research article on distributed reduced-order observer-based...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 793276 13 pageshttpdxdoiorg1011552013793276

Research ArticleOn Distributed Reduced-Order Observer-Based Protocol forLinear Multiagent Consensus under Switching Topology

Yan Zhang Lixin Gao and Changfei Tong

Institute of Intelligent Systems and Decision Wenzhou University Zhejiang 325027 China

Correspondence should be addressed to Lixin Gao gao-lixin163com

Received 21 January 2013 Accepted 18 March 2013

Academic Editor H G Enjieu Kadji

Copyright copy 2013 Yan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We discuss linear multiagent systems consensus problem with distributed reduced-order observer-based protocol under switchingtopology We use Jordan decomposition method to prove that the proposed protocols can solve consensus problem under directedfixed topology By constructing a parameter-dependent common Lyapunov function we prove that the distributed reduced-order observer-based protocol can also solve the continuous-time multi-agent consensus problem under the undirected switchinginterconnection topology Then we investigate the leader-following consensus problem and propose a reduced-order observer-based protocol for each following agent By using similar analysis method we can prove that all following agents can track theleader under a class of directed interaction topologies Finally the given simulation example also shows the effectiveness of ourobtained result

1 Introduction

Recently a great number of researchers pay much attentionto the coordination control of the multiagent systems whichhave various subject background such as biology physicsmathematics information science computer science andcontrol science in [1ndash4] Consensus problem is one of themost basic problems of the coordination control of themultiagent systems and the main idea is to design the dis-tributed protocols which enable a group of agents to achievean agreement on certain quantities of interest The well-known early work [1] was done in the control systems com-munity which gave the theoretical explanation of the consen-sus behavior of the very famous Vicsek model [2] Till nowmany interesting results for solving similar or generalizedconsensus problems have been obtained

The interaction topologies among agents include fixedand switching cases Fixed topology may be easy to be han-dled by using eigenvalue decomposition method [5 6]Saber and Murray established a general model for consensusproblems of themultiagent systems by introducing Lyapunovmethod to reveal the contract with the connectivity of thegraph theory and the stability of the system in [7] The

Lyapunov-based approach is often chosen to solve high-order consensus problem [8 9] In most existing works thedynamics of agents is assumed to be first- second- andsometimes high-order integrators and the proposed con-sensus protocols are based on information of relative statesamong neighboring agents [10ndash13] However the interactingtopology between agents may change dynamically due to thechanges of environment the unreliable communication linksand time-delay It is more difficult to deal with switchinginteraction topology in mathematics than fixed interactiontopology The common Lyapunov method is fit to probe theswitching interaction topology [8] Too many results havebeen established for multiagent consensus under switchingtopology [14ndash18] Some other relevant research topics havealso been addressed such as oscillator network [19] clus-ter synchronization [20 21] mean square consensus [22]fractional-order multiagent systems [23] descriptor multia-gent system [24] randomnetworks [25] and time-delay [26]

The leader-following configuration is very useful todesign the multiagent systems Leader-following consensusproblems with first-order dynamics under jointly con-nected interacting topology were investigated by [1] Honget al [8] considered a multiagent consensus problem with

2 Abstract and Applied Analysis

a second-order active leader and variable interconnectiontopology A distributed consensus protocol was proposed forfirst-order agent with distributed estimation of the generalactive leaderrsquos unmeasurable state variables in [15] while [16]extended the results of [15] to the case of communicationdelays among agents under switching topology In [17] theauthors considered leader-following problem in the mul-tiagent system with general linear dynamics in both fixedtopology case and switching topology case respectively Thecooperative output regulation of linear multiagent systemscan be viewed as a generalization of some results of the leader-following consensus problem of multiagent systems [27 28]

In many practical systems the state variables cannot beobtained directly To achieve the state consensus the agenthas to estimate those unmeasurable state variables by outputvariables In [8] the authors proposed a distributed observer-based tracking protocols for each first-order following agentUnder the assumption that the active leaderrsquos velocity cannotbe measured directly [29] proposed a distributed observer-based tracking protocol for each second-order following-agent To track the accelerated motion leader [30] proposedan observer-based tracking protocol for each second-orderfollower agent to estimate the acceleration of the leaderA robust adaptive observer based on the response systemwas constructed to practically synchronize a class of uncer-tain chaotic systems [31] In [23] the author proposed anobserver-type consensus protocol to the consensus problemfor a class of fractional-order uncertain multiagent systemswith general linear dynamics For the multiagent system withgeneral linear dynamics [32] established a unified frameworkand proposed an observer-type consensus protocol and [33]proposed a framework including full state feedback controlobserver design and dynamic output feedback control forleader-following consensus problem The leader-followingconsensus problem was investigated under a class of directedswitching topologies in [34] In [35] distributed reduced-order observer-based consensus protocols were proposedfor both continuous- and discrete-time linear multiagentsystems Other observer-based previous works include [36ndash38]

Motivated by the previousworks especially by [35] we dosome further investigations on the reduced-order observer-based consensus protocol problem which had been studiedby [35] under directed fixed interconnection topology Wefirst correct some errors and propose a new proof of the mainresult established in the aforementioned paper based on theJordan decomposition method Moreover by constructing aparameter-dependent common Lyapunov function we provethat the proposed protocol can guarantee the multiagentconsensus system to achieve consensus under undirectedswitching topology Although the Lyapunov functionmethodis conservative and is not easy to be constructed it is fitto solve the problem under the switching interconnectiontopology We propose distributed protocol to solve leader-following consensus with a little simple modification to thereduced-order observer-based consensus protocol Similarlywe can prove that all following agents can track the leaderunder a class of directed interaction topologies As the specialcases the consensus conditions for balanced and undirected

interconnection topology cases can be obtained directly Al-though the leader-following consensus problem in this paperhas been studied in many papers with the aid of internalmodel principle we obtain a low-dimensional controller inour model

The paper is organized as follows In Section 2 some no-tations and preliminaries are introduced Then in Section 3and Section 4 the main results on the consensus stabilityare obtained for both leaderless and leader-following casesrespectively Following that Section 5 provides a simulationexample to illustrate the established results and finally theconcluding remarks are given in Section 6

2 Preliminaries

To make this paper more readable we first introduce somenotations and preliminaries most of which can be found in[35] Let 119877119898times119899 and 119862119898times119899 be the set of119898times 119899 real matrices andcomplex matrices respectively Re(120585) denotes the real part of120585 isin 119862 119868 is the identity matrix with compatible dimension119860

119879 and 119860119867 represent transpose and conjugate transpose ofmatrix 119860 isin 119862

119898times119899 respectively 1119899= [1 1]

119879

isin 119877119899 For

symmetric matrices 119860 and 119861 119860 gt (ge) 119861 means that 119860 minus 119861is positive (semi-) definite otimes denotes the Kronecker productwhich satisfies (119860otimes119861)(119862otimes119863) = (119860119862)otimes (119861119863) and (119860otimes119861)119879 =119860

119879

otimes 119861119879 A matrix is said to be Hurwitz stable if all of its

eigenvalues have negative real partsA weighted digraph is denoted by G = V 120576 119860 where

V = V1 V

2 V

119899 is the set of vertices 120576 sub V times V is the

set of edges and a weighted adjacency matrix 119860 = [119886119894119895] has

nonnegative elements 119886119894119895 The set of all neighbor nodes of

node V119894is defined byN

119894= 119895 | (V

119894 V

119895) isin 120576The degreematrix

119863 = 1198891 119889

2 119889

119899 isinR119899times119899 of digraphG is a diagonalmatrix

with diagonal elements 119889119894= sum

119895isinN119894119886119894119895 Then the Laplacian

matrix of G is defined as 119871 = 119863 minus 119860 isin 119877119899times119899 which satisfies1198711

119899= 0 The Laplacian matrix has following interesting

property

Lemma 1 (see [14]) The Laplacian matrix 119871 associated withweighted digraph G has at least one zero eigenvalue and allof the non-zero eigenvalues are located on the open right halfplane Furthermore 119871 has exactly one zero eigenvalue if andonly if the directed graph G has a directed spanning tree

A weighted graph is called undirected graph if for all(V

119894 V

119895) isin 120576 we have (V

119895 V

119894) isin 120576 and 119886

119894119895= 119886

119895119894 It is well known

that Laplacian matrix of weighted undirected graph is sym-metric positive semidefinite which can be derived fromLemma 1 by noticing the fact that all eigenvalues of symmet-ric matrix are real and nonnegative Furthermore Laplacianmatrix 119871 has exactly one zero eigenvalue if and only if theundirected graphG is connected

To establish our result the well-known Schur Comple-ment Lemma is introduced

Lemma 2 (see [39]) Let 119878 be a symmetric matrix of the par-titioned form 119878 = [119878

119894119895] with 119878

11isin 119877

119903times119903 11987812isin 119877

119903times(119899minus119903) and

Abstract and Applied Analysis 3

11987822isin 119877

(119899minus119903)times(119899minus119903) Then 119878 lt 0 if and only if

11987811lt 0 119878

22minus 119878

21119878minus1

1111987812lt 0 (1)

or equivalently

11987822lt 0 119878

11minus 119878

12119878minus1

2211987821lt 0 (2)

3 Multiagent Consensus Problem

Consider a multiagent system consisting of 119873 identicalagents whose dynamics are modeled by

119894= 119860119909

119894+ 119861119906

119894 119910

119894= 119862119909

119894 119894 = 1 119873 (3)

where 119909119894isin 119877

119899 is the agent 119894rsquos state 119906119894isin 119877

119901 agent 119894rsquos controlinput and 119910

119894isin 119877

119902 the agent 119894rsquos measured output 119860 119861 and119862 are constant matrices with compatible dimensions It isassumed that (119860 119861) is stabilizable (119860 119862) is observable and119862 has full row rank

To solve consensus problem a weighted counterpart ofthe distributed reduced-order observer-based consensus pro-tocol proposed in [35] for agent 119894 is given as follows

V119894= 119865V

119894+ 119866119910

119894+ T119861119906

119894

119906119894= 120581119870119876

1sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895)

+ 1205811198701198762sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895)

(4)

where V119894isin 119877

119899minus119902 is the protocol state the weight 119886119894119895(119905) is

chosen as

119886119894119895(119905) =

120572119894119895 if agent 119894 is connected to agent 119895

0 otherwise(5)

120581 is the coupling strength and 119865 isin 119877(119899minus119902)times(119899minus119902) 119866 isin 119877(119899minus119902)times119902119879 isin 119877

(119899minus119902)times119899 1198761isin 119877

119899times119902 and 1198762isin 119877

119899times(119899minus119902) are constantmatrices which will be designed later

For the multiagent system under consideration the inter-connection topology may be dynamically changing which isassumed that there are only finite possible interconnectiontopologies to be switched The set of all possible topologydigraphs is denoted as S = G

1G

2 G

119872 with index set

P = 1 2 119872 The switching signal 120590 [0infin) rarr P isused to represent the index of topology digraph that is ateach time 119905 the underlying graph isG

120590(119905) Let 0 = 119905

1 119905

2 119905

3

be an infinite time sequence at which the interconnectiongraph switches Certainly it is assumed that chattering doesnot occur when the switching interconnection topology isconsidered The main objective of this section is to designprotocol (4) which is used to solve the consensus problemunder switching interconnection topology

Let 119909 = [119909119879

1 119909

119879

2 119909

119879

119873]119879 and V = [V119879

1 V119879

2 V119879

119873]119879

Then after manipulation with combining (3) and (4) theclosed-loop system can be expressed as

119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 + 119871

120590(119905)otimes (120581119861119870119876

1119862) 119871

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119871

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119871

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(6)

We first discuss consensus problem under fixed intercon-nection topology which has been investigated by [35] In thiscase the subscript 120590(119905) in closed-loop system (6) should bedropped Algorithm 31 in [35] is slightly modified to presentas follows which is used to choose control parameters inprotocol (4)

Algorithm 3 Given (119860 119861 119862) with properties that (119860 119861) isstabilizable (119860 119862) is observable and 119862 has full row rank 119902the control parameters in the distributed consensus protocol(4) are selected as follows

(1) Select a Hurwitz matrix 119865 isin 119877(119899minus119902)times(119899minus119902) with a set ofdesired eigenvalues that contains no eigenvalues in commonwith those of119860 Select119866 isin R(119899minus119902)times119902 randomly such that (119865 119866)is controllable

(2) Solve Sylvester equation

119879119860 minus 119865119879 = 119866119862 (7)

to get the unique solution 119879 which satisfies that [ 119862119879] is

nonsingular If [ 119862119879] is singular go back to Step 2 to select

another 119866 until [ 119862119879] is nonsingular Compute matrices 119876

1isin

119877119899times119902 and 119876

2isin 119877

119899times(119899minus119902) by [1198761119876

2] = [

119862

119879]minus1

(3) For a given positive definite matrix 119876 solve the fol-lowing Riccati equation

119860119879

119875 + 119875119860 minus 119875119861119861119879

119875 + 119876 = 0 (8)

to obtain the unique positive definitematrix119875Then the gainmatrix119870 is chosen by119870 = minus119861119879

119875(4) Select the coupling strength 120581 ge 1(2min

120582119894 = 0Re(120582

119894))

where 120582119894is the 119894th eigenvalue of Laplacian matrix 119871

Remark 4 According to Theorem 8M6 in [40] if 119860 and 119865have no common eigenvalues then the matrix [ 119862

119879] is nonsin-

gular only if (119860 119862) is observable and (119865 119866) is controllableThus the assumptions that (119860 119862) is detectable and (119865 119866)is stabilizable in Algorithm 31 of [35] may be questionableStep 3 of Algorithm 31 in [35] is an LMI inequality Here wepropose the Riccati equation to replace the LMI inequality inStep 3 of the algorithm If we use the LMI design approachproposed by [35] all our following analysis process is alsoright as long as we do some sight modification The Riccatiequation has been widely studied in the subsequent centuriesand has known an impressive range of applications in control


theory which will make the condition expressed in Riccatiequation easy to be generalized to other cases such as descrip-tor multiagent system If (119860 119861) is stabilizable and 119876 is apositive definite matrix the Riccati equation (8) has a uniquepositive definite matrix119875 which can be found inmany bookssuch as [41] On the other hand it is convenient for us to solveRiccati equation by using Matlab toolbox

The following theorem is a modified vision of Theo-rem 33 in [35] which is the main result of [35] To proveTheorem 33 in [35] the authors mainly use the followingassumption Let 119880 isin 119877

119873times119873 be such a unitary matrix that119880

119879

119871119880 = Λ = [0 0

0 Δ] where the diagonal entries of Δ are the

nonzero eigenvalues of 119871 Unfortunately this assumption isnot right

Since Laplacian matrix 119871 of directed topology graph Gis not symmetric it can only be assumed that there exists aunitary matrix 119880 satisfying 119880119867

119871119880 = [0 lowast

0 Δ] where Δ is an

upper triangular matrix and lowast is a nonzero row vector (see[39]) Thus I think the proof in [35] is not strict too On theother hand the limit function of V

119894(119905) in Theorem 33 of [35]

is not right which should be 119879120603(119905) Now we propose a strictproof which is based on Jordan decomposition and may beeasier to be understood Before giving our proof we presentthe theorem as follows

Theorem 5 For the multiagent system (3) whose interconnec-tion topology graph G contains a directed spanning tree thedynamic protocol (4) constructed by Algorithm 3 can solve theconsensus problem Moreover

119909119894(119905) 997888rarr 120603 (119905) ≜ (119903

119879

otimes 119890119860119905

)[[

[

1199091(0)

119909119873(0)

]]

]

= 119890119860119905

119873

sum

119895=1

119903119895119909119895(0)

V119894997888rarr 119879120603 (119905) 119894 = 1 2 119873 as 119905 997888rarr infin

(9)

where 119903 = (1199031 119903

2 119903

119873)119879

isin 119877119873 is a nonnegative vector such

that 119903119879119871 = 0 and 1199031198791 = 1

Proof By Lemma 1 the assumption that G contains adirected spanning tree means that zero is a simple eigenvalueof 119871 and all other eigenvalues of 119871 have positive real partsFrom Jordan decomposition of 119871 let 119878 be nonsingular matrixsuch 119878minus1119871119878 = 119869 = [

0 0

0 1198691] where the Jordan matrix 119869

1isin

119862(119873minus1)times(119873minus1) is an upper triangular matrix and the diagonal

entries of 1198691are the nonzero eigenvalues of 119871

Let 119909 = (119878minus1 otimes 119868119899)119909 and V = (119878minus1 otimes 119868

119899minus119902)V Then system

(6) can be represented in terms of 119909 and V as follows119889

119889119905[119909

V]

=[

[

119868119873otimes 119860 + 119869 otimes (120581119861119870119876

1119862) 119869 otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119869 otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119869 otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(10)

Certainly system (10) can be divided into the following twosubsystems one is

119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (11)

and the other one is119889

119889119905[1199091

V1]

= [

[

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870119876

1119862) 119869

1otimes (120581119861119870119876

2)

119868119873minus1

otimes (119866119862) + 1198691otimes (120581119879119861119870119876

1119862) 119868

119873minus1otimes 119865 + 119869

1otimes (120581119879119861119870119876

2)

]

]

times [1199091

V1]

(12)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and v0

being their first 119899 and 119899 minus 119901 column respectivelyDenote 119911

119894(119896) for agent 119894 as

119911119894= 119909

119894minus

119873

sum

119895=1

119903119895119909119895 119894 = 1 119873 (13)

Obviously for all 119894 119895 = 1 2 119873 119911119894= 0 if and only if

119909119894= 119909

119895 that is the consensus is achieved Let 119911 = [z

1

119879

1199112

119879

119911119873

119879

]119879 Then we have 119911 = ((119868

119873minus 1119903

119879

) otimes 119868119899)119909 Let

119911 = (119878minus1

otimes119868119899)119911 We know that 119911 = 0 if and only if 119911 = 0 Since

119871119878 = 119878119869 the first column of 119878 is right zero eigenvector 119908119903of

119871 Similarly 119878minus1119871 = 119869119878minus1 implies that the first row of 119878minus1 isis left zero eigenvector 119908119879

119897 Set 119878 = [119908

119903 119878

1] and 119878minus1 = [ 119908

119879

119897

1198841

]Due to 119878minus1119878 = 119868 we have 119908119879

119897119908

119903= 1 119908119879

1198971198781= 0 and 119884

1119908

119903= 0

Since 1 and 119903119879 are the right and left zero eigenvectors of 119871respectively and zero is simple eigenvalue of 119871 there existsconstant 120572 = 0 such that 119908

119903= 1205721 and 119908

119897= (1120572)119903 Then we

can verify directly that

119878minus1

(119868119873minus 1119903119879) 119878 = 119878minus1119878 minus 119878minus11119903119879119878 = 119878minus1119878 minus 119878minus1119908

119903119908

119879

119897119878

= [0 0

0 119868119873minus1

]

(14)

Thus we have

119911 = (119878minus1

otimes 119868119899) 119911 = (119878

minus1

otimes 119868119899) ((119868

119873minus 1119903119879) otimes 119868

119899) (119878 otimes 119868

119899) 119909

= ([0 0

0 119868119873minus1

] otimes 119868119899)119909 = [

0

1199091]

(15)

From the previous analysis we know that 1199091= 0 hArr 119911 = 0 hArr

119911 = 0Thus the stability of system (12) implies thatmultiagentsystem (6) can achieve consensus Let 119879 = [ 119868119873minus1otimes119868119899 0

minus119868119873minus1otimes119879 119868119873minus1otimes119868119899minus119902]

which is nonsingular and 119879minus1

= [119868119873minus1otimes119868119899 0

119868119873minus1otimes119879 119868119873minus1otimes119868119899minus119902] Let 120577 =

119879 [1199091

V1] By Step (2) of Algorithm 3 system (12) is equivalent

to the following system

120577 = [

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870) 119869

1otimes (120581119861119870119876

2)

0 I119873minus1

otimes 119865

] 120577 ≜ 119865120577 (16)


The matrix 119865 is block upper triangular matrix with diagonalblockmatrix entries119860+120581120582

119894119861119870 (119894 = 2 3 119873) and119865 By Step

(3) and Step (4) of Algorithm 3 the unique positive definitesolution 119875 of Riccati equation (8) satisfies

(119860 + 120582119894120581119861119870)

119867

119875 + 119875 (119860 + 120582119894120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2Re (120582119894) 120581119875119861119861

119879

119875 le minus119876

(17)

that is119860+120581120582119894119861119870 (119894 = 2 3 119873) are stableThus119865 is stable

Moreover system (12) is asymptotically stable that is theconsensus problem can be solved by protocol (4)

From the first equation of system (11) we have 1199090(119905) =119890119860119905

1199090

(0) = 119890119860119905

(119908119879

119897otimes 119868

119899) times [119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879 Inaddition the solution of system (6) under fixed topologysatisfies 119909(119905) = (119878 otimes 119868

119899)119909(119905) = [1205721 otimes 119868

119899 119878

1otimes 119868

119899] [

1199090(119905)

1199091(119905)

] rarr

[1205721 otimes 119868119899 119878

1otimes 119868

119899] [ 1199090(119905)

0

] = [

1205721199090(119905)

1205721199090(119905)

] as 119905 rarr infin Thus

119909119894(119905) rarr 120572119890

119860119905

(119908119879

119897otimes 119868

119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= 119890119860119905

(119903119879

otimes

119868119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= (119903119879

otimes 119890119860119905

) [

1199091(0)119909119873(0)

] as 119905 rarr

infin From the second equation of system (11) we have

119889

119889119905(V

0

(119905) minus 1198791199090

(119905))

= 119865V0

(119905) + 1198661198621199090

(119905) minus 1198791198601199090

(119905) = 119865 (V0

(119905) minus 1198791199090

(119905))

(18)

Since 119865 is Hurwitz stable we know that lim119905rarrinfin

(V0(119905) minus

1198791199090

(119905)) = 0 Noticing that V = (119878 otimes 119868119899minus119902)V we can also obtain

V119894(119905) rarr 120572V0(119905) as 119905 rarr infinThus we have V

119894(119905) rarr 120572119879119909

0

(119905) =

119879120603(119905) as 119905 rarr infin The proof is now completed

Next we probe the consensus problem under switchinginterconnection topology For the switching interconnectiontopology case we always assume that all interconnectiontopology graphs G

119894 119894 isin P are undirected and connected

Choose an orthogonal matrix with form 119880 = [(1radic119873)1 1198801]

with 1198801isin 119877

119873times(119873minus1) Noticing that the Laplacian matrix 119871119894of

G119894(119894 isin P) is symmetric and 119871

1198941 = 0 we have

119880119879

119871119894119880 = (

0 0

0 1119894

) = 119894 (19)

where 1119894is an (119873 minus 1) times (119873 minus 1) symmetric matrix Since all

G119894are undirected and connected 119871

119894is positive semidefinite

and 1119894is positive definite

Then we can define

120582 = min119894isinP

1205822(119871

119894) | G

119894is undirected and connected gt 0

=max119894isinP

120582max (119871 119894) | G

119894is undirected and connectedgt0

(20)

where 1205822(119871

119894) is the second small eigenvalue of 119871

119894 SinceP is

finite set 120582 and are fixed and positiveTo measure the disagreement of 119909

119894(119905) to the average state

of all agents denote 119911119894(119905) for agent 119894 as

119911119894= 119909

119894minus1

119873

119873

sum

119895=1

119909119895 (21)

Obviously 119911119894= 0 for any 119894 = 1 2 119873 if and only if 119909

119894= 119909

119895

for any 119894 119895 = 1 2 119873 that is the consensus is achievedLet 119911 = [119911119879

1 119911

119879

2 119911

119879

119873]119879 Then we have119911 = (119871

119900otimes 119868

119899) 119909 (22)

where

119871119900= 119868

119873minus1

11987311198731119879119873 (23)

which satisfies 1198711199001119873= 0 It can be verified that 1198801198791

119873=

[radic119873 0 0]119879 from which we have

119880119879

119871119900119880 = 119880

119879

119880 minus1

119873119880

11987911119879119880 = [0 0

0 119868119873minus1

] ≜ 119900 (24)

Let 119909 = (119880119879

otimes 119868119899)119909 V = (119880119879

otimes 119868119899minus119902)V and = (119880119879

otimes

119868119899)119911 Then system (6) can be expressed in terms of 119909 and V as

follows119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 +

120590(t) otimes (1205811198611198701198761119862)

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) +

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 +

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(25)Similarly system (25) can be divided into the following twosubsystems

119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (26)

119889

119889119905[1199091

V1]

=[

[

119868119873minus1 otimes 119860 + 1120590(119905) otimes (1205811198611198701198761119862) 1120590(119905) otimes (1205811198611198701198762)

119868119873minus1 otimes (119866119862) + 1120590(119905) otimes (1205811198791198611198701198761119862) 119868119873minus1 otimes 119865 + 1120590(119905) otimes (1205811198791198611198701198762)

]

]

times [1199091

V1]

(27)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and V0

being their first 119899 and 119899 minus 119901 columns respectivelyWe know that = 0 if and only if 119911 = 0 On the other

hand we have

= (119880119879

otimes 119868119899) 119911 = (119880

119879

otimes 119868119899) (119871

119900otimes 119868

119899) 119909

= (119900otimes 119868

119899) 119909 = [

0

1199091]

(28)

from which we know that 1199091 = 0 is equivalent to 119911 = 0


Thus the stability of the switching system (27) implies thatmultiagent system (6) can achieve consensus

Denote that 120585 = [119868119873minus1otimes119868119899 0

minus119868119873minus1otimes119879 119868119873minus1otimes119868119899minus119902] [

1199091

V1] By Step (2) of

Algorithm 3 system (27) is equivalent to the followingswitching system

120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)

Next we investigate consensus problem of multiagentsystem under switching interconnection topology based onconvergence analysis of the switching system (29) and presentour main result as follows

Theorem 6 For the multiagent system (3) whose interconnec-tion topology graphG

120590(119905)associated with any interval [119905

119895 119905

119895+1)

is assumed to be undirected and connected suppose that theparameter matrices 119865 119866 119879 119870 119876

1 and 119876

2used in control

protocol (4) are constructed by Steps (1)ndash(3) of Algorithm 3 andthe coupling strength 120581 is satisfied as

120581 ge1

2120582

(31)

Then the distributed control protocol (4) can guarantee thatthe multiagent system achieves consensus from any initialcondition Moreover

119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]

V119894(119905) 997888rarr 119879120603 (119905) 119894 = 1 2 119873 as 119905 997888rarr infin

(32)

Proof Till now we know that the multiagent system achievesconsensus if the state 120585 of systems (29) satisfies lim

119905rarrinfin120585 =

0 Although system (29) is switching in [0infin) it is time-invariant in any interval [119905

119894 119905

119894+1) Assume that 120590(119905) = 119901

which belongs to P Since G119901is undirected and connected

1119901

is positive definite Let 119880119901be an orthogonal matrix such

that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)

where 120582119894119901is 119894th eigenvalue of matrix

1119901The unique positive

definite solution 119875 gt 0 of Riccati equation (8) satisfies

(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)

from which we can obtain

[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)

times [119868 otimes 119860 + Λ119901otimes (120581119861119870)] le minus119868 otimes 119876 lt 0

(35)

Pre- and postmultiplying inequality (63) by119880119901otimes119868 and its

transpose respectively we get

[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)

times [119868 otimes 119860 + 1119901otimes (120581119861119870)] le minus119868 otimes 119876 lt 0

(36)

In addition for stable matrix 119865 there exist positive defi-nite matrices 119876 and 119875 satisfying the Lyapunov equation

119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)

Consider the parameter-dependent Lyapunov functionfor dynamic system (29)

119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)

where matrix has the form

= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)

with positive parameter120596 In interval [119905119894 119905

119894+1) the time deriv-

ative of this Lyapunov function along the trajectory of system(29) is

119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)

From (64) and (65) we have

119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

minus119868 otimes 119876

)

(43)

Since the constant 120596 can be chosen large enough to satisfy

120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]

times (119868 otimes 119876minus1

) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)

According to Lemma 2 we know that 119876120590is positive definite

Because there are only finite interconnection topology graphsto be switched we know that system (29) is asymptoticallystable that is system (6) achieves consensus Similarly we canprove that

119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)

The proof is now completed

Remark 7 Here the topological graph is assumed to be undi-rected for convenience If all graphs G

120590(119905)are directed and

balanced with a directed spanning tree we also have

119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)

is positive definite (see [18]) Define

120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G

119897is balanced and

has a directed spanning tree gt 0(48)

where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897

Following the similar line to analyze the directed topology asTheorem 11 in the next section it is not difficult to establishsimilar condition to guarantee that the multiagent systemachieves consensus

4 Multiagent Consensus Problemwith a Leader

In this section we consider the multiagent system consistingof119873 identical agent and a leaderThe dynamics of the follow-ing agents are described by system (3) and the dynamics ofthe leader are given as

0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)

where 1199090is the state of the leader and 119910

0is the measured

output of the leaderOur aim is to construct the distributed control protocol

for each agent to track the leader that is 119909119894rarr 119909

0 119905 rarr infin

for any 119894 = 1 2 119873 To this endwe propose a reduced-order

observer-based consensus protocol for each agent as follows

V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)


119899minus119902 is the protocol state 120581 gt 0 is the couplingstrength 119886

119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =

120573119894 if agent 119894 is connected to the leader at time 119905

0 otherwise(51)

where 120573119894is positive connected weight of edge (119894 0)

Remark 8 The leaderrsquos dynamics is only based on itself butits system matrices are the same as all following agents In[17] the authors investigated this leader-following consensusproblem by using the distributed state feedback controlprotocol In this paper we will solve the problem via the dis-tributed reduced-order observer-based protocol (50) whichneeds to be assumed that only the neighbors of leader canobtain the state information of the leader

In what follows the digraph G of order 119873 + 1 is intro-duced to model interaction topology of the leader-followingmultiagent system whose nodes V

119894 119894 = 1 2 119873 are used

to label 119873 following agents and V0is labeled leader In fact

G contains graph G which models the topology relation ofthese119873 followers and V

0with the directed edges from some

agents to the leader describes the topology relation among allagents Node V

0is said to be globally reachable if there is a

directed path from every other node to node V0in digraph G

Let 119871120590(119905)

be the Laplacian matrix of the interaction graphG

120590(119905) and let 119861

120590(119905)be an 119873 times 119873 diagonal matrix whose 119894th

diagonal element is 119889119894(119905) at time 119905 For convenience denote

that119867120590(119905)

= 119871120590(119905)+119861

120590(119905)Thematrix119867 has the following prop-

erty

Lemma 9 (see [9]) Matrix119867 = 119871 + 119861 is positive stable if andonly if node 0 is globally reachable in G

Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909

0 Then the dynamics of

120576119894and 120576

119894are described as follows

120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)

According to (7) and (53) we can obtain

120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879

120576)119879 From (52) and (54) the error dynamics can be rep-

resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0

minus119868119873otimes119879 119868119873otimes119868119899minus119902] 120585 Similarly system (55) is equiva-

lent to the following switching system

120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)

From the previous transformation we know that the sta-bility of error system (29) means that all following agents cantrack the leader First we consider fixed topology case andgive the result as follows

Theorem 10 For the leader-following multiagent systems (3)and (49) whose interconnection topology graph G is fixed andhas a globally reachable node V

0 suppose that the parameter

matrices 119865 119866 119879 119870 1198761 and 119876

2in the dynamic protocol

(50) are constructed by Steps (1)ndash(3) of Algorithm 3 and thecoupling strength 120581 is satisfied as

120581 ge1

2min119894Re (120582

119894(119867))

(58)

Then the distributed control protocol (50) guarantees that allfollowing agents can track the leader from any initial condition

Proof Of course 120590(119905) in dynamical equation can also beremoved According to Lemma 9 we know that119867 is positivestable that is all eigenvalues120582

119894(119867) of119867 satisfy Re(120582

119894(119867))gt0

Similarly we can do similarity transformation to matrix 119865which can be similar to block upper triangle matrix withdiagonal block matrix entries119860+120581120582

119894(119867)119861119870 and 119865 Based on

Step (3) of Algorithm 3 and (58) we know that119860+120581120582119894(119867)119861119870

is stable Thus the error system 120578 = 119865120578 is stableNext we probe the leader-following consensus problem

under switching interconnection topologyUnlike undirectedswitching topology assumption in the previous section weassume that the interconnection topology switches in a classof directed topologies For convenience the interconnectiontopology switches in finite set which is defined by

Γ = G119894| V

0is globally reachable node

in graph G119894and 119867119879

(G119894) + 119867(G

119894)

is positive definite 119894 isin P0

(59)

with index setP0= 1 2 119872

0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)

Noticing that the set Γ is finite set and 119867119879

(G) + 119867(G) ispositive definite we know that is well defined which ispositive and depends directly on the constants all constants119886119894119895and 120573

119894(119894 119895 = 1 2 119899) given in (5) and (51)

Theorem 11 For the leader-following multiagent system (3)and (49) whose interconnection topology graph G

120590(119905)associ-

ated with any interval [119905119895 119905

119895+1) is assumed to have a globally

reachable node V0 suppose that the parameter matrices 119865 119866

119879 119870 1198761 and 119876

2used in control protocol (50) are constructed

by Steps (1)ndash(3) of Algorithm 3 and the coupling strength 120581 issatisfied as

120581 ge1

2

(61)


Proof In time interval [119905119894 119905

119894+1) assume that 120590(119905) = 119901 isin

P0 There exists an orthogonal transformation 119880

119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879

119901is a diagonal matrixΛ

119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582

119894119901is 119894th eigenvalue of matrix119867119879

119901+ 119867

119901

According to Algorithm 3 and condition (61) we canknow that the unique solution 119875 gt 0 of Riccati equationsatisfies

119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)

form which we can obtain the following inequality

(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870)) (119868 otimes 119875) le minus119868 otimes 119876 lt 0

(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663

Figure 1 Three interconnection topology graphs

0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)

Figure 2 Trajectories 119909119894(119895)(119905) (119895 = 1 2 3) of four agents


0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)

Figure 3 Trajectories V(119895)119894(119905) and V

119894(119905) minus 119879120603(119905) of four agents

1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663


By pre- and postmultiplying (63)with119880119901otimes119868 and its transpose

respectively we have

(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)) (119868 otimes 119875) le minus119868 otimes 119876 lt 0

(64)

Since 119875119861119870 is a symmetric matrix we know that the followinginequality holds

(119868 otimes 119875) (119868 otimes 119860 + 119867119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 + 119867119901otimes (120581119861119870)) (119868 otimes 119875)

= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)

le minus119868 otimes 119876 lt 0

(65)

Similarly as the proof of the stability of system (29) in theproof of Theorem 6 we can prove that the error system 120578 =

119865120590(119905)120578 is stable

5 Simulation Example

The multiagent system contains four agents with each onemodeled by the following linear dynamics


0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)

Figure 5 Trajectories 119909119894(119895)(119905) minus 119909

0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)

The interconnection topologies are arbitrarily switchedwith period 01 s among three graphs G

119894(119894 = 1 2 3) which

are shown in Figure 1 For simplicity all nonzero weightingfactors of the graphs are taken as (1)

Each agent uses the reduced-order observer-based pro-tocol (4) Take 119865 = minus08 and 119866 = [09 01] To solvethe Sylvester equation (7) one can get 119879 = [91676 minus

46282 minus 19688] which satisfies that [ 119862119879] is nonsingu-

lar Then we know that 1198761= [

10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079

] Taking a positive definite matrix 119876 = 4119868 we have

119870 = [31532 minus 32555 minus 27856] by solving the Riccatiequation (8) Take 120581 = 10243 which satisfies condition (31)

Let 119909119894(119895)(119905) (119895 = 1 2 3) denote the 119895th component of

119909119894 The trajectories of 119909

119894(119895)(119905) are depicted in Figure 2 which

shows that the multiagent system can achieve consensusThetrajectories of V

119894(119905) and V

119894(119905) minus 119879120603(119905) are depicted in Figure 3

which is also shown that V119894(119905) rarr 119879120603(119905) as 119905 rarr infin

For simplicity we consider the group to consist of fourfollowers and one leader that is 119873 = 4 Assume that theinterconnection topologies are arbitrarily switched amongthree graphs G

120590(119894 = 1 2 3) and the communication graph

given by Figure 4 All nonzero weighting factors of the graphsare taken as (1)

Take 120581 = 186 which satisfies condition (61) All otherparameters used in protocol (50) are chosen as aforemen-tionedThe trajectories of 119909

119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted

in Figure 5 which shows that the follower agents can trackthe leader agent


6 Conclusion

In this paper we investigate the consensus problem for agroup of agents with high-dimensional linear couplingdynamics under undirected switching interaction topologyWe propose a strict proof of main result in [35] based on theJordan decomposition method and generalize the proposedprotocol to solve the consensus problem under undirectedswitching interconnection topology To solve the leader-following consensus problem we propose a neighbor-basedtrack law for each following agent with a little simplemodification to the reduced-order observer-based consensusprotocol The control parameters used in protocol can beconstructed by solving the Riccati equation and the Sylv-ester equationTheparameter-dependent commonLyapunovfunction method is involved to analyze the consensus prob-lems under undirected switching topology Since commonLyapunov function method is conservative the less con-servative method should be probed Of course more gen-eralized and interesting cases such as switching directedinteraction topology random interaction topology jointlyconnected convergence condition and the effect of timedelays arising in the communication between agents will beinvestigated in our future work

Acknowledgments

This work was supported by the National Nature ScienceFunction of China under Grant nos 61074123 and 61174063and the open project of State Key Laboratory of IndustrialControl Technology in Zhejiang University China underGrant no ICT1218

References

[1] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003

[2] T Vicsek A Czirk E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash12291995

[3] N Corson M A Aziz-Alaoui R Ghnemat S Balev andC Bertelle ldquoModeling the dynamics of complex interactionsystems from morphogenesis to controlrdquo International Journalof Bifurcation and Chaos vol 22 no 2 Article ID 1250025 20pages 2012

[4] W Ren and R W Beard Distributed Consensus in MultivehicleCooperative Control Theory and Applications Springer BerlinGermany 2008

[5] W Yu G Chen and M Cao ldquoSome necessary and sufficientconditions for second-order consensus in multi-agent dynami-cal systemsrdquo Automatica vol 46 no 6 pp 1089ndash1095 2010

[6] F Xiao and L Wang ldquoConsensus problems for high-dimensional multi-agent systemsrdquo IET Control Theory andApplications vol 1 no 3 pp 830ndash837 2007

[7] R Olfati-Saber A A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[8] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006

[9] J Hu and Y Hong ldquoLeader-following coordination of multi-agent systems with coupling time delaysrdquo Physica A vol 374no 2 pp 853ndash863 2007

[10] W Ren ldquoOn consensus algorithms for double-integratordynamicsrdquo IEEE Transactions on Automatic Control vol 53 no6 pp 1503ndash1509 2008

[11] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[12] WRen K LMoore andYChen ldquoHigh-order andmodel refer-ence consensus algorithms in cooperative control of multivehi-cle systemsrdquo ASME Journal of Dynamic Systems Measurementand Control vol 129 no 5 pp 678ndash688 2007

[13] F Jiang and L Wang ldquoConsensus seeking of high-orderdynamic multi-agent systems with fixed and switching topolo-giesrdquo International Journal of Control vol 83 no 2 pp 404ndash420 2010

[14] W Ren and R W Beard ldquoConsensus seeking in multiagentsystems under dynamically changing interaction topologiesrdquoIEEE Transactions on Automatic Control vol 50 no 5 pp 655ndash661 2005

[15] Y Hong and X Wang ldquoMulti-agent tracking of a high-dimensional active leader with switching topologyrdquo Journal ofSystems Science amp Complexity vol 22 no 4 pp 722ndash731 2009

[16] L Gao Y TangW Chen and H Zhang ldquoConsensus seeking inmulti-agent systems with an active leader and communicationdelaysrdquo Kybernetika vol 47 no 5 pp 773ndash789 2011

[17] W Ni and D Cheng ldquoLeader-following consensus of multi-agent systems under fixed and switching topologiesrdquo Systems ampControl Letters vol 59 no 3-4 pp 209ndash217 2010

[18] L Gao J Zhang and W Chen ldquoSecond-order consensus formultiagent systems under directed and switching topologiesrdquoMathematical Problems in Engineering vol 2012 Article ID273140 21 pages 2012

[19] R Yamapi H G Enjieu Kadji and G Filatrella ldquoStability of thesynchronizationmanifold in nearest neighbor nonidentical vander Pol-like oscillatorsrdquoNonlinear Dynamics vol 61 no 1-2 pp275ndash294 2010

[20] Y Chen J Lu F Han and X Yu ldquoOn the cluster consensus ofdiscrete-time multi-agent systemsrdquo Systems amp Control Lettersvol 60 no 7 pp 517ndash523 2011

[21] J Zhao M A Aziz-Alaoui and C Bertelle ldquoCluster synchro-nization analysis of complex dynamical networks by input-to-state stabilityrdquo Nonlinear Dynamics vol 70 no 2 pp 1107ndash11152012

[22] Y Sun ldquoMean square consensus for uncertain multiagentsystems with noises and delaysrdquo Abstract and Applied Analysisvol 2012 Article ID 621060 18 pages 2012

[23] H Li ldquoObserver-type consensus protocol for a class offractional-order uncertain multiagent systemsrdquo Abstract andApplied Analysis vol 2012 Article ID 672346 18 pages 2012

[24] X-R Yang and G-P Liu ldquoNecessary and sufficient consensusconditions of descriptor multi-agent systemsrdquo IEEE Transac-tions on Circuits and Systems I vol 59 no 11 pp 2669ndash26772012

[25] F Sun Z-H Guan X-S Zhan and F-S Yuan ldquoConsensus ofsecond-order and high-order discrete-timemulti-agent systemswith random networksrdquo Nonlinear Analysis Real World Appli-cations vol 13 no 5 pp 1979ndash1990 2012


[26] U Munz A Papachristodoulou and F Allgower ldquoConsensusin multi-agent systems with coupling delays and switchingtopologyrdquo IEEE Transactions on Automatic Cotrol vol 56 no12 pp 2976ndash2982 2011

[27] X Wang Y Hong J Huang and Z-P Jiang ldquoA distributedcontrol approach to a robust output regulation problem formulti-agent linear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 12 pp 2891ndash2895 2010

[28] Y Su and J Huang ldquoCooperative output regulation of linearmulti-agent systemsrdquo IEEE Transactions on Automatic Controlvol 57 no 4 pp 1062ndash1066 2012

[29] Y Hong G Chen and L Bushnell ldquoDistributed observersdesign for leader-following control of multi-agent networksrdquoAutomatica vol 44 no 3 pp 846ndash850 2008

[30] L Gao X Zhu and W Chen ldquoLeader-following consensusproblem with an accelerated motion leaderardquo InternationalJournal of Control Automation and Systems vol 10 no 5 pp931ndash939 2012

[31] S Bowong and R Yamapi ldquoAdaptive observer based synchro-nization of a class of uncertain chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 18 no 8 pp 2425ndash2435 2008

[32] Z Li Z Duan G Chen and L Huang ldquoConsensus ofmultiagent systems and synchronization of complex networksa unified viewpointrdquo IEEE Transactions on Circuits and SystemsI vol 57 no 1 pp 213ndash224 2010

[33] H Zhang F L Lewis and A Das ldquoOptimal design for synchro-nization of cooperative systems state feedback observer andoutput feedbackrdquo IEEE Transactions on Automatic Control vol56 no 8 pp 1948ndash1952 2011

[34] L Gao X ZhuW Chen andH Zhang ldquoLeader-following con-sensus of linear multi-agent systems with state-observer underswitching topologiesrdquo Mathematical Problems in Engineeringvol 2013 Article ID 873140 12 pages 2013

[35] Z Li X Liu P Lin and W Ren ldquoConsensus of linear multi-agent systems with reduced-order observer-based protocolsrdquoSystems amp Control Letters vol 60 no 7 pp 510ndash516 2011

[36] J H Seo H Shim and J Back ldquoConsensus of high-order linearsystems using dynamic output feedback compensator low gainapproachrdquo Automatica vol 45 no 11 pp 2659ndash2664 2009

[37] Z Li Z Duan and G Chen ldquoDynamic consensus of linearmulti-agent systemsrdquo IET Control Theory amp Applications vol5 no 1 pp 19ndash28 2011

[38] W Dong ldquoDistributed observer-based cooperative control ofmultiple nonholonomic mobile agentsrdquo International Journal ofSystems Science vol 43 no 5 pp 797ndash808 2012

[39] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press New York NY USA 1985

[40] C Chen Linear System Theory and Design Oxford UniversityPress New York NY USA 1999

[41] W M Wonham Linear Multivariable Control Springer NewYork NY USA 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a second-order active leader and variable interconnectiontopology A distributed consensus protocol was proposed forfirst-order agent with distributed estimation of the generalactive leaderrsquos unmeasurable state variables in [15] while [16]extended the results of [15] to the case of communicationdelays among agents under switching topology In [17] theauthors considered leader-following problem in the mul-tiagent system with general linear dynamics in both fixedtopology case and switching topology case respectively Thecooperative output regulation of linear multiagent systemscan be viewed as a generalization of some results of the leader-following consensus problem of multiagent systems [27 28]

In many practical systems the state variables cannot beobtained directly To achieve the state consensus the agenthas to estimate those unmeasurable state variables by outputvariables In [8] the authors proposed a distributed observer-based tracking protocols for each first-order following agentUnder the assumption that the active leaderrsquos velocity cannotbe measured directly [29] proposed a distributed observer-based tracking protocol for each second-order following-agent To track the accelerated motion leader [30] proposedan observer-based tracking protocol for each second-orderfollower agent to estimate the acceleration of the leaderA robust adaptive observer based on the response systemwas constructed to practically synchronize a class of uncer-tain chaotic systems [31] In [23] the author proposed anobserver-type consensus protocol to the consensus problemfor a class of fractional-order uncertain multiagent systemswith general linear dynamics For the multiagent system withgeneral linear dynamics [32] established a unified frameworkand proposed an observer-type consensus protocol and [33]proposed a framework including full state feedback controlobserver design and dynamic output feedback control forleader-following consensus problem The leader-followingconsensus problem was investigated under a class of directedswitching topologies in [34] In [35] distributed reduced-order observer-based consensus protocols were proposedfor both continuous- and discrete-time linear multiagentsystems Other observer-based previous works include [36ndash38]

Motivated by the previousworks especially by [35] we dosome further investigations on the reduced-order observer-based consensus protocol problem which had been studiedby [35] under directed fixed interconnection topology Wefirst correct some errors and propose a new proof of the mainresult established in the aforementioned paper based on theJordan decomposition method Moreover by constructing aparameter-dependent common Lyapunov function we provethat the proposed protocol can guarantee the multiagentconsensus system to achieve consensus under undirectedswitching topology Although the Lyapunov functionmethodis conservative and is not easy to be constructed it is fitto solve the problem under the switching interconnectiontopology We propose distributed protocol to solve leader-following consensus with a little simple modification to thereduced-order observer-based consensus protocol Similarlywe can prove that all following agents can track the leaderunder a class of directed interaction topologies As the specialcases the consensus conditions for balanced and undirected

interconnection topology cases can be obtained directly Al-though the leader-following consensus problem in this paperhas been studied in many papers with the aid of internalmodel principle we obtain a low-dimensional controller inour model

The paper is organized as follows In Section 2 some no-tations and preliminaries are introduced Then in Section 3and Section 4 the main results on the consensus stabilityare obtained for both leaderless and leader-following casesrespectively Following that Section 5 provides a simulationexample to illustrate the established results and finally theconcluding remarks are given in Section 6

2 Preliminaries

To make this paper more readable we first introduce somenotations and preliminaries most of which can be found in[35] Let 119877119898times119899 and 119862119898times119899 be the set of119898times 119899 real matrices andcomplex matrices respectively Re(120585) denotes the real part of120585 isin 119862 119868 is the identity matrix with compatible dimension119860

119879 and 119860119867 represent transpose and conjugate transpose ofmatrix 119860 isin 119862

119898times119899 respectively 1119899= [1 1]

119879

isin 119877119899 For

symmetric matrices 119860 and 119861 119860 gt (ge) 119861 means that 119860 minus 119861is positive (semi-) definite otimes denotes the Kronecker productwhich satisfies (119860otimes119861)(119862otimes119863) = (119860119862)otimes (119861119863) and (119860otimes119861)119879 =119860

119879

otimes 119861119879 A matrix is said to be Hurwitz stable if all of its

eigenvalues have negative real partsA weighted digraph is denoted by G = V 120576 119860 where

V = V1 V

2 V

119899 is the set of vertices 120576 sub V times V is the

set of edges and a weighted adjacency matrix 119860 = [119886119894119895] has

nonnegative elements 119886119894119895 The set of all neighbor nodes of

node V119894is defined byN

119894= 119895 | (V

119894 V

119895) isin 120576The degreematrix

119863 = 1198891 119889

2 119889

119899 isinR119899times119899 of digraphG is a diagonalmatrix

with diagonal elements 119889119894= sum

119895isinN119894119886119894119895 Then the Laplacian

matrix of G is defined as 119871 = 119863 minus 119860 isin 119877119899times119899 which satisfies1198711

119899= 0 The Laplacian matrix has following interesting

property

Lemma 1 (see [14]) The Laplacian matrix 119871 associated withweighted digraph G has at least one zero eigenvalue and allof the non-zero eigenvalues are located on the open right halfplane Furthermore 119871 has exactly one zero eigenvalue if andonly if the directed graph G has a directed spanning tree

A weighted graph is called undirected graph if for all(V

119894 V

119895) isin 120576 we have (V

119895 V

119894) isin 120576 and 119886

119894119895= 119886

119895119894 It is well known

that Laplacian matrix of weighted undirected graph is sym-metric positive semidefinite which can be derived fromLemma 1 by noticing the fact that all eigenvalues of symmet-ric matrix are real and nonnegative Furthermore Laplacianmatrix 119871 has exactly one zero eigenvalue if and only if theundirected graphG is connected

To establish our result the well-known Schur Comple-ment Lemma is introduced

Lemma 2 (see [39]) Let 119878 be a symmetric matrix of the par-titioned form 119878 = [119878

119894119895] with 119878

11isin 119877

119903times119903 11987812isin 119877

119903times(119899minus119903) and


11987822isin 119877


11987811lt 0 119878

22minus 119878

21119878minus1

1111987812lt 0 (1)

or equivalently

11987822lt 0 119878

11minus 119878

12119878minus1

2211987821lt 0 (2)



119894= 119860119909

119894+ 119861119906

119894 119910

119894= 119862119909

119894 119894 = 1 119873 (3)

where 119909119894isin 119877



119894isin 119877



V119894= 119865V

119894+ 119866119910

119894+ T119861119906

119894

119906119894= 120581119870119876

1sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895)

+ 1205811198701198762sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895)

(4)



chosen as

119886119894119895(119905) =


0 otherwise(5)


(119899minus119902)times119899 1198761isin 119877

119899times119902 and 1198762isin 119877



1G

2 G



120590(119905) Let 0 = 119905

1 119905

2 119905

3


Let 119909 = [119909119879

1 119909

119879

2 119909

119879

119873]119879 and V = [V119879

1 V119879

2 V119879

119873]119879


119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 + 119871

120590(119905)otimes (120581119861119870119876

1119862) 119871

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119871

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119871

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(6)





119879119860 minus 119865119879 = 119866119862 (7)




1isin

119877119899times119902 and 119876

2isin 119877

119899times(119899minus119902) by [1198761119876

2] = [

119862

119879]minus1


119860119879

119875 + 119875119860 minus 119875119861119861119879

119875 + 119876 = 0 (8)



120582119894 = 0Re(120582

119894))



119879] is nonsin-






119879

119871119880 = Λ = [0 0




119871119880 = [0 lowast



119894(119905) in Theorem 33 of [35]



119909119894(119905) 997888rarr 120603 (119905) ≜ (119903

119879

otimes 119890119860119905

)[[

[

1199091(0)

119909119873(0)

]]

]

= 119890119860119905

119873

sum

119895=1

119903119895119909119895(0)


(9)

where 119903 = (1199031 119903

2 119903

119873)119879


that 119903119879119871 = 0 and 1199031198791 = 1


0 0


1isin






119889119905[119909

V]

=[

[

119868119873otimes 119860 + 119869 otimes (120581119861119870119876

1119862) 119869 otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119869 otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119869 otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(10)


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (11)


119889119905[1199091

V1]

= [

[

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870119876

1119862) 119869

1otimes (120581119861119870119876

2)

119868119873minus1

otimes (119866119862) + 1198691otimes (120581119879119861119870119876

1119862) 119868

119873minus1otimes 119865 + 119869

1otimes (120581119879119861119870119876

2)

]

]

times [1199091

V1]

(12)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and v0


119894(119896) for agent 119894 as

119911119894= 119909

119894minus

119873

sum

119895=1

119903119895119909119895 119894 = 1 119873 (13)


119909119894= 119909


1

119879

1199112

119879

119911119873

119879

]119879 Then we have 119911 = ((119868

119873minus 1119903

119879

) otimes 119868119899)119909 Let

119911 = (119878minus1




119897 Set 119878 = [119908

119903 119878

1] and 119878minus1 = [ 119908

119879

119897

1198841


119897119908

119903= 1 119908119879

1198971198781= 0 and 119884

1119908

119903= 0


119903= 1205721 and 119908

119897= (1120572)119903 Then we


119878minus1


119903119908

119879

119897119878

= [0 0

0 119868119873minus1

]

(14)

Thus we have

119911 = (119878minus1

otimes 119868119899) 119911 = (119878

minus1

otimes 119868119899) ((119868

119873minus 1119903119879) otimes 119868

119899) (119878 otimes 119868

119899) 119909

= ([0 0

0 119868119873minus1

] otimes 119868119899)119909 = [

0

1199091]

(15)





= [119868119873minus1otimes119868119899 0


119879 [1199091



120577 = [

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870) 119869

1otimes (120581119861119870119876

2)

0 I119873minus1

otimes 119865

] 120577 ≜ 119865120577 (16)



119894119861119870 (119894 = 2 3 119873) and119865 By Step


(119860 + 120582119894120581119861119870)

119867

119875 + 119875 (119860 + 120582119894120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2Re (120582119894) 120581119875119861119861

119879

119875 le minus119876

(17)




1199090

(0) = 119890119860119905

(119908119879

119897otimes 119868

119899) times [119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]


119899)119909(119905) = [1205721 otimes 119868

119899 119878

1otimes 119868

119899] [

1199090(119905)

1199091(119905)

] rarr

[1205721 otimes 119868119899 119878

1otimes 119868

119899] [ 1199090(119905)

0

] = [

1205721199090(119905)

1205721199090(119905)


119909119894(119905) rarr 120572119890

119860119905

(119908119879

119897otimes 119868

119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= 119890119860119905

(119903119879

otimes

119868119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= (119903119879

otimes 119890119860119905

) [

1199091(0)119909119873(0)

] as 119905 rarr


119889

119889119905(V

0

(119905) minus 1198791199090

(119905))

= 119865V0

(119905) + 1198661198621199090

(119905) minus 1198791198601199090

(119905) = 119865 (V0

(119905) minus 1198791199090

(119905))

(18)


(V0(119905) minus

1198791199090



119894(119905) rarr 120572119879119909

0

(119905) =





with 1198801isin 119877



1198941 = 0 we have

119880119879

119871119894119880 = (

0 0

0 1119894

) = 119894 (19)





Then we can define

120582 = min119894isinP

1205822(119871

119894) | G


=max119894isinP

120582max (119871 119894) | G


(20)

where 1205822(119871


119894 SinceP is




119911119894= 119909

119894minus1

119873

119873

sum

119895=1

119909119895 (21)


119894= 119909

119895


1 119911

119879

2 119911

119879

119873]119879 Then we have119911 = (119871

119900otimes 119868

119899) 119909 (22)

where

119871119900= 119868

119873minus1

11987311198731119879119873 (23)


119873=


119880119879

119871119900119880 = 119880

119879

119880 minus1

119873119880

11987911119879119880 = [0 0

0 119868119873minus1

] ≜ 119900 (24)

Let 119909 = (119880119879

otimes 119868119899)119909 V = (119880119879


otimes


follows119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 +

120590(t) otimes (1205811198611198701198761119862)

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) +

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 +

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (26)

119889

119889119905[1199091

V1]

=[

[



]

]

times [1199091

V1]

(27)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and V0


hand we have

= (119880119879

otimes 119868119899) 119911 = (119880

119879

otimes 119868119899) (119871

119900otimes 119868

119899) 119909

= (119900otimes 119868

119899) 119909 = [

0

1199091]

(28)






1199091

V1] By Step (2) of


120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)




119895 119905

119895+1)


1 and 119876

2used in control


120581 ge1

2120582

(31)


119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(32)


119905rarrinfin120585 =


119894 119905

119894+1) Assume that 120590(119905) = 119901


1119901


that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)




(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)


[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(35)



[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(36)


119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)


119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)


= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)




119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)


119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879


)

(43)


120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11987822isin 119877


11987811lt 0 119878

22minus 119878

21119878minus1

1111987812lt 0 (1)

or equivalently

11987822lt 0 119878

11minus 119878

12119878minus1

2211987821lt 0 (2)



119894= 119860119909

119894+ 119861119906

119894 119910

119894= 119862119909

119894 119894 = 1 119873 (3)

where 119909119894isin 119877



119894isin 119877



V119894= 119865V

119894+ 119866119910

119894+ T119861119906

119894

119906119894= 120581119870119876

1sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895)

+ 1205811198701198762sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895)

(4)



chosen as

119886119894119895(119905) =


0 otherwise(5)


(119899minus119902)times119899 1198761isin 119877

119899times119902 and 1198762isin 119877



1G

2 G



120590(119905) Let 0 = 119905

1 119905

2 119905

3


Let 119909 = [119909119879

1 119909

119879

2 119909

119879

119873]119879 and V = [V119879

1 V119879

2 V119879

119873]119879


119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 + 119871

120590(119905)otimes (120581119861119870119876

1119862) 119871

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119871

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119871

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(6)





119879119860 minus 119865119879 = 119866119862 (7)




1isin

119877119899times119902 and 119876

2isin 119877

119899times(119899minus119902) by [1198761119876

2] = [

119862

119879]minus1


119860119879

119875 + 119875119860 minus 119875119861119861119879

119875 + 119876 = 0 (8)



120582119894 = 0Re(120582

119894))



119879] is nonsin-






119879

119871119880 = Λ = [0 0




119871119880 = [0 lowast



119894(119905) in Theorem 33 of [35]



119909119894(119905) 997888rarr 120603 (119905) ≜ (119903

119879

otimes 119890119860119905

)[[

[

1199091(0)

119909119873(0)

]]

]

= 119890119860119905

119873

sum

119895=1

119903119895119909119895(0)


(9)

where 119903 = (1199031 119903

2 119903

119873)119879


that 119903119879119871 = 0 and 1199031198791 = 1


0 0


1isin






119889119905[119909

V]

=[

[

119868119873otimes 119860 + 119869 otimes (120581119861119870119876

1119862) 119869 otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119869 otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119869 otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(10)


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (11)


119889119905[1199091

V1]

= [

[

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870119876

1119862) 119869

1otimes (120581119861119870119876

2)

119868119873minus1

otimes (119866119862) + 1198691otimes (120581119879119861119870119876

1119862) 119868

119873minus1otimes 119865 + 119869

1otimes (120581119879119861119870119876

2)

]

]

times [1199091

V1]

(12)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and v0


119894(119896) for agent 119894 as

119911119894= 119909

119894minus

119873

sum

119895=1

119903119895119909119895 119894 = 1 119873 (13)


119909119894= 119909


1

119879

1199112

119879

119911119873

119879

]119879 Then we have 119911 = ((119868

119873minus 1119903

119879

) otimes 119868119899)119909 Let

119911 = (119878minus1




119897 Set 119878 = [119908

119903 119878

1] and 119878minus1 = [ 119908

119879

119897

1198841


119897119908

119903= 1 119908119879

1198971198781= 0 and 119884

1119908

119903= 0


119903= 1205721 and 119908

119897= (1120572)119903 Then we


119878minus1


119903119908

119879

119897119878

= [0 0

0 119868119873minus1

]

(14)

Thus we have

119911 = (119878minus1

otimes 119868119899) 119911 = (119878

minus1

otimes 119868119899) ((119868

119873minus 1119903119879) otimes 119868

119899) (119878 otimes 119868

119899) 119909

= ([0 0

0 119868119873minus1

] otimes 119868119899)119909 = [

0

1199091]

(15)





= [119868119873minus1otimes119868119899 0


119879 [1199091



120577 = [

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870) 119869

1otimes (120581119861119870119876

2)

0 I119873minus1

otimes 119865

] 120577 ≜ 119865120577 (16)



119894119861119870 (119894 = 2 3 119873) and119865 By Step


(119860 + 120582119894120581119861119870)

119867

119875 + 119875 (119860 + 120582119894120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2Re (120582119894) 120581119875119861119861

119879

119875 le minus119876

(17)




1199090

(0) = 119890119860119905

(119908119879

119897otimes 119868

119899) times [119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]


119899)119909(119905) = [1205721 otimes 119868

119899 119878

1otimes 119868

119899] [

1199090(119905)

1199091(119905)

] rarr

[1205721 otimes 119868119899 119878

1otimes 119868

119899] [ 1199090(119905)

0

] = [

1205721199090(119905)

1205721199090(119905)


119909119894(119905) rarr 120572119890

119860119905

(119908119879

119897otimes 119868

119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= 119890119860119905

(119903119879

otimes

119868119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= (119903119879

otimes 119890119860119905

) [

1199091(0)119909119873(0)

] as 119905 rarr


119889

119889119905(V

0

(119905) minus 1198791199090

(119905))

= 119865V0

(119905) + 1198661198621199090

(119905) minus 1198791198601199090

(119905) = 119865 (V0

(119905) minus 1198791199090

(119905))

(18)


(V0(119905) minus

1198791199090



119894(119905) rarr 120572119879119909

0

(119905) =





with 1198801isin 119877



1198941 = 0 we have

119880119879

119871119894119880 = (

0 0

0 1119894

) = 119894 (19)





Then we can define

120582 = min119894isinP

1205822(119871

119894) | G


=max119894isinP

120582max (119871 119894) | G


(20)

where 1205822(119871


119894 SinceP is




119911119894= 119909

119894minus1

119873

119873

sum

119895=1

119909119895 (21)


119894= 119909

119895


1 119911

119879

2 119911

119879

119873]119879 Then we have119911 = (119871

119900otimes 119868

119899) 119909 (22)

where

119871119900= 119868

119873minus1

11987311198731119879119873 (23)


119873=


119880119879

119871119900119880 = 119880

119879

119880 minus1

119873119880

11987911119879119880 = [0 0

0 119868119873minus1

] ≜ 119900 (24)

Let 119909 = (119880119879

otimes 119868119899)119909 V = (119880119879


otimes


follows119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 +

120590(t) otimes (1205811198611198701198761119862)

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) +

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 +

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (26)

119889

119889119905[1199091

V1]

=[

[



]

]

times [1199091

V1]

(27)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and V0


hand we have

= (119880119879

otimes 119868119899) 119911 = (119880

119879

otimes 119868119899) (119871

119900otimes 119868

119899) 119909

= (119900otimes 119868

119899) 119909 = [

0

1199091]

(28)






1199091

V1] By Step (2) of


120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)




119895 119905

119895+1)


1 and 119876

2used in control


120581 ge1

2120582

(31)


119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(32)


119905rarrinfin120585 =


119894 119905

119894+1) Assume that 120590(119905) = 119901


1119901


that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)




(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)


[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(35)



[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(36)


119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)


119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)


= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)




119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)


119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879


)

(43)


120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119879

119871119880 = Λ = [0 0




119871119880 = [0 lowast



119894(119905) in Theorem 33 of [35]



119909119894(119905) 997888rarr 120603 (119905) ≜ (119903

119879

otimes 119890119860119905

)[[

[

1199091(0)

119909119873(0)

]]

]

= 119890119860119905

119873

sum

119895=1

119903119895119909119895(0)


(9)

where 119903 = (1199031 119903

2 119903

119873)119879


that 119903119879119871 = 0 and 1199031198791 = 1


0 0


1isin






119889119905[119909

V]

=[

[

119868119873otimes 119860 + 119869 otimes (120581119861119870119876

1119862) 119869 otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119869 otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119869 otimes (120581119879119861119870119876

2)

]

]

times [119909

V]

(10)


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (11)


119889119905[1199091

V1]

= [

[

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870119876

1119862) 119869

1otimes (120581119861119870119876

2)

119868119873minus1

otimes (119866119862) + 1198691otimes (120581119879119861119870119876

1119862) 119868

119873minus1otimes 119865 + 119869

1otimes (120581119879119861119870119876

2)

]

]

times [1199091

V1]

(12)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and v0


119894(119896) for agent 119894 as

119911119894= 119909

119894minus

119873

sum

119895=1

119903119895119909119895 119894 = 1 119873 (13)


119909119894= 119909


1

119879

1199112

119879

119911119873

119879

]119879 Then we have 119911 = ((119868

119873minus 1119903

119879

) otimes 119868119899)119909 Let

119911 = (119878minus1




119897 Set 119878 = [119908

119903 119878

1] and 119878minus1 = [ 119908

119879

119897

1198841


119897119908

119903= 1 119908119879

1198971198781= 0 and 119884

1119908

119903= 0


119903= 1205721 and 119908

119897= (1120572)119903 Then we


119878minus1


119903119908

119879

119897119878

= [0 0

0 119868119873minus1

]

(14)

Thus we have

119911 = (119878minus1

otimes 119868119899) 119911 = (119878

minus1

otimes 119868119899) ((119868

119873minus 1119903119879) otimes 119868

119899) (119878 otimes 119868

119899) 119909

= ([0 0

0 119868119873minus1

] otimes 119868119899)119909 = [

0

1199091]

(15)





= [119868119873minus1otimes119868119899 0


119879 [1199091



120577 = [

119868119873minus1

otimes 119860 + 1198691otimes (120581119861119870) 119869

1otimes (120581119861119870119876

2)

0 I119873minus1

otimes 119865

] 120577 ≜ 119865120577 (16)



119894119861119870 (119894 = 2 3 119873) and119865 By Step


(119860 + 120582119894120581119861119870)

119867

119875 + 119875 (119860 + 120582119894120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2Re (120582119894) 120581119875119861119861

119879

119875 le minus119876

(17)




1199090

(0) = 119890119860119905

(119908119879

119897otimes 119868

119899) times [119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]


119899)119909(119905) = [1205721 otimes 119868

119899 119878

1otimes 119868

119899] [

1199090(119905)

1199091(119905)

] rarr

[1205721 otimes 119868119899 119878

1otimes 119868

119899] [ 1199090(119905)

0

] = [

1205721199090(119905)

1205721199090(119905)


119909119894(119905) rarr 120572119890

119860119905

(119908119879

119897otimes 119868

119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= 119890119860119905

(119903119879

otimes

119868119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= (119903119879

otimes 119890119860119905

) [

1199091(0)119909119873(0)

] as 119905 rarr


119889

119889119905(V

0

(119905) minus 1198791199090

(119905))

= 119865V0

(119905) + 1198661198621199090

(119905) minus 1198791198601199090

(119905) = 119865 (V0

(119905) minus 1198791199090

(119905))

(18)


(V0(119905) minus

1198791199090



119894(119905) rarr 120572119879119909

0

(119905) =





with 1198801isin 119877



1198941 = 0 we have

119880119879

119871119894119880 = (

0 0

0 1119894

) = 119894 (19)





Then we can define

120582 = min119894isinP

1205822(119871

119894) | G


=max119894isinP

120582max (119871 119894) | G


(20)

where 1205822(119871


119894 SinceP is




119911119894= 119909

119894minus1

119873

119873

sum

119895=1

119909119895 (21)


119894= 119909

119895


1 119911

119879

2 119911

119879

119873]119879 Then we have119911 = (119871

119900otimes 119868

119899) 119909 (22)

where

119871119900= 119868

119873minus1

11987311198731119879119873 (23)


119873=


119880119879

119871119900119880 = 119880

119879

119880 minus1

119873119880

11987911119879119880 = [0 0

0 119868119873minus1

] ≜ 119900 (24)

Let 119909 = (119880119879

otimes 119868119899)119909 V = (119880119879


otimes


follows119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 +

120590(t) otimes (1205811198611198701198761119862)

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) +

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 +

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (26)

119889

119889119905[1199091

V1]

=[

[



]

]

times [1199091

V1]

(27)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and V0


hand we have

= (119880119879

otimes 119868119899) 119911 = (119880

119879

otimes 119868119899) (119871

119900otimes 119868

119899) 119909

= (119900otimes 119868

119899) 119909 = [

0

1199091]

(28)






1199091

V1] By Step (2) of


120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)




119895 119905

119895+1)


1 and 119876

2used in control


120581 ge1

2120582

(31)


119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(32)


119905rarrinfin120585 =


119894 119905

119894+1) Assume that 120590(119905) = 119901


1119901


that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)




(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)


[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(35)



[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(36)


119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)


119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)


= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)




119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)


119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879


)

(43)


120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119861119870 (119894 = 2 3 119873) and119865 By Step


(119860 + 120582119894120581119861119870)

119867

119875 + 119875 (119860 + 120582119894120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2Re (120582119894) 120581119875119861119861

119879

119875 le minus119876

(17)




1199090

(0) = 119890119860119905

(119908119879

119897otimes 119868

119899) times [119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]


119899)119909(119905) = [1205721 otimes 119868

119899 119878

1otimes 119868

119899] [

1199090(119905)

1199091(119905)

] rarr

[1205721 otimes 119868119899 119878

1otimes 119868

119899] [ 1199090(119905)

0

] = [

1205721199090(119905)

1205721199090(119905)


119909119894(119905) rarr 120572119890

119860119905

(119908119879

119897otimes 119868

119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= 119890119860119905

(119903119879

otimes

119868119899)[119909

119879

1(0) 119909

119879

2(0) 119909

119879

119873(0)]

119879

= (119903119879

otimes 119890119860119905

) [

1199091(0)119909119873(0)

] as 119905 rarr


119889

119889119905(V

0

(119905) minus 1198791199090

(119905))

= 119865V0

(119905) + 1198661198621199090

(119905) minus 1198791198601199090

(119905) = 119865 (V0

(119905) minus 1198791199090

(119905))

(18)


(V0(119905) minus

1198791199090



119894(119905) rarr 120572119879119909

0

(119905) =





with 1198801isin 119877



1198941 = 0 we have

119880119879

119871119894119880 = (

0 0

0 1119894

) = 119894 (19)





Then we can define

120582 = min119894isinP

1205822(119871

119894) | G


=max119894isinP

120582max (119871 119894) | G


(20)

where 1205822(119871


119894 SinceP is




119911119894= 119909

119894minus1

119873

119873

sum

119895=1

119909119895 (21)


119894= 119909

119895


1 119911

119879

2 119911

119879

119873]119879 Then we have119911 = (119871

119900otimes 119868

119899) 119909 (22)

where

119871119900= 119868

119873minus1

11987311198731119879119873 (23)


119873=


119880119879

119871119900119880 = 119880

119879

119880 minus1

119873119880

11987911119879119880 = [0 0

0 119868119873minus1

] ≜ 119900 (24)

Let 119909 = (119880119879

otimes 119868119899)119909 V = (119880119879


otimes


follows119889

119889119905[119909

V]

= [

[

119868119873otimes 119860 +

120590(t) otimes (1205811198611198701198761119862)

120590(119905)otimes (120581119861119870119876

2)

119868119873otimes (119866119862) +

120590(119905)otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 +

120590(119905)otimes (120581119879119861119870119876

2)

]

]

times [119909

V]


119889

119889119905[1199090

V0] = [

119860 0

119866119862 119865] [1199090

V0] (26)

119889

119889119905[1199091

V1]

=[

[



]

]

times [1199091

V1]

(27)

where 119909 = [1199090119879

1199091119879

]119879 and V = [V0119879 V1119879]119879 with 1199090 and V0


hand we have

= (119880119879

otimes 119868119899) 119911 = (119880

119879

otimes 119868119899) (119871

119900otimes 119868

119899) 119909

= (119900otimes 119868

119899) 119909 = [

0

1199091]

(28)






1199091

V1] By Step (2) of


120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)




119895 119905

119895+1)


1 and 119876

2used in control


120581 ge1

2120582

(31)


119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(32)


119905rarrinfin120585 =


119894 119905

119894+1) Assume that 120590(119905) = 119901


1119901


that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)




(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)


[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(35)



[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(36)


119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)


119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)


= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)




119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)


119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879


)

(43)


120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199091

V1] By Step (2) of


120585 = 119865120590(119905)120585 (29)

where

119865120590(119905)

= [

119868119873minus1

otimes 119860 + 1120590(119905)

otimes (120581119861119870) 1120590(119905)

otimes (1205811198611198701198762)

0 119868119873minus1

otimes 119865

]

(30)




119895 119905

119895+1)


1 and 119876

2used in control


120581 ge1

2120582

(31)


119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(32)


119905rarrinfin120585 =


119894 119905

119894+1) Assume that 120590(119905) = 119901


1119901


that

1198801199011119901119880

119879

119901= Λ

119901≜ diag 120582

1119901 120582

2119901 120582

(119873minus1)119901 (33)




(119860 + 120582119894119901120581119861119870)

119879

119875 + 119875 (119860 + 120582119894119901120581119861119870)

= minus119876 + 119875119861119861119879

119875 minus 2120582119894119901120581119875119861119861

119879

119875 le minus119876

(34)


[119868 otimes 119860 + Λ119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(35)



[119868 otimes 119860 + 1119901otimes (120581119861119870)]

119879

(119868 otimes 119875) + (119868 otimes 119875)


(36)


119865119879

119875 + 119875119865 = minus119876 (37)

or equivalently

(119868119873minus1

otimes 119865)119879

(119868119873minus1

otimes 119875) + (119868119873minus1

otimes 119875) (119868119873minus1

otimes 119865)

= minus (119868119873minus1

otimes 119876)

(38)


119881 (120585 (119905)) = 120585(119905)119879

120585 (119905) (39)


= (

1

120596119868 otimes 119875 0

0 119868 otimes 119875

) (40)




119889

119889119905119881 (120585) = 120585

119879

(119865119879

120590 + 119865

120590) 120585 ≜ 120585

119879

119876120590120585 (41)

where

119876120590= (

1

120596119876

1120590

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

119868 otimes (119865119879

119875 + 119875119865119879

)

)

1198761120590=1

120596[(119868 otimes 119860 +

1120590(119905)otimes (120581119861119870))

119879

(119868 otimes 119875)

+ (119868 otimes 119875) (119868 otimes 119860 + 1120590(119905)

otimes (120581119861119870)) ]

(42)


119876120590le (

minus1

120596119868 otimes 119876

1

1205961120590(119905)

otimes (1205811198751198611198701198762)

1

120596119879

1120590(119905)otimes (120581119875119861119870119876

2)119879


)

(43)


120596 gt 2

1205812

120582max (119876minus1

(1198751198611198701198762)119879

119876minus1

(1198751198611198701198762)) (44)


this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










this implies that

minus1

120596(119868 otimes 119876) +

1

1205962[

1120590(119905)otimes (120581119875119861119870119876

2)]


) [119879

1120590(119905)otimes (120581119875119861119870119876

2)119879

] lt 0

(45)



119909119894(119905) 997888rarr 120603 (119905) ≜ 119890

119860119905 [

[

1

119873

119873

sum

119895=1

119909119895(0)]

]


(46)





119880119879

119871120590(119905) 119880 = (

0 0

0 1120590(119905)

) (47)

where 1120590(119905)


120582 = min119897isinP

1

21205822(119871

119879

119897+ 119871

119897) | G



where 1205822(119871

119879

119897+ 119871


119897+ 119871

119897




0= 119860119909

0 119910

0= 119862119909

0 119909

0isin 119877

119899

1199100isin 119877

119902

(49)


0is the measured



0 119905 rarr infin



V119894= 119865V

119894+ 119866119910

119894+ 119879119861119906

119894

119906119894= 120581119870119876

1

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (119910

119894minus 119910

119895) + 119889

119894(119905) (119910

119894minus 119910

0)]

]

+ 1205811198701198762

[

[

sum

119895isinN119894(119905)

119886119894119895(119905) (V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

(50)



119894119895(119905) is chosen by (5) and 119889

119894(119905) is chosen by

119889119894(119905) =


0 otherwise(51)




119894 119894 = 1 2 119873 are used







Let 119871120590(119905)


120590(119905) and let 119861



that119867120590(119905)

= 119871120590(119905)+119861


erty


Let 120576119894= 119909

119894minus 119909

0 and 120576

119894= V

119894minus 119879119909


120576119894and 120576


120576119894= 119860120576

119894+ 120581119861119870119876

1119862 sum

119895isin119873119894(119905)

119886119894119895(119905) [(120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(52)


120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120576119894= V

119894minus 119879

0= 119865V

119894+ 119866119862119909

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119909

119894minus 119909

119895) + 119889

119894(119905) (119909

119894minus 119909

0)]

]

+ 1205811198701198762

times [

[

sum

119895isin119873119894(119905)

119886119894119895(V

119894minus V

119895) + 119889

119894(119905) (V

119894minus 119879119909

0)]

]

minus 1198791198601199090

(53)


120576119894= 119865120576

119894+ 119866119862120576

119894+ 120581119861119870119876

1119862

times [

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) 120576

119894

]

]

+ 1205811198701198762

[

[

sum

119895isin119873119894(119905)

119886119894119895(119905) (120576

119894minus 120576

119895) + 119889

119894(119905) (120576

119894)]

]

(54)

Let 120576 = (1205761198791 120576

119879

2 120576

119879

119873)119879 120576 = (120576119879

1 120576

119879

2 120576

119879

119873)119879 and 120585119879 =

(120576119879


resented as

120585 =

[

[

119868119873otimes 119860 +119867

120590otimes (120581119861119870119876

1119862) 119867

120590otimes (120581119861119870119876

2)

119868119873otimes (119866119862) + 119867

120590otimes (120581119879119861119870119876

1119862) 119868

119873otimes 119865 + 119867

120590otimes (120581119879119861119870119876

2)

]

]

120585

(55)

Let 120578 = [ 119868119873otimes119868119899 0



120578 = 119865120590(119905)120578 (56)

where

119865120590(119905)

= [119868119873otimes 119860 + 119867

120590otimes (120581119861119870) 119867

120590otimes (120581119861119870119876

2)

0 119868119873otimes 119865

] (57)




matrices 119865 119866 119879 119870 1198761 and 119876



120581 ge1

2min119894Re (120582

119894(119867))

(58)



119894(119867) of119867 satisfy Re(120582

119894(119867))gt0


119894(119867)119861119870 and 119865 Based on




Γ = G119894| V


in graph G119894and 119867119879

(G119894) + 119867(G

119894)


(59)


0

Therefore define

= minGisinΓ

120582 (119867119879

(G) + 119867(G)) (60)



119894(119894 119895 = 1 2 119899) given in (5) and (51)


120590(119905)associ-




119879 119870 1198761 and 119876



120581 ge1

2

(61)



119894+1) assume that 120590(119905) = 119901 isin


119901such that

119880119901(119867

119879

119901+119867

119901)119880

119879


119901= diag120582

1119901 120582

2119901

120582119899119901 where 120582


119901+ 119867

119901


119875(119860 +1

2120582119894119901120581119861119870)

119879

+ (119860 +1

2120582119894119901120581119861119870)119875

= minus119876 + 119875119861119861119879

119875 minus 120582119894119901120581119875119862

119879

119862119875 le minus119876

(62)


(119868 otimes 119875) (119868 otimes 119860 +1

2Λ

119901otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2Λ


(63)


1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

32

4

(a) 1198661

1

32

4

(b) 1198662

1

32

4

(c) 1198663


0 5 10 151

2

3

4

5

6

7

8

9

10

Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(1)

for a

ll ag

ents

(a)

2

3

4

5

6

7

8

9

10

11

12

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(2)

for a

ll ag

ents

(b)

3

4

5

6

7

8

9

10

11

12

13

0 5 10 15Time

Agent 1Agent 2

Agent 3Agent 4

Traj

ecto

ries o

f119909119894(3)

for a

ll ag

ents

(c)



0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

Agent 1Agent 2

Agent 3Agent 4

0 5 10 15Time

minus15

minus10

minus5

Traj

ecto

ries o

f119894

for a

ll ag

ents

(a)

0 5 10 15

0

5

10

15

20

Time

minus10

minus5

1 minus 119879120603

2 minus 119879120603

3 minus 119879120603

4 minus 119879120603

Traj

ecto

ries o

f119894minus119879120603

for a

ll ag

ents

(b)



1

3

0

2

4

(a) 1198661

1

3

0

2

4

(b) 1198662

1

3

0

2

4

(c) 1198663




(119868 otimes 119875) (119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870))

119879

+ (119868 otimes 119860 +1

2(119867

119879

119901+ 119867


(64)



119879


= (119868 otimes 119875) [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)]

119879

+ [119868 otimes 119860 +1

2(119867

119879

119901+ 119867

119901) otimes (120581119861119870)] (119868 otimes 119875)


(65)


119865120590(119905)120578 is stable




0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(1)minus1199090(1)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(a)

0 5 10 15

0

5

10

15

minus15

minus10

minus5

Traj

ecto

ries119909119894(2)minus1199090(2)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(b)

10

0

2

4

6

8

minus10

minus8

minus6

minus4

minus2

0 5 10 15

Traj

ecto

ries119909119894(3)minus1199090(3)

of fo

ur ag

ents

Agent 1Agent 2

Agent 3Agent 4

119905 (s)

(c)


0(119895)(119895 = 1 2 3) of four agents

119894(119905) = [

[

minus21622 11469 06921

minus20497 09585 08159

minus19817 11558 05047

]

]

119909119894(119905) + [

[

0

0

1

]

]

119906119894(119905)

119910119894(119905) = [

1 0 0

0 1 0] 119909

119894(119905)

(66)


119894(119894 = 1 2 3) which





10000 0

0 10000

46564 minus23508

] and 1198762=

[0

0

minus05079







119894(119905) and V







119894(119895)minus119909

0(119895) 119895 = 1 2 3 are depicted



6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Conclusion


Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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