Resilient Consensus of Second-Order Agent Networks: Asynchronous Update Rules with Delays

0.59 MB·

by  Seyed Mehran Dibaji

ResilientConsensusofSecond-OrderAgentNetworks:
AsynchronousUpdateRuleswithDelays
SeyedMehranDibajia and HideakiIshiib
aDepartmentofMechanicalEngineering,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA
bDepartmentofComputerScience,TokyoInstituteofTechnology,Yokohama226-8502,Japan
7
1
0
2
Abstract
n
a
Westudytheproblemofresilientconsensusofsampled-datamulti-agentnetworkswithdouble-integratordynamics.Theterm
J
resilientpointstoalgorithmsconsideringthepresenceofattacksbyfaulty/maliciousagentsinthenetwork.Eachnormalagent
4
updatesitsstatebasedonapredeterminedcontrollawusingitsneighbors’informationwhichmaybedelayedwhilemisbehaving
1
agentsmakeupdatesarbitrarilyandmightthreatentheconsensuswithinthenetwork.Assumingthatthemaximumnumber

ofmaliciousagentsinthesystemisknown,wefocusonalgorithmswhereeachnormalagentignoreslargeandsmallposition
]
Y valuesamongitsneighborstoavoidbeinginfluencedbymaliciousagents.Themaliciousagentsareassumedtobeomniscient
inthattheyknowtheupdatingtimesanddelaysandcancolludewitheachother.Wedealwithbothsynchronousandpartially
S asynchronouscaseswithdelayedinformationandderivetopologicalconditionsintermsofgraphrobustness.
.
s
c Keywords: Multi-agentSystems;Cyber-security;ConsensusProblems
[

2
v
1 Introduction the normal agents from reaching consensus. This type
0
3 ofproblemshasarichhistoryindistributedalgorithms
4 Inrecentyears,muchattentionhasbeendevotedtothe intheareaofcomputerscience(see,e.g.,(Lynch1996))
3 wheretheagents’valuesareoftendiscreteandfinite.It
studyofnetworkedcontrolsystemswithanemphasison
0 isinterestingthatrandomizationsometimesplayacru-
cyber security. Due to communications through shared
. cialrole;seealso(MotwaniandRaghavan1995,Tempo
1 networks, there are many vulnerabilities for potential
0 attacks, which can result in irreparable damages. Con- andIshii2007,Dibajietal.2016).
7 ventional control approaches are often not applicable
1 for resiliency against such unpredictable but probable Insuchproblems,thenon-faultyagentscooperatebyin-
v: misbehaviorsinnetworks(e.g.,(Sandbergetal.2015)). teracting locally with each other to achieve agreement.
i Oneofthemostessentialproblemsinnetworkedmulti- There are different techniques to mitigate the effects of
X agentsystemsisconsensuswhereagentsinteractlocally attacks.Insomesolutions,eachagenthasabankofob-
r to achieve the global goal of reaching a common value. servers to identify the faulty agents within the network
a
HavingawidevarietyofapplicationsinUAVformations, using their past information. Such solutions are formu-
sensor networks, power systems, and so on, consensus latedasakindoffaultdetectionandisolationproblems
problems have been studied extensively (Mesbahi and (Pasqualetti et al. 2012, Shames et al. 2011, Sundaram
Egerstedt 2010, Ren and Cao 2011). Resilient consen- and Hadjicostis 2011). However, identifying the mali-
sus pointstothecasewheresomeagentsinthenetwork cious agents can be challenging and requires much in-
anonymously try to mislead the others or are subject formation processing at the agents. In particular, these
to failures. Such malicious agents do not comply with techniques usually necessitate each agent to know the
the predefined interaction rule and might even prevent topology of the entire network. This global information
typically is not desirable in distributed algorithms. To
(cid:63) overrule the effects of f malicious agents, the network
ThisworkwassupportedinpartbytheJapanScienceand
hastobeatleast(2f +1)-connected.
Technology Agency under the EMS-CREST program and
by JSPS under Grant-in-Aid for Scientific Research Grant
No.15H04020. Thereisanotherclassofalgorithmsforresilientconsen-
Emailaddresses: [email protected](SeyedMehran sus where each normal agent disregards the most de-
Dibaji),[email protected](HideakiIshii). viated agents in the updates. In this case, they simply
neglect the information received from suspicious agents datingrules:Synchronousandpartiallyasynchronous1.
or those with unsafe values whether or not they are Inthesynchronouscase,allagentssimultaneouslymake
trulymisbehaving.Thisclassofalgorithmshasbeenex- updatesateachtimestepusingthecurrentinformation
tensively used in computer science (Azadmanesh and oftheirneighbors.Bycontrast,intheasynchronouscase,
Kieckhafer2002,AzevedoandBlough1998,Bouzidetal. normal agents may decide to update only occasionally
2010, Lynch 1996, Plunkett and Fekete 1998, Vaidya et andmoreover,theneighbors’datamaybedelayed.This
al.2012)aswellascontrol(DibajiandIshii2015a,Dibaji is clearly a more vulnerable situation, allowing the ad-
andIshii2015d,LeBlancandKoutsoukos2012,LeBlanc versaries to take advantage by quickly moving around.
et al. 2013); see also (Feng et al. 2016, Khanafer et We consider the worst-case scenarios where the mali-
al. 2012) for related problems. They are often called cious agents are aware of the updating times and even
Mean Subsequence Reduced (MSR) algorithms, which thedelaysintheinformationofnormalagents.Thenor-
was coined in (Kieckhafer and Azadmanesh 1993). Un- malagentsontheotherhandareunawareoftheupdat-
tilrecently,thisstrategyhadbeenstudiedmostlyinthe ingtimesoftheirneighborsandhencecannotpredictthe
case where the agent networks form complete graphs. plans of adversaries. For both cases, we develop graph
The authors of (LeBlanc et al. 2013) have given a thor- robustnessconditionsfortheoverallnetworktopologies.
ough study for the non-complete case and have shown Itwillbeshownthatthesynchronousupdatingrulesre-
thatthetraditionalconnectivitymeasureisnotadequate quire less connectivity than the asynchronous counter-
forMSR-typealgorithmstoachieveresilientconsensus. part; see also (Dibaji and Ishii 2015d) regarding corre-
Theythenintroducedanewnotioncalledgraphrobust- spondingresultsforfirst-orderagentsystems.
ness. We note that most of these works have dealt with
single-integratorandsynchronousagentnetworks. Themainfeaturesofthisworkarethree-fold:(i)Wedeal
with second-order agents, which are more suitable for
In this paper, we consider agents having second-order modelingnetworksofvehicles,butexhibitmorecompli-
dynamics,whichisacommonmodelforautonomousmo- cated dynamics in comparison to the single-order case.
bile robots and vehicles. Such applications in fact pro- (ii) For the malicious agents, we consider the f-total
vide motivations different from those in computer sci- model,whichislessstringentthanthef-localcase,but
ence as we will see. In our previous paper (Dibaji and the analysis is more involved. (iii) In the asynchronous
Ishii 2015a), an MSR-type algorithm has been applied case with delayed information, we introduce a new up-
to sampled-data second-order agent networks. We have datescheme,whichismorenaturalinviewofthecurrent
consideredtheproblemofresilientconsensuswheneach research in the area of multi-agent systems than those
agentisaffectedbyatmostf maliciousagentsamongits based on the so-called rounds, commonly employed in
neighbors. Such a model is called f-local malicious. We computerscienceaswediscusslater.
haveestablishedasufficientconditionontheunderlying
graphstructuretoreachconsensus.Itisstatedinterms Thepaperisorganizedasfollows.Section2presentspre-
of graph robustness and is consistent with the result in liminariesforintroducingtheproblemsetting.Section3
(LeBlancetal.2013)forthefirst-orderagentcase. focusesonresilientconsensusbasedonsynchronousup-
date rules. Section 4 is devoted to the problem of par-
Here, the focus of our study is on the so-called f-total tial asynchrony with delayed information. We illustrate
model, where the total number of faulty agents is at the results through a numerical example in Section 5.
mostf,whichhasbeendealtwithin,e.g.,(Azadmanesh Finally, Section 6 concludes the paper. The material of
and Kieckhafer 2002, Bouzid et al. 2010, Kieckhafer thispaperappearsin(DibajiandIshii2015b,Dibajiand
and Azadmanesh 1993, LeBlanc and Koutsoukos 2012, Ishii 2015c) in preliminary forms; here, we present im-
LeBlancetal.2013,Lynch1996,Vaidyaetal.2012).We provedresultswithfullproofsandmorediscussions.
derive a necessary and sufficient condition to achieve
resilient consensus by an MSR-like algorithm. Again,
weshowthatgraphrobustnessinthenetworkistherel- 2 ProblemSetup
evantnotion.However,thef-totalmodelassumesfewer
malicious agents in the system, and hence, the condi- 2.1 GraphTheoryNotions
tionwillbeshowntobelessrestrictivethanthatforthe
f-local case. The works (Bouzid et al. 2010, Kieckhafer
We recall some concepts on graphs (Mesbahi and
andAzadmanesh1993,Lynch1996,Vaidyaetal.2012)
Egerstedt 2010). A directed graph (or digraph) with n
have studied this model for the first-order agents case,
butbasedontheByzantinemaliciousagents,whichare
1 The term partially asynchronous refers to the case where
allowedtosenddifferentvaluestotheirneighbors.Such
agents share some level of synchrony by having the same
attacks may be impossible, e.g., if the measurements
sampling times; however, they make updates at different
aremadebyon-boardsensorsinmobilerobots.
timesbasedondelayedinformation(BertsekasandTsitsiklis
1989). This is in contrast to the fully asynchronous case
Underthef-totalmodel,wesolvetheresilientconsensus whereagentsmustbefacilitatedwiththeirownclocks;such
problemusingMSR-typealgorithmsfortwodifferentup- settingsarestudiedin,e.g.,(Qinetal.2012).
2
nodes (n > 1) is defined as G = (V,E) with the node 1
set V = {1,...,n} and the edge set E ⊆ V ×V. The
edge (j,i) ∈ E means that node i has access to the in- 5 2
formation of node j. If E = {(i,j) : i,j ∈ V, i (cid:54)= j},
the graph is said to be complete. For node i, the set of
its neighbors, denoted by N = {j : (j,i) ∈ E}, con- 4 3
i
sists of all nodes having directed edges toward i. The
Fig.1.Agraphwhichis(2,2)-robustbutnot3-robust.
degree of node i is the number of its neighbors and is
denotedbyd =|N |.TheadjacencymatrixA=[a ]is
i i ij robust,butnot3-robust;further,removinganyedgede-
given by a ∈ [γ,1) if (j,i) ∈ E and otherwise a = 0,
ij ij stroysits(2,2)-robustness.Ingeneral,todetermineifa
where γ > 0 is a fixed lower bound. We assume that
given graph has a robustness property is computation-
(cid:80)n a ≤1. Let L=[l ] be the Laplacian matrix
j=1,j(cid:54)=i ij ij ally difficult since the problem involves combinatorial
of G, whose entries are defined as l = (cid:80)n a aspects.Itisknownthatrandomgraphsbecomerobust
ii j=1,j(cid:54)=i ij
and l = −a , i (cid:54)= j; we can see that the sum of the whentheirsizetendstoinfinity(Zhangetal.2015).
ij ij
elementsofeachrowofLiszero.
2.2 Second-OrderConsensusProtocol
A path from node v to v is a sequence (v ,v ,...,v )
1 p 1 2 p
in which (vi,vi+1) ∈ E for i = 1,...,p−1. If there is Consideranetworkofagentswhoseinteractionsarerep-
a path between each pair of nodes, the graph is said to resented by the directed graph G. Each agent i∈V has
be strongly connected. A directed graph is said to have adouble-integratordynamicsgivenby
a directed spanning tree if there is a node from which
thereisapathtoeveryothernodeinthegraph. x˙ (t)=v (t), v˙ (t)=u (t), i=1,...,n,
i i i i
For the MSR-type resilient consensus algorithms, the where x (t) ∈ R and v (t) ∈ R are its position and ve-
i i
critical topological notion is graph robustness, which is locity, respectively, and u (t) is the control input. We
i
a connectivity measure of graphs. Robust graphs were discretizethesystemwithsamplingperiodT as
introduced in (LeBlanc et al. 2013) for the analysis of
resilientconsensusoffirst-ordermulti-agentsystems. T2
x [k+1]=x [k]+Tv [k]+ u [k],
i i i 2 i (1)
Definition2.1 ThedigraphGis(r,s)-robust(r,s<n) v [k+1]=v [k]+Tu [k], i=1,...,n,
i i i
ifforeverypairofnonemptydisjointsubsetsS ,S ⊂V,
1 2
atleastoneofthefollowingconditionsissatisfied: where x [k], v [k], and u [k] are, respectively, the posi-
i i i
tion, the velocity, and the control input of agent i at
1.XSr1 =S1, 2.XSr2 =S2, 3.|XSr1|+|XSr2|≥s, t=kT fork ∈Z+.Ourdiscretizationisbasedoncontrol
inputs generated by zeroth order holds; other methods
where XSr(cid:96) is the set of all nodes in S(cid:96) which have at areemployedin,e.g.,(LinandJia2009,Qinetal.2012).
leastr incomingedgesfromoutsideofS .Inparticular,
(cid:96)
graphswhichare(r,1)-robustarecalledr-robust. At each time step k, the agents update their positions
andvelocitiesbasedonthetime-varyingtopologyofthe
Thefollowinglemmahelpstohaveabetterunderstand- graphG[k],whichisasubgraphofGandisspecifiedlater.
ingof(r,s)-robustgraphs(LeBlanc2012). Inparticular,thecontrolusestherelativepositionswith
itsneighborsanditsownvelocity(RenandCao2011):
Lemma2.2 Foran(r,s)-robustgraphG,thefollowing
(cid:88)
hold: u [k]=− a [k][(x [k]−δ )−(x [k]−δ )]−αv [k],
i ij i i j j i
(i) Gis(r(cid:48),s(cid:48))-robust,where0≤r(cid:48) ≤rand1≤s(cid:48) ≤s, j∈Ni (2)
andinparticular,itisr-robust. where a [k] is the (i,j) entry of the adjacency matrix
ij
(ii) G is(r−1,s+1)-robust. A[k]∈Rn×ncorrespondingtoG[k],αisapositivescalar,
(iii) G isatleastr-connected,butanr-connectedgraph andδ ∈Risaconstantrepresentingthedesiredrelative
i
isnotnecessarilyr-robust. positionofagentiinaformation.
(iv) G hasadirectedspanningtree.
(v) r ≤(cid:100)n/2(cid:101).Also,ifG isacompletegraph,thenitis Theagents’objectiveisconsensusinthesensethatthey
(r(cid:48),s)-robustforall0<r(cid:48) ≤(cid:100)n/2(cid:101)and1≤s≤n. cometoformationandthenstopasymptotically:
Moreover,agraphis(r,s)-robustifitis(r+s−1)-robust. x [k]−x [k]→δ −δ , v [k]→0ask →∞, ∀i,j ∈V.
i j i j i
It is clear that (r,s)-robustness is more restrictive than In(RenandCao2011),itisshownthatifthereissome
r-robustness.ThegraphwithfivenodesinFig.1is(2,2)- (cid:96) ∈ Z such that for any nonnegative integer k , the
0 + 0
3
unionofG[k]acrossk ∈[k ,k +(cid:96) ]hasadirectedspan- We introduce the notion of resilient consensus for the
0 0 0
ningtree,thenconsensuscanbeobtainedunderthecon- networkofsecond-orderagents(DibajiandIshii2015a).
trollaw (2)byproperlychoosingα andT.
Definition2.5 If for any possible set of malicious
Inthispaper,westudythecasewheresomeagentsmal- agents,anyinitialpositionsandvelocities,andanyma-
functionduetofailure,disturbances,orattacks.Insuch licious inputs, the following conditions are met, then
circumstances, they may not follow the predefined up- thenetworkissaidtoreachresilientconsensus:
date rule (2). In the next subsection, we introduce nec-
essary definitions and then formulate the resilient con- 1. Safety: There exists a bounded interval S deter-
sensusprobleminthepresenceofmaliciousagents. mined by the initial positions and velocities of the
normal agents such that xˆ [k] ∈ S, i ∈ V\M,k ∈
i
Finally,werepresenttheagentsysteminavectorform. Z+.ThesetS iscalledthesafetyinterval.
Let xˆ [k] = x [k] − δ , xˆ[k] = [xˆ [k] ··· xˆ [k]]T, and 2. Agreement: For some c ∈ S, it holds that
i i i 1 n lim xˆ [k]=candlim v [k]=0,i∈V\M.
v[k]=[v [k] ··· v [k]]T.Forthesakeofsimplicity,here- k→∞ i k→∞ i
1 n
after,theagents’positionsrefertoxˆ[k]andnotx[k].The
Afewremarksareinorderregardingthesafetyinterval
system(1)thenbecomes
S. (i) The malicious agents may or may not be in S,
while the normal agents must stay inside though they
T2 may still be influenced by the malicious agents staying
xˆ[k+1]=xˆ[k]+Tv[k]+ u[k],
2 (3) inS.(ii)Weimposethesafetyconditiontoensurethat
v[k+1]=v[k]+Tu[k], thebehaviorofthenormalagentsremainsclosetothat
whennomaliciousagentispresent.(iii)Wedonothave
andthecontrollaw (2)canbewrittenas asafetyintervalforvelocityofnormalagentsandhence
theymayevenmovefasterthantheirinitialspeeds.
u[k]=−L[k]xˆ[k]−αv[k], (4)
3 SynchronousNetworks
whereL[k]istheLaplacianmatrixforthegraphG[k].
3.1 DP-MSRAlgorithm
2.3 ResilientConsensus
Wefirstoutlinethealgorithmforachievingconsensusin
the presence of misbehaving agents in the synchronous
We introduce notions related to malicious agents and
case,whereallagentsmakeupdatesateverytimestep.
consensus in the presence of such agents (LeBlanc et
The algorithm is called DP-MSR, which stands for
al.2013,Lynch1996,Vaidyaetal.2012).
Double-Integrator Position-Based Mean Subsequence
Reduced algorithm. It was proposed in (Dibaji and
Definition2.3 Agent i is called normal if it updates
Ishii2015a)forthef-localmaliciousmodel.
itsstatebasedonthepredefinedcontrol(2).Otherwise,
it is called malicious and may make arbitrary updates.
Thealgorithmhasthreestepsasfollows:
TheindexsetofmaliciousagentsisdenotedbyM⊂V.
Thenumbersofnormalagentsandmaliciousagentsare
1. Ateachtimestepk,eachnormalagentireceivesthe
denotedbyn andn ,respectively.
N M relativepositionvaluesxˆ [k]−xˆ [k]ofitsneighbors
j i
j ∈N [k]andsortstheminadecreasingorder.
i
Weassumethatanupperboundisavailableforthenum-
2. If there are less than f agents whose relative posi-
ber of misbehaving agents in the entire network or at
tion values are greater than or equal to zero, then
leastineachnormalagent’sneighborhood.
thenormalagentiignorestheincomingedgesfrom
those agents. Otherwise, it ignores the incoming
Definition2.4 The network is f-total malicious if the edgesfromf agentscountingfromthosehavingthe
number nM of faulty agents is at most f, i.e., nM ≤ f. largest relative position values. Similarly, if there
On the other hand, the network is f-local malicious if arelessthanf agentswhosevaluesaresmallerthan
the number of malicious agents in the neighborhood of orequaltozero,thenagentiignorestheincoming
eachnormalagentiisboundedbyf,i.e.,|Ni∩M|≤f. edges from those agents. Otherwise, it ignores the
f incoming edges counting from those having the
According to the model of malicious agents considered, smallestrelativepositionvalues.
the difference between normal agents and malicious 3. Applythecontrolinput(2)bysubstitutinga [k]=
ij
agents lies in their control inputs u : For the normal 0foredges(j,i)whichareignoredinstep2.
i
agents, it is given by (2) while for the malicious agents,
it is arbitrary. On the other hand, the position and The main feature of this algorithm lies in its simplicity.
velocitydynamicsforallagentsremainthesameas(1). Eachnormalagentignorestheinformationreceivedfrom
4
itsneighborswhichmaybemisleading.Inparticular,it Forthesampling period T andthe parameter α,weas-
ignoresuptof edgesfromneighborswhosepositionsare sume2
large, and f edges from neighbors whose positions are T2 T2
1+ ≤αT ≤2− . (9)
small.TheunderlyinggraphG[k]attimekisdetermined 2 2
bytheremainingedges.TheadjacencymatrixA[k]and Thefollowinglemmafrom(DibajiandIshii2015a)plays
theLaplacianmatrixL[k]aredeterminedaccordingly. akeyroleintheanalysis.
The problem for the synchronous agent network can be Lemma3.1 Underthecontrolinputs(6),theposition
stated as follows: Under the f-total malicious model, vectorxˆ[k]oftheagentsfork ≥1canbeexpressedas
find a condition on the network topology such that the
normalagentsreachresilientconsensusbasedontheDP- (cid:34) (cid:35)
MSRalgorithm. (cid:104) (cid:105) xˆ[k]
xˆ[k+1]= Φ Φ
1k 2k
xˆ[k−1]
3.2 MatrixRepresentation
(cid:34) (cid:35)
+ T2 0 (cid:0)uM[k]+uM[k−1](cid:1),
We provide a modified system model when malicious 2 I
agentsarepresent.Tosimplifythenotation,theagents’ nM
indicesarereordered.Letthenormalagentstakeindices
where
1,...,n andletthemaliciousagentsben +1,...,n.
N N
Thus, the vectors representing the positions, velocities, (cid:34) (cid:35)
T2 LN[k]
andcontrolinputsofallagentsconsistoftwopartsas
Φ =R+I − ,
1k n 2 0
(cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35)
xˆN[k] vN[k] uN[k] (cid:34) (cid:35)
xˆ[k]= , v[k]= , u[k]= , T2 LN[k−1]
xˆM[k] vM[k] uM[k] Φ2k =−R− 2 0 .
(5)
where the superscript N stands for normal and M
(cid:2) (cid:3)
for malicious. Regarding the control inputs uN[k] and Moreover,under(9),thematrix Φ1k Φ2k isnonnega-
uM[k],thenormalagentsfollow (2)whilethemalicious tive,andthesumofeachofitsfirstnN rowsisone.
agentsmaynot.Hence,theycanbeexpressedas
Remark3.2 It is clear from the lemma that the con-
(cid:104) (cid:105) trolsuM[k]anduM[k−1]donotdirectlyenterthenew
uN[k]=−LN[k]xˆ[k]−α I 0 v[k],
nN (6) positionsxˆN[k+1]ofthenormalagents.Moreover,the
uM[k]:arbitrary, positions of the normal agents xˆN[k+1] for k ≥ 1 are
obtainedviatheconvexcombinationofthecurrentposi-
where LN[k]∈RnN×n is the matrix formed by the first tionsxˆ[k]andthosefromtheprevioustimestepxˆ[k−1].
n rowsofL[k]associatedwithnormalagents.Therow
N
sumsofthismatrixLN[k]arezeroasinL[k]. 3.3 ANecessaryandSufficientCondition
With the control inputs of (6), we obtain the model for We are now ready to state the main result for the syn-
theoverallsystem(3)as chronouscase.LettheintervalS begivenby
(cid:32) T2 (cid:34)LN[k](cid:35)(cid:33) (cid:20) (cid:26) (cid:18) αT2(cid:19) (cid:27)
xˆ[k+1]= I − xˆ[k]+Qv[k] S = minxˆN[0]+min 0, T − vN[0] ,
n 2 0 2
(cid:26) (cid:18) αT2(cid:19) (cid:27)(cid:21)
T2 (cid:34) 0 (cid:35) maxxˆN[0]+max 0, T − vN[0] , (10)
+ uM[k], (7) 2
2 I
nM
(cid:34) (cid:35) (cid:34) (cid:35) 2 The condition (9) on T and α ensures that the matrix
v[k+1]=−T LN[k] xˆ[k]+Rv[k]+T 0 uM[k], (cid:2)Φ1k Φ2k(cid:3)possessesthepropertiesstatedinLemma3.1.We
0 InM lmesasytrhealanx1i.t,Wfohrileexa(9m)pclae,nbbyenfoutlfiimllepdosbiyngan(cid:80)ynjT=1,athije[kc]otnotrboel
law (2) may make the agents exhibit undesired oscillatory
where the partitioning in the matrices is in accordance
movements. This type of property is also seen in previous
withthevectorsin(5),andQandR aregivenby
works on consensus of agents with second-order dynamics;
see, e.g., (Ren 2008, Ren and Cao 2011). We remark that
(cid:34) (cid:35) (cid:34) (cid:35)
αT2 I 0 I 0 the condition (9) is less restrictive than that in (Qin and
Q=TI − nN , R=I −αT nN . (8) Gao2012).Ingeneral,shortcomingsduetothisassumption
n 2 0 0 n 0 0 shouldbefurtherstudiedinfutureresearch.
5
where the minimum and the maximum are taken over In what follows, we show that x[k] is a nonincreasing
allentriesofthevectors.Notethattheintervalisdeter- function of k ≥ 1. For k ≥ 2, Lemma 3.1 and Re-
minedonlybytheinitialstatesofthenormalagents. mark 3.2 indicate that the positions of normal agents
are convex combinations of those of its neighbors from
Theorem3.3 Under the f-total malicious model, the time k −1 and k −2. If any neighbors of the normal
networkofagentswithsecond-orderdynamicsusingthe agents at those time steps are malicious and are out-
control in (6) and the DP-MSR algorithm reaches re- side the range of the normal agents’ position values,
silient consensus if and only if the underlying graph is they are ignored by step 2 in DP-MSR. Hence, we have
(f+1,f+1)-robust.Thesafetyintervalisgivenby(10). maxxˆN[k] ≤ max(cid:0)xˆN[k−1],xˆN[k−2](cid:1). It also easily
followsthatmaxxˆN[k−1]≤max(cid:0)xˆN[k−1],xˆN[k−2](cid:1).
Proof.(Necessity)Weprovebycontradiction.Suppose Thus,wearriveat
thatthenetworkisnot(f+1,f+1)-robust.Then,there
arenonemptydisjointsetsV1,V2 ⊂V suchthatnoneof x[k]=max(cid:0)xˆN[k],xˆN[k−1](cid:1)
the conditions 1–3 in Definition 2.1 holds. Suppose all ≤max(cid:0)xˆN[k−1],xˆN[k−2](cid:1)=x[k−1].
agentsinV haveinitialpositionsataandallagentsinV
1 2
haveinitialpositionsatbwitha<b.Letallotheragents
have initial positions taken from the interval (a,b) and Similarly,x[k]isanondecreasingfunctionoftime.Thus,
every agent has 0 as initial velocity. From condition 3, wehavethatfork ≥2,normalagentssatisfyxˆi[k]∈S,
we have that |Xf+1| + |Xf+1| ≤ f. Suppose that all i∈V \M.Thesafetyconditionhasnowbeenproven.
V1 V2
agents in Xf+1 and Xf+1 are malicious and keep their
V1 V2 Itremainstoestablishtheagreementcondition.Westart
values constant. There is at least one normal agent in
with the proof of agreement in the position values. Be-
V1 and one normal agent in V2 by |XVf1+1| < |V1| and cause x[k] and x[k] are bounded and monotone, their
|Xf+1| < |V | because conditions 1 and 2 do not hold. limits exist, which are denoted by x(cid:63) and x(cid:63), respec-
ThVe2se norma2l agents have f or fewer neighbors outside tively. If x(cid:63) = x(cid:63), then resilient consensus follows. We
oftheirownsetsbecausetheyarenotinXf+1 orXf+1. provebycontradiction,andthusassumex(cid:63) >x(cid:63).
V1 V2
Asaresult,allnormalagentsinV andV updatebased
1 2
only on the values inside V and V by removing the Denote by β the minimum nonzero entry of the matrix
1 2 (cid:2) (cid:3)
values received from outside of their sets. This makes Φ1k Φ2k over all k. This matrix is determined by the
their positions at a and b unchanged. Hence, there will structure of the graph G[k] and thus can vary over a
benoagreementamongthenormalagents. finite number of candidates; by (9), we have β ∈ (0,1).
Let(cid:15) >0and(cid:15)>0besufficientlysmallthat
0
(Sufficiency) Wefirstestablishthesafetyconditionwith
S given in (10), i.e., xˆi[k] ∈ S for all k and i ∈ V \M. x(cid:63)+(cid:15) <x(cid:63)−(cid:15) , (cid:15)< βnN(cid:15)0 . (13)
For k = 0, it is obvious that the condition holds. For 0 0 1−βnN
k =1,thepositionsofnormalagentsaregivenby(7)as
Weintroducethesequence{(cid:15) }definedby
(cid:18)(cid:104) (cid:105) T2 (cid:19) (cid:18) αT2(cid:19) (cid:96)
xˆN[1]= I 0 − LN[0] xˆ[0]+ T− vN[0].
nN 2 2 (cid:15)(cid:96)+1 =β(cid:15)(cid:96)−(1−β)(cid:15), (cid:96)=0,1,...,nN −1.
(11)
While the initial velocities of the malicious agents do It is easy to see that (cid:15) < (cid:15) for i = 0,1,...,n −1.
(cid:96)+1 (cid:96) N
notappearinxˆN atthistime,theirinitialpositionsmay Moreover,by (13),theyarepositivebecause
have influences. However, for a normal agent, if some
otefrvitasln[meiignhxˆbNor[0s],amreaxmxˆaNlic[0io]]u,sthaenndtahreeyowuitllsibdeeitghneoriend- (cid:15) =βnN(cid:15) −n(cid:88)N−1β(cid:96)(1−β)(cid:15)=βnN(cid:15) −(cid:0)1−βnN(cid:1)(cid:15)>0.
nN 0 0
by step 2 in DP-MSR because there are at most f such
(cid:96)=0
agents.Thematrix[I 0]−(T2/2)LN[0]in(11)isnon-
nN
negative and its row sums are one because LN[0] con- Wealsotakek ∈Z suchthatfork ≥k ,itholdsthat
(cid:15) + (cid:15)
sists of the first n rows of the Laplacian L[0]. As a x[k]<x(cid:63)+(cid:15),x[k]>x(cid:63)−(cid:15).Duetoconvergenceofx[k]
N
result, the first term on the right-hand side of (11) is andx[k],suchk exists.Forthesequence{(cid:15) },let
(cid:15) (cid:96)
a vector whose entries are convex combinations of val-
ueswithin[minxˆN[0],maxxˆN[0]].Thus,foreachnormal X (k +(cid:96),(cid:15) )={j ∈V : xˆ [k +(cid:96)]>x(cid:63)−(cid:15) },
agenti∈V \M,itholdsthatxˆi[1]∈S. X1(k(cid:15)+(cid:96),(cid:15)(cid:96))={j ∈V : xˆj[k(cid:15)+(cid:96)]<x(cid:63)+(cid:15)(cid:96)}. (14)
2 (cid:15) (cid:96) j (cid:15) (cid:96)
Next,tofurtheranalyzethenormalagents,let
Foreachfixed(cid:96),thesesetsaredisjointby(13)and(cid:15) <
(cid:96)+1
(cid:15) .Here,weclaimthatinafinitenumberofsteps,oneof
x[k]=max(cid:0)xˆN[k],xˆN[k−1](cid:1), (cid:96)
these sets will contain no normal agent. Note that this
(12)
x[k]=min(cid:0)xˆN[k],xˆN[k−1](cid:1). contradictstheassumptionofx(cid:63) andx(cid:63) beinglimits.
6
Suppose that the set X (k ,(cid:15) ) has this normal agent;
1 (cid:15) 0
xˆ[k]
denoteitsindexbyi.Becauseateachtime,eachnormal
x[k] agent ignores at most f smallest neighbors and all of
thesef +1neighborsareupperboundedbyx(cid:63)−(cid:15) ,at
0
leastoneoftheagentsaffectingihasavaluesmallerthan
(cid:15)0 (cid:15) orequaltox(cid:63)−(cid:15)0.FromRemark3.2,everynormalagent
updates by a convex combination of position values of
current and previous time steps. By (9) and the choice
x(cid:63)
Normalagentsin Normalagentsin of β, one of the normal neighbors must be used in the
(cid:15) (cid:15)1 X1(k(cid:15),(cid:15)0) xˆ X1(k(cid:15)+1,(cid:15)1) updatewithDP-MSR.Thus,foragenti,
0 i
xˆN[k +1]≤(1−β)x[k ]+β(x(cid:63)−(cid:15) ).
i (cid:15) (cid:15) 0
xˆ
j
By (15),xˆN[k +1]isupperboundedbyx(cid:63)−(cid:15) .Thus,
k(cid:15) k(cid:15)+1 k at least, onie o(cid:15)f the normal agents in X (k ,(cid:15) )1has de-
1 (cid:15) 0
creased to x(cid:63)−(cid:15) , and the number of normal agents in
Fig.2.AsketchfortheproofofTheorem3.3,whereasetof 1
normalagentsaroundx(cid:63) becomessmallerastimeproceeds. X1(k(cid:15) + 1,(cid:15)1) is less than X1(k(cid:15),(cid:15)0) (see Fig. 2). The
same argument holds for X (k ,(cid:15) ). Hence, it follows
2 (cid:15) 0
that the number of normal agents in X (k + (cid:96),(cid:15) ) is
WestartbyconsideringX (k ,(cid:15) )(seeFig.3.3).Because 1 (cid:15) (cid:96)
1 (cid:15) 0
ofthelimitx(cid:63),oneormorenormalagentsarecontained less than that in X1(k(cid:15) +(cid:96)−1,(cid:15)(cid:96)−1) and/or the num-
berofnormalagentsinX (k +(cid:96),(cid:15) )islessthanthatin
in this set or the set X (k +1,(cid:15) ) from the next step. 2 (cid:15) (cid:96)
1 (cid:15) 1
X (k +(cid:96)−1,(cid:15) ).Becausethenumberofnormalagents
In what follows, we prove that X (k ,(cid:15) ) is nonempty. 2 (cid:15) (cid:96)−1
1 (cid:15) 0 is finite, there is a time (cid:96)(cid:63) ≤ n where the set of nor-
We can in fact show that each normal agent j outside N
malagentsinX (k +(cid:96),(cid:15) )and/orthatinX (k +(cid:96),(cid:15) )
of X (k ,(cid:15) ) at time k will remain outside of X (k + 1 (cid:15) (cid:96) 2 (cid:15) (cid:96)
1,(cid:15) )1. H(cid:15)er0e, agent j sa(cid:15)tisfies xˆN[k ] ≤ x(cid:63) − (cid:15) .1Fr(cid:15)om is empty for (cid:96) ≥ (cid:96)(cid:63). This fact contradicts the existence
1 j (cid:15) 0 of the two limits x(cid:63) and x(cid:63). Thus, we conclude that
Remark 3.2, each normal agent updates its position by
x(cid:63) =x(cid:63),i.e.,allnormalagentsreachpositionconsensus.
takingaconvexcombinationoftheneighbors’positions
at the current and previous time steps. Thus, by the
choiceofβ,thepositionofagentj isboundedas Itisfinallyshownthatallnormalagentsstopasymptot-
ically, which is agreement in the velocity values. When
xˆN[k +1]≤(1−β)x[k ]+β(x(cid:63)−(cid:15) ) thenormalagentsreachagreementintheirpositions,the
j (cid:15) (cid:15) 0 controls (2) become uN[k] → −αvN[k] as k → ∞. By
≤(1−β)(x(cid:63)+(cid:15))+β(x(cid:63)−(cid:15) ) i i
0 thedynamics(1)oftheagents,itholdsthatxˆN[k+1]→
≤x(cid:63)−β(cid:15) +(1−β)(cid:15)=x(cid:63)−(cid:15) . (15) i
0 1 xˆN[k]+T (1−αT/2)vN[k] as k → ∞. Noting (9), we
i i
arriveatvN[k]→0ask →∞. (cid:3)
i
Hence,agentj isnotinX (k +1,(cid:15) ).Similarly,wecan
1 (cid:15) 1
showthatX (k ,(cid:15) )isnonempty.
2 (cid:15) 0 In(DibajiandIshii2015a),wehavestudiedthef-local
model,whereeachnormalagenthasatmostf malicious
Sincethegraphis(f +1,f +1)-robust,oneofthecon- agents as neighbors. Clearly, there may be more mali-
ditions1–3fromDefinition2.1holds: cious agents overall than the f-total case. There, a suf-
ficient condition for resilient consensus is that the net-
1. All agents in X (k ,(cid:15) ) have f +1 neighbors from work is (2f +1)-robust. From Lemma 2.2, such graphs
1 (cid:15) 0
outsideofX (k ,(cid:15) ). requiremoreedgesthan(f+1,f+1)-robustgraphs,as
1 (cid:15) 0
2. All agents in X (k ,(cid:15) ) have f +1 neighbors from giveninthetheoremabove.
2 (cid:15) 0
outsideofX (k ,(cid:15) ).
2 (cid:15) 0
3. Inthetwosetsintotal,therearef+1agentshaving
Theresultisconsistentwiththatforthefirst-orderagent
atleastf+1neighborsfromoutsidetheirownsets.
case in (LeBlanc et al. 2013). More from the technical
side, difficulties in dealing with second-order agent dy-
In the first case, based on the definition of x(cid:63) and x[k] namics can be described as follows: In the proof above,
there exists at least one normal agent inside the set an important step is to establish that x[k] and x[k] de-
X (k ,(cid:15) ) having f +1 incoming links from outside of fined in (12) are monotonically nonincreasing and non-
1 (cid:15) 0
X (k ,(cid:15) ).Similarly,inthesecondcase,thereexistsone decreasing, respectively. These properties do not hold
1 (cid:15) 0
normal agent having f +1 incoming links from outside forthemaximumpositionmax xˆN[k]andtheminimum
X (k ,(cid:15) ).Inthethirdcase,becausethemaximumnum- position min xˆN[k] as in the first-order case. Further-
2 (cid:15) 0
berofmaliciousagentsisf,thereisonenormalagentin more, as a consequence of this fact, it is more involved
X (k ,(cid:15) ) or X (k ,(cid:15) ) which has f +1 neighbors from toshowthatthesetsX (k +(cid:96),(cid:15) )andX (k +(cid:96),(cid:15) )in
1 (cid:15) 0 2 (cid:15) 0 1 (cid:15) (cid:96) 2 (cid:15) (cid:96)
outsidethesetitbelongsto. (14)becomesmalleras(cid:96)increases.
7
AsacorollarytoTheorem3.3,weshowthattheconver- instants.Moreover,theymakeupdatesbasedontherel-
genceratetoachieveconsensusisexponential. ativelocationsoftheirneighborswithoutanytimedelay
inthemodel.Inpractice,however,theagentsmightnot
Corollary3.4 UndertheassumptionsinTheorem3.3, be synchronized nor have access to the current data of
the network of agents with second-order dynamics all neighbors simultaneously. In this section, we extend
reaches resilient consensus with an exponential conver- the setup so that the agents are allowed to update at
gencerate. differenttimeswithdelayedinformation.
Proof. Weoutlinetheproofwhichfollowsasimilarline Wewouldliketoemphasizethedifferenceinthepartially
ofargumenttothatofTheorem3.3,butwiththeknowl- asynchronousagentmodelemployedherefromthosein
edge that the agents come to agreement. Let V(k) = theresilientconsensusliterature.Inparticular,wefollow
x[k]−x[k].Weshowthatthisfunctiondecreasestozero the approach generally assumed in asynchronous con-
exponentiallyfastask →∞. sensus for the case without malicious agents; see, e.g.,
(Mesbahi and Egerstedt 2010, Su et al. 1998, Xiao and
Takeanarbitraryconstantη ∈(0,1).Weintroducetwo Wang2006)forsingle-integratornetworksand(Linand
setsasfollows:Fork =0,1,...,n ,let Jia 2009, Liu and Liu 2012, Qin and Gao 2012, Qin et
N
al. 2012) for the double-integrator case. That is, at the
X(cid:48)(k)=(cid:8)j ∈V : xˆ [k]>(cid:0)1−βkη(cid:1)(cid:0)x[0]−x(cid:63)(cid:1)+x(cid:63)(cid:9), time for an update, each agent uses the most recently
1 j
X(cid:48)(k)=(cid:8)j ∈V : xˆ [k]<βkη(cid:0)x[0]−x(cid:63)(cid:1)+x[0](cid:9). receivedpositionsofitsneighbors.Thisisanaturalset-
2 j tingespeciallyforautonomousmobilerobotsorvehicles
usingsensorstolocatetheirneighborsinrealtime.
Clearly,foreachk,thesetsX(cid:48)(k)andX(cid:48)(k)aredisjoint,
1 2
soby(f+1,f+1)robustness,thereisonenormalagent
In contrast, the works (Azadmanesh and Kieckhafer
i in one of the sets having f +1 incoming links from
2002, Kieckhafer and Azadmanesh 1993, LeBlanc and
outsidethesettowhichitbelongs.IfagentiisinX(cid:48)(k),
1 Koutsoukos 2012, Vaidya et al. 2012) from the area
then,similarto(15),wecanupperbounditspositionas
of computer science consider asynchronous MSR-type
algorithms based on the notion of rounds (for the case
xˆ [k+1]≤(1−βk+1)x[0]+βk+1(cid:2)(1−η)(x[0]−x(cid:63))+x(cid:63)(cid:3).
i with first-order agents). There, when each agent makes
a transmission, it broadcasts its state together with its
Hence, in this case, agent i is not in X(cid:48)(k+1) at time roundr,representingthenumberoftransmissionsmade
1
k+1.Ontheotherhand,ifagentiisinX(cid:48)(k),then so far. The agent makes an update to obtain the new
2
statecorrespondingtoroundr+1byusingthestatesof
xˆ [k+1]≥(1−βk+1)x[0]+βk+1(cid:2)η(x(cid:63)−x[0])+x[0](cid:3), neighbors,butonlywhenasufficientnumberofthosela-
i
beledwiththesameroundrarereceived.Duetodelays
implyingthatxˆ [k+1]isoutsideofX(cid:48)(k+1).Sincethe in communication, the states labeled with round r may
i 2
number of normal agents is n , at time k = n , both be received at various times, causing potentially large
N N
ofthesetsX(cid:48)(n )andX(cid:48)(n )donotcontainanynor- delaysinmakingthe(r+1)thupdateforsomeagents.
1 N 2 N
mal agent. Hence, we can conclude that the maximum
position x[k] and the minimum position x[k] of normal Comparedtotheresultsintheprevioussection,theanal-
agents,respectively,decreasedandincreasedsincetime ysis in the partially asynchronous model studied here
0. More concretely, we have V(nN) ≤ (1−βnNη)V(0). becomes more complicated. Moreover, the derived con-
By repeating this argument, we can establish that dition is more restrictive because there are additional
V(knN)≤(1−βnNη)kV(0). (cid:3) waysforthemaliciousagentstodeceivethenormalones.
Forexample,theymayquicklymovesothattheyappear
Thiscorollaryalsoindicatesthatconventionalconsensus to be at different positions for different normal agents,
algorithmswithoutmaliciousagentsinthenetworkhave whichmaypreventthemfromcomingtogether.
exponentialconvergencerate.IntheproposedMSR-type
algorithm,therateofconvergenceisaffectedbytwofac- 4.1 AsynchronousDP-MSRAlgorithm
tors.Oneisthatanumberofedgesareignoredandnot
usedfortheupdates,whichwillreducetheconvergence
Here,weemploythecontrolinputtakingaccountofpos-
rate.Theotheristhatifmaliciousagentsstaytogether
sibledelaysinthepositionvaluesfromtheneighborsas
with some of the normal agents, they can still influence
theratetoachieveconsensusandslowitdown. (cid:88) (cid:0) (cid:1)
u [k]= a [k] xˆ [k−τ [k]]−xˆ [k] −αv [k],
i ij j ij i i
j∈Ni
4 NetworkswithPartialAsynchronyandDelay (16)
whereτ [k]∈Z denotesthedelayintheedge(j,i)at
ij +
Sofar,theunderlyingassumptioninthemodelhasbeen time k. From the viewpoint of agent i, the most recent
that all agents exchange their states at the same time information regarding agent j at time k is the position
8
at time k −τ [k] relative to its own current position. Then, let D[k] be a diagonal matrix whose ith entry is
ij
Thedelaysaretimevaryingandmaybedifferentateach given by d [k] = (cid:80)n a [k]. Now, the n × (τ + 1)n
i j=1 ij
edge,butweassumethecommonupperboundτ as matrixL [k]isdefinedas
τ
0≤τ [k]≤τ, (j,i)∈E, k ∈Z . (17) (cid:104) (cid:105)
ij + L [k]= D[k]−A [k] ··· −A [k] .
τ 0 τ
Hence,eachnormalagentbecomesawareoftheposition
ofeachofitsneighborsatleastonceinτ timesteps,but It is clear that the summation of each row is zero as in
possibly at different time instants. In other words, nor- theLaplacianmatrixL[k].
malagentsmustupdateandtransmittheirinformation
oftenenoughtomeet(17).Itisalsoassumedin(16)that Now,thecontrolinput(16)canbeexpressedas
agentiusesitsowncurrentvelocity.Weemphasizethat
thevalueofτ in(17)canbearbitraryandmoreoverneed uN[k]=−LN[k]z[k]−α(cid:104)I 0(cid:105)v[k],
notbeknowntotheagentssincethisinformationisnot τ nN (18)
used in the update rule. In (Gao and Wang 2010, Liu uM[k]: arbitrary,
andLiu2012,Qinetal.2012),timedelaysforpartially
asynchronous cases have been studied for agents with where z[k] = [xˆ[k]Txˆ[k −1]T ···xˆ[k −τ]T]T is a (τ +
second-orderdynamics. 1)n-dimensional vector for k ≥0 and LN[k] is a matrix
τ
formed by the first n rows of L [k]. Here, z[0] is the
N τ
As in the synchronous case, the malicious agents are given initial position values of the agents and can be
assumedtobeomniscient.Here,itmeansthattheyhave chosen arbitrarily. By (3) and (18), the agent dynamics
prior knowledge of update times and τij[k] for all links canbewrittenas
andk ≥0.Themaliciousagentsmighttakeadvantageof
thisknowledgetodecidehowtheyshouldmakeupdates (cid:34) (cid:35)
T2 0
toconfuseandmisleadthenormalagents. xˆ[k+1]=Γ[k]z[k]+Qv[k]+ uM[k],
2 I
nM
To achieve resilient consensus, we employ a modified (cid:34) (cid:35) (cid:34) (cid:35)
version of the algorithm in Section 3, called the asyn- v[k+1]=−T LNτ [k] z[k]+Rv[k]+T 0 uM[k],
chronousDP-MSR,outlinedbelow. 0 I
nM
(19)
1. At each time step k, each normal agent i decides
whethertomakeanupdateornot.
whereΓ[k]isann×(τ +1)nmatrixgivenby
2. Ifitdecidestodoso,thenitusestherelativeposi-
tionvaluesofitsneighborsj ∈N basedonthemost
i (cid:34) (cid:35)
recentvaluesintheformofxˆ [k−τ [k]]−xˆ [k]and T2 LN[k]
j ij i Γ[k]=[I 0]− τ
thenfollowsstep2oftheDP-MSRalgorithmbased n 2 0
on these values. Afterwards, it applies the control
input(16)bysubstitutinga [k]=0foredges(j,i)
ij
whichareignoredinstep2ofDP-MSR. and Q and R are given in (8). Based on the expression
3. Otherwise,itappliesthecontrol(16)wherethefirst (19),wecanderivearesultcorrespondingtoLemma3.1
termofpositionvaluesofitsneighborsremainsthe forthepartiallyasynchronousanddelayedprotocol.The
sameastheprevioustimestep,andforthesecond proof is omitted since it is by direct calculation similar
term,itsowncurrentvelocityisused. toLemma3.1shownin(DibajiandIshii2015a).
The asynchronous version of the resilient consensus Lemma4.1 Underthecontrolinputs(18),theposition
problemisstatedasfollows:Underthef-totalmalicious vectorxˆ[k]oftheagentsfork ≥1canbeexpressedas
model,findaconditiononthenetworktopologysothat
(cid:34) (cid:35)
the normal agents reach resilient consensus using the z[k]
(cid:2) (cid:3)
asynchronousDP-MSRalgorithm. xˆ[k+1]= Λ1k Λ2k
z[k−1]
4.2 MatrixRepresentation (cid:34) (cid:35)
+ T2 0 (cid:0)uM[k]+uM[k−1](cid:1),
2 I
Beforepresentingthemainresultofthissection,wein- nM
troducesomenotationtorepresenttheequationsinthe
matrixform.DefinethematricesA [k],0≤(cid:96)≤τ,by where
(cid:96)
(cid:0)A [k](cid:1) =(cid:26)aij[k] if(j,i)∈E[k]andτij[k]=(cid:96), Λ =(cid:2)R 0(cid:3)+Γ[k], Λ =−RΓ[k−1]−QT (cid:34)LNτ [k](cid:35).
(cid:96) ij 0 otherwise. 1k 2k 0
9
(cid:2) (cid:3)
Furthermore, the matrix Λ Λ is nonnegative and Here,weclaimthatx [k]isanonincreasingfunctionof
1k 2k τ
thesumofeachofitsfirstn rowsisone. k ≥1.ByLemma4.1,attimek ≥2,eachnormalagent
N
updates its position based on a convex combination of
theneighbors’positionsfromk−1tok−τ−1.Ifsome
4.3 ResilientConsensusAnalysis
neighbors are malicious and stay outside of the interval
determined by the normal agents’ positions, then they
The following theorem is the main result of the paper,
areignoredinstep2ofDP-MSR.Hence,weobtainxˆ [k+
i
addressingresilientconsensusviatheasynchronousDP- 1] ≤ max(cid:0)xˆN[k],xˆN[k−1],...,xˆN[k−τ −1](cid:1) for i ∈
MSRinthepresenceofdelayedinformation.Thesafety
V \M.Italsofollowsthat
intervaldiffersfromthepreviouscaseandisgivenby
(cid:20) (cid:26) (cid:18) αT2(cid:19) (cid:27) xˆi[k]≤max(cid:0)xˆN[k],...,xˆN[k−τ −1](cid:1),
Sτ = minzN[0]+min 0, T − 2 vN[0] , xˆi[k−1]≤max(cid:0)xˆN[k],...,xˆN[k−τ −1](cid:1),
(cid:26) (cid:18) αT2(cid:19) (cid:27)(cid:21) ..
maxzN[0]+max 0, T − vN[0] . (20) .
2
xˆ [k−τ]=xˆ [k+1−(τ +1)]
i i
≤max(cid:0)xˆN[k],...,xˆN[k−τ −1](cid:1)
Theorem4.2 Under the f-total malicious model, the
networkofagentswithsecond-orderdynamicsusingthe
controlin(18)andtheasynchronousDP-MSRalgorithm fori∈V \M.Hence,wehave
reaches resilient consensus only if the underlying graph
(f+1,f+1)-robust.Moreover,iftheunderlyinggraph x [k+1]=max(cid:0)xˆN[k+1],...,xˆN[k+1−(τ +1)](cid:1)
τ
is (2f +1)-robust, then resilient consensus is attained ≤max(cid:0)xˆN[k],...,xˆN[k−(τ +1)](cid:1)=x [k].
τ
withasafetyintervalgivenby (20).
We can similarly prove that x [k] is nondecreasing in
The proof of this theorem given below follows an argu- τ
time. This indicates that for k ≥ 2, we have xˆ [k] ∈ S
i τ
mentsimilartothatofTheorem3.3.However,theprob-
fori∈V\M.Thus,wehaveshownthesafetycondition.
lemismoregeneralwiththedelayboundτ ≥0in(17).
Thisinturnresultsinmoreinvolvedanalysiswithsub-
In the rest of the proof, we must show the agreement
tledifferences.Weprovidefurtherdiscussionslater.
condition. As x [k] and x [k] are monotone functions
τ τ
andcontainedin[x [1],x [1]],bothoftheirlimitsexist,
τ τ
Proof. (Necessity) The synchronous network is a spe- whicharedenotedbyx(cid:63) andx(cid:63),respectively.Weclaim
τ τ
cial case of partially asynchronous ones with τ = 0. thatthelimitsinfactsatisfyx(cid:63) =x(cid:63),i.e.,thepositions
τ τ
Thus,thenecessaryconditioninTheorem3.3isvalid. ofthenormalagentscometoconsensus.Theproofisby
contradiction,soweassumethatx(cid:63) >x(cid:63).
τ τ
(Sufficiency) We first show that the safety condition
holds. For k = 0, by (20), we have xˆi[0] ∈ Sτ for i ∈ First, let β be the minimum nonzero element over all
V\M.Fork =1,by(19),thepositionsofnormalagents possible cases of (cid:2)Λ Λ (cid:3). From (9) and the bound γ
1k 2k
canbeexpressedas on a [k], it holds that β ∈ (0,1). Choose (cid:15) > 0 and
ij 0
(cid:15)>0smallenoughthat
(cid:18)(cid:104) (cid:105) T2 (cid:19) (cid:18) αT2(cid:19)
xˆN[1]= I 0 − LN[0] z[0]+ T− vN[0].
nN 2 τ 2 x(cid:63) +(cid:15) <x(cid:63) −(cid:15) , (cid:15)< β(τ+1)nN(cid:15)0 . (23)
(21) τ 0 τ 0 1−β(τ+1)nN
This vector may be affected by the malicious agents
throughtheirinitialpositions.However,bystep2inDP- Wenexttakethesequence{(cid:15) }via
(cid:96)
MSR,foranynormalagent,ifsomeneighborsaremali-
ciousandareoutsideof[minzN[0],maxzN[0]],thenthey (cid:15) =β(cid:15) −(1−β)(cid:15), (cid:96)=0,1,...,(τ +1)n −1.
(cid:96)+1 (cid:96) N
willbeignored.In(21),thematrix[I 0]−(T2/2)LN[0]
nN τ
isnonnegativeanditsrowsumsareone.Hence,thefirst Itcanbeshownthat0<(cid:15) <(cid:15) forall(cid:96).Inparticular,
(cid:96)+1 (cid:96)
termontheright-handsideof(21)becomesconvexcom- theyarepositivebecauseby (23),itholdsthat
binationsofvaluesintheinterval[minzN[0],maxzN[0]].
Thus,wehavexˆ [1]∈S fori∈V \M.
i τ (τ+1(cid:88))nN−1
(cid:15) =β(τ+1)nN(cid:15) − βk(1−β)(cid:15)
Next,fork ≥1,definetwovariablesby (τ+1)nN 0
k=0
x [k]=max(cid:0)xˆN[k],xˆN[k−1],...,xˆN[k−τ −1](cid:1), =β(τ+1)nN(cid:15)0−(cid:0)1−β(τ+1)nN(cid:1)(cid:15)>0.
τ
x [k]=min(cid:0)xˆN[k],xˆN[k−1],...,xˆN[k−τ −1](cid:1). We also take k ∈ Z such that x [k] < x(cid:63) + (cid:15) and
τ (cid:15) + τ τ
(22) x [k] > x(cid:63) −(cid:15) for k ≥ k . Due to convergence of x [k]
τ τ (cid:15) τ
10