Control of Networked Multiagent Systems with Uncertain Graph Topologies

0.22 MB·English

by  Tansel Yucelen

January2015
Control of Networked Multiagent Systems
†
with Uncertain Graph Topologies
5
1
0
2

n ∗
TanselYucelen andJohnDanielPeterson
a
J DepartmentofMechanicalandAerospaceEngineering

4 MissouriUniversityofScienceandTechnology
1

] KevinL.Moore
C
DepartmentofElectricalEngineeringandComputerScience
O
ColoradoSchoolofMines
.
h
t
a
m
Abstract—Multiagentsystemsconsistofagentsthatlocallyexchangeinformationthrough
[
a physical network subject to a graph topology. Current control methods for networked mul-

1
tiagentsystemsassumetheknowledgeofgraphtopologiesinordertodesigndistributedcon-
v
9 trollawsforachievingdesiredglobalsystembehaviors. However,thisassumptionmaynotbe
3
4 valid for situations where graph topologies are subject to uncertaintieseither due to changes
3 in the physical network or the presence of modeling errors especially for multiagent systems
0
. involving a large number of interacting agents. Motivating from this standpoint, this paper
1
0 studiesdistributedcontrolofnetworked multiagentsystemswithuncertaingraphtopologies.
5
Theproposedframeworkinvolvesacontrollerarchitecturethathasanabilitytoadaptitsfeed-
1
: backgainsinresponsetosystemvariations.Specifically,weanalyticallyshowthattheproposed
v
i controllerdrivesthetrajectoriesofanetworkedmultiagentsystemsubjecttoagraphtopology
X
r withtime-varyinguncertaintiestoacloseneighborhoodofthetrajectoriesofagivenreference
a
modelhavingadesiredgraphtopology.Asaspecialcase,wealsoshowthatanetworkedmulti-
agentsystemsubjecttoagraphtopologywithconstantuncertaintiesasymptoticallyconverges
tothetrajectoriesofagivenreferencemodel.Althoughthemainresultofthispaperispresented
in the context of average consensus problem, the proposed framework can be used for many
otherproblemsrelatedtonetworkedmultiagentsystemswithuncertaingraphtopologies.
Keywords — Networked multiagent systems; uncertain graph topologies; distributed con-
trol;adaptivecontrol;stability
† ThisresearchwassupportedinpartbytheUniversityofMissouriResearchBoard.
∗ Correspondingauthor: 400West,13thStreet,Rolla,MO65409(Address);[phone](Phone);+1573341
6899(Fax);[email protected](Email).
1. Introduction
Multiagent systems consist of agents that locally exchange information through a physical
network subject to a graph topology. Current control methods for networked multiagentsys-
temsassumetheknowledgeofgraphtopologiesinordertodesigndistributedcontrollawsfor
achievingdesiredglobalsystembehaviors(see,forexample,[1]andreferencestherein). How-
ever,thisassumptionmaynotbevalidforsituationswheregraphtopologiesaresubjecttoun-
certainties either due to changes in the physical network or the presence of modeling errors
especiallyformultiagentsystemsinvolvingalargenumberofinteractingagents.
Uncertainnatureofnetworkedmultiagentsystemshasreceivedaconsiderableattentionre-
cently,includingnotableresults[2–10]. Forachievingdesiredmultiagentsystembehavior,[2,3]
make a specific assumptionon thenetwork connectivityother thanthestandard assumption
ontheconnectednessofnetworkedagents. Theauthorsof[4]excitethemultiagentsystemin
ordertodetectandisolatetheuncertainagentsfromthenetworktopology. Like[2,3],acom-
putationallyexpensiveandnotscalablealgorithmisproposedin[5,6]basedoninputobservers
technique,wheretheeffectofuncertainagentsontheoverallmultiagentsystemperformance
is quantified. An extension of this work is also given in [7] that focuses on the detection and
isolationofuncertainagents.Theauthorsof[8,9]useanadaptivecontrolapproachinorderto
suppresstheeffectofuncertainagentsontheoverallmultiagentsystemperformancewithout
making specific assumptions on the fraction of misbehaving agents. A common similarityof
theapproachesdocumentedin[2–9]isthattheymodeluncertaintiesintheagentdynamicsas
additive perturbationsthat do not depend on the state of agents, where these results are not
applicabletothenetworkedmultiagentsystemswithgraphtopologyuncertaintiessincesuch
uncertaintiesdependonthestateofagents.
Onerelevantworktotheresultsofthispaperisrecentlyappearedin[10],wheretheauthors
utilizeadaptiveandslidingmodecontrolmethodologiesinordertoenforceanetworkedmul-
tiagentsystemsubjecttoanuncertaingraphtopologytofollowagivenreferencemodelhaving
a desired graph topology. However, the result in [10] may require a centralized information
1
exchangeamongnetworkedagentsingeneralduetothestructureoftheproposedcontrolal-
gorithm (see (8) or (18) of [10]). Other important results, which are related to this paper, are
presentedin[11–13]withoutrequiringacentralizedinformationexchange. However,thesere-
sultsholdforgraphtopologiessubjecttoconstantuncertaintiesonly.
Inthispaper,we studydistributedcontrolofnetworked multiagentsystemswithuncertain
graph topologies. The proposed framework involves a novel controller architecture that has
anabilitytoadaptitsfeedbackgainsinresponsetosystemvariations. Specifically,weanalyti-
callyshowthattheproposedcontrollerdrivesthetrajectoriesofanetworkedmultiagentsystem
subjecttoagraphtopologywithtime-varyinguncertaintiestoacloseneighborhoodofthetra-
jectoriesofagivenreferencemodelhavingadesiredgraphtopology. Asaspecialcase,wealso
showthatanetworkedmultiagentsystemsubjecttoagraphtopologywithconstantuncertain-
tiesasymptoticallyconvergestothetrajectoriesofagivenreferencemodel. Althoughthemain
result of this paper is presented in the context of average consensus problem, the proposed
frameworkcanbeusedformanyotherproblemsrelatedtonetworkedmultiagentsystemswith
uncertaingraphtopologies.
2. Notation,Definitions,andGraph-TheoreticNotions
Thenotationusedinthispaperisfairlystandard. Specifically,Rdenotesthesetofrealnum-
bers,Rndenotesthesetofn×1realcolumnvectors,Rn×m denotesthesetofn×mrealmatrices,
R denotesthesetofpositiverealnumbers,Rn×n (resp.,Rn×n)denotesthesetofn×npositive-
+ + +
definite(resp.,nonnegative-definite)realmatrices,Sn×n (resp.,Sn×n)denotesthesetofn×n
+ +
symmetricpositive-definite(resp.,symmetricnonnegative-definite)realmatrices,0 (resp.,1
n n
)denotesthen×1vectorofallzeros(resp.,ones),andI denotesthen×nidentitymatrix.Fur-
n
thermore,wewrite(·)T fortranspose,k·k fortheEuclidiannorm,λ (A)(resp.,λ (A))for
2 min max
theminimum(resp.,maximum)eigenvalueoftheHermitianmatrixA,λ (A)forthei-theigen-
i
valueof A(Aissymmetricandtheeigenvaluesareorderedfromleasttogreatestvalue),diag(a)
forthediagonalmatrixwiththevectora onitsdiagonal,and[A] fortheentryofthematrix A
ij
onthei-throwand j-thcolumn.
2
Next, we recall some of the basic notions from graph theory, where we refer to [1] for fur-
therdetails. Inthemultiagentliterature,graphsarebroadlyadoptedtoencodeinteractionsin
networkedsystems. AnundirectedgraphG isdefinedbyasetVG ={1,...,n}ofnodesandaset
EG ⊂VG×VG of edges. If (i,j)∈EG, then thenodes i and j areneighbors and theneighboring
relation is indicated with i ∼ j. The degree of a node is given by the number of its neighbors.
Letting d be the degree of node i, then the degree matrix of a graph G, D(G)∈Rn×n, is given
i
by D(G) , diag(d), d = [d ,...,d ]T. A path i i ...i is a finite sequence of nodes such that
1 n 0 1 L
i ∼i , k =1,...,L,anda graphG is connected ifthereisa pathbetweenanypairofdistinct
k−1 k
nodes. Theadjacency matrixofagraphG,A(G)∈Rn×n,isgivenby
[A(G)] , 1, if(i,j)∈EG,
ij
½ 0, otherwise.
The Laplacian matrix of a graph, L(G)∈Sn×n, playing a central role in many graph theoretic
+
treatmentsof multiagentsystems, is given by L(G),D(G)−A(G), where thespectrumof the
Laplacianofaconnected,undirectedgraphG canbeorderedas
0=λ (L(G))<λ (L(G))≤···≤λ (L(G)), (1)
1 2 n
with1 astheeigenvectorcorrespondingtothezeroeigenvalueλ (L(G))andL(G)1 =0 and
n 1 n n
eL(G)1 =1 .
n n
3. ProblemFormulation
Consider a multiagent system consisting of n agents that locally exchange information ac-
cording to a connected, undirected uncertain graph G with nodes and edges representing
u
agents and interagent information exchange links, respectively. We assume that the network
isstatic,andhence,agentevolutionwillnotcauseedgestoappearordisappearinthenetwork.
Specifically,letx (t)∈Rdenotethestateofnodei attimet ∈R ,whosedynamicsisdescribed
i +
by
x˙ (t) = −α (t)x (t)+ β (t)x (t)+u (t), x (0)=x , (2)
i i i ij j i i i0
iX∼j
3
(a) (b)
1.1 1.1
1 1
0.9 0.9
ts0.8 ts0.8
n n
e0.7 e0.7
g g
A0.6 A0.6
0.5 0.5
0.4 0.4
0.3 0.3
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Time (sec) Time (sec)
(c) (d)
2
1 1.8
0.8 1.6
s 0.6 s1.4
t t
en 0.4 en1.2
Ag 0.2 Ag 1
0.8
0
0.6
−0.2
0.4
−0.4
0 1 2 3 4 5 6 0 1 2 3 4 5 6
Time (sec) Time (sec)
Figure1:Trajectoriesofthreeagentsonalinegraphsubjecttoinitialconditions(x ,x ,x =
10 20 30
(0.2,0.4,1.2) for (a) (α ,α ,α ) = (1,2,1) and (β ,β ,β ,β ) = (1,1,1,1), (b)
1 2 3 12 21 23 32
(α ,α ,α ) = (1,1.1,1) and (β ,β ,β ,β ) = (1,0.1,1,1), (c) (α ,α ,α ) =
1 2 3 12 21 23 32 1 2 3
(1,2,1) and (β ,β ,β ,β ) = (−1,−1,1,1), and (d) (α ,α ,α ) = (1,1.5,1) and
12 21 23 32 1 2 3
(β ,β ,β ,β )=(1,1,1,1).
12 21 23 32
whereα (t)∈Randβ (t)∈RareunknownboundedcoefficientsofthegraphG withbounded
i ij u
timederivatives,andu (t)∈R,t ∈R ,isthecontrolinputofnodei. Inthispaper,weareinter-
i +
estedtodesignadistributedcontrolinputu (t),t ∈R ,suchthat(2)achievesaverageconsen-
i +
susapproximately(orasymptotically,i.e.,x(t)→(1 1T/n)x ast →∞,x(t)=[x (t),...,x (t)]T∈
n n 0 1 n
Rn)inthepresenceofanuncertaingraphtopology.
Remark1. In the absence of proper control inputs u (t)∈R, t ∈R , (2) cannot necessarily
i +
achieveaverageconsensus.Toelucidatethispoint,letunknowncoefficientsofthegraphG be
u
constant,i.e.,(α (t),β (t))=(α ,β ),andconsiderfourcasesgiveninFigure1thatshowtra-
i ij i ij
jectoriesofthreeagentsonalinegraphsubjecttoinitialconditions(x ,x ,x )=(0.2,0.4,1.2).
10 20 30
Since α = β and β = β in case (a), this case results in average consensus at point
i i∼j ij ij ji
P
(1 1T/n)x =0.6. Sinceβ 6=β incase(b),thiscasedoesnotresultinaverageconsensusat
n n 0 12 21
point0.6.Case(c)considersamultiagentsystemwithantagonisticinteractions[14],andhence,
4
itdoesnotresultinaverageconsensusduetotheexistenceofmultipleequilibriumpoints. Fi-
nally, since α 6= β for i = 2 in case (d), this case does not result in average consensus
i i∼j ij
P
as well. In summary, (2) results in average consensus if αi = i∼jβij and βij =βji ∈R+ [15].
P
However,thiscannotbejustifiedduetounknowncoefficientsofthegraphG ,andhence,one
u
needstodesignpropercontrolinputsu (t)∈R,t ∈R .
i +
Next, we propose a control input u (t)∈R, t ∈R , to drive the trajectories of (2) to a close
i +
neighborhood of a given reference model having a desired graph topology without requiring
a centralized informationexchange among networked agents. For this purpose, consider the
referencemodelthatlocallyexchangeinformationaccordingtoaconnected,undirectedgraph
G givenby
r˙ (t) = − r (t)−r (t) , r (0)=x , (3)
i i j i i0
iX∼j¡ ¢
wherer (t)∈R,t ∈R ,denotesthestateofthereferencemodelfornodei. Notethat
i +
lim r(t)=(1 1T/n)x , (4)
n n 0
t→∞
wherer(t)=[r (t),...,r (t)]T∈Rn.Throughoutthispaperweassumethatthenodesandedges
1 n
ofgraphsG andG coincide,howeverthegraphG issubjecttounknowncoefficientsα (t)and
u u i
β (t),asdiscussedearlier.
ij
Remark2. Thereferencemodelgivenby(3)canbeeasilyextendedto
r˙ (t) = − ξ r (t)−r (t) , r (0)=x , (5)
i ij i j i i0
iX∼j ¡ ¢
withoutchangingthefollowingresultsofthispaper,whereξii = i∼jξij andξij =ξji ∈R+.
P
Remark3. Notethat(3)canbeequivalentlywrittenas
r˙ (t) = −d r (t)+ r (t), r (0)=x , (6)
i i i j i i0
iX∼j
whered isthedegreeofnodei ongraphG. Therefore,ifoneknowsthecoefficientsα (t)and
i i
β (t)of2,thenthecontrolinput
ij
u (t) = − d −α (t) x (t)+ 1−β (t) x (t), (7)
i i i i ij j
¡ ¢ iX∼j¡ ¢
5
resultsinaverageconsensusatpoint(1 1T/n)x .
n n 0
Sincethecontrolinput(7)giveninRemark3isnotfeasibleduetounknowncoefficientsα (t)
i
andβ (t),weproposetheadaptivecontrolinputgivenby
ij
u (t) = −k x (t)−r (t) −wˆ (t)x (t)− wˆ (t)x (t), (8)
i i i i i i ij j
¡ ¢ iX∼j
where k ∈R for at leastone agentor a subsetof agents(and k =0 for others), and theesti-
i + i
mateswˆ (t)∈Randwˆ (t)∈R,t ∈R ,satisfytheupdatelaws
i ij +
w˙ˆ (t) = γ Proj wˆ (t), x (t) x (t)−r (t) , wˆ (0)=wˆ , (9)
i i i i i i i i0
³ ´
¡ ¢
w˙ˆ (t) = γ Proj wˆ (t), x (t) x (t)−r (t) , wˆ (0)=wˆ , (10)
ij ij ij j i i ij ij0
³ ´
¡ ¢
withγi ∈R+andγij ∈R+beingthecorrespondinglearningrates.Intheupdatelawsgivenby(9)
and(10),Projdenotestheprojectionoperator[16,17],whichisusedtokeeptheestimateswˆ (t)
i
and wˆ (t)boundedfor all t ∈R . In thenextsection,we analyticallyshowthattheproposed
ij +
adaptivecontrolinputgivenby(8)alongwiththeupdatelaws(9)and(10)drivesthetrajectories
of(2)toacloseneighborhoodofthereferencemodeltrajectoriesgivenby(5).
4. StabilityAnalysis
Inthissection,weestablishstabilitypropertiesoftheproposedadaptivecontrolinputgiven
by(8)alongwiththeupdatelaws(9)and(10). Forthispurpose,let
e (t),x (t)−r (t), t ∈R , (11)
i i i +
denotethelocalerrordynamicsthatsatisfy
e˙ (t) = −α (t)x (t)+ β (t)x (t)+u (t)+d r (t)− r (t)
i i i ij j i i i j
iX∼j iX∼j
= −α (t)x (t)+ β (t)x (t)+u (t)+d r (t)− r (t)+d x (t)
i i ij j i i i j i i
iX∼j iX∼j
−d x (t)+ x (t)− x (t)
i i j j
iX∼j iX∼j
= −d e (t)+ e (t)+ d −α (t) x (t)+ β (t)−1 x (t)+u (t)
i i j i i i ij j i
iX∼j ¡ ¢ iX∼j¡ ¢
6
= − e (t)−e (t) +w (t)x (t)+ w (t)x (t)+u (t)
i j i i ij j i
iX∼j¡ ¢ iX∼j
= − e (t)−e (t) +w (t)x (t)+ w (t)x (t)−k e (t)
i j i i ij j i i
iX∼j¡ ¢ iX∼j
−wˆ (t)x (t)− wˆ (t)x (t)
i i ij j
iX∼j
= −k e (t)− e (t)−e (t) −w˜ (t)x (t)− w˜ (t)x (t), e (0)=0, (12)
i i i j i i ij j i
iX∼j¡ ¢ iX∼j
where
w˜ (t) , wˆ (t)−w (t), t ∈R , (13)
i i i +
w˜ (t) , wˆ (t)−w (t), t ∈R , (14)
ij ij ij +
wi(t),di−αi(t),t ∈R+,andwij(t),βij(t)−1,t ∈R+. Inaddition,itfollowsfrom(9)and(10)
that
w˙˜ (t) = γ Proj wˆ (t), x (t) x (t)−r (t) −w˙ (t), w˜ (0)=w˜ , (15)
i i i i i i i i i0
³ ´
¡ ¢
w˙˜ (t) = γ Proj wˆ (t), x (t) x (t)−r (t) −w˙ (t), w˜ (0)=w˜ , (16)
ij ij ij j i i ij ij ij0
³ ´
¡ ¢
where w˜ ,wˆ −w and w˜ ,wˆ −w . Note that w˙ (t)and w˙ (t) areboundedsinceit
i0 i0 i ij0 ij0 ij i ij
isassumedthatunknownboundedcoefficientsα (t)andβ (t)haveboundedtimederivates.
i ij
Wenowstatethefollowinglemmanecessaryfortheresultsofthissection.
Lemma1. Let K =diag(k), k =[k ,k ,...,k ]T, k ∈R , i =1,...,n, andassumethatatleast
1 2 n i +
oneelementofk isnonzero. Then,fortheLaplacianofaconnected,undirectedgraph,
F(G),L(G)+K ∈Sn×n, (17)
+
anddet(F(G))6=0.
Proof. Consider the decomposition K =K +K , where K ,diag([0,...,0,φ ,0,...,0]T) and
1 2 1 i
K ,K −K , where φ denotes the smallest nonzero diagonal element of K appearing on its
2 1 i
i-thdiagonal,sothatK ∈Sn×n. FromtheRayleigh’sQuotient[18],theminimumeigenvalueof
2 +
L(G)+K canbegivenby
1
λ (L(G)+K )=min{xT L(G)+K x|xTx=1}, (18)
min 1 1
x
¡ ¢
7
where x istheeigenvectorcorrespondingtothisminimumeigenvalue. NotethatsinceL(G)∈
Sn×n andK ∈Sn×n, andhence, L(G)+K isrealandsymmetric, x isarealeigenvector. Now,
+ 1 + 1
expanding(18)as
xT(L(G)+K )x = (x −x )2+φ x2, (19)
1 i j i i
iX∼j
andnotingthattherighthandsideof(19)iszeroonlyifx≡0,itfollowsthatλ (L(G)+K )>0,
min 1
and hence, L(G)+K ∈Sn×n. Finally, let λ be an eigenvalue of F(G)= L(G)+K +K . Since
1 + 1 2
λ (L(G)+K )>0 and λ (K )=0, it follows from Fact 5.11.3of [19] thatλ (L(G)+K )+
min 1 min 2 min 1
λ (K )≤λ,andhence,λ>0,whichimpliesthat(17)holdsanddet(F(G))6=0. (cid:4)
min 2
Thenexttheorempresentsthefirstresultofthissection.
Theorem1. Considerthenetworkedmultiagentsystemgivenby(2)subjecttoanuncertain
graph topology, thereference modelgiven by (3), theadaptivecontrol inputgiven by(8), and
theupdatelawsgivenby(9)and(10). Then,thesolution e (t),w˜ (t),w˜ (t) oftheclosed-loop
i i ij
¡ ¢
dynamicalsystemgivenby(12),(15),and(16)isboundedforall 0,w˜ ,w˜ ,t ∈R ,and(i,j).
i0 ij0 +
¡ ¢
Proof. Firstconsiderthequadraticfunctiongivenby
1
V (e ,w˜ ,w˜ ) = e2+γ−1w˜2+ γ−1w˜2 , (20)
i i i ij 2³ i i i iX∼j ij ij´
and notethatV (0,0,0)=0 andV (e ,w˜ ,w˜ )∈R , (e ,w˜ ,w˜ )6=(0,0,0). Furthermore,V (e ,
i i i i ij + i i ij i i
w˜ ,w˜ ) is radially unbounded. Differentiating (20) along the closed-loop trajectories of (12),
i ij
(15),and(16)yields
V˙ e (t),w˜ (t),w˜ (t) ≤ −e (t) e (t)−e (t) −k e2(t)+w∗, (21)
i i i ij i i j i i i
¡ ¢ iX∼j¡ ¢
where w∗ is an upper bound satisfying γ−1 wˆ (t)−w (t) w˙ (t)+ γ−1 wˆ (t)−w (t)
i i i i i i∼j ij ij ij
¯¯ ¡ ¢ P ¡ ¢
·w˙ (t) ≤ w∗, t ∈ R . Note that w∗ exi¯s¯ts since all the terms inside the norm operator are
ij 2 i + i
¯¯
¯¯
boundedandprojectionoperatorisusedfortheestimateswˆ (t)andwˆ (t). Now,considerthe
i ij
Lyapunovfunctioncandidategivenby
n
V(·) = V (e ,w˜ ,w˜ ). (22)
i i i ij
iX=1
8
Thetimederivativeof(22)isgivenusing(21)as
n
V˙(·) ≤ −eT(t) L(G)+K e(t)+w∗, w∗, w∗, (23)
i
¡ ¢ iX=1
where L(G) denotes theLaplacian matrixof (3), K ,diag(k), k =[k ,k ,...,k ]T, k ∈R , and
1 2 n i +
e(t) = [e (t),...,e (t)]T. From the definition of the adaptive control input in (8), notice that
1 n
at least one element of k is nonzero. This implies from Lemma 1 that L(G)+K ∈ Sn×n and
+
det(L(G)+K)6=0,andhence,λ L(G)+K ke(t)k ≤eT(t) L(G)+K e(t). Now, sinceV˙(·)≤0
min 2
¡ ¢ ¡ ¢
when ke(t)k ≥w∗/λ L(G)+K , itfollows thattheclosed-loop dynamicalsystem givenby
2 min
¡ ¢
(12),(15),and(16)isboundedforall 0,w˜ ,w˜ ,t ∈R ,and(i,j). (cid:4)
i0 ij0 +
¡ ¢
Remark4. In order to drivethe trajectories of (2) to a close neighborhoodof the reference
∗
modeltrajectoriesgivenby(5),theperturbationtermw in(25)needstobesmall. Thisholds
ifthetimederivativeofunknowncoefficients α (t)andβ (t)issmall. Ifthisisnottrue,then
i ij
∗
onecanincreasethelearningratesγ andγ tomakew small.
i ij
As a special case when theunknown coefficients are constant, i.e., (α (t),β (t))=(α ,β ),
i ij i ij
the next theorem shows that the proposed adaptive control input given by (8) along with the
updatelaws(9)and(10)asymptoticallydrivesthetrajectoriesof(2)tothereferencemodeltra-
jectoriesgivenby(5).
Theorem2. Considerthenetworkedmultiagentsystemgivenby(2)subjecttoanuncertain
graph topology, thereference modelgiven by (3), theadaptivecontrol inputgiven by(8), and
theupdatelawsgivenby(9)and(10). Then,thesolution e (t),w˜ (t),w˜ (t) oftheclosed-loop
i i ij
¡ ¢
dynamical system given by (12), (15), and (16) is Lyapunov stable for all 0,w˜ ,w˜ , t ∈R ,
i0 ij0 +
¡ ¢
and(i,j),andlim e (t)=0foralli. Inaddition,lim x(t)=(1 1T/n)x .
t→∞ i t→∞ n n 0
Proof. To show the Lyapunov stability of the closed-loop dynamical system given by (12),
(15),and(16),firstconsiderthequadraticfunctiongivenby(20). Differentiating(20)alongthe
closed-looptrajectoriesof(12),(15),and(16)yields
V˙ e (t),w˜ (t),w˜ (t) ≤ −e (t) e (t)−e (t) −k e2(t). (24)
i i i ij i i j i i
¡ ¢ iX∼j¡ ¢
Now,considertheLyapunovfunctioncandidategivenby(22),wherethetimederivativeof(22)
9