Research ArticleFast Consensus Tracking of Multiagent Systems with DiverseCommunication Delays and Input Delays

Chun-xi Yang1 Wei-xing Hong1 Ling-yun Huang2 and Hua Wang3

1 Faculty of Chemical Engineering Ministry of Education and Faculty of Metallurgical and Energy EngineeringKunming University of Science and Technology Kunming 650500 China

2 Faculty of Land Resource Engineering Ministry of Education and Faculty of Metallurgical and Energy EngineeringKunming University of Science and Technology Kunming 650500 China

3 Engineering Research Center of Metallurgical Energy Conservation amp Emission ReductionMinistry of Education and Faculty of Metallurgical and Energy Engineering Kunming University of Science and TechnologyKunming 650500 China

Correspondence should be addressed to Ling-yun Huang hly0510126com

Received 20 November 2013 Accepted 6 February 2014 Published 19 March 2014

Academic Editor Huaicheng Yan

Copyright copy 2014 Chun-xi Yang 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

The consensus tracking problem for discrete-timemultiagent systems with input and communication delays is studied A sufficientcondition is obtained over a directed graph based on the frequency-domain analysis Furthermore a fast decentralized consensustracking conditions based on increment PID algorithm are discussed for improving convergence speed of the multiagent systemsBased on this result genetic algorithm is introduced to construct increment PID based on genetic algorithm for obtainingoptimization consensus tracking performance Finally a numerable example is given to compare convergence speed of threetracking algorithms in the same condition Simulation results show the effectiveness of the proposed algorithm

1 Introduction

As an effective method to solve decentralized multiagentcooperative control which is widely applied into manyfields such as flocking [1 2] formation control [3 4] andunmanned air vehicles [5] consensus algorithms designingof multiagent systems has attracted great attention in recentyears A key task for consensus algorithms is to achieve aglobal common behavior through designing a distributedprotocol based on local information Reference [6] proposeda simple model for phase transition of self-driven of themodel A simple consensus protocol to solve the averageconsensus problem was discussed in [7] Furthermore twosurvey papers which introduce the basic concepts of con-sensus of multiagent systems the methods of convergenceand performance analysis for the protocols and recent devel-opment can be seen in [8 9] In real applications whenlocal information data travel along channels in a multiagentnetwork a communication delay exists due to the physical

characteristics of the medium transmitting the informationAnd at the same time each agent also needs computing timeto process its information As a result time-delay problemincluding communication delays and computing delays (alsocalled input delays) is not avoided in designing consensusalgorithms Reference [10] discusses the consensus problemformultiagent systems with input and communication delaysbased on the frequency-domain in discrete-time formulationand a conclusionwhere the consensus condition is dependenton input delays but independent of communication delays isobtained

Convergence rate or speed is an important performanceindex in the analysis of consensus problems For examplesensors need to reach fast consensus on the estimates betweensensor observing intervals in distributed estimation problemIn this field main research works focus on fast consensus[11ndash13] and finite-time consensus [14ndash16] Reference [11] pro-poses a new consensus protocol which considers the average

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 253806 10 pageshttpdxdoiorg1011552014253806

2 Mathematical Problems in Engineering

information of the agentsrsquo states in a certain time intervaland increases consensus speed ofmultiagent systems throughdetermining suitable upper limit of time interval based onthe frequently domain analysis and matrix theory Reference[12] proposes a class of pinning predictive controllers for con-sensus networks to substantially increase their convergencespeed towards consensus In [13] an optimal synchronizationprotocol was designed for the fastest convergence speedwhenthe protocol is perturbed by an additive measurement andprocess noise As to finite-time consensus algorithms design[14] designed continuous distributed control algorithms fordouble integrators leaderless and leader-follower multiagentsystems with external disturbances based on the finite-timecontrol technique Based on positive or negative values oferrors between their neighborrsquos values and their own state val-ues in multiagent systems a simple distributed continuous-time protocol is introduced by [15] that guarantees finite-time consensus in networks of autonomous agents when thenetwork has directed switching network topologies and time-delayed communications

Although fast or finite-time consensus without a virtualleader is interesting it is sometimes more meaningful andinteresting to study consensus tracking problem when thevirtual leaderrsquos state (also called reference) may representthe state of interest for these systems In [17] a coordinatedtracking algorithm with a time-varying leader for first-orderdynamics is proposed and bounded control and directedswitching interaction topologies are considered when a time-varying consensus reference state is available to only a subsetof a team However this algorithm requires the estimatesof the neighborsrsquo velocities In [18] distributed coordinatedtracking algorithms are studied when only partial measure-ments of the states of the virtual leader and the followersare available Reference [19] studies the issues associatedwith distributed coordinated tracking formultiple networkedEuler-Lagrange systems where only a subset of the followershas access to the leader As to discrete-time formulation[20] considers consensus tracking problem when locationinformation of the active leader is completely known but theacceleration information may not be measured a neighbor-based pinning control law and a neighbor-based state estima-tion rule are proposed Although consensus tracking problemis discussed widely few works focus on fast consensustracking with time-delays in discrete-time formulation

Motivated by these topics fast consensus tracking prob-lemof discrete-timemultiagent systemswith communicationdelays and input delays is discussed in this paper The maincontribution of this paper is to establish a simple protocol inorder to guarantee consensus tracking convergence in generaldirected network topology based on the frequency-domainanalysis Then an increment PID algorithm is introducedto improve the convergent speed and an inequity condi-tion which can describe relations of controller gain inputdelays communications delays and topology structure isobtained Furthermore genetic algorithm [21] is introducedto construct increment PID based on genetic algorithmfor obtaining optimization consensus tracking performanceThis makes the proposed protocol more practical for applica-tion to real-time applications

This paper is organized as follows In the next sec-tion preliminary notions and multiagent systems modelare provided A conventional 119875-like discrete-time consensustracking algorithm for a single-integrator systems is statedin Section 3 and a fast discrete-time consensus trackingalgorithmbased on increment PID is established in Section 4By applying to genetic algorithms the fast consensus trackingalgorithm mentioned above is optimized in order to obtainan optimal cost in Section 5 Simulation example is shown inSection 6 Finally concluding remarks are stated in Section 7

2 Preliminaries and MultiagentSystems Modeling

21 Graph Theory Notions

Notations The notation used in this paper for graph theoryis quite standard For a system with 119873 agents the commu-nication graph among these agents is modeled by a directedweighted graph 119866 = (119881119882119860) where 119881 = V

1 V2 V

119899

119882 sube 1198812 and 119860 = 119886

119894119895 isin R119899times119899 represent the set of

agents the edge set and the weighted adjacency matrixrespectively The agent indexes belong to a finite index setΦ = 1 2 119873 An edge denoted as (V

119894 V119895) means that

the V119895th agent can access the information of the V

119894th agent

We assume that the adjacency elements associated with theedges of the digraph are positive That means 119886

119894119895gt 0 if

agent 119894 receives information from agent 119895 otherwise 119886119894119895= 0

Moreover we assume feedback gain 119886119894119894gt 0 for all 119894 isin Φ if 119894th

agent has feedback control loop and 119886119894119894= 0 otherwise Define

the set of neighbors of agent V119894as119873119894= V119895isin 119881 (V

119894 V119895) isin 119882

For the directed digraph the outdegree of agent 119894 is definedas degout(V119894) = sum

119899

119895=1119886119894119895 Let 119863 be the degree matrix of 119866

which is defined as a diagonal matrix with the degree of eachagent along its diagonalThe Laplacianmatrix of theweighteddigraph is defined as 119871 = 119863 minus 119860 satisfying zero-row sum

In multiagent systems each agent can be considered asa node in a digraph and the information flow between twoagents can be regarded as a directed path between the nodesThus a directed graph has a directed spanning tree if thereexists at least one agent called a globally reachable agent thathas a directed path to all other agents

22 Multiagent System Modeling Consider agents with asingle-integrator kinematics in discrete-time formulationgiven by

119909119894(119896 + 1) = 119909

119894(119896) + 119906

119894(119896) 119894 isin Φ (1)

where 119909119894isin R and 119906

119894isin R denote the state and the

control input of agent 119894 respectivelyThe following consensustracking protocol for the multiagent systems (1) is a classi-cal formulation mentioned by literature [10] which can bedescribed by

119906119894(119896) = minus 119892

119894(0)119886119894119894(119909119894(119896) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896) minus 119909

119894(119896))

(2)

Mathematical Problems in Engineering 3

where 120577119903(119896) is a time-varying reference state or a virtual leaderwith the states named agent 0 and the other agents indexedby 1 2 119873 are referred to as followers without loss ofgenerality (Especially if 120577119903(119896) = 120577119903 this reference state canbe simplified to a constant one) 119892

119894(0)is 1 if agent 119894 has access

to 120577119903(119896) and 0 otherwise 119873119894denotes the neighbors of agent

119894 and 119886119894119895gt 0 is the adjacency element of 119860 in the directed

digraph119866 = (119881119882119860) 119886119894119894gt 0 denotes the feedback control

gain of agent 119894 and 119886119894119894= 0 otherwise

When agent 119894 is subjected to a time-varying input delay119879119894119894(119905) system (1) can be rewritten as follows

119909119894(119896 + 1) = 119909

119894(119896) + 119906

119894(119896 minus 119879

119894119894(119905)) 119894 isin Φ (3)

Consider the total delay where an agent receives datafrom its neighbors is sum of time-varying input delay andtime-varying diverse communication delay so the consensustracking protocol becomes

119906119894(119896 minus 119879

119894119894(119905))

= minus119892119894(0)119886119894119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119879

119895119895(119905) minus 119879

119894119895(119905)) minus 119909

119894(119896 minus 119879

119894119894(119905)))

(4)

where119879119894119894(119905) 119879119895119895(119905) denotes time-varying input delays of agent

119894 119895 and 119879119894119895(119905) denotes time-varying communication delay

from agent 119895 to agent 119894 respectively It is assumed that eachagent has similar computer capacity so the time-varyinginput delay of each agent can be treated with the sametime-delay value that is 119879

119894119894(119905) = 119879

119895119895(119905) To simplify the

complexity of calculation we assume that agent 119894 needs topossess memory capability such that 119909

119894(119896minus119879

119894119894(119905) minus119879

119894119895(119905)) can

be used in the consensus tracking protocol Substitute state119909119894(119896 minus119879

119894119894(119905)) in coupling terms of (4) for 119909

119894(119896 minus119879

119894119894(119905) minus119879

119894119895(119905))

and let the total delay 119894119895(119905) = 119879

119894119894(119905) + 119879

119894119895(119905) which satisfies

119879119894119894(119905) le

119894119895(119905) As a result (4) can be rewritten as

119906119894(119896 minus 119879

119894119894(119905)) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus

119894119895(119905)) minus 119909

119894(119896 minus

119894119895(119905)))

(5)

Moreover these two classes of delays can be approximatedby119879119894119894(119905) = (119898

119894minus1)119879+120576

119894and 119894119895(119905) = (119899

119894119895minus1)119879+120576

119894119895 respectively

Where 119879119894119894(119905) 119894119895(119905) ge 0 119879 denotes sample period of this

discrete-time system 119898119894 119899119894119895are all nonnegative integers and

120576119894 120576119894119895are unknown-but-bounded variables which belong to

interval (0 119879) So it is reasonable that 119879119894119894(119905) and

119894119895(119905) are

approximated by 119898119894and 119899

119894119895although some artificial delays

are included In the end (4) can be rewritten as

119906119894(119896 minus 119898

119894) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119898

119894) minus 120577119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895))

(6)

Substituting protocol (6) to the system (3) we have

119909119894(119896 + 1) = 119909

119894(119896) minus 119892

119894(0)119870119894[119909119894(119896 minus 119898

119894) minus 120577119903(119896)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 isin Φ

(7)

Using algorithm (6) each agent essentially updates its nextstate based on its past state with limited time delay and itsneighborsrsquo current as well as the referencersquos current if thereference is a neighbor of the agent As a result (6) can beeasily implemented in practice

3 119875-Like Discrete-Time ConsensusTracking with Input Delays andCommunication Delays

In this section we consider consensus tracking problem ofmultiagent systems with both communication delays andinput delays Firstly two lemmas related to this topic needto be introduced Then a sufficient consensus trackingcondition of multiagent systems (7) with conventional 119875-like algorithmn is proposed based on the frequency-domainanalysis and matrix theory

Lemma 1 (Gershgorinrsquos disk theorem) LetΛ = (119886119894119895) be a119873times

119873 complex matrix then all eigenvalues of matrix Λ belong tothe union set of119873 circular disc on the complex plane that is

119863119894(Λ) = 119911|

1003816100381610038161003816119911 minus 1198861198941198941003816100381610038161003816 le 119877119894 (119894 = 1 2 119873) (8)

where 119877119894= |1198861198941| + |1198861198942| + sdot sdot sdot |119886

119894119894minus1| + |119886119894119894+1| + sdot sdot sdot + |119886

119894119873|

Lemma 2 (see [10]) The following inequalitysin (((2119863 + 1) 2) 120596)

sin (1205962)le 2119863 + 1 (9)

holds for all nonnegative integers 119863 and all 120596 isin [minus120587 120587]

In the following we apply Lemmas 1 and 2 to derive ourmain result

Theorem 3 Consider multiagent systems (3) with algorithm(6) Assume that the interconnection topology digraph119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receivereference information Then the system achieves a consensustracking asymptotically if

119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 1 (10)

where 119870119894119894= 119892119894(0)119870119894denotes the feedback control gain of agent

119894 119870119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree of agent 119894

Proof Themultiagent systems of (3) with (6) are given by (7)Taking the 119911-transformation of the system (7) we get

119911119883119894(119911) = 119883

119894(119911) minus 119892

119894(0)119870119894[119911minus119898119894119883119894(119911) minus 120577

119903(119911)]

+ sum

V119895isin119873119894

119886119894119895[119911minus119899119894119895119883119895(119911) minus 119911

minus119899119894119895119883119894(119911)]

(11)

4 Mathematical Problems in Engineering

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911) =

minus119886119894119895119911minus119899119894119895 V

119895isin 119873119894

119892119894(0)119870119894119911minus119898119894 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119894 = 119895

0 otherwise

(12)

and 119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = diag(119892

1(0)1198701

119892119873(0)119870119873) and then (11) becomes

119911119883 (119911) = 119883 (119911) + 119876 (1 otimes 120577119903(119911)) minus (119911)119883 (119911) (13)

Let 120577119903(119911) denote input signal and let 119883(119911) denote outputsignal then the transform function is denoted by

119866 (119911) =119883 (119911)

120577119903 (119911)=

119876

(119911 minus 1) 119868 + (119911)

(14)

and correspondent characteristic equation of mulitagentsystem (13) is 119901(119911) = det[119868 + (1(119911 minus 1))(119911)] Then we willprove that all the zeros of 119901(119911) have modulus less than unityin the following

Based on the general Nyquist stability criterion themodulus of all roots satisfying 119901(119911) = 0 should be less thanunity if all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596)do not enclose the point (minus1 1198950) for120596 isin [minus120587 120587] By Lemma 1all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119892119894(0)119870119894119890minus119895120596119898119894 + sumV119895isin119873119894 119886119894119895119890

minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(15)

To simplify we define 119870119894119894= 119892119894(0)119870119894 119870119894119895= sumV119895isin119873119894 119886119894119895 and

let

119866119894(120596) =

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1 (16)

It is easy to see that 119866119894(120596) is the center of the disc 119863

119894

Thus the point (minus1 1198950) does not enclose in disc 119863119894for all

120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 As aresult when 120596 isin [minus120587 120587] 119886 ge 1 we have1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then this inequality can be rewritten as1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(119886 minus

119870119894119894 sin (((2119898119894 + 1) 2) 120596) + 119870119894119895 sin (((2119899119894119895 + 1) 2) 120596)sin (1205962)

)

gt 0

(18)

According to Lemma 2 we have sin(((2119898119894+ 1)2)120596)

sin(1205962) le 2119898119894+ 1 sin(((2119899

119894119895+ 1)2)120596) sin(1205962) le 2119899

119894119895+ 1

for 120596 isin [minus120587 120587] Then the following inequality is obtained by

119870119894119894sin (((2119898

119894+ 1) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

sin (1205962)

le 119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(19)

Let 119886 = 1 then all disc 119863119894do not enclose the point

(minus1 1198950) As a result multiagent systems (7) can achievea consensus tracking asymptotically That is end of thisproof

Remark 4 If we rewrite (19) as (119870119894119894+119870119894119895)(2119898119894+1)+2119870

119894119895(119899119894119895minus

119898119894) lt 1 it is easy to know that consensus tracking problem

of multiagent systems is more sensitive to input delays thancommunication delays Moreover Theorem 3 is also suitablefor the case when it only has input delays if we let119898

119894= 119899119894119895

4 Fast Discrete-Time Consensus TrackingAlgorithm Based on Increment 119875119868119863

Considering feedback control gains 119870119894 119894 = 1 2 119899 is the

similar with conventional 119875-like controllers an incrementPID algorithm is proposed consequently to accelerate theconvergence speed of multiagent systems (7) Discrete PIDalgorithm is described by

ℎ119894(119896) = 119870

119901[

[

119890119894(119896) +

119879

119879119894

119896

sum

119895=0

119890119894(119895) + 119879

119889

119890119894(119896) minus 119890

119894(119896 minus 1)

119879

]

]

(20)

where ℎ119894(119896) is feedback control signal of agent 119894 at time

interval 119896 119890119894(119896) denotes the error of current state between

agent 119894 and current reference state and 119890119894(119896 minus 1) denotes

the error of the value between agent 119894 and reference stateat time interval 119896 minus 1 respectively 119870

119901denotes proportional

coefficient 119879119894denotes integral time 119879

119889denotes derivative

time and sampling period is described by 119879Through ℎ

119894(119896) minus ℎ

119894(119896 minus 1) we obtain increment PID

algorithm as

Δℎ119894(119896) = ℎ

119894(119896) minus ℎ

119894(119896 minus 1)

= 119860119890119894(119896) minus 119861119890

119894(119896 minus 1) + 119862119890

119894(119896 minus 2)

(21)

where 119860 = 119870119901(1 + 119879119879119894+ 119879119889119879) 119861 = 119870

119901(1 + 2(119879

119889119879)) and

119862 = 119870119901(119879119889119879)

Remark 5 From (21) we know that if sampling period 119879and coefficient 119860 119861 119862 are chosen control signal Δℎ

119894(119896) will

be obtained by only using three adjacent deviation valuesBecause this algorithm is easily realized in the agent withlimited computing capacity it is very suitable for multiagentsystem

Mathematical Problems in Engineering 5

Let 119890119894(119896 minus 119898

119894) = 119909

119894(119896 minus 119898

119894) minus 120577119903(119896) in (7) and substitute

feedback control gains 119870119894into increment PID algorithm as

(21) (7) can be rewritten by

119909119894(119896 + 1)

= 119909119894(119896)

minus 119892119894(0)[119860119890119894(119896 minus 119898

119894)

minus119861119890119894(119896 minus 119898

119894minus 1) + 119862119890

119894(119896 minus 119898

119894minus 2)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 119895 isin Φ

(22)

Then we obtain sufficient fast consensus tracking con-dition of multiagent system (22) based on increment PIDalgorithm as follows

Theorem 6 Consider multiagent systems (3) with algorithm(5) Assume that the interconnection topology digraph 119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receive ref-erence information Then the system achieves a fast consensustracking asymptotically if

119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5)

+ 119870119894119895(2119899119894119895+ 1) le 1 119894 119895 isin Φ

(23)

where119860 = 119892119894(0)119870119901(1+119879119879

119894+119879119889119879) 119861 = 119892

119894(0)119870119901(1+2(119879

119889119879))

and 119862 = 119892119894(0)119870119901(119879119889119879) and119870

119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree

of agent 119894

Proof The multiagent systems of (3) with (5) and (21) aregiven by (22) Taking the 119911-transformation of the system (22)we get

119911119883119894(119911) = 119883

119894(119911) minus (119860119911

minus119898119894 + 119861119911minus119898119894minus1 + 119862119911

minus119898119894minus2)119883119894(119911)

+ (119860 + 119861119911minus1+ 119862119911minus2) 120577119903(119911)

+ sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 [119883

119895(119911) minus 119883

119894(119911)]

(24)

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911)

=

minus119886119894119895119911minus119899 V

119895isin 119873119894

119860119911minus119898119894 + 119861119911

minus119898119894minus1 + 119862119911minus119898119894minus2 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119895 = 119894

0 otherwise(25)

and119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = 119860 + 119861119911minus1 + 119862119911minus2 then

(24) becomes

119911119883 (119911) = 119883 (119911) + 119876120577119903(119911) 1119873minus (119911)119883 (119911) (26)

Let 120577119903(119911) denote input and let 119883(119911) denote output and 1119873

as unit column vector with 119873 times 1 dimensions then thecharacteristic equation of system (26) becomes119901(119911) = det[119868+(1(119911 minus 1))(119911)] In consequence we will prove the all rootsof 119901(119911) = 0 whose module is less than unity

According to general Nyquist stability criterion modulusof all roots satisfied 119901(119911) = 0 should be less than unity ifall poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596) do notenclose the point (minus1 1198950) for 120596 isin [minus120587 120587] By Lemma 1 allpoles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + sumV119895isin119873119894

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(27)

Moreover we have

119863119894

=

1003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

lt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(28)

where119870119894119895= sumV119895isin119873119894 119886119894119895

Define 119866119894(120596) = (119860119890

minus119895120596119898119894 + 119861119890minus119895120596(119898119894+1) + 119862119890

minus119895120596(119898119894+2) +

119870119894119895119890minus119895120596119899119894119895)(119890

119895120596minus 1) it is easy to see that 119866

119894(120596) is center of

disc 119863119894 Then the point (minus1 1198950) does not enclose in disc 119863

119894

for all 120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 Asa result when isin [minus120587 120587] we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(29)

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

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Mathematical Problems in Engineering

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

information of the agentsrsquo states in a certain time intervaland increases consensus speed ofmultiagent systems throughdetermining suitable upper limit of time interval based onthe frequently domain analysis and matrix theory Reference[12] proposes a class of pinning predictive controllers for con-sensus networks to substantially increase their convergencespeed towards consensus In [13] an optimal synchronizationprotocol was designed for the fastest convergence speedwhenthe protocol is perturbed by an additive measurement andprocess noise As to finite-time consensus algorithms design[14] designed continuous distributed control algorithms fordouble integrators leaderless and leader-follower multiagentsystems with external disturbances based on the finite-timecontrol technique Based on positive or negative values oferrors between their neighborrsquos values and their own state val-ues in multiagent systems a simple distributed continuous-time protocol is introduced by [15] that guarantees finite-time consensus in networks of autonomous agents when thenetwork has directed switching network topologies and time-delayed communications

Although fast or finite-time consensus without a virtualleader is interesting it is sometimes more meaningful andinteresting to study consensus tracking problem when thevirtual leaderrsquos state (also called reference) may representthe state of interest for these systems In [17] a coordinatedtracking algorithm with a time-varying leader for first-orderdynamics is proposed and bounded control and directedswitching interaction topologies are considered when a time-varying consensus reference state is available to only a subsetof a team However this algorithm requires the estimatesof the neighborsrsquo velocities In [18] distributed coordinatedtracking algorithms are studied when only partial measure-ments of the states of the virtual leader and the followersare available Reference [19] studies the issues associatedwith distributed coordinated tracking formultiple networkedEuler-Lagrange systems where only a subset of the followershas access to the leader As to discrete-time formulation[20] considers consensus tracking problem when locationinformation of the active leader is completely known but theacceleration information may not be measured a neighbor-based pinning control law and a neighbor-based state estima-tion rule are proposed Although consensus tracking problemis discussed widely few works focus on fast consensustracking with time-delays in discrete-time formulation

Motivated by these topics fast consensus tracking prob-lemof discrete-timemultiagent systemswith communicationdelays and input delays is discussed in this paper The maincontribution of this paper is to establish a simple protocol inorder to guarantee consensus tracking convergence in generaldirected network topology based on the frequency-domainanalysis Then an increment PID algorithm is introducedto improve the convergent speed and an inequity condi-tion which can describe relations of controller gain inputdelays communications delays and topology structure isobtained Furthermore genetic algorithm [21] is introducedto construct increment PID based on genetic algorithmfor obtaining optimization consensus tracking performanceThis makes the proposed protocol more practical for applica-tion to real-time applications

This paper is organized as follows In the next sec-tion preliminary notions and multiagent systems modelare provided A conventional 119875-like discrete-time consensustracking algorithm for a single-integrator systems is statedin Section 3 and a fast discrete-time consensus trackingalgorithmbased on increment PID is established in Section 4By applying to genetic algorithms the fast consensus trackingalgorithm mentioned above is optimized in order to obtainan optimal cost in Section 5 Simulation example is shown inSection 6 Finally concluding remarks are stated in Section 7

2 Preliminaries and MultiagentSystems Modeling

21 Graph Theory Notions

Notations The notation used in this paper for graph theoryis quite standard For a system with 119873 agents the commu-nication graph among these agents is modeled by a directedweighted graph 119866 = (119881119882119860) where 119881 = V

1 V2 V

119899

119882 sube 1198812 and 119860 = 119886

119894119895 isin R119899times119899 represent the set of

agents the edge set and the weighted adjacency matrixrespectively The agent indexes belong to a finite index setΦ = 1 2 119873 An edge denoted as (V

119894 V119895) means that

the V119895th agent can access the information of the V

119894th agent

We assume that the adjacency elements associated with theedges of the digraph are positive That means 119886

119894119895gt 0 if

agent 119894 receives information from agent 119895 otherwise 119886119894119895= 0

Moreover we assume feedback gain 119886119894119894gt 0 for all 119894 isin Φ if 119894th

agent has feedback control loop and 119886119894119894= 0 otherwise Define

the set of neighbors of agent V119894as119873119894= V119895isin 119881 (V

119894 V119895) isin 119882

For the directed digraph the outdegree of agent 119894 is definedas degout(V119894) = sum

119899

119895=1119886119894119895 Let 119863 be the degree matrix of 119866

which is defined as a diagonal matrix with the degree of eachagent along its diagonalThe Laplacianmatrix of theweighteddigraph is defined as 119871 = 119863 minus 119860 satisfying zero-row sum

In multiagent systems each agent can be considered asa node in a digraph and the information flow between twoagents can be regarded as a directed path between the nodesThus a directed graph has a directed spanning tree if thereexists at least one agent called a globally reachable agent thathas a directed path to all other agents

22 Multiagent System Modeling Consider agents with asingle-integrator kinematics in discrete-time formulationgiven by

119909119894(119896 + 1) = 119909

119894(119896) + 119906

119894(119896) 119894 isin Φ (1)

where 119909119894isin R and 119906

119894isin R denote the state and the

control input of agent 119894 respectivelyThe following consensustracking protocol for the multiagent systems (1) is a classi-cal formulation mentioned by literature [10] which can bedescribed by

119906119894(119896) = minus 119892

119894(0)119886119894119894(119909119894(119896) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896) minus 119909

119894(119896))

(2)

Mathematical Problems in Engineering 3

where 120577119903(119896) is a time-varying reference state or a virtual leaderwith the states named agent 0 and the other agents indexedby 1 2 119873 are referred to as followers without loss ofgenerality (Especially if 120577119903(119896) = 120577119903 this reference state canbe simplified to a constant one) 119892

119894(0)is 1 if agent 119894 has access

to 120577119903(119896) and 0 otherwise 119873119894denotes the neighbors of agent

119894 and 119886119894119895gt 0 is the adjacency element of 119860 in the directed

digraph119866 = (119881119882119860) 119886119894119894gt 0 denotes the feedback control

gain of agent 119894 and 119886119894119894= 0 otherwise

When agent 119894 is subjected to a time-varying input delay119879119894119894(119905) system (1) can be rewritten as follows

119909119894(119896 + 1) = 119909

119894(119896) + 119906

119894(119896 minus 119879

119894119894(119905)) 119894 isin Φ (3)

Consider the total delay where an agent receives datafrom its neighbors is sum of time-varying input delay andtime-varying diverse communication delay so the consensustracking protocol becomes

119906119894(119896 minus 119879

119894119894(119905))

= minus119892119894(0)119886119894119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119879

119895119895(119905) minus 119879

119894119895(119905)) minus 119909

119894(119896 minus 119879

119894119894(119905)))

(4)

where119879119894119894(119905) 119879119895119895(119905) denotes time-varying input delays of agent

119894 119895 and 119879119894119895(119905) denotes time-varying communication delay

from agent 119895 to agent 119894 respectively It is assumed that eachagent has similar computer capacity so the time-varyinginput delay of each agent can be treated with the sametime-delay value that is 119879

119894119894(119905) = 119879

119895119895(119905) To simplify the

complexity of calculation we assume that agent 119894 needs topossess memory capability such that 119909

119894(119896minus119879

119894119894(119905) minus119879

119894119895(119905)) can

be used in the consensus tracking protocol Substitute state119909119894(119896 minus119879

119894119894(119905)) in coupling terms of (4) for 119909

119894(119896 minus119879

119894119894(119905) minus119879

119894119895(119905))

and let the total delay 119894119895(119905) = 119879

119894119894(119905) + 119879

119894119895(119905) which satisfies

119879119894119894(119905) le

119894119895(119905) As a result (4) can be rewritten as

119906119894(119896 minus 119879

119894119894(119905)) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus

119894119895(119905)) minus 119909

119894(119896 minus

119894119895(119905)))

(5)

Moreover these two classes of delays can be approximatedby119879119894119894(119905) = (119898

119894minus1)119879+120576

119894and 119894119895(119905) = (119899

119894119895minus1)119879+120576

119894119895 respectively

Where 119879119894119894(119905) 119894119895(119905) ge 0 119879 denotes sample period of this

discrete-time system 119898119894 119899119894119895are all nonnegative integers and

120576119894 120576119894119895are unknown-but-bounded variables which belong to

interval (0 119879) So it is reasonable that 119879119894119894(119905) and

119894119895(119905) are

approximated by 119898119894and 119899

119894119895although some artificial delays

are included In the end (4) can be rewritten as

119906119894(119896 minus 119898

119894) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119898

119894) minus 120577119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895))

(6)

Substituting protocol (6) to the system (3) we have

119909119894(119896 + 1) = 119909

119894(119896) minus 119892

119894(0)119870119894[119909119894(119896 minus 119898

119894) minus 120577119903(119896)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 isin Φ

(7)

Using algorithm (6) each agent essentially updates its nextstate based on its past state with limited time delay and itsneighborsrsquo current as well as the referencersquos current if thereference is a neighbor of the agent As a result (6) can beeasily implemented in practice

3 119875-Like Discrete-Time ConsensusTracking with Input Delays andCommunication Delays

In this section we consider consensus tracking problem ofmultiagent systems with both communication delays andinput delays Firstly two lemmas related to this topic needto be introduced Then a sufficient consensus trackingcondition of multiagent systems (7) with conventional 119875-like algorithmn is proposed based on the frequency-domainanalysis and matrix theory

Lemma 1 (Gershgorinrsquos disk theorem) LetΛ = (119886119894119895) be a119873times

119873 complex matrix then all eigenvalues of matrix Λ belong tothe union set of119873 circular disc on the complex plane that is

119863119894(Λ) = 119911|

1003816100381610038161003816119911 minus 1198861198941198941003816100381610038161003816 le 119877119894 (119894 = 1 2 119873) (8)

where 119877119894= |1198861198941| + |1198861198942| + sdot sdot sdot |119886

119894119894minus1| + |119886119894119894+1| + sdot sdot sdot + |119886

119894119873|

Lemma 2 (see [10]) The following inequalitysin (((2119863 + 1) 2) 120596)

sin (1205962)le 2119863 + 1 (9)

holds for all nonnegative integers 119863 and all 120596 isin [minus120587 120587]

In the following we apply Lemmas 1 and 2 to derive ourmain result

Theorem 3 Consider multiagent systems (3) with algorithm(6) Assume that the interconnection topology digraph119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receivereference information Then the system achieves a consensustracking asymptotically if

119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 1 (10)

where 119870119894119894= 119892119894(0)119870119894denotes the feedback control gain of agent

119894 119870119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree of agent 119894

Proof Themultiagent systems of (3) with (6) are given by (7)Taking the 119911-transformation of the system (7) we get

119911119883119894(119911) = 119883

119894(119911) minus 119892

119894(0)119870119894[119911minus119898119894119883119894(119911) minus 120577

119903(119911)]

+ sum

V119895isin119873119894

119886119894119895[119911minus119899119894119895119883119895(119911) minus 119911

minus119899119894119895119883119894(119911)]

(11)

4 Mathematical Problems in Engineering

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911) =

minus119886119894119895119911minus119899119894119895 V

119895isin 119873119894

119892119894(0)119870119894119911minus119898119894 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119894 = 119895

0 otherwise

(12)

and 119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = diag(119892

1(0)1198701

119892119873(0)119870119873) and then (11) becomes

119911119883 (119911) = 119883 (119911) + 119876 (1 otimes 120577119903(119911)) minus (119911)119883 (119911) (13)

Let 120577119903(119911) denote input signal and let 119883(119911) denote outputsignal then the transform function is denoted by

119866 (119911) =119883 (119911)

120577119903 (119911)=

119876

(119911 minus 1) 119868 + (119911)

(14)

and correspondent characteristic equation of mulitagentsystem (13) is 119901(119911) = det[119868 + (1(119911 minus 1))(119911)] Then we willprove that all the zeros of 119901(119911) have modulus less than unityin the following

Based on the general Nyquist stability criterion themodulus of all roots satisfying 119901(119911) = 0 should be less thanunity if all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596)do not enclose the point (minus1 1198950) for120596 isin [minus120587 120587] By Lemma 1all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119892119894(0)119870119894119890minus119895120596119898119894 + sumV119895isin119873119894 119886119894119895119890

minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(15)

To simplify we define 119870119894119894= 119892119894(0)119870119894 119870119894119895= sumV119895isin119873119894 119886119894119895 and

let

119866119894(120596) =

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1 (16)

It is easy to see that 119866119894(120596) is the center of the disc 119863

119894

Thus the point (minus1 1198950) does not enclose in disc 119863119894for all

120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 As aresult when 120596 isin [minus120587 120587] 119886 ge 1 we have1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then this inequality can be rewritten as1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(119886 minus

119870119894119894 sin (((2119898119894 + 1) 2) 120596) + 119870119894119895 sin (((2119899119894119895 + 1) 2) 120596)sin (1205962)

)

gt 0

(18)

According to Lemma 2 we have sin(((2119898119894+ 1)2)120596)

sin(1205962) le 2119898119894+ 1 sin(((2119899

119894119895+ 1)2)120596) sin(1205962) le 2119899

119894119895+ 1

for 120596 isin [minus120587 120587] Then the following inequality is obtained by

119870119894119894sin (((2119898

119894+ 1) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

sin (1205962)

le 119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(19)

Let 119886 = 1 then all disc 119863119894do not enclose the point

(minus1 1198950) As a result multiagent systems (7) can achievea consensus tracking asymptotically That is end of thisproof

Remark 4 If we rewrite (19) as (119870119894119894+119870119894119895)(2119898119894+1)+2119870

119894119895(119899119894119895minus

119898119894) lt 1 it is easy to know that consensus tracking problem

of multiagent systems is more sensitive to input delays thancommunication delays Moreover Theorem 3 is also suitablefor the case when it only has input delays if we let119898

119894= 119899119894119895

4 Fast Discrete-Time Consensus TrackingAlgorithm Based on Increment 119875119868119863

Considering feedback control gains 119870119894 119894 = 1 2 119899 is the

similar with conventional 119875-like controllers an incrementPID algorithm is proposed consequently to accelerate theconvergence speed of multiagent systems (7) Discrete PIDalgorithm is described by

ℎ119894(119896) = 119870

119901[

[

119890119894(119896) +

119879

119879119894

119896

sum

119895=0

119890119894(119895) + 119879

119889

119890119894(119896) minus 119890

119894(119896 minus 1)

119879

]

]

(20)

where ℎ119894(119896) is feedback control signal of agent 119894 at time

interval 119896 119890119894(119896) denotes the error of current state between

agent 119894 and current reference state and 119890119894(119896 minus 1) denotes

the error of the value between agent 119894 and reference stateat time interval 119896 minus 1 respectively 119870

119901denotes proportional

coefficient 119879119894denotes integral time 119879

119889denotes derivative

time and sampling period is described by 119879Through ℎ

119894(119896) minus ℎ

119894(119896 minus 1) we obtain increment PID

algorithm as

Δℎ119894(119896) = ℎ

119894(119896) minus ℎ

119894(119896 minus 1)

= 119860119890119894(119896) minus 119861119890

119894(119896 minus 1) + 119862119890

119894(119896 minus 2)

(21)

where 119860 = 119870119901(1 + 119879119879119894+ 119879119889119879) 119861 = 119870

119901(1 + 2(119879

119889119879)) and

119862 = 119870119901(119879119889119879)

Remark 5 From (21) we know that if sampling period 119879and coefficient 119860 119861 119862 are chosen control signal Δℎ

119894(119896) will

be obtained by only using three adjacent deviation valuesBecause this algorithm is easily realized in the agent withlimited computing capacity it is very suitable for multiagentsystem

Mathematical Problems in Engineering 5

Let 119890119894(119896 minus 119898

119894) = 119909

119894(119896 minus 119898

119894) minus 120577119903(119896) in (7) and substitute

feedback control gains 119870119894into increment PID algorithm as

(21) (7) can be rewritten by

119909119894(119896 + 1)

= 119909119894(119896)

minus 119892119894(0)[119860119890119894(119896 minus 119898

119894)

minus119861119890119894(119896 minus 119898

119894minus 1) + 119862119890

119894(119896 minus 119898

119894minus 2)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 119895 isin Φ

(22)

Then we obtain sufficient fast consensus tracking con-dition of multiagent system (22) based on increment PIDalgorithm as follows

Theorem 6 Consider multiagent systems (3) with algorithm(5) Assume that the interconnection topology digraph 119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receive ref-erence information Then the system achieves a fast consensustracking asymptotically if

119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5)

+ 119870119894119895(2119899119894119895+ 1) le 1 119894 119895 isin Φ

(23)

where119860 = 119892119894(0)119870119901(1+119879119879

119894+119879119889119879) 119861 = 119892

119894(0)119870119901(1+2(119879

119889119879))

and 119862 = 119892119894(0)119870119901(119879119889119879) and119870

119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree

of agent 119894

Proof The multiagent systems of (3) with (5) and (21) aregiven by (22) Taking the 119911-transformation of the system (22)we get

119911119883119894(119911) = 119883

119894(119911) minus (119860119911

minus119898119894 + 119861119911minus119898119894minus1 + 119862119911

minus119898119894minus2)119883119894(119911)

+ (119860 + 119861119911minus1+ 119862119911minus2) 120577119903(119911)

+ sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 [119883

119895(119911) minus 119883

119894(119911)]

(24)

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911)

=

minus119886119894119895119911minus119899 V

119895isin 119873119894

119860119911minus119898119894 + 119861119911

minus119898119894minus1 + 119862119911minus119898119894minus2 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119895 = 119894

0 otherwise(25)

and119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = 119860 + 119861119911minus1 + 119862119911minus2 then

(24) becomes

119911119883 (119911) = 119883 (119911) + 119876120577119903(119911) 1119873minus (119911)119883 (119911) (26)

Let 120577119903(119911) denote input and let 119883(119911) denote output and 1119873

as unit column vector with 119873 times 1 dimensions then thecharacteristic equation of system (26) becomes119901(119911) = det[119868+(1(119911 minus 1))(119911)] In consequence we will prove the all rootsof 119901(119911) = 0 whose module is less than unity

According to general Nyquist stability criterion modulusof all roots satisfied 119901(119911) = 0 should be less than unity ifall poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596) do notenclose the point (minus1 1198950) for 120596 isin [minus120587 120587] By Lemma 1 allpoles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + sumV119895isin119873119894

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(27)

Moreover we have

119863119894

=

1003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

lt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(28)

where119870119894119895= sumV119895isin119873119894 119886119894119895

Define 119866119894(120596) = (119860119890

minus119895120596119898119894 + 119861119890minus119895120596(119898119894+1) + 119862119890

minus119895120596(119898119894+2) +

119870119894119895119890minus119895120596119899119894119895)(119890

119895120596minus 1) it is easy to see that 119866

119894(120596) is center of

disc 119863119894 Then the point (minus1 1198950) does not enclose in disc 119863

119894

for all 120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 Asa result when isin [minus120587 120587] we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(29)

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where 120577119903(119896) is a time-varying reference state or a virtual leaderwith the states named agent 0 and the other agents indexedby 1 2 119873 are referred to as followers without loss ofgenerality (Especially if 120577119903(119896) = 120577119903 this reference state canbe simplified to a constant one) 119892

119894(0)is 1 if agent 119894 has access

to 120577119903(119896) and 0 otherwise 119873119894denotes the neighbors of agent

119894 and 119886119894119895gt 0 is the adjacency element of 119860 in the directed

digraph119866 = (119881119882119860) 119886119894119894gt 0 denotes the feedback control

gain of agent 119894 and 119886119894119894= 0 otherwise

When agent 119894 is subjected to a time-varying input delay119879119894119894(119905) system (1) can be rewritten as follows

119909119894(119896 + 1) = 119909

119894(119896) + 119906

119894(119896 minus 119879

119894119894(119905)) 119894 isin Φ (3)

Consider the total delay where an agent receives datafrom its neighbors is sum of time-varying input delay andtime-varying diverse communication delay so the consensustracking protocol becomes

119906119894(119896 minus 119879

119894119894(119905))

= minus119892119894(0)119886119894119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119879

119895119895(119905) minus 119879

119894119895(119905)) minus 119909

119894(119896 minus 119879

119894119894(119905)))

(4)

where119879119894119894(119905) 119879119895119895(119905) denotes time-varying input delays of agent

119894 119895 and 119879119894119895(119905) denotes time-varying communication delay

from agent 119895 to agent 119894 respectively It is assumed that eachagent has similar computer capacity so the time-varyinginput delay of each agent can be treated with the sametime-delay value that is 119879

119894119894(119905) = 119879

119895119895(119905) To simplify the

complexity of calculation we assume that agent 119894 needs topossess memory capability such that 119909

119894(119896minus119879

119894119894(119905) minus119879

119894119895(119905)) can

be used in the consensus tracking protocol Substitute state119909119894(119896 minus119879

119894119894(119905)) in coupling terms of (4) for 119909

119894(119896 minus119879

119894119894(119905) minus119879

119894119895(119905))

and let the total delay 119894119895(119905) = 119879

119894119894(119905) + 119879

119894119895(119905) which satisfies

119879119894119894(119905) le

119894119895(119905) As a result (4) can be rewritten as

119906119894(119896 minus 119879

119894119894(119905)) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119879

119894119894(119905)) minus 120577

119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus

119894119895(119905)) minus 119909

119894(119896 minus

119894119895(119905)))

(5)

Moreover these two classes of delays can be approximatedby119879119894119894(119905) = (119898

119894minus1)119879+120576

119894and 119894119895(119905) = (119899

119894119895minus1)119879+120576

119894119895 respectively

Where 119879119894119894(119905) 119894119895(119905) ge 0 119879 denotes sample period of this

discrete-time system 119898119894 119899119894119895are all nonnegative integers and

120576119894 120576119894119895are unknown-but-bounded variables which belong to

interval (0 119879) So it is reasonable that 119879119894119894(119905) and

119894119895(119905) are

approximated by 119898119894and 119899

119894119895although some artificial delays

are included In the end (4) can be rewritten as

119906119894(119896 minus 119898

119894) = minus 119892

119894(0)119870119894(119909119894(119896 minus 119898

119894) minus 120577119903(119896))

+ sum

V119895isin119873119894

119886119894119895(119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895))

(6)

Substituting protocol (6) to the system (3) we have

119909119894(119896 + 1) = 119909

119894(119896) minus 119892

119894(0)119870119894[119909119894(119896 minus 119898

119894) minus 120577119903(119896)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 isin Φ

(7)

Using algorithm (6) each agent essentially updates its nextstate based on its past state with limited time delay and itsneighborsrsquo current as well as the referencersquos current if thereference is a neighbor of the agent As a result (6) can beeasily implemented in practice

3 119875-Like Discrete-Time ConsensusTracking with Input Delays andCommunication Delays

In this section we consider consensus tracking problem ofmultiagent systems with both communication delays andinput delays Firstly two lemmas related to this topic needto be introduced Then a sufficient consensus trackingcondition of multiagent systems (7) with conventional 119875-like algorithmn is proposed based on the frequency-domainanalysis and matrix theory

Lemma 1 (Gershgorinrsquos disk theorem) LetΛ = (119886119894119895) be a119873times

119873 complex matrix then all eigenvalues of matrix Λ belong tothe union set of119873 circular disc on the complex plane that is

119863119894(Λ) = 119911|

1003816100381610038161003816119911 minus 1198861198941198941003816100381610038161003816 le 119877119894 (119894 = 1 2 119873) (8)

where 119877119894= |1198861198941| + |1198861198942| + sdot sdot sdot |119886

119894119894minus1| + |119886119894119894+1| + sdot sdot sdot + |119886

119894119873|

Lemma 2 (see [10]) The following inequalitysin (((2119863 + 1) 2) 120596)

sin (1205962)le 2119863 + 1 (9)

holds for all nonnegative integers 119863 and all 120596 isin [minus120587 120587]

In the following we apply Lemmas 1 and 2 to derive ourmain result

Theorem 3 Consider multiagent systems (3) with algorithm(6) Assume that the interconnection topology digraph119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receivereference information Then the system achieves a consensustracking asymptotically if

119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 1 (10)

where 119870119894119894= 119892119894(0)119870119894denotes the feedback control gain of agent

119894 119870119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree of agent 119894

Proof Themultiagent systems of (3) with (6) are given by (7)Taking the 119911-transformation of the system (7) we get

119911119883119894(119911) = 119883

119894(119911) minus 119892

119894(0)119870119894[119911minus119898119894119883119894(119911) minus 120577

119903(119911)]

+ sum

V119895isin119873119894

119886119894119895[119911minus119899119894119895119883119895(119911) minus 119911

minus119899119894119895119883119894(119911)]

(11)

4 Mathematical Problems in Engineering

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911) =

minus119886119894119895119911minus119899119894119895 V

119895isin 119873119894

119892119894(0)119870119894119911minus119898119894 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119894 = 119895

0 otherwise

(12)

and 119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = diag(119892

1(0)1198701

119892119873(0)119870119873) and then (11) becomes

119911119883 (119911) = 119883 (119911) + 119876 (1 otimes 120577119903(119911)) minus (119911)119883 (119911) (13)

Let 120577119903(119911) denote input signal and let 119883(119911) denote outputsignal then the transform function is denoted by

119866 (119911) =119883 (119911)

120577119903 (119911)=

119876

(119911 minus 1) 119868 + (119911)

(14)

and correspondent characteristic equation of mulitagentsystem (13) is 119901(119911) = det[119868 + (1(119911 minus 1))(119911)] Then we willprove that all the zeros of 119901(119911) have modulus less than unityin the following

Based on the general Nyquist stability criterion themodulus of all roots satisfying 119901(119911) = 0 should be less thanunity if all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596)do not enclose the point (minus1 1198950) for120596 isin [minus120587 120587] By Lemma 1all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119892119894(0)119870119894119890minus119895120596119898119894 + sumV119895isin119873119894 119886119894119895119890

minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(15)

To simplify we define 119870119894119894= 119892119894(0)119870119894 119870119894119895= sumV119895isin119873119894 119886119894119895 and

let

119866119894(120596) =

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1 (16)

It is easy to see that 119866119894(120596) is the center of the disc 119863

119894

Thus the point (minus1 1198950) does not enclose in disc 119863119894for all

120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 As aresult when 120596 isin [minus120587 120587] 119886 ge 1 we have1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then this inequality can be rewritten as1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(119886 minus

119870119894119894 sin (((2119898119894 + 1) 2) 120596) + 119870119894119895 sin (((2119899119894119895 + 1) 2) 120596)sin (1205962)

)

gt 0

(18)

According to Lemma 2 we have sin(((2119898119894+ 1)2)120596)

sin(1205962) le 2119898119894+ 1 sin(((2119899

119894119895+ 1)2)120596) sin(1205962) le 2119899

119894119895+ 1

for 120596 isin [minus120587 120587] Then the following inequality is obtained by

119870119894119894sin (((2119898

119894+ 1) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

sin (1205962)

le 119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(19)

Let 119886 = 1 then all disc 119863119894do not enclose the point

(minus1 1198950) As a result multiagent systems (7) can achievea consensus tracking asymptotically That is end of thisproof

Remark 4 If we rewrite (19) as (119870119894119894+119870119894119895)(2119898119894+1)+2119870

119894119895(119899119894119895minus

119898119894) lt 1 it is easy to know that consensus tracking problem

of multiagent systems is more sensitive to input delays thancommunication delays Moreover Theorem 3 is also suitablefor the case when it only has input delays if we let119898

119894= 119899119894119895

4 Fast Discrete-Time Consensus TrackingAlgorithm Based on Increment 119875119868119863

Considering feedback control gains 119870119894 119894 = 1 2 119899 is the

similar with conventional 119875-like controllers an incrementPID algorithm is proposed consequently to accelerate theconvergence speed of multiagent systems (7) Discrete PIDalgorithm is described by

ℎ119894(119896) = 119870

119901[

[

119890119894(119896) +

119879

119879119894

119896

sum

119895=0

119890119894(119895) + 119879

119889

119890119894(119896) minus 119890

119894(119896 minus 1)

119879

]

]

(20)

where ℎ119894(119896) is feedback control signal of agent 119894 at time

interval 119896 119890119894(119896) denotes the error of current state between

agent 119894 and current reference state and 119890119894(119896 minus 1) denotes

the error of the value between agent 119894 and reference stateat time interval 119896 minus 1 respectively 119870

119901denotes proportional

coefficient 119879119894denotes integral time 119879

119889denotes derivative

time and sampling period is described by 119879Through ℎ

119894(119896) minus ℎ

119894(119896 minus 1) we obtain increment PID

algorithm as

Δℎ119894(119896) = ℎ

119894(119896) minus ℎ

119894(119896 minus 1)

= 119860119890119894(119896) minus 119861119890

119894(119896 minus 1) + 119862119890

119894(119896 minus 2)

(21)

where 119860 = 119870119901(1 + 119879119879119894+ 119879119889119879) 119861 = 119870

119901(1 + 2(119879

119889119879)) and

119862 = 119870119901(119879119889119879)

Remark 5 From (21) we know that if sampling period 119879and coefficient 119860 119861 119862 are chosen control signal Δℎ

119894(119896) will

be obtained by only using three adjacent deviation valuesBecause this algorithm is easily realized in the agent withlimited computing capacity it is very suitable for multiagentsystem

Mathematical Problems in Engineering 5

Let 119890119894(119896 minus 119898

119894) = 119909

119894(119896 minus 119898

119894) minus 120577119903(119896) in (7) and substitute

feedback control gains 119870119894into increment PID algorithm as

(21) (7) can be rewritten by

119909119894(119896 + 1)

= 119909119894(119896)

minus 119892119894(0)[119860119890119894(119896 minus 119898

119894)

minus119861119890119894(119896 minus 119898

119894minus 1) + 119862119890

119894(119896 minus 119898

119894minus 2)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 119895 isin Φ

(22)

Then we obtain sufficient fast consensus tracking con-dition of multiagent system (22) based on increment PIDalgorithm as follows

Theorem 6 Consider multiagent systems (3) with algorithm(5) Assume that the interconnection topology digraph 119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receive ref-erence information Then the system achieves a fast consensustracking asymptotically if

119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5)

+ 119870119894119895(2119899119894119895+ 1) le 1 119894 119895 isin Φ

(23)

where119860 = 119892119894(0)119870119901(1+119879119879

119894+119879119889119879) 119861 = 119892

119894(0)119870119901(1+2(119879

119889119879))

and 119862 = 119892119894(0)119870119901(119879119889119879) and119870

119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree

of agent 119894

Proof The multiagent systems of (3) with (5) and (21) aregiven by (22) Taking the 119911-transformation of the system (22)we get

119911119883119894(119911) = 119883

119894(119911) minus (119860119911

minus119898119894 + 119861119911minus119898119894minus1 + 119862119911

minus119898119894minus2)119883119894(119911)

+ (119860 + 119861119911minus1+ 119862119911minus2) 120577119903(119911)

+ sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 [119883

119895(119911) minus 119883

119894(119911)]

(24)

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911)

=

minus119886119894119895119911minus119899 V

119895isin 119873119894

119860119911minus119898119894 + 119861119911

minus119898119894minus1 + 119862119911minus119898119894minus2 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119895 = 119894

0 otherwise(25)

and119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = 119860 + 119861119911minus1 + 119862119911minus2 then

(24) becomes

119911119883 (119911) = 119883 (119911) + 119876120577119903(119911) 1119873minus (119911)119883 (119911) (26)

Let 120577119903(119911) denote input and let 119883(119911) denote output and 1119873

as unit column vector with 119873 times 1 dimensions then thecharacteristic equation of system (26) becomes119901(119911) = det[119868+(1(119911 minus 1))(119911)] In consequence we will prove the all rootsof 119901(119911) = 0 whose module is less than unity

According to general Nyquist stability criterion modulusof all roots satisfied 119901(119911) = 0 should be less than unity ifall poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596) do notenclose the point (minus1 1198950) for 120596 isin [minus120587 120587] By Lemma 1 allpoles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + sumV119895isin119873119894

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(27)

Moreover we have

119863119894

=

1003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

lt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(28)

where119870119894119895= sumV119895isin119873119894 119886119894119895

Define 119866119894(120596) = (119860119890

minus119895120596119898119894 + 119861119890minus119895120596(119898119894+1) + 119862119890

minus119895120596(119898119894+2) +

119870119894119895119890minus119895120596119899119894119895)(119890

119895120596minus 1) it is easy to see that 119866

119894(120596) is center of

disc 119863119894 Then the point (minus1 1198950) does not enclose in disc 119863

119894

for all 120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 Asa result when isin [minus120587 120587] we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(29)

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

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Mathematical Problems in Engineering

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911) =

minus119886119894119895119911minus119899119894119895 V

119895isin 119873119894

119892119894(0)119870119894119911minus119898119894 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119894 = 119895

0 otherwise

(12)

and 119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = diag(119892

1(0)1198701

119892119873(0)119870119873) and then (11) becomes

119911119883 (119911) = 119883 (119911) + 119876 (1 otimes 120577119903(119911)) minus (119911)119883 (119911) (13)

Let 120577119903(119911) denote input signal and let 119883(119911) denote outputsignal then the transform function is denoted by

119866 (119911) =119883 (119911)

120577119903 (119911)=

119876

(119911 minus 1) 119868 + (119911)

(14)

and correspondent characteristic equation of mulitagentsystem (13) is 119901(119911) = det[119868 + (1(119911 minus 1))(119911)] Then we willprove that all the zeros of 119901(119911) have modulus less than unityin the following

Based on the general Nyquist stability criterion themodulus of all roots satisfying 119901(119911) = 0 should be less thanunity if all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596)do not enclose the point (minus1 1198950) for120596 isin [minus120587 120587] By Lemma 1all poles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119892119894(0)119870119894119890minus119895120596119898119894 + sumV119895isin119873119894 119886119894119895119890

minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(15)

To simplify we define 119870119894119894= 119892119894(0)119870119894 119870119894119895= sumV119895isin119873119894 119886119894119895 and

let

119866119894(120596) =

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1 (16)

It is easy to see that 119866119894(120596) is the center of the disc 119863

119894

Thus the point (minus1 1198950) does not enclose in disc 119863119894for all

120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 As aresult when 120596 isin [minus120587 120587] 119886 ge 1 we have1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

Then this inequality can be rewritten as1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119870119894119894119890minus119895120596119898119894 + 119870119894119895119890

minus119895120596119899119894119895

119890119895120596minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(119886 minus

119870119894119894 sin (((2119898119894 + 1) 2) 120596) + 119870119894119895 sin (((2119899119894119895 + 1) 2) 120596)sin (1205962)

)

gt 0

(18)

According to Lemma 2 we have sin(((2119898119894+ 1)2)120596)

sin(1205962) le 2119898119894+ 1 sin(((2119899

119894119895+ 1)2)120596) sin(1205962) le 2119899

119894119895+ 1

for 120596 isin [minus120587 120587] Then the following inequality is obtained by

119870119894119894sin (((2119898

119894+ 1) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

sin (1205962)

le 119870119894119894(2119898119894+ 1) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(19)

Let 119886 = 1 then all disc 119863119894do not enclose the point

(minus1 1198950) As a result multiagent systems (7) can achievea consensus tracking asymptotically That is end of thisproof

Remark 4 If we rewrite (19) as (119870119894119894+119870119894119895)(2119898119894+1)+2119870

119894119895(119899119894119895minus

119898119894) lt 1 it is easy to know that consensus tracking problem

of multiagent systems is more sensitive to input delays thancommunication delays Moreover Theorem 3 is also suitablefor the case when it only has input delays if we let119898

119894= 119899119894119895

4 Fast Discrete-Time Consensus TrackingAlgorithm Based on Increment 119875119868119863

Considering feedback control gains 119870119894 119894 = 1 2 119899 is the

similar with conventional 119875-like controllers an incrementPID algorithm is proposed consequently to accelerate theconvergence speed of multiagent systems (7) Discrete PIDalgorithm is described by

ℎ119894(119896) = 119870

119901[

[

119890119894(119896) +

119879

119879119894

119896

sum

119895=0

119890119894(119895) + 119879

119889

119890119894(119896) minus 119890

119894(119896 minus 1)

119879

]

]

(20)

where ℎ119894(119896) is feedback control signal of agent 119894 at time

interval 119896 119890119894(119896) denotes the error of current state between

agent 119894 and current reference state and 119890119894(119896 minus 1) denotes

the error of the value between agent 119894 and reference stateat time interval 119896 minus 1 respectively 119870

119901denotes proportional

coefficient 119879119894denotes integral time 119879

119889denotes derivative

time and sampling period is described by 119879Through ℎ

119894(119896) minus ℎ

119894(119896 minus 1) we obtain increment PID

algorithm as

Δℎ119894(119896) = ℎ

119894(119896) minus ℎ

119894(119896 minus 1)

= 119860119890119894(119896) minus 119861119890

119894(119896 minus 1) + 119862119890

119894(119896 minus 2)

(21)

where 119860 = 119870119901(1 + 119879119879119894+ 119879119889119879) 119861 = 119870

119901(1 + 2(119879

119889119879)) and

119862 = 119870119901(119879119889119879)

Remark 5 From (21) we know that if sampling period 119879and coefficient 119860 119861 119862 are chosen control signal Δℎ

119894(119896) will

be obtained by only using three adjacent deviation valuesBecause this algorithm is easily realized in the agent withlimited computing capacity it is very suitable for multiagentsystem

Mathematical Problems in Engineering 5

Let 119890119894(119896 minus 119898

119894) = 119909

119894(119896 minus 119898

119894) minus 120577119903(119896) in (7) and substitute

feedback control gains 119870119894into increment PID algorithm as

(21) (7) can be rewritten by

119909119894(119896 + 1)

= 119909119894(119896)

minus 119892119894(0)[119860119890119894(119896 minus 119898

119894)

minus119861119890119894(119896 minus 119898

119894minus 1) + 119862119890

119894(119896 minus 119898

119894minus 2)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 119895 isin Φ

(22)

Then we obtain sufficient fast consensus tracking con-dition of multiagent system (22) based on increment PIDalgorithm as follows

Theorem 6 Consider multiagent systems (3) with algorithm(5) Assume that the interconnection topology digraph 119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receive ref-erence information Then the system achieves a fast consensustracking asymptotically if

119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5)

+ 119870119894119895(2119899119894119895+ 1) le 1 119894 119895 isin Φ

(23)

where119860 = 119892119894(0)119870119901(1+119879119879

119894+119879119889119879) 119861 = 119892

119894(0)119870119901(1+2(119879

119889119879))

and 119862 = 119892119894(0)119870119901(119879119889119879) and119870

119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree

of agent 119894

Proof The multiagent systems of (3) with (5) and (21) aregiven by (22) Taking the 119911-transformation of the system (22)we get

119911119883119894(119911) = 119883

119894(119911) minus (119860119911

minus119898119894 + 119861119911minus119898119894minus1 + 119862119911

minus119898119894minus2)119883119894(119911)

+ (119860 + 119861119911minus1+ 119862119911minus2) 120577119903(119911)

+ sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 [119883

119895(119911) minus 119883

119894(119911)]

(24)

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911)

=

minus119886119894119895119911minus119899 V

119895isin 119873119894

119860119911minus119898119894 + 119861119911

minus119898119894minus1 + 119862119911minus119898119894minus2 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119895 = 119894

0 otherwise(25)

and119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = 119860 + 119861119911minus1 + 119862119911minus2 then

(24) becomes

119911119883 (119911) = 119883 (119911) + 119876120577119903(119911) 1119873minus (119911)119883 (119911) (26)

Let 120577119903(119911) denote input and let 119883(119911) denote output and 1119873

as unit column vector with 119873 times 1 dimensions then thecharacteristic equation of system (26) becomes119901(119911) = det[119868+(1(119911 minus 1))(119911)] In consequence we will prove the all rootsof 119901(119911) = 0 whose module is less than unity

According to general Nyquist stability criterion modulusof all roots satisfied 119901(119911) = 0 should be less than unity ifall poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596) do notenclose the point (minus1 1198950) for 120596 isin [minus120587 120587] By Lemma 1 allpoles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + sumV119895isin119873119894

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(27)

Moreover we have

119863119894

=

1003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

lt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(28)

where119870119894119895= sumV119895isin119873119894 119886119894119895

Define 119866119894(120596) = (119860119890

minus119895120596119898119894 + 119861119890minus119895120596(119898119894+1) + 119862119890

minus119895120596(119898119894+2) +

119870119894119895119890minus119895120596119899119894119895)(119890

119895120596minus 1) it is easy to see that 119866

119894(120596) is center of

disc 119863119894 Then the point (minus1 1198950) does not enclose in disc 119863

119894

for all 120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 Asa result when isin [minus120587 120587] we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(29)

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Let 119890119894(119896 minus 119898

119894) = 119909

119894(119896 minus 119898

119894) minus 120577119903(119896) in (7) and substitute

feedback control gains 119870119894into increment PID algorithm as

(21) (7) can be rewritten by

119909119894(119896 + 1)

= 119909119894(119896)

minus 119892119894(0)[119860119890119894(119896 minus 119898

119894)

minus119861119890119894(119896 minus 119898

119894minus 1) + 119862119890

119894(119896 minus 119898

119894minus 2)]

+ sum

V119895isin119873119894

119886119894119895[119909119895(119896 minus 119899

119894119895) minus 119909119894(119896 minus 119899

119894119895)] 119894 119895 isin Φ

(22)

Then we obtain sufficient fast consensus tracking con-dition of multiagent system (22) based on increment PIDalgorithm as follows

Theorem 6 Consider multiagent systems (3) with algorithm(5) Assume that the interconnection topology digraph 119866 =

(119881119882119860) of the system has no less than a globally reachableagent and at least one globally reachable agent can receive ref-erence information Then the system achieves a fast consensustracking asymptotically if

119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5)

+ 119870119894119895(2119899119894119895+ 1) le 1 119894 119895 isin Φ

(23)

where119860 = 119892119894(0)119870119901(1+119879119879

119894+119879119889119879) 119861 = 119892

119894(0)119870119901(1+2(119879

119889119879))

and 119862 = 119892119894(0)119870119901(119879119889119879) and119870

119894119895= sumV119895isin119873119894 119886119894119895 denotes outdegree

of agent 119894

Proof The multiagent systems of (3) with (5) and (21) aregiven by (22) Taking the 119911-transformation of the system (22)we get

119911119883119894(119911) = 119883

119894(119911) minus (119860119911

minus119898119894 + 119861119911minus119898119894minus1 + 119862119911

minus119898119894minus2)119883119894(119911)

+ (119860 + 119861119911minus1+ 119862119911minus2) 120577119903(119911)

+ sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 [119883

119895(119911) minus 119883

119894(119911)]

(24)

where 119883119894(119911) and 120577119903(119911) are the 119911-transformation of 119909

119894(119896) and

120577119903(119896) respectively Define a 119873 times 119873 matrix (119911) = 119897

119894119895(119911) as

follows

119894119895(119911)

=

minus119886119894119895119911minus119899 V

119895isin 119873119894

119860119911minus119898119894 + 119861119911

minus119898119894minus1 + 119862119911minus119898119894minus2 + sum

V119895isin119873119894

119886119894119895119911minus119899119894119895 119895 = 119894

0 otherwise(25)

and119883(119911) = [1198831(119911) 119883

119873(119911)]119879 119876 = 119860 + 119861119911minus1 + 119862119911minus2 then

(24) becomes

119911119883 (119911) = 119883 (119911) + 119876120577119903(119911) 1119873minus (119911)119883 (119911) (26)

Let 120577119903(119911) denote input and let 119883(119911) denote output and 1119873

as unit column vector with 119873 times 1 dimensions then thecharacteristic equation of system (26) becomes119901(119911) = det[119868+(1(119911 minus 1))(119911)] In consequence we will prove the all rootsof 119901(119911) = 0 whose module is less than unity

According to general Nyquist stability criterion modulusof all roots satisfied 119901(119911) = 0 should be less than unity ifall poles 120582((1(119890119895120596 minus 1))(119890119895120596)) of (1(119890119895120596 minus 1))(119890119895120596) do notenclose the point (minus1 1198950) for 120596 isin [minus120587 120587] By Lemma 1 allpoles 120582((1(119890119895120596 minus 1))(119890119895120596)) belong to the union set of 119873circular disks that is

119863119894

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + sumV119895isin119873119894

119886119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

V119895isin119873119894

1003816100381610038161003816100381610038161003816100381610038161003816

119886119894119895119890minus119895120596119899

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(27)

Moreover we have

119863119894

=

1003816100381610038161003816100381610038161003816100381610038161003816

120585 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

lt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(28)

where119870119894119895= sumV119895isin119873119894 119886119894119895

Define 119866119894(120596) = (119860119890

minus119895120596119898119894 + 119861119890minus119895120596(119898119894+1) + 119862119890

minus119895120596(119898119894+2) +

119870119894119895119890minus119895120596119899119894119895)(119890

119895120596minus 1) it is easy to see that 119866

119894(120596) is center of

disc 119863119894 Then the point (minus1 1198950) does not enclose in disc 119863

119894

for all 120596 isin [minus120587 120587] as long as the point (minus119886 1198950) with 119886 ge 1 Asa result when isin [minus120587 120587] we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

gt

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

(29)

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Through several trivial transform we have

1003816100381610038161003816100381610038161003816100381610038161003816

minus119886 + 1198950 minus

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

minus

1003816100381610038161003816100381610038161003816100381610038161003816

119860119890minus119895120596119898119894 + 119861119890

minus119895120596(119898119894+1) + 119862119890minus119895120596(119898119894+2) + 119870

119894119895119890minus119895120596119899119894119895

119890119895120596 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 119886(minus119860 sin (((2119898

119894+1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596)

2 sin (1205962)minus

119862 sin (((2119898119894+ 5) 2) 120596)+119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)) gt 0

(30)

As for Lemma 2 we have sin(((2119898119894+1)2)120596) sin(1205962) le

2119898119894+ 1 sin(((2119898

119894+ 3)2)120596) sin(1205962) le 2119898

119894+ 3 sin

(((2119898119894+ 5)2)120596) sin(1205962) le 2119898

119894+ 5 and sin(((2119899

119894119895+

1)2)120596) sin(1205962) le 2119899119894119895+ 1 Then the following inequality

is obtained by

119860 sin (((2119898119894+ 1) 2) 120596) + 119861 sin (((2119898

119894+ 3) 2) 120596) + 119862 sin (((2119898

119894+ 5) 2) 120596) + 119870

119894119895sin (((2119899

119894119895+ 1) 2) 120596)

2 sin (1205962)

le 119860 (2119898119894+ 1) + 119861 (2119898

119894+ 3) + 119862 (2119898

119894+ 5) + 119870

119894119895(2119899119894119895+ 1) lt 119886

(31)

Similar to (19) this inequality holds under the conditionsof 120596 isin [minus120587 120587] and 119886 = 1 That is end of this proof

Remark 7 Because an inequality is used in (28) the result ofTheorem 6 is considerable conservatism If119860 = 119870

119894119894 119861 119862 = 0

we can obtainTheorem 3That is to say this conservatism canbe ignored if 119861 119862 are very small

Remark 8 By using increment PID algorithm the maximumallowable time delay of consensus tracking ofmultiagents sys-tem become larger even input delays119898

119894and communication

delay 119899119894119895in Theorem 6 are chosen to disobey the inequality

this multiagent systems is still converged to its reference inmany cases

5 Optimization Consensus Tracking PIDAlgorithm Based on Genetic Algorithm

The main result in Section 4 gives a consensus trackingrange whose multiagent systems can converge to referenceHowever our interesting is how fast these multiagent systemscan converge to reference or are there optimalPIDparameterswhich make these systems track reference with optimalperformances Here a new optimization increment PIDalgorithm based on genetic algorithm (GA-PID) is proposedfor optimization cost including rise time output energy ofcontrollers and tracking error

Here the fast consensus tracking problem of multiagentsystem is described as follows finding the optimal PID

parameters 119870119901 119879119894 119879119889of Theorem 6 which can make the

system achieve faster consensus tracking It is well knownthat the genetic algorithm is an effective method that can findthe global optimal solution so we improve the conventionalgenetic algorithm in order to solve this optimization trackingproblem The basic design steps of self-adjusting PID con-troller based on the genetic algorithm are as follows

(1) To ascertain parameters To ascertain the values ofPID parameters according to the mathematic modelof the system so as to narrow the searching scopeand improve the efficiency of optimization here let119870119901 119879119894 119879119889le 1

(2) To select the initial population Here 50 initial popu-lations are chosen at random so populations size119872is equal to 50

(3) To ascertain the adaptation parameter Combingthree control performances stability raPIDity andaccuracy the target functions shown as below canbe used as the optimal index for the selection ofparameters

119869119904= sum119869

119894

119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896)) 119889119905 + 120596

3119896119894

119906 119894 isin Φ

(32)

where 119869119904is the global optimal index 119869

119894is the local opti-

mal index of agent 119894 for tracking reference 119890119894(119896) 119906119894(119896)

and 119896119894119906are the error controller input and rising time

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Comparison of convergence time among three algorithms in different topologies (119898 = 1 119899 = 3)

Agents receivedconstant reference

119875-like algorithm119875 = 01

Increment 119875119868119863 algorithm119875 = 01 119868 = 2119863 = 01

Increment 119875119868119863algorithm based on GA

All 84 45 24Agent 6 155 87 60Agent 2 179 76 56Agent 3 123 78 56Agents (1 4 5 and 6) 153 87 76Agents (1 4 5 and 3) 119 78 64Agents (1 4 5 and 2) 175 70 56Agents (2 6) 89 45 32Agents (2 3) 85 41 43Agents (3 6) 91 47 68Agents (2 3 and 6) 84 41 33

of agent 119894 respectively 1205961 1205962 1205963denotes weighted

values of these three parametersIn order to avoid overshooting a punishment mech-anism is introduced That is if the overshootinghappened this overshoot should become a term oflocal optimal index of agent 119894 As a result the localoptimal index becomes

If 119890119894(119896) lt 0

then 119869119894= int

infin

0

(1205961

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816 + 1205962119906

2

119894(119896) + 120596

4

1003816100381610038161003816119890119894 (119896)1003816100381610038161003816) 119889119905

+ 1205963119896119894

119906

(33)

where1205964is a weighted value satisfying120596

4≫ 1205961 Here

related weighted values are 1205961= 0999 120596

2= 0001

and 1205963= 20 The weighted value of punishment

mechanism is 1205964= 100 The fitness function is 119865 =

1119869119904

(4) Design of genetic operator Designing genetic oper-ator is a basic operation of genetic algorithm topopulations including selection operator crossoveroperator and mutation operator Selection operatoris determined by its selection probability describedby 119901119894= 119865119894sum119872

1119865119894 where 119865

119894is fitness of agent

119894 Crossover operator is determined by crossoverprobability 119875

119888 Mutation operator is determined by

mutation probability 119875119898

(5) To ascertain evolution parameters Here let initialpopulation 119872 = 50 end-up generation 119866 = 100crossover probability 119875

119888= 09 and initial mutation

probability 119875119898= 01

6 Simulation Example

Example 9 Consider the multiagent systems which arecomposed of one virtue leader and 6 following agents withan interconnection digraph shown by Figure 1 The weightsof the directed paths are 119886

12= 01 119886

16= 005 119886

23=

2

0

1

3

4

56

Figure 1 Digraph of a group of seven agents

015 11988636= 01 119886

43= 005 119886

45= 01 and 119886

56= 015 119886

62=

015 Agent ldquo0rdquo is consensus tracking reference 120577119903(119896) andinterconnection with dotted line denotes a communicationexisting between reference agent and neighbors of this ref-erence agent Without loss of generality the virtual leader0 can also be known only for part of agents and differencestopologies could be seen in Table 1 Initial state of all agents is119909(0) = (8 6 5 3 0 minus2)

119879 and sample period of these systemsis 119879 = 1 second For simplify here let 119898

119894= 119898 119899

119894119895= 119899 119894 119895 isin

(1 2 6)

It is easy to see that there exist three globally reachableagents named agent 2 agent 3 and agent 6 in this directedgraph Firstly all agents which can receive reference areconsideredwhen this reference is constant (120577119903(119896) = 4) or timevarying (120577119903(119896) = sin(015119896) + 4) respectively and consensustracking responds could be seen in Figures 2 3 4 5 and 6under the condition of 119898 = 1 119899 = 3 From Figures 2ndash4we can see that the multiagent systems can converge to theconstant reference when 119875-like (119870

119894119894= 1198701= 1198702= sdot sdot sdot =

1198706= 01) Increment 119875119868119863(119870

119901= 01 119879

119894= 05 119879

119889= 01)

and GA minus 119875119868119863 (119870119901= 05122 119879

119894= 24149 119879

119889= 00515) and

GA-PID algorithm has faster convergent speed than that ofthe two If we compare maximum allowable time delay GA-PID has the largest consensus tracking allowable time delaysknown as 119898 = 1 119899 = 21 in contrast to 119898 = 1 119899 = 5

as to conventional 119875-like algorithm and 119898 = 1 119899 = 12 asto increment PID algorithm Figure 5 show the optimizationprocess of Best 119869 based on genetic algorithm

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

7

8

Time (s)

Reference

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 2 States trajectory achieving consensus by 119875-like algorithm

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 3 States trajectory achieving consensus by increment PIDalgorithm

In fact the multiagent systems can converge to thereference on the condition that at least one globally reachableagent can receive the constant reference Similar results canalso be obtained in different topologies Details could be seenin Table 1 Comparing these three algorithms we can see that119875-like algorithm has slowest convergence speed incrementPID algorithm has strongest robustness performance andincrement PID based on GA has fastest convergence speedin almost all cases What is more a very interesting thing

0 10 20 30 40 50 60 70 80Time (s)

0

1

2

3

4

5

6

7

8

minus2

minus1

Reference

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 4 States trajectory achieving consensus by GA-PID algo-rithm

0 20 40 60 80 100890

900

910

920

930

940

950

960

970

Time (s)

Best J

Figure 5 Best 119869 for GA-PID algorithm

is also deduced from Table 1 That is convergence speedis similar between all agents receiving the reference andonly globally reachable agents receiving the reference Thismeans that only globally reachable agents instead of all agentsreceive reference and a similar convergence speed can also beobtained

From [9] we know that conventional 119875-like algorithmis not sufficient for consensus tracking when all agentsreceive a time-varying consensus reference However if ourincrement PID algorithm is adopted consensus trackingcan be achieved through choosing suitable PID parametersComparing Figure 6 with Figure 7 we can see that suitablePID parameters not only decrease errors between referenceand current states of agents but also increase the convergence

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 20 40 60 80 100 120 140Time (s)

Reference

0

1

2

3

4

5

6

7

8

minus2

minus1

ReX

1X

2X

3X

4X

5X

6

X1

X2

X3

X4

X5

X6

Figure 6 States trajectory with time-varying reference by 119875-likealgorithm when 119875 = 01

0 20 40 60 80 100 120 140

0

2

4

6

8

10

Time (s)

Reference

minus2

X4

X5

X6

X1

X2

X3

ReX

1X

2X

3X

4X

5X

6

Figure 7 States trajectory achieving consensus with time-varyingreference by increment PID algorithm when 119875 = 02 119868 = 256 and119863 = 0125

speed Regretfully these three algorithms cannot be usedto track a time-varying reference that is available to onlya subset of the team members for only receiving time-varying reference state If time-varying reference changes ina piecewise constant case increment PID algorithm basedon genetic algorithm can track this time-varying referencewhether the reference is available to all team members or toa subset of the team member From Figure 8 we can see that

0 20 40 60 80 100 120

0

2

4

6

8

10

Time (s)

Reference

minus2

X1

X2

X3

X4

X5

X6

ReX

1X

2X

3X

4X

5X

6Figure 8 Consensus tracking in piecewise constant by GA-PIDalgorithm only (2 3 6) receiving reference

the multiagent systems can converge to a piecewise constantreference within about 40 seconds when only agents (2 3 6)receive this time-varying referenceThis characteristic can beapplied to many fields such as synchronizing a network ofclocks [13]

7 Conclusions

In this paper three consensus tracking algorithms named119875-like algorithm increment PID algorithm and incrementPID algorithm based on genetic algorithm respectively fordiscrete-time multiagent systems with time-varying inputdelays and communication delays are proposed based onthe frequency-domain analysis Firstly a consensus track-ing sufficient condition of conventional 119875-like algorithm isobtained Secondly a new increment PID algorithm based onsimilar frequency-domain method is designed for improvingconsensus convergence speed and an inequality conditionis also deduced Finally considering three control perfor-mances stability rapidity and accuracy an increment PIDalgorithm based on genetic algorithm is designed to findoptimal PID parameters within an inequality allowable spanfor achieving optimization cost These three algorithms cansolve tracking problem of multiagent systems with a constantreference effectively when reference state is available to all theteam members If the reference state might only be availableto a portion of the agents in the team the convergence speedmay increase in the same condition As for a time-varyingreference case if the reference state has a directed path toall team agents increment PID algorithm and increment PIDalgorithm based on genetic algorithm can realize consensustracking through choosing suitable PID parameters whileconventional 119875-like algorithm fails to track the reference

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

However these three algorithms cannot be used to track atime-varying reference state when the reference is available toonly a subset of teammembers In the future research we willfocus onmore complex issues in the controller design such asactuator delay and fault 119867

infincontroller design with control

delay quantized control and global consensus problem withsaturated control [22ndash29]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China under Grant nos 61004033 and61364002 Foundation of 2012 Jinchuan School-EnterpriseCooperation and the Yunnan Natural Science Foundation ofYunnan Province China under Grant no 2010ZC035

References

[1] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory and Applications vol 6 no 4 pp 560ndash569 2012

[2] H Su X Wang and Z Lin ldquoFlocking of multi-agents with avirtual leaderrdquo IEEE Transactions on Automatic Control vol 54no 2 pp 293ndash307 2009

[3] R Sepulchre D A Paley and N E Leonard ldquoStabilization ofplanar collective motion with limited communicationrdquo IEEETransactions on Automatic Control vol 53 no 3 pp 706ndash7192008

[4] J A Fax and R M Murray ldquoInformation flow and cooperativecontrol of vehicle formationsrdquo IEEE Transactions on AutomaticControl vol 49 no 9 pp 1465ndash1476 2004

[5] RW Beard TWMcLainM A Goodrich and E P AndersonldquoCoordinated target assignment and intercept for unmanned airvehiclesrdquo IEEE Transactions on Robotics and Automation vol18 no 6 pp 911ndash922 2002

[6] 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

[7] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[8] R MMurray ldquoRecent research in cooperative control of multi-vehicle systemsrdquo Journal ofDynamic SystemsMeasurement andControl vol 129 no 5 pp 571ndash583 2007

[9] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Theory and Applications Communicationsand Control Engineering Springer 2008

[10] Y-P Tian and C-L Liu ldquoConsensus of multi-agent systemswith diverse input and communication delaysrdquo IEEE Transac-tions on Automatic Control vol 53 no 9 pp 2122ndash2128 2008

[11] Y She and H Fang ldquoFast distributed consensus control forsecond-ordermulti-agent systemsrdquo inProceedings of theChinese

Control and Decision (CCDC rsquo10) pp 87ndash92 Xuzhou ChinaMay 2010

[12] H-T ZhangM Z Q Chen and G-B Stan ldquoFast consensus viapredictive pinning controlrdquo IEEE Transactions on Circuits andSystems I vol 58 no 9 pp 2247ndash2258 2011

[13] R Carli A Chiuso L Schenato and S Zampieri ldquoOptimalsynchronization for networks of noisy double integratorsrdquo IEEETransactions on Automatic Control vol 56 no 5 pp 1146ndash11522011

[14] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[15] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 B pp21ndash34 2011

[16] Y Zheng and LWang ldquoFinite-time consensus of heterogeneousmulti-agent systems with and without velocity measurementsrdquoSystems amp Control Letters vol 61 pp 871ndash878 2012

[17] W Ren ldquoConsensus tracking under directed interaction topolo-gies algorithms and experimentsrdquo IEEE Transactions on Con-trol Systems Technology vol 18 no 1 pp 230ndash237 2010

[18] Y Cao and W Ren ldquoDistributed coordinated tracking withreduced interaction via a variable structure approachrdquo IEEETransactions onAutomatic Control vol 57 no 1 pp 33ndash48 2012

[19] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple euler-lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[20] Z Chen L Xiang and Z Yuan ldquoA tracking control scheme forleader based multi-agent consensus for discrete-time caserdquo inProceedings of the 27th Chinese Control Conference (CCC rsquo08)pp 494ndash498 Kunming China July 2008

[21] D Whitley ldquoA genetic algorithm tutorialrdquo Statistics and Com-puting vol 4 no 2 pp 65ndash85 1994

[22] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[23] H Li X Jing and H R Karimi ldquoOutput-feedback basedH-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[24] Z Su H Zhang and F W Yang ldquoObserver-based H-infinitycontrol for discrete-time stochastic systems with quantizationand random communication delaysrdquo IET Control Theory andApplications vol 7 no 3 pp 372ndash379 2013

[25] H Zhang Q Chen H Yan and J Liu ldquoRobust Hinfin filtering forswitched stochastic system with missing measurementsrdquo IEEETransactions on Signal Processing vol 57 no 9 pp 3466ndash34742009

[26] Q Wang and H Gao ldquoGlobal consensus of multiple integratoragents via saturated controlsrdquo Journal of the Franklin Institutevol 350 no 8 pp 2261ndash2276 2013

[27] H Zhang H Yan F Yang and Q Chen ldquoQuantized controldesign for impulsive fuzzy networked systemsrdquo IEEE Transac-tions on Fuzzy Systems vol 19 no 6 pp 1153ndash1162 2011

[28] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

[29] Q Wang C Yu and H Gao ldquoSemiglobal synchronization ofmultiple generic linear agents with input saturationrdquo Interna-tional Journal of Robust and Nonlinear Control 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of