Gain-scheduled Leader-follower Tracking Control for Interconnected Parameter Varying Systems

1.7 MB·English

by  Yi Cheng

Gain-scheduled Leader-follower Tracking Control for
Interconnected Parameter Varying Systems
∗†
YiCheng and V. Ugrinovskii‡
SchoolofEngineeringandInformationTechnology,
TheUniversityofNewSouthWalesattheAustralianDefenceForceAcademy,Canberra,Australia,
5 Email: yi.cheng985,v.ugrinovskii @gmail.com
1 { }
0
2

n
Abstract
a
J This paper considers the gain-scheduled leader-follower tracking control problem for a parameter

4 varying complex interconnected system with directed communication topology and uncertain norm-
2 bounded coupling between theagents. A gain-scheduled consensus-type control protocol isproposed
andasufficientconditionisobtainedwhichguaranteesasuboptimalboundonthesystemtrackingper-

] formanceunderthisprotocol. Aninterpolationtechniqueisusedtoobtainaprotocolschedulewhichis
Y
continuousintheschedulingparameter.Theeffectivenessoftheproposedmethodisdemonstratedusing
S
asimulationexample.
.
s
c
[
Key words: Gain scheduling; leader-follower tracking control; interconnected parameter varying
1
systems;interpolationtechnique
v
2
3
1 Introduction
0
6
0 In recent years, the topic of cooperative control has attracted much attention. The objective of the
. cooperativecontrolproblemistoproposedistributedcontrollawstoachieveadesiredsystembehavior[1].
1
0 Arelatedproblemisthatofsynchronizationofcomplexdynamicalsystemswhereallthecomponentsare
5 controlledtoexhibitasimilarbehaviorbyinterconnectingthemintoanetwork[2,3].
1 Thereareseveralapproachestothesynchronizationproblemforcomplexsystemsconsistingofmany
:
v dynamic subsystems-agents. In the average consensus problem, which has received considerable atten-
i tion in the last decade [4, 5], the objective is to synchronize all the agents to a common state. Another
X
approachis to employ a suitable internal modelfor the system. For instance, it is shown in [6] that the
r
a existence of such an implicit internal model is necessary and sufficient for synchronization of a system
of heterogeneouslinear agents consideredin that paper. However, the internalmodelapproachdoesnot
directlyaddresstheoverallsystemperformance,sinceeachsystemiscontrolledtofollowitsowninternal
model dynamics. In general, it may be difficult to determine an internal model that guarantees a good
synchronizationperformance. Yetanotherapproachistodesignateoneoftheagentstoserveasaleader,
anddesigninterconnectionswithinthesystemsothattherestofthesystemfollowstheleader[7];also,see
[8]forarecentexample. Thisidealeadstotheleader-followertrackingproblem,whichisthemainfocus
ofthispaper.
A common feature of many papers that consider the leader-follower problem is that the dynamics
of agents are usually assumed to be dynamically decoupled [22, 23, 24, 25, 26, 27]. In many complex
systems, however, interactions between subsystems are inevitable and must be taken into account [11].
∗AcceptedforpublicationinInternationalJournalofRobustandNonlinearControl
†ThisworkwassupportedbytheAustralianResearchCouncilunderDiscoveryProjectsfundingscheme(projectsDP0987369
andDP120102152).
‡Correspondingauthor.
1
Examplesofsystemswithdynamicalinteractionsbetweensubsystemsincludespacecraftcontrolsystems
andpowersystems[9].Forexample,itwasnotedin[10]thatweakuncertaincouplingsbetweengenerators
isoneofthereasonsfordynamicinstabilityinpowersystemsand,therefore,theyshouldnotbeneglected.
Thismotivatesustoconsidertheleader-followertrackingproblemforinterconnectedsystems. Whilewe
do not consider a specific application, our approachmakes a step towardsusing consensus feedbackfor
synchronizationofsuchsystems,comparedtoothercontributionsintheareaofnetworkedcontrolsystems
whichdonotaddressthepresenceofinterconnections.
Inthispaper,weareconcernedwiththeleader-followertrackingproblemforinterconnectedsystems
that depend on a time varying parameter. Analysis and control of linear parameter varying systems has
attracted much attention in the last two decades due to their applications in flight control [12], turbofan
enginesandwindturbinesystems[13,14]. Forexample,anapplicationofthedistributedcontrolapproach
tocontrolandsynchronizationofwindgenerationsystemsmodeledasparametervaryingsystemshasbeen
presentedin [14]. Even thougha discrete time modelwas consideredin [14], parametervaryingsystem
modelingwasmotivatedbythefactthatthewindenergysourcedrivingwindturbinesexhibitstime-varying
nature.Inordertointegrateawindgenerationsystemintoapowergridinagrid-friendlymanner,thetotal
poweroutputof the wind turbinesmustbe regulatedto conformto a constantoutput. Each turbinethen
servesasanodeandiscontrolledwithrespecttoitspoweroutputbyturningthebladepitchangle,andthe
valueofthebladepitchangleispropagatedthroughthecommunicationnetworkusingtheleader-follower
algorithm. Thanks to this and many other potential applications, the theory of cooperative control for
parametervaryingsystemshasbeengainingattentioninrecentyears[15,16].
Themaincontributionofthispaperconcernstheleader-followercontrolofparametervaryingintercon-
nectedsystems. Arelatedproblem,fromthesynchronizationviewpoint,hasbeenstudiedin[18, 19]. In
thefirstreference,thesynchronizationproblemissolvedforheterogeneoussystemswhichdependonthe
parametersinanaffinefashion,undertheassumptionthatthenetworkofLPVagentsallowsforaninternal
modelforsynchronizationwhiletheagentsaredecoupledfromeachother.Thatis,theagentsinteractover
thecontrolprotocolonly. In[19],whilethelineardependencyontheparameterisnotrequired,theleader
isassumedtobegivenandbecompletelydecoupledfromtheagents. Incontrastwiththesereferences,we
assumetheleadertobechosenfromthegroupofparametervaryingagents;itisinterconnectedtotherest
ofthenetwork,andthelineardependencyontheparameterisnotrequired.
Thefactthattheleaderisinterconnectedwiththefollowersmakesitdifficulttoapplyregularcentral-
ized tracking techniquesto the problem under consideration. Many tracking techniquesassume that the
referencetrajectoryisgeneratedbyanexosystemorisaprioriknown,andisindependentofthefollowers.
Thisisnotthecaseinthispaper.Alternatively,acentralizedtrackingcontrollercanbeobtainedbysolving
astabilizationproblemforalarge-scalesystemcomprisedofsubsystemsdescribingdynamicsofindividual
trackingerrors.Thiscanbedonebyapplyingoneoftheexistingcentralizedgain-schedulingrobustcontrol
techniques,e.g.,usingthetechniquefrom[21]. However,ingeneralthecentralizedcontrollerobtainedthis
way will not have a desired information structure. Indeed, a controller obtained in such a way will not
generallyguaranteethattheinformationfromsubsystemsthatarenotobservedbyanodeisnotrequired
forfeedback. OnewaytoenforcesuchaninformationconstraintistoemployablockdiagonalLyapunov
function. Thisisthesolutionapproachundertakeninthispaperandisitsmaindifferencefromsolutions
whichcouldbeobtainedinacentralizedsetup.
Different from many papers that study synchronization of decoupled systems (including the above
mentioned references [18, 19], also see [2, 3]), in the case of coupled systems, it is essential to distin-
guish the network representing the existing interactions between the subsystems (including interactions
with the leader) from the network which realizes communicationand control. This leads us to consider
a two-network structure in this paper. The rationale for this is twofold. Firstly, our aim is to construct
synchronization protocols for all agents excluding the leader, as we wish to avoid perturbing the leader
dynamics(otherthanthroughanunavoidablephysicalcoupling). Hence,thecommunicationgraphofthe
networkmustbe differentfromthe interconnectiongraph. Thesecondreasonisthatthe interconnection
graphdescribesdynamicalcouplingsbetweensubsystemsandthusmaybedifferentfromthecommunica-
tionandcontrolgraph.
Our main result is a sufficient condition for the design of a gain-scheduled leader-follower control
protocol for parameter varying multi-agent systems with directed control network topology and linear
uncertaincouplingssubject to norm-boundedconstraints. The conditioninvolvescheckingfeasibility of
2
parameterized linear matrix inequalities (LMIs) at several operating points of the system. These LMIs
serveasthebasisforthedesignofacontinuous(intheschedulingparameter)controlprotocolforcoupled
parameter varying systems, by interpolating consensus control protocols computed for those operating
points; cf. [29, 19, 21]. Interpolationallows usto mitigatedetrimentaleffectsof transientsarising when
thesystemtraversesfromoneoperatingconditiontoanother.
Theremainderofthepaperproceedsasfollows.InSection2,weformulatetheleaderfollowercontrol
problemforparametervaryingcoupledmulti-agentsystemsandgivesomepreliminaries.Themainresults
are given in Section 3. Section 4 gives an example which illustrates the theory presented in the paper.
Finally,theconclusionsaregiveninSection5.
2 Problem Formulation and Preliminaries
2.1 Interconnection and communicationgraphs
Asstatedintheintroduction,inthispaperwedrawadistinctionbetweenthecommunicationtopology
ofthesystem usedforcontrolandthe topologyofinteractionsbetweensubsystems. Theexampleofthe
two-graph structure is shown in Fig. 1, where the edges of the interconnection graph are indicated by
the solid lines and the edgesof the communicationgraphare shown by the dashed lines. Both coupling
andcommunicationtopologiesaredescribedintermsofdirectedgraphsdefinedonthecommonnodeset
= 0,...,N . Without loss of generality, node 0 will be assigned to be the leader of the network,
V { }
while thenodesfromthe set = 1,...,N willrepresentthefollowers. Thecouplinggraphand the
0
V { }
communicationgraphwillbedenotedas ϕ and c,respectively.
G G
Theedgesetsofbothgraphsaresubsetsoftheset andconsistofpairsofnodes. Thepair(j,i)
V ×V
ineachedgesetdenotesthedirectededgewhichoriginatesatnodej andendsatnodei. Edge(j,i)inthe
edgesetofthedirectedcouplinggraph ϕ,denoted ϕ,describesthefactthatnodeiisinfluencedbynode
G E
j throughadirectedinteractionbetweennodesj andi.
Also, c denotesanedgesetofthecommunicationgraph c,consistingoforderedpairsof
E ⊆ V ×V G
nodes. Eachsuchedgeindicatestheinformationflowbetweenthenodes,thatis,(j,i) c ifandonlyif
∈ E
nodeiobtainsinformationfromnodej,whichitcanuseforcontrol.
Theadjacencymatrixofthedirectedinterconnectiongraph ϕ isdenotedas ϕ, its(i,j)-thentryis
G A
1ifandonlyif(j,i) ϕ. Theadjacencymatrix c ofthedirectedcommunicationgraph c isdefined
∈ E A G
inthesamemanner.Sinceaccordingtoastandardconventionweassumethatboththecouplinggraphand
communicationgraphhavenoself-loops,thediagonalentriesof ϕ and c areallequaltozero.
Todistinguishbetweentheleaderandtherestofthenetwork,AwedefinAesubgraphs ϕ, cofthegraphs
ϕ, c definedonthenodeset = 1,...,N ,withtheedgesets ϕ = (j,i) G0ϕ:Gi0,j and
Gc =G (j,i) c: i,j ,reVs0pecti{vely.Selec}tingthesubgraphs ϕE0and {cindu∈cesEtheparti∈tioVn0o}fthe
E0 { ∈E ∈V0} G0 G0
matrices ϕ, c,
A A
0 d¯ 0 0
ϕ = , c = ,
A (cid:20)d ϕ(cid:21) A (cid:20)g c(cid:21)
A0 A0
whered=[d ... d ]′,d¯=[d¯ ... d¯ ],g =[g ... g ]′and ϕ, c aretheadjacencymatricesofthe
subgraphs ϕ1and cN,respective1ly. TheNzerorow1oftheNmatrix Ac0reAfle0ctsourassumptionthattheleader
G0 G0 A
nodedoesnotreceivethestateinformationfromothernodesofthenetwork. However,thecorresponding
rowof ϕ maybenonzerosincetheleadernodemaybephysicallycoupledwithsomeofthefollowers.
A
Inthispaperweareconcernedwiththecasewhereonlysomeofthefollowersreceivethestateinfor-
mationdirectlyfromtheleader. Werefertosuchnodesi aspinnednodes;thecorrespondingentries
0
∈V
oftheadjacencymatrix c,g =1,andg =0ifnodeiisnotpinned.ThematrixG=diag g N×N
i i i
A { }∈ℜ
isreferredtoasapinningmatrix.
TheLaplacianmatrixofthesubgraph cisdefinedas c = c,where =diag p ,...,p
G0 L0 P−A0 P { 1 N}∈
RN×N isthein-degreematrixof c,i.e.,thediagonalmatrix,whosediagonalelementsarethein-degrees
G0
ofthecorrespondingnodesofthegraph c,p = N ac fori = 1,...,N,whereac aretheelements
G0 i j=1 ij ij
oftheithrowofthematrix c. P
A0
Finally,wegivethedefinitionandthenotationforneighborhoodsintheabovegraphs.
3
0
Interactionedges
d1 g2 d¯3
Communicationedges
1 2 3
4
Figure1:Anexampleoftheinterconnectionandcommunicationgraphs.
Definition1 c = ( , c) is a communication graph, each node i represents a subsystem in the
G0 V E ∈ V
interconnectedsystemandedge(j,i) c meansthatnodeiobtaininformationfromnodej.
∈E
Definition2 φisaninterconnectiongraph,eachnodei representsasubsystemintheinterconnected
G ∈V
system and edge (j,i) φ indicates that subsystem i is influenced by subsystem j through a directed
∈ E
interactionbetweensubsystemsiandj.
Node j is called a neighbor of node i in the graph ϕ (or ϕ, c, c, respectively) if (j,i) ϕ
( ϕ, c or c, respectively). The sets of neighbors of nGode i iGn0theGgraGp0hs ϕ and c are denot∈edEas
NE0ϕ =E j (Ej0,i) ϕ , and Nc = j (j,i) c , respectively. The sets ofGneighborGsof node i in the
suibgrap{hs| ϕ an∈d Ec a}redenoteidas{Sϕ| = j∈(jE,i)} ϕ andSc = j (j,i) c ,respectively.
G0 G0 i { | ∈E0} i { | ∈E0}
Weconcludethissubsectionbyspecifyingourstandingrequirementsonthecommunicationtopology
ofthenetworkwhichareassumedtoholdthroughoutthepaper.
Assumption1 Thecommunicationsubgraph ccontainsaspanningtree,whoserootnodei isthepinned
G0 r
node,i.e.,g >0.
ir
Remark1 Assumption1ensuresthattheinformationflowsfromtheleadertoitspinnedfollowers,aswell
asbetweenotherfollowers[25,28].
2.2 Notation
Throughoutthepaper,thefollowingnotationwillbeused.
nand n×m arearealEuclideann-dimensionalvectorspaceandaspaceofrealn mmatrices.
ℜ ℜ ×
Γdenotesaninterval[ρ ,ρ ] . Thesymbolsρ,ρ , ρ¯ ,ρ , etc.,willrepresentvariouspoints
min max ⊂ ℜ s s s
withintheintervalΓ,andΓ ,Γ willdenotesetsofpointswithinΓwhichwillbedefinedlaterinthepaper.
ℓ 0
Also, ρ(t) is a scalar function, defined on [0, ) and taking values in the interval Γ. This function will
∞
describeatime-varyingschedulingparameterofthemulti-agentsystemunderconsideration.
Unless stated otherwise, the notationsA(ρ), Y , etc., will refer to matrices of appropriatedimension
ρ
parameterizedbyρ Γ.
∈
Forq n,diag q denotesthediagonalmatrixwiththeentriesofqasitsdiagonalelements.
∈ℜ { }
denotestheKronneckerproductoftwomatrices.
⊗
λ ()andλ ()willdenotethelargestandthesmallesteigenvaluesofarealsymmetricmatrix.
max min
I ist·heN N id·entitymatrix,and1 N isthevectorwhoseallentriesareequalto1.Whenthe
N N
× ∈ℜ
dimensionisclearfromthecontext,thesubscriptN willbesuppressed,andI willrepresenttheidentity
matrixofanappropriatedimension.
FortwosymmetricmatricesX andY ofthesamedimensions,X Y,(X >Y)ifandonlyifX Y
≥ −
ispositivesemidefinite(positivedefinite).
Consider the Laplacian matrix of the subgraph c, c. According to [28], under Assumption 1, the
G0 L0
matrix c + G is nonsingular (also see [27]), and all the entries of the vector ϑ = [ϑ ,...,ϑ ]′ =
L0 1 N
4
( c +G)−11 arepositive. Also,thematrixΘ−1( c +G)+( c +G)′Θ−1 ispositivedefinite,where
ΘL,0 diag ϑ N.ThefollowingtwoconstantswillbeuLs0edintheprLoo0fsofourresults:
{ }
1
σ = λ Θ−1( c +G)+( c +G)′Θ−1 ,
2 min L0 L0
(cid:0) (cid:1)
λˆ = λ ( c +G)′Θ−2( c +G) . (1)
max L0 L0
(cid:0) (cid:1)
2.3 Problem Formulation
ThesystemunderconsiderationconsistsofN +1parameter-varyingdynamicalagents,coupledwith
theirneighbors; thetopologyofinterconnectionsis capturedbythe directedgraph ϕ. Dynamicsof the
G
ithagentaredescribedbythelinearequation
x˙ =A(ρ(t))x +B u +B ϕ (t,x x ), i=0,...,N, (2)
i i 1 i 2 ij j i
−
jX∈Nϕ
i
wherex n isthestateofagenti,u p isthecontrolinputandρ(t)isthetime-varyingparameter,
i i
∈ ℜ ∈ ℜ
whichisavailabletoallagents. Itisassumedthatρ: [0, ) Γ isacontinuousfunction. Alsoin
∞ → ⊂ ℜ
equation(2),B andB arerealmatricesofappropriatedimensions,andA(ρ(t))isthecompositionofa
1 2
continuousmatrix-valuedfunctionA(): Γ n×nandthefunctionρ(t)definedabove.
· →ℜ
Thefunctionsϕ : [0, ) n m,
ij
∞ ×ℜ →ℜ
ϕ (t,x)=∆ (t)C x, x n,
ij ij ij
∈ℜ
describe how agent j influences the dynamics of agent i. Here Cij qij×n and ∆ij m×qij are
∈ ℜ ∈ ℜ
respectivelyconstantandtime-varyingmatrices. Accordingtotheabovemodel,wefocusonlineartime-
varyinginteractionsbetweenthe subsystemswhosestrengthsdependonthe relativestate ofsubsystemi
withrespecttosubsystemj,x x . Also,wewillassumethatthecoefficients∆ (t)areuncertain,and
j i ij
−
satisfytheconstraint
∆′ (t)∆ (t) I, t [0, ). (3)
ij ij ≤ ∀ ∈ ∞
Thatis,weconsideraclassofnorm-boundeduncertaininteractions,whichwillbedenotedbyΞ.
Since we have designated agent 0 to be the leader, and the rest of the agents to be controlled are to
followtheleader,thenaccordingto(2)dynamicsoftheithfolloweraredescribedbytheequation
x˙ =A(ρ(t))x +B u +B ϕ (t,x x )+d ϕ (t,x x ) , i=1,...,N, (4)
i i 1 i 2 ij j i i i0 0 i
(cid:16)jX∈Sϕ − − (cid:17)
i
whiletheleaderdynamicsaregivenbytheequation
x˙ =A(ρ(t))x +B ϕ (t,x x ). (5)
0 0 2 0k k 0
−
k:Xd¯k=1
Unlikeagentsi, i = 1,...N, the leaderis notcontrolled,i.e., u 0. On thecontrary,allotheragents
0
≡
willbecontrolledtotracknode0.
Remark2 In this paper, no a prioristability assumptionsare made aboutthe matrices A(ρ). This is in
contrastto,forexample,thesynchronizationproblemconsideredin[22],wherethedynamicsoftheleader
were required to be stable or marginally stable. In the sequel, however, certain conditions, in an LMI
form, will be imposed onthe coefficientsofthe leaderand followersdynamics. While we formally make
no special provisions regarding these coefficients, except for the feasibility of these LMI conditions, for
suchLMIstobefeasibleonecanreasonablyexpecttheoverallsystemdynamicstohavecertaincollective
stabilizabilityanddetectabilityproperties;seeRemark3below.
5
Definetrackingperformanceassociatedwiththecontrolinputu()=[u ()′ ... u ()′]′as
1 N
· · ·
N ∞
(u)= ((x x )′Q(x x )+u′Ru )dt, (6)
J Z 0− i 0− i i i
Xi=1 0
whereQ = Q′ > 0andR = R′ > 0aregivenweightingmatrices. Inthispaperweareconcernedwith
thefollowingproblem.
Problem1 Foreachfolloweri,findgainschedulefunctionsK : Γ p×n sothatthecontrolprotocols
i
→ℜ
oftheform
u = K (ρ(t)) (x x )+g (x x ), (7)
i i j i i 0 i
− jX∈Sc − − 
i
ensureaboundedworst-casetrackingperformance, 
sup (u)<const, (8)
J
Ξ
whereconstdenotesaconstantwhichcandependonx (0)andx (0).
0 i
3 The Main Results
In this section, we first revisit the leader follower tracking control problem for multi-agent systems
withfixedparametersconsideredin[17]toobtainasufficientconditionforthedesignofacontrolprotocol
foramoregeneralclassofsystemsinvolvingcouplingbetweentheleaderandtheagents. Thissufficient
conditionwillthenbeappliedtodevelopaninterpolationtechniquetoobtainacontinuousgainscheduling
trackingcontrolprotocolforthesystem(2). Theinterpolationtechniqueisthemainresultofthispaper.
3.1 Leader followercontrol forfixedparameter systems
Wenowrevisittheresultsof[17]. Thisrevisionispromptedbyamoregeneralstructureofthesystem
(2) whichallowsforphysicalconnectionsbetweentheleaderandthe followers. As a result, theformof
thecontrolprotocolissomewhatdifferenthereinthattheresultingcontrollergainsdependoni.
Considerafixed-parameterversionofthesystem(2)describedbytheequation
x˙ =A(ρ)x +βξ (t,x )+B u +B ϕ (t,x x ), i=0,...,N, (9)
i i i i 1 i 2 ij j i
−
jX∈Nϕ
i
whereρ Γisfixed,andβ isapositiveconstant. Comparedto(2),thesystem(9)includesanadditional
∈
uncertaintyelementξ (t,x ),whichsatisfiesthefollowingconstraint
i i
ξ (t,x) ξ (t,y) 2 x y 2, x,y n. (10)
0 i
k − k ≤k − k ∀ ∈ℜ
Inthesequel,wewillshowthatwhenthedifferenceA(ρ(t)) A(ρ)issufficientlysmall,thesystem(2)
−
canberepresentedasthesystem(9)subjectto(10).
Wenowderiveadistributedprotocoloftheform(7)underwhichthefixed-parameteruncertainsystem
(9)satisfiestheperformancerequirement(8).
Definetheleadertrackingerrorvectorsase = x x , i = 1,...,N. Dynamicsofthevariablee
i 0 i i
−
satisfytheequation
e˙ =A(ρ)e B u +βξ˜(t) B ϕ (t,e )
i i 1 i i 2 0k k
− −
k:Xd¯k=1
B ϕ (t,e ) ϕ (t,e ) B d ϕ (t,e ), (11)
2 ij i ij j 2 i i0 i
− − −
jX∈Sϕ(cid:0) (cid:1)
i
6
whereξ˜(t),ξ (t,x (t)) ξ (t,x (t)). Itfollowsfrom(10)that ξ˜ 2 e 2forallt 0.
i 0 0 i i i i
Fornodeiofthesubgra−ph ϕ,introducematricesCˆ =[C′ ..k.Ck′ ≤]′k,C¯k=[C′ ..≥.C′ ]′,where
j ,...,j arethe elementsofGt0he neighborhoodset Sϕi, andirj1,...,rijκiare tihenodre1siwithrthχieiproperty
1 κi i 1 χi
(i,r ) ϕ;κ andχ are,respectively,thein-degreeandtheout-degreeofnodeiinthegraph ϕ. Also,
ι ∈E0 i i G0
letQ¯ =(σ2/λˆ)Q,whereσ,λˆaretheconstantsdefinedin(1).
InordertoformulatetheextensionofTheorem1in[17],witheachnodei,i=1,...,N,weassociate
acollectionofpositiveconstantsν ,µ , j Sϕ,π ,ν (onlyforthosenodesiforwhichd = 1),and
µ (onlyfor those nodesi for whiicjhd¯ij= 1∈). Ai lsoi, leit0Y = Y′ > 0 be an n n matrix. Uising these
0i i
constantsandthematrix,foreachnodeiintroduceamatrixΠ defineddepending×ond ,d¯ asfollows.
i i i
Case1. d =0andd¯ =0. Foreachsuchnodei,definethematrixΠ as
i i i
6 6
Zi YQ¯1/2 YCˆi′ YC¯i′ Y YCi′0 YC0′i
Q¯1/2Y −I 0 0 0 0 0
 
CˆiY 0 −Φi 0 0 0 0
Πi = C¯iY 0 0 −Ωi 0 0 0 , (12)
 Y 0 0 0 −Λi 0 0 
 CC0i0iYY 00 00 00 00 −ν01i0I −N0µ10iI
where
1 1 1
Φ =diag[ I,j Sϕ], Ω =diag[ I,j: i Sϕ], Λ = I,
i ν ∈ i i µ ∈ j i π
ij ji i
1
Z =A(ρ)Y +YA(ρ)′ B R−1B′ + β2I
i − 1 1 π
i
1 1 1 1
+ ( + )+ + B B′. (13)
ν µ ν µ 2 2
(cid:0)jX∈Siϕ ij ij i0 k:Xd¯k=1 0k(cid:1)
Case2. d = 0andd¯ = 0. Foreachsuchnodei, theconstantν isnotdefined. Accordingly,we
i i i0
6
definethematrixΠ byremovingthesecondlastcolumnandrowfromthematrixin(12):
i
Z YQ¯1/2 YCˆ′ YC¯′ Y YC′
i i i 0i
Q¯1/2Y I 0 0 0 0 
−
Cˆ Y 0 Φ 0 0 0
Π = i i , (14)
i  C¯ Y 0 0 Ω 0 0 
 i i 
 Y 0 0 0 Λ 0 
 i 
 C Y 0 0 0 0 1 I
 0i −Nµ0i 
wherethematrixZ ismodifiedtobe
i
1 1 1 1
Z =A(ρ)Y +YA(ρ)′ B R−1B′ + β2I+ ( + )+ B B′. (15)
i − 1 1 π ν µ µ 2 2
i (cid:0)jX∈Siϕ ij ij k:Xd¯k=1 0k(cid:1)
Case3.d =0andd¯ =0. Foreachsuchnodetheconstantµ isnotdefined,hencethecorresponding
i i 0i
6
matrixΠ willbedefinedbyremovingthelastcolumnandrowfromthematrixin(12):
i
Z YQ¯1/2 YCˆ′ YC¯′ Y YC′
i i i i0
Q¯1/2Y I 0 0 0 0 
−
Cˆ Y 0 Φ 0 0 0
Πi = C¯iY 0 −0 i Ω 0 0 , (16)
 i i 
 Y 0 0 −0 Λ 0 
 i 
 Ci0Y 0 0 0 −0 −ν1i0I
ThematrixZ isthesameasin(13).
i
7
Case 4. d = 0 and d¯ = 0. In this case, both ν and µ are not defined, and the corresponding
i i i0 0i
matrixΠ isdefinedbyremovingtwolastcolumnsandrowsfromthematrixin(12):
i
Z YQ¯1/2 YCˆ′ YC¯′ Y
i i i
Q¯1/2Y I 0 0 0 
−
Πi = CCˆ¯iYY 00 −0Φi 0Ω 00 , (17)
 i − i 
 Y 0 0 0 Λ 
 − i
ThematrixZ isthesameasin(15).
i
ThefollowingtheoremisanextensionofTheorem1in[17].
Theorem1 Under Assumption 1, let a matrix Y = Y′ > 0, Y n×n, constants ν > 0, µ > 0,
ij ij
π > 0,j Sϕ,i = 1,...,N,andconstantsν > 0(forthosei∈wiℜthd = 0),µ > 0(forthoseiwith
d¯i=0)exi∈stsuichthatthefollowingLMIsaresai0tisfiedsimultaneously i 6 0i
i
6
Π <0, i=1,...,N. (18)
i
Thenthecontrolprotocol(7)with
K (ρ)= (ϑ σ)−1R−1B′Y−1 (19)
i − i 1
solvestheleaderfollowertrackingcontrolproblemforthefixedparametersystem(9). Furthermore,this
protocolguaranteesthefollowingperformancebound
λˆ N
sup (u) (x (0) x (0))′Y−1(x (0) x (0)) (20)
J ≤ σ2 0 − i 0 − i
Ξ Xi=1
foralluncertaintiesξ forwhich(10)holds.
i
Remark3 The feasibility of the LMIs (18) in Theorem 1 is a sufficient condition for the existence of a
controlschemetoguaranteetheperformancebound(20). TheLMIsarenumericallytractableandcanbe
solvedusingtheexistingsoftware.Additionally,usingthestandardtoolsoftheH controltheory,suchas
∞
theStrictBoundedRealLemmaandtheassociatedRiccatiequation,thefeasibilityoftheLMIs(18)canbe
relatedtocollectivestabilizabilityand/ordetectabilitypropertiesofthecoefficientsoftheinterconnected
system (9). Sincethissystem consistsofidenticalagents, onecanexpectthateachagentmust necessar-
ily have correspondingstabilizability anddetectability properties for these collective properties to hold;
cf.[20].
Proof: Using the Schur complement, each LMI (18) can be transformed into the following Riccati
inequality:
1 1 1 1
A(ρ)Y +YA(ρ)′ B R−1B′ + ( + )+ B B′ + β2I
− 1 1 ν µ µ 2 2 π
(cid:0)jX∈Siϕ ij ij k:Xd¯k=1 0k(cid:1) i
+Y Q¯+ ν C′ C + µ C′ C +π I Y
ij ij ij ji ji ji i
(cid:0) jX∈Sϕ j:Xi∈Sϕ (cid:1)
i j
1
+ B B′ +ν YC′ C Y
(cid:18)ν 2 2 i0 i0 i0 (cid:19)
i0
(thistermispresentonlyifd =1)
i
+Nµ YC′ C Y <0 (21)
0i 0i 0i
(thistermispresentonlyifd¯ =1).
i
NotethatthelastandthesecondlastlinesintheRiccatiinequality(21)arepresentonlyforthosenodesi
forwhichd =0and/ord¯ =0,respectively.
i i
6 6
8
After pre- and post-multiplying (21) by Y−1, and then using the expression (19) for K (ρ) in the
i
resultinginequality,weobtain
Y−1(A(ρ)+σϑ B K (ρ))+(A(ρ)+σϑ B K (ρ))′Y−1+ ν C′ C + µ C′ C
i 1 i i 1 i ij ij ij ji ji ji
jX∈Sϕ j:Xi∈Sϕ
i j
1 1 1 1
+Y−1 ( + )+ B B′ + β2I Y−1+Y−1B R−1B′Y−1+Q¯+π I
ν µ µ 2 2 π 1 1 i
(cid:16)(cid:0)jX∈Siϕ ij ij k:Xd¯k=1 0k(cid:1) i (cid:17)
1
+ Y−1B B′Y−1+ν C′ C
(cid:18)ν 2 2 i0 i0 i0(cid:19)
i0
(thistermispresentonlyifd =1)
i
+Nµ C′ C <0 (22)
0i 0i 0i
(thistermispresentonlyifd¯ =1).
i
Define e = [e′,...,e′ ]′ and consider the following Lyapunov function candidate for the intercon-
1 N
nectedsystemconsistingofthesubsystems(11):
N
V(e)= e′Y−1e . (23)
i i
Xi=1
Then
N
dV(e)
= 2e′Y−1 A(ρ)e +B K (ρ)( (e e )+g e )
dt Xi=1 i (cid:16) i 1 i jX∈Sc i− j i i (cid:17)
i
N N N
2 e′Y−1B ϕ (t,e )+2 e′Y−1βξ˜(t,e ) 2 d e′Y−1B ϕ (t,e )
− i 2 0k k i i i − i i 2 i0 i
Xi=1k:Xd¯k=1 Xi=1 Xi=1
N N
2 e′Y−1B ϕ (t,e )+2 e′Y−1B ϕ (t,e ). (24)
− i 2 ij i i 2 ij j
Xi=1jX∈Sϕ Xi=1jX∈Sϕ
i i
Notethefollowinginequality:
N
2e′Y−1B K (ρ)( (e e )+g e )
i 1 i i− j i i
Xi=1 jX∈Sc
i
= 2e′(Θ( c +G) (Y−1B (σR)−1B′Y−1))e
− L0 ⊗ 1 1
= y′ (Θ( c +G)+( c +G)′Θ) I y
− L0 L0 ⊗ p
2σ(cid:0)y′ I I y (cid:1)
N p
≤− ⊗
= 2σe′(cid:0)I Y−(cid:1)1B (σR)−1B′Y−1 e
− N ⊗ 1 1
N(cid:0) (cid:1)
= 2 e′Y−1B R−1B′Y−1e , (25)
− i 1 1 i
Xi=1
wherey =(I (σR)−1/2B′Y−1)e.
N ⊗ 1
9
From(24)and(25),onehas
N N
dV(e)
2e′Y−1 A(ρ)+σϑ B K (ρ) e +2 e′Y−1βξ˜(t,e )
dt ≤Xi=1 i (cid:16) i 1 i (cid:17) i Xi=1 i i i
N N
2 e′Y−1B ϕ (t,e ) 2 e′Y−1B ϕ (t,e )
− i 2 ij i − i 2 0k k
Xi=1jX∈Siϕ Xi=1k:Xd¯k=1
N N
2 d e′Y−1B ϕ (t,e )+2 e′Y−1B ϕ (t,e ). (26)
− i i 2 i0 i i 2 ij j
Xi=1 Xi=1jX∈Sϕ
i
UsingtheRiccatiinequality(22),itfollowsfrom(26)that
N
dV(e)
e′ Y−1B R−1B′Y−1+Q¯+ ν C′ C + µ C′ C
dt ≤−Xi=1 i(cid:16) 1 1 jX∈Sϕ ij ij ij j:Xi∈Sϕ ji ji ji
i j
1 1 1 1
+π I+Y−1 ( ( + )+ )B B′ + β2I Y−1 e
i ν µ µ 2 2 π i
(cid:0) jX∈Siϕ ij ij k:Xd¯k=1 0k i (cid:1) (cid:17)
N N
1
+ e′ Y−1B B′Y−1+ν C′ C e +N e′µ C′ C e
i ν 2 2 i0 i0 i0 i i 0i 0i 0i i
i:Xdi=1 (cid:16) i0 (cid:17) i:Xd¯i=1
N N N
+2 e′Y−1βξ˜(t,e ) 2 e′Y−1B ϕ (t,e ) 2 d e′Y−1B ϕ (t,e )
i i i − i 2 ij i − i i 2 i0 i
Xi=1 Xi=1jX∈Sϕ Xi=1
i
N N
2 e′Y−1B ϕ (t,e )+2 e′Y−1B ϕ (t,e ). (27)
− i 2 0k k i 2 ij j
Xi=1k:Xd¯k=1 Xi=1jX∈Siϕ
Usingthefollowingidentities,
N N
µ e′C′ C e = µ e′C′ C e ,
ij j ij ij j ji i ji ji i
Xi=1jX∈Sϕ Xi=1j:Xi∈Sϕ
i j
N N
N e′µ C′ C e′ =N e′µ C′ C e
i 0i 0i 0i i k 0k 0k 0k k
i:Xd¯i=1 k:Xd¯k=1
N
d e′Y−1B ϕ (t,e )= e′Y−1B ϕ (t,e ),
i i 2 i0 i i 2 i0 i
Xi=1 i:Xdi=1
10