t

)

T

ψ

i

(

i

)

+



i



ei , (13) ˙ ˆ W_i =P_i

ψ

_i

(

_i

)

e_i , (14)



i =

κ

i

(

t

)

sgn

(

ei

)

+

γ

n j∈Ni(G σ (t)) a_{i j}

(

t

)(

s˜_i − ˜s_j

)

, (15)

inwhichtheadaptivegainmatrixPiissymmetricpositivedefiniteand ˙

κ

i =

α

i 2

|

ei

|

. (16)

where

α

_i 2∈R+.

Proof.Thefirsttimederivativeof(5) isobtainedby ˙

si =

_γ

1 n



x_i (n )+

γ

2x(_i n −1)+...+

γ

n −1¨xi +

γ

n x˙i



whichbyusing(2) andu_i =

v

i +

ui ,itcanbewrittenby

6 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 Bysubstitutingv_i from(12) into(17),onecanget

˙ si =

_γ

1 n



fi

(

Xi

)

+ n −1 i =1

γ

i +1x(i n −i)+bi

ui +di



+bi N

_γ

m

(

ξ

i

)

n



ˆ Wi

(

t

)

T

ψ

i

(

Xi

)

+



i



. (18)

Bytakingderivativeofe_i =s_i −

τ

i − zi andusing(6),(9),and(18),onecanobtain ˙ e_i =

_γ

1 n



f_i

(

Xi

)

+ n−1 i =1

γ

i +1x(i n −i)+

βγ

n

τ

i +bi

ui +di



+bi N

_γ

m

(

ξ

i

)

n



ˆ W_i

(

t

)

T

ψ

_i

(

Xi

)

+



i



− ki

γ

n sgn

(

ei

)

|

ui

|

+ j∈Ni(G σ (t)) ai j

(

t

)(

s˜i − ˜sj

)

. (19)

Bysubstituting(10) into(19),onecanget ˙ e_i =1

γ

n



WT i

ψ

i

(

i

)

+bi

ui +di +



i



+bi Nm

(

ξ

i

)

γ

n



ˆ W_i

(

t

)

T

ψ

_i

(

Xi

)

+



i



− ki

γ

n sgn

(

ei

)

|

ui

|

+ j∈Ni(G σ (t)) a_{i j}

(

t

)(

s˜_i − ˜s_j

)

. (20)

Now,letusconsiderthepositivedefinitefunctionasV_{i =}V_{i 1+}V_{i 2}with Vi 1= 1 2e 2 i , V_i 2= 1 2

γ

n



˜ W_i T P_i −1W˜_i + 1

α

i 1 ˜ k2 i + 1

α

i 2 ˜

κ

2 i



, (21)

whereW˜_i

(

t

)

=Wˆ_i

(

t

)

− Wi ,k˜i

(

t

)

=ki

(

t

)

−

|

bi

|

,and

κ

˜i

(

t

)

=

κ

i

(

t

)

−

(

Di +

ε

i +

βγ

n

)

.ThereasonforconsideringV_i 2asapartof

theoverallLyapunovfunctionistoderivetheadaptivelawsforneuralnetworkweightsandestimatorsk_i and

κ

_i .Sincethe boundsonthesignals|b_i | andDi+

ε

i+

βγ

n arenot known,theadaptivegainski and

κ

i are respectivelyimplementedto estimatethesebounds.

Byusing(13) andtakingthetimederivativeofV_i 1along(20),onecanconcludethat

˙ V_i 1= 1

γ

n



W_i T

ψ

_i

(

_i

)

+

γ

n j∈Ni(G σ (t)) a_{i j}

(

t

)(

s˜_i − ˜s_j

)



e_i +b_i Nm

(

ξ

i

)

ξ

˙i −

_γ

ki n

|

ei

ui

|

+ bi

γ

n ei

ui + di +



i

γ

n ei . (22)

Bysubtractingandadding

ξ

˙_i totheright-handsideof(22) andusing(13) and(15),itcanbeconcludedthat ˙ Vi 1= 1

γ

n



W_i T

ψ

i

(

i

)

+

γ

n j∈Ni(G σ (t)) ai j

(

t

)(

s˜i − ˜sj

)



ei +

_ξ

˙_{i −} 1

γ

n



ˆ W_i

(

t

)

T

ψ

_i

(

_i

)

+

κ

i

(

t

)

sgn

(

ei

)

+

γ

n j∈Ni(G σ (t)) a_{i j}

(

t

)(

s˜_i − ˜s_j

)



e_i +b_i Nm

(

ξ

i

)

ξ

˙i − ki

γ

n

|

ei

ui

|

+ b_i

γ

n ei

ui + d_i +



i

γ

n ei =−

_γ

1 n ˜ W_i T

ψ

i

(

i

)

ei +



bi Nm

(

ξ

i

)

+1



ξ

˙i −

_γ

ki n

|

ei

ui

|

+ bi

γ

n ei

ui −

κ

i

γ

n

|

ei

|

+ di +



i

γ

n ei .

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 7 NowwecanobtainthetimederivativeofV_i as

˙ Vi ≤

_γ

1 n ˜ W_i T P_i −1



W˜˙i − Pi

ψ

i

(

i

)

ei



+



bi Nm

(

ξ

i

)

+1



ξ

˙i + ˜ ki

γ

n

α

i 1



_˜˙ ki −

α

i 1

|

ei

ui

|



+

_γ

κ

˜i n

α

i 2



_˙ ˜

κ

i −

α

i 2

|

ei

|



−

β|

ei

|

. (23)

BecauseW_i ,

θ

_i ,|b_i |,andD_i +

ε

i +

βγ

n areconstant,wehaveW˜˙_i =_Wˆ˙_{i ,}

_θ

˜˙_{i =}

_θ

ˆ˙_{i ,k}˜˙_=k˙_{i ,}

_κ

_˜˙ ₌

_κ

_{˙i ,andhence,substituting(8),(14),} and(16) into(23) yields

˙

V_i ≤ −

β|

e_i

|

+



b_i Nm

(

ξ

i

)

+1



ξ

˙i . (24)

Hence,from(24),itfollowsthat: ˙

V_i ≤



b_i Nm

(

ξ

i

)

+1



ξ

˙i . (25)

Bytakingthetimeintegralofbothsidesof(25),onecanget 0≤ Vi

(

t

)

≤ Vi

(

0

)

+ t 0 bi Nm



ξ

i

(

τ

)



ξ

˙i

(

τ

)

d

τ

+

ξ

i

(

t

)

where

ξ

i

(

0

)

=0.Since _t 0 Nm



ξ

i

(

τ

)



ξ

˙i

(

τ

)

d

τ

= _ξi(t) 0 Nm

(

ξ

i

)

d

ξ

i , wehave 0≤ Vi

(

t

)

≤ Vi

(

0

)

+ _ξi(t) 0 b_i Nm

(

ξ

i

)

d

ξ

i +

ξ

i

(

t

)

(26) orequivalently −

ξ

i

(

t

)

− Vi

(

0

)

≤ _ξi(t) 0 b_i Nm

(

ξ

i

)

d

ξ

i . (27)

Accordingto(4),Nm

(

ξ

i

)

hasthefollowingproperties: lim ξi(t)→±∞ sup 1

ξ

i

(

t

)

_ξi(t) 0 bi Nm

(

ξ

i

)

d

ξ

i =∞ (28) and lim ξi(t)→±∞ inf 1

ξ

i

(

t

)

_ξi(t) 0 b_i Nm

(

ξ

i

)

d

ξ

i =−∞. (29)

Bythemethodofcontradiction,itcanbeprovedthat

ξ

_i (t)∈L_∞.Supposethat

ξ

_i (t)becomesunbounded,thentherearetwo cases.

1.If

ξ

_i (t)→∞,thenbyutilizing(27),onecanget lim ξi(t)→∞ −

ξ

i

(

t

)

+Vi

(

0

)

ξ

i

(

t

)

≤ 1

ξ

i

(

t

)

_ξi(t) 0 b_i Nm

(

ξ

i

)

d

ξ

i anditcanbeobtainedthat

−1≤ lim ξi(t)→∞ 1

ξ

i

(

t

)

_ξi(t) 0 b_i Nm

(

ξ

i

)

d

ξ

i . (30)

Easilyobservedthat(30) contradictswith(29). 2.If

ξ

_i

(

t

)

→−∞,thenbyusing(27),onecanobtain

lim ξi(t)→−∞ −

ξ

i

(

t

)

_ξ

+Vi

(

0

)

i

(

t

)

≥ 1

ξ

i

(

t

)

_ξi(t) 0 bi Nm

(

ξ

i

)

d

ξ

i anditcanbeconcludedthat

−1≥ lim ξi(t)→−∞ 1

ξ

i

(

t

)

_ξi(t) 0 bi Nm

(

ξ

i

)

d

ξ

i . (31)

8 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 Therefore,

ξ

_i (t) isbounded,andξi(t)

0 biNm

(

ξ

i

)

d

ξ

i is alsobounded.By exploiting(26),Vi is bounded.Accordingto Vi=

V_i 1+Vi 2,where Vi 1and Vi 2defined in(21), one can conclude that ei , ˜Wi ,Wˆi ,k˜i ,ki ,

κ

˜i ,and

κ

i arebounded. From(24),onecanget

˙

Vi ≤ −

β|

ei

|

+



bi Nm

(

ξ

i

)

+1



ξ

˙i . (32)

Then,thetimeintegrationof(32) over[0,_∞)becomes

_∞ 0

|

e_i

|

≤

_β

1



V_i

(

0

)

− Vi

(

∞

)

+ _ξi(∞) 0 b_i Nm

(

ξ

i

)

d

ξ

i +

ξ

i

(

∞

)



. (33)

Sincetherightsideof(33) isbounded,wecanconcludethate_i ∈L1.

From(9) ande_i =s˜_i − zi ,onecanget ˙ z_i =− j∈Ni(G σ (t)) a_{i j}

(

t

)(

z_i − zj

)

+ wi (34) where wi=− j∈Ni(G σ (t)) ai j

(

t

)(

ei− ej

)

. (35)

Since e_i ∈L1 for i=1,...,N, the equation (35) implies that wi ∈L1. According to Lemma 1, one can conclude that zi is bounded and limt→∞



zi

(

t

)

− zj

(

t

)



=0. Based on the facts that zi ∈L∞, ei ∈L∞, and ei =s˜i − zi , the boundedness of s˜i can be concluded. According to s˜_{i ∈}L_∞ for i₌1_,...,N_, Wˆ_{i ,}

κ

_{i ,}

ξ

_{i ∈}L_∞, 0_<

ψ

_ij (

_i )_{≤ 1,} and (12), it can be said that v_i is bounded.Sincev_i ∈L_∞ and(7),

u_i isbounded.Based ons˜_i ∈L_∞ fori=1,...,N,

u_i ,

ξ

i ,Wˆi ,

κ

i ,ei ,ki ∈L∞,|di |≤ Di ,|



i |≤

ε

i , 0_<

ψ

_ij (

_i )_{≤ 1,}and(20),one canconcludethate˙_i isbounded.Barbalat’slemmacanalsobeappliedinthiscasebecausee_i ande˙_i areboundedande_i ∈L1.Therefore,onecanconcludethat

lim

t→∞ei

(

t

)

=0.

Toprooftheboundednessof

τ

_i ,letusrewrite(6) as ˙

τ

i =−

βτ

i +hi (36)

where hi=

(

ki/

γ

n

)

sgn

(

ei

)

|

ui

|

. Since ki,

ui∈L∞, hi is bounded. Hence, (36) can be regarded as a linear systems with bounded input h_i . It is obvious that

τ

_i ∈L_∞ since input h_i is bounded. Based on limt→∞ei

(

t

)

=0 and hi =

(

k_i /

γ

n

)

sgn

(

e_i

)

|

u_i

|

,onecan conclude thatlimt→∞hi

(

t

)

=0,whichresultsinlimt→∞

τ

i

(

t

)

=0.Basedon limt→∞

τ

i

(

t

)

=0, limt→∞ei

(

t

)

=0,andei=si−

τ

i− zi,onecanget

lim

t→∞si

(

t

)

=tlim→∞zi

(

t

)

.

Now,(34) canbedescribedforthewholeMASas: ˙

z=−LG σ (t)z− L_G σ (t)e (37)

wherez=[z1,z2,...,zN ]T ande=[e1,e2,...,eN ]T .SinceGσ(t )isbalancedatanytimeinstant,onehave1T N LG σ (t) =0T _N .Based

onthisandbymultiplyingbothsidesof(37) by1T _N ,onecanconcludethat1_N T z˙=0T _N .Then,takingintegrationyields1T _N z

(

t

)

= 1T Nz

(

0

)

.Sincelimt→∞si

(

t

)

=limt→∞zi

(

t

)

,limt→∞z1

(

t

)

=z2

(

t

)

=...=zN

(

t

)

,andzi

(

0

)

=xi

(

0

)

,onecanconcludethat

lim t→∞si

(

t

)

=tlim→∞zi

(

t

)

= 1 N N i =1 x_i

(

0

)

.

BytakingtheLaplacetransformof(5) andaftersomemathematicalmanipulations,onecanobtain

X_i

(

s

)

= 1 sn −1₊

_γ

_2sn −2_+...+

_γ

_n ×



γ

n Si

(

s

)

+ n −1 k =1 sn −1−kx(_i k −1)

(

0

)

+ n −1 h =2 n −h k =1

γ

h sn −h−kx(i k −1)

(

0

)



(38) whereX_i (s)andS_i (s)denoteLaplacetransformofx_i (t)ands_i (t),respectively.Byusing(38),limt→∞si

(

t

)

=

(

1/N

)

N i =1xi

(

0

)

, thefinal valuetheorem,i.e.,limt→∞xi

(

t

)

=lims →0sXi

(

s

)

andlims →0sSi

(

s

)

=limt→∞si

(

t

)

,andthefact thatthepolynomial

sn −1+

γ

2sn −2+...+

γ

n hasrootsintheopen-lefthalfplane,onecanconcludethat

lim t→∞xi

(

t

)

= 1 N N i =1 xi

(

0

)

andtherefore,limt→∞x_i (k )

(

t

)

=0,k∈

{

1,...,n− 1

}

,andtheproofiscompleted. 

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 9

Table 2

Initial states of the agents .

Agent φi1 , φi2 , φi3 , φi4 Agent φi1 , φi2 , φi3 , φi4 1 1, 0.5, 1, 0.8 4 0, 0.2, 0.5, 0.2 2 0.5, 0.5, 0.2, 0.1 5 0.2, 5, 0.2, 0, -0.5, 1 3 1, 0, 2, 0.5 6 -0.5, 0.8, 0.5, -0.4

controldirectionisdecoupled tothetwo tasks:consensus ofthefirst-orderMASs(9) andthestabilizationcontrol ofthe unknownnonlinearsystemwithinputsaturationandexternaldisturbancefore_i withthedynamicsobtainedin(19).Asit isdiscussed in[1,3,13,30],extendingconventionalNussbaumgain techniquetothecooperativetaskisquiteinvolved. Hav-ingintroducedtheproposedstructure,wefacilitateextendingtheNussbaum-typefunctionforanindividualsystemtothe cooperativecaseandalsorelax all the limitingconditions.Ourmethodjustrequiresthe control effectsto havenon-zero values.

Remark2. Intheproposedframework,eachagentissupposedtotransmitthesignals˜itoitsneighbors.Thissignalcanbe calculatedonlineoneachagent.

Remark 3. It is noteworthy that an important issue, which is very important and crucial especially in MASs is event-triggered control. In this approach,the sensors andcontrollers are updated when a specific eventhappens. This frame-workcomes up withseveraladvantagessuch asreducing the communicationbandwidth andthecontrol effort. Because, inpractice,thecontrolsystemsandembeddedsensorsare resourceconstrained,theeventtriggeredcontrolholds promis-ingpotentialinpractical implementationofdistributedcontrolsystemsandhence,worthtobetakenintoaccountinthis context.Thereadersarereferredto[8–10,15–17] for more information.

Remark 4. The final value of the agreement is dependent on the initial conditions of the auxiliary systems (9), i.e., x_i →N i =1zi

(

0

)

/N.Therefore,theaverageconsensusisaccomplishedbylettingzi

(

0

)

=xi

(

0

)

.Inthedirectedcase,thegraph shouldbeuniformlyjointlyquasi-stronglyconnectedandbalancedinordertoaverageconsensusisachieved.Ifthenetwork topologyisonly uniformlyjointlyquasi-stronglyconnectedandnot balanced,thenthe consensusisstill acquiredbutthe finalvalueoftheagreementisnottheaverageoftheinitialconditions.

5. Simulationresults

Cooperationofthemanipulatorsplaysakey roleintheassemblyautomationandproductionprocesseswithhigh flex-ibility.One ofthe typicaltask inforce control is thegrasping taskvia robotmanipulatorsin whichall manipulators are requiredtoreachacommonconfiguration.Tostudytheeffectivenessofthepresentedcontrol,weconductnumerical simu-lationsforconsensusofmultiplesingle-linkflexiblejointmanipulators.Thisisdonebyapplyingbothourmethodandthat of[37].Consider agroupofsixsingle-linkflexible jointmanipulatorswithDCmotoractuatorswhichthedynamicsofthe ithoneisgovernedbythefollowingequations[37]

Motor



_˙

φ

1i =

φ

2i ˙

φ

2i = gi

(

φ

1i ,

φ

2i ,

φ

3i

)

+

ρ

i ui +di

10 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14

Fig. 2. Trajectories of the six joints angular rotations.

Joint



_˙

φ

3i =

φ

4i ˙

φ

4i = 19.5

(

φ

1i −

φ

3i

)

− 3.33sin

(

φ

3i

)

where

φ

1i and

φ

2i denote,respectively,theangularrotationandangularvelocityofthemanipulatormotor,

φ

3i and

φ

4i de-notetheangularrotationandangularvelocityofthemanipulatorjoint,respectively,u_i isthecontrolinput,g_i

(

φ

1i ,

φ

2i ,

φ

3i

)

= 48.6

(

φ

3i −

φ

1i

)

− 1.25

φ

2i , and

ρ

i =21.6.Intheprocess ofcontrol design,in[37],

ρ

i issupposed to bean unknown posi-tiveconstantwhileinourmethoditisassumedthat

ρ

_i isanunknown constant.Asanotherdifference,in[37],g_i (

φ

1i ,

φ

2i ,

φ

3i )isdescribedbyaknownnonlinearregressorvectormultipliedbyanunknownconstantparametervector.However,in ourframework, thisnonlinear functionisconsideredto becompletelyunknown. d_i ischosento be0.1isin(t). Considering x_i =

φ

3i ,onecanhave

xi =

φ

3i ˙ xi =

φ

4i ¨xi =19.5

φ

1i − 19.5

φ

3i − 3.33sin

(

φ

3i

)

... x_i =19.5

φ

2i − 19.5

φ

4i − 3.33

φ

4i cos

(

φ

3i

)

.

Nowtheequationofeachmanipulatorcanbetransformedtothefollowingfourth-orderdynamics x(_i 4)=fi

(

Xi

)

+biui+19.5di

withb_{i =}19_.5

ρ

_{i ,Xi =}[x_{i ,}x˙_{i ,}¨x_{i ,}...x_i ]T_{.In[37], f}

i

(

Xi

)

isdescribedbyaknownnonlinearregressorvector

ϕ

T

i =



˙

x_i ¨x_i ...x_i sin

(

x_i

)

x˙2

i sin

(

xi

)

x˙i cos

(

xi

)

¨xi cos

(

xi

)



multipliedbyanunknownconstantparametervector

ψ

_{i ∈R}7 _{as f}

i

(

Xi

)

=

ϕ

i T

ψ

i .However,inourframework, fi

(

Xi

)

is com-pletelyunknownfunctionas:

fi

(

Xi

)

=19.5gi − 19.5¨xi +3.33x˙2i sin

(

xi

)

− 3.33¨xi cos

(

xi

)

.

The interaction topology of the manipulators via which the agents exchange their informationis jointly connected. The network topology switches between the four graphs shown in Fig. 1.

σ

(

t

)

=fix



mod

(

2t,4

)

+1



. The initial conditions, (

φ

_i 1(0),

φ

i 2(0),

φ

i 3(0),

φ

i 4(0)),aregiveninTable2.Thecontrollerparameters oftheproposed controlschemeareselected

as



=0.02,s3₊

_γ

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 11

Fig. 3. Trajectories of the six motors angular rotations.

Fig. 4. Trajectories of the six joints angular velocities.

Toconduct simulationsforthe approachof [37],thecontrol parameters are selectedthesame asthosein thenumerical exampleofthatpaper.

12 M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14

Fig. 5. Trajectories of the six motors angular velocities.

Fig. 6. Control input signals.

6. Conclusion

M.H. Rezaei et al. / Information Sciences 0 0 0 (2018) 1–14 13 References

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