Research ArticleConsensus of Multiagent Systems Described by VariousNoninteger Derivatives

G Nava-Antonio 1 G Fernaacutendez-Anaya1 E G Hernaacutendez-Mart-nez 2

J J Flores-Godoy 3 and E D Ferreira-Vaacutezquez 4

1Departamento de Fısica y Matematicas Universidad Iberoamericana Mexico City Mexico2Departamento de Estudios en Ingenierıa para la Innovacion Universidad Iberoamericana Mexico City Mexico3Departamento de Matematica Universidad Catolica del Uruguay Montevideo Uruguay4Departamento de Ingenierıa Electrica Universidad Catolica del Uruguay Montevideo Uruguay

Correspondence should be addressed to J J Flores-Godoy josefloresucueduuy

Received 5 November 2018 Accepted 6 January 2019 Published 26 February 2019

Guest Editor Carlos-Arturo Loredo-Villalobos

Copyright copy 2019 G Nava-Antonio et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we unify and extend recent developments in Lyapunov stability theory to present techniques to determine theasymptotic stability of six types of fractional dynamical systemsThese differ by being modeled with one of the following fractionalderivatives the Caputoderivative the Caputodistributed order derivative the variable order derivative the conformable derivativethe local fractional derivative or the distributed order conformable derivative (the latter defined in this work) Additionally weapply these results to study the consensus of a fractional multiagent system considering all of the aforementioned fractionaloperators Our analysis covers multiagent systems with linear and nonlinear dynamics affected by bounded external disturbancesand described by fixed directed graphs Lastly examples which are solved analytically and numerically are presented to validateour contributions

1 Introduction

The concept of fractional calculus arose more than threecenturies ago thanks to a question posed by LrsquoHopital toLeibniz where the meaning of derivatives of order 12 wasasked [1] However this discipline has gained popularity onlyin the last decades in which new methods to solve andanalyze fractional differential equations have appeared andresearchers have made great efforts to study real phenomenausing these tools This modern boom has occurred mainlybecause of the capability of fractional order calculus tomodel certain systems more accurately in comparison withtraditional integer order calculus This greater precision isdue to the liberty that fractional calculus gives us to considernoninteger orders for the differential and integral operators

Throughout the life of this branch of mathematicsvarious definitions for the fractional derivative have been

proposed A survey of the most common of these can befound in Kilbas Srivastava and Trujillo [2] Petras [3]Podlubny [4] along with a rich overview of interestingapplications and simulation techniques In this paper wewill focus on six different fractional derivatives The firstof them is the Caputo fractional derivative which is widelystudied in the already-mentioned references and is preferredby many because the Caputo derivative of a constant is zero(which is not true for all fractional derivatives) and theinitial conditions of a Caputo fractional system have the samephysical interpretation as the integer order case

The rest of the fractional derivatives addressed in thiswork are of more recent origin In the fractional variable-order (also known as time-varying order) derivative intro-duced in Samko and Ross [5] the orders of differentiationcan be functions of the independent variable or even of otherparameters In Sun Chen Wei and Chen [6] it is argued

HindawiComplexityVolume 2019 Article ID 3297410 14 pageshttpsdoiorg10115520193297410

2 Complexity

that variable-order calculus allows to better describe certainsystems with memory properties that change for examplewith time or position

The Caputo distributed order derivative originally pre-sented in Caputo [7] acquired relevance in problems withultra-slow diffusion where it has been applied with physicaljustification for instance in Chechkin Klafter and Sokolov[8] Naber [9] Regarding its meaning a possible conceptualinterpretation for this fractional derivative is suggested inLorenzo and Hartley [10] in systems where nonhomogenousor anisotropic properties are involved itmight be appropriateto consider that each differential element of the system shouldhave its own differentiation order

The fractional derivatives discussed in the above linesdo not satisfy the product or the chain rules Furthermorethe monotonicity of a function is not specified by the signof those derivatives The conformable derivative emergedin response to these inconveniences as explained in KhalilAl Horani Yousef and Sababheh [11] nevertheless thisfractional operator which is defined as a limit loses thememory properties and global character of the others whichare built with integrals A very similar but more generalfractional derivative was proposed in Almeida Guzowskaand Odzijewicz [12] That operator is characterized by akernel function that can be tuned to better represent agiven physical system In the same spirit of the Caputodistributed derivative in this article we introduce the dis-tributed conformable derivative a further generalization ofthe fractional differential operator defined in Khalil et al[11]

The broad variety of applications that the aforementionedfractional derivatives have had is remarkable from quantummechanics Laskin [13] to control theory Baleanu Machadoand Luo [14] and study of human memory and emotionsTabatabaei Yazdanpanah Jafari and Sprott [15] In generalthese and other problems involving differential equationsof fractional order are complicated to approach and inmany cases there are no analytical or numerical schemes tosolve them Consequently the qualitative theory of fractionaldynamical systems has become an important line of researchWithin this field the Lyapunov direct method is a tool thatallows us to determine the stability and long-term behaviorof a certain system without the need of solving it

In recent years Lyapunov stability theory has integratedto fractional calculus see Li Chen and Podlubny [16]Souahi Makhlouf and Hammami [17] Tabatabaei Talebiand Tavakoli [18] Taghavian and Tavazoei [19] Wang andLi [20] One of the objectives of the present article is totake advantage of the similarities between those papersand unify them into a generalized fractional Lyapunovmethod useful for systems of differential equations with thefractional derivatives mentioned in the above paragraphsAs an application of thismdash and with the purpose of com-paring the performance of different fractional derivativesin the same problemmdash we also study in this article theconsensus problem of a generalized fractional multiagentsystem

A multiagent system is an arrangement of various agentsthat are organized to accomplish group objectives by means

of their local interactions A multiagent system is said toachieve consensus when the dynamics of the agents convergeto a certain desired value We can appreciate the relevance ofthis concept by noticing its multiple applications includingthe study of the formation of multivehicle systems [21] thesynchronization of coupled oscillators [22] or the distributedsensor fusion in sensor networks [23]

The consensus of multiagent systems has been mostlyinvestigated under the framework of integer order calcu-lus Extensive introductory reviews of this topic can befound in W Ren and Beard [21] W Ren and Cao [24]Some of the ideas presented in those references have beengeneralized for fractional order systems solely using theCaputo fractional derivative for example in Yu Jiang Huand Yu [25] an adaptive pinning control is used to realizeleader-following consensus in a fractional multiagent systemYin Yue and Hu [26] studied the consensus problem forfractional heterogeneous systems made up of agents withdifferent dynamics Song Cao and Liu [27] proposed adistributed protocol to accomplish robust consensus basedon the information of second-order neighborsNava-Antonioet al [28] present sufficient conditions for consensus ofmultiagent systems with distributed fractional order andG Ren and Yu [29] gave conditions for fractional multia-gent systems to achieve robust consensus via Mittag-Lefflerstability methods That last article is the main inspirationof the second half of this paper where we will extend theresults of G Ren and Yu [29] to be used in multiagentsystems with five other fractional differentiation orders withlinear or nonlinear dynamics and in the presence of externalperturbations

The order of this text is described next In Section 2fundamental preliminary concepts are introduced Section 3contains in two parts our main results firstly we presentthe generalized fractional Lyapunov direct method and thenwe apply it to study the consensus of multiagent systemsmodeled with different fractional derivatives AfterwardsSection 4 gives examples where we verify the validity of thedeveloped theory Lastly Section 5 contains the conclusionsof the present work

2 Preliminary Concepts

In the following section we present the definitions of var-ious fractional derivatives and discuss certain properties ofsystems of equations with these operators All the definitionsbelow are given considering orders of differentiation 120572 isin(0 1)Definition 1 (Aguila-Camacho Duarte-Mermoud and Gal-legos [30]) The Caputo fractional derivative of order 120572 isdefined as

1198791205721 119909 (119905) = 1Γ (1 minus 120572) int119905

1199050

1199091015840 (120591)(119905 minus 120591)120572 119889120591 119905 gt 1199050 (1)

where 1199091015840(119905) is the integer derivative of 119909(119905) We suppose that119909(119905) is a differentiable function for all 119905 ge 1199050 and for alloperators in this work

Complexity 3

Definition 2 (Tabatabaei et al [18]) The modified initializedCaputo fractional derivative of time-varying order 120572(119905) isdefined as follows

1198791205722 119909 (119905) = 1Γ (1 minus 120572 (119905)) int119905

0

1199091015840 (120577)(119905 minus 120577)120572(119905) 119889120577 + Ψ119909119888 (119905) forall119905 ge 0

(2)

where Ψ119909119888 (119905) = (1Γ(1 minus 120572(119905))) int0minus119888(119905 minus 120577)minus120572(119905)1199091015840(120577)119889120577 captures

the behavior of 119909 before 119905 = 0 assuming that 119909 begins fromminus119888 lt 0 Since we will focus in this paper on systems which areat rest at 119905 lt 0 Ψ119909119888 (119905) = 0Definition 3 (Jiao Chen and Podlubny [31]) The distributedorder fractional derivative in the Caputo sense with respectto the density function 119888 [120583 1] 997888rarr [0 +infin) for some 1 gt120583 gt 0 such that int1

120583119888(120572)119889120572 = 0 is defined as follows

1198791205723 119909 (119905) = int1120583119888 (120572) 1198791205721 119909 (119905) 119889120572 (3)

The Laplace transform of a distributed order derivativewhich will appear in the derivation of our main results is

L 1198791205723 119909 (119905) = 119862 (119904)119883 (119904) minus 1119904 119862 (119904) 119909 (0+) (4)

where 119862(119904) = int119898119898minus1

119888(120572)119904120572119889120572Definition 4 (Souahi et al [17]) The conformable fractionalderivative starting from 119886 of a function 119909 defined on [119886infin) is

1198791205724 119909 (119905) = lim120598997888rarr0

119909 (119905 + 120598 (119905 minus 119886)1minus120572) minus 119909 (119905)120598 (5)

for all 119905 gt 119886 If lim119905997888rarr119886+1198791205724 119909(119905) exists then1198791205724 119909 (119886) = lim

119905997888rarr119886+1198791205724 119909 (119905) (6)

Definition 5 (Almeida et al [12]) Let 119896 [119886 119887] 997888rarr R be acontinuous nonnegative map such that 119896(119905) = 0 whenever119905 gt 119886 and 119909 [119886 119887] 997888rarr R By definition 119909 is 120572-differentiableat 119905 gt 119886 with respect to kernel 119896 if the limit

1198791205725 119909 (119905) = lim120598997888rarr0

119909 (119905 + 120598119896 (119905)1minus120572) minus 119909 (119905)120598 (7)

exists The local fractional derivative at 119905 = 119886 is defined by

1198791205725 119909 (119886) = lim119905997888rarr119886+

1198791205725 119909 (119905) (8)

if the limit exists

Definition 6 Let 119896 and 119909 be functions as in Definition 5and 119888 a density function as in Definition 3 The distributedconformable fractional derivative is defined as

1198791205726 119909 (119905) = int10119888 (120572) 1198791205725 119909 (119905) 119889120572 (9)

Theorem 7 (Almeida et al [12]) A function 119909 [119886 119887] 997888rarr Ris 120572-differentiable at 119905 gt 119886 if and only if it is differentiable at 119905In that case we have the relation

1198791205725 119909 (119905) = 119896 (119905)1minus120572 1199091015840 (119905) 119905 gt 119886 (10)

Notice that Definition 5 is a particular case of Defini-tion 6 Then byTheorem 7 if we take 119896(119905) = 119905 minus 119886 we obtain

1198791205724 119909 (119905) = (119905 minus 119886)1minus120572 1199091015840 (119905) 119905 gt 119886 (11)

In a similar form by substituting (10) in (9) we have

1198791205726 119909 (119905) = 1199091015840 (119905) int10119888 (120572) 119896 (119905)1minus120572 119889120572 (12)

Consider the generalized system of fractional differentialequations of order 120572 isin (0 1)

119879120572119895 119909 = 119891 (119905 119909 (119905)) 119895 = 1 2 3 4 5 or 6 (13)

where 119909 isin R119899 119909(1199050) = 1199090 and 119891 R+ times R119899 997888rarrR119899 is a given nonlinear function is Lipzchitz with respectto the second argument For simplicity and without loss ofgenerality we will consider that the equilibrium points of thesystems analyzed hereafter are at the origin ie 119891(119905 0) = 0forall119905 ge 0

Throughout this paper we will assume that the studiedsystems have unique solutions The existence and uniquenessof the solution of system (13) is discussed in Podlubny [4] Xuand He [32] Ford and Morgado [33] and Bayour and Torres[34] for the cases 119895 = 1 2 3 and 4 respectively The theoryof existence and uniqueness of solutions when 119895 = 5 or 6 canbe easily generalized from Bayour and Torres [34] by takinginto account (11) and (12)

The Final Value Theorem and an important Laplacetransform associated with fractional calculus both used inthe following sections of this article are presented next

Theorem 8 (Duffy [35]) Let 119865(119904) = L119891(119905) If all poles of119904119865(119904) are in the open left-half complex plane then

lim119905997888rarrinfin

119891 (119905) = lim119904997888rarr0

119904119865 (119904) (14)

Definition 9 (Podlubny [4]) A two-parameter function of theMittag-Leffler type is defined by

119864120572120573 (119911) = infinsum119896=0

119911119896Γ (120572119896 + 120573) 120572 gt 0 120573 gt 0 (15)

Lemma 10 (Podlubny [4]) TheMittag-Leffler function of twoparameters satisfies the following relationship

Lminus1 [119904minus(120572minus120573)119904120573 minus 119886 ] = 119905120572minus1119864120573120572 (119886119905120573) 10038161003816100381610038161003816119904120573 minus 11988610038161003816100381610038161003816 lt 1 (16)

3 Lyapunov Stability for GeneralizedFractional Systems

The two Theorems in this section summarize the knownresults for Lyapunov stability theory for nonlinear systems

4 Complexity

of (A) Li et al [16] and Wang and Li [20] (Definition 1)(B) Souahi et al [17] (Definition 2) (C) Tabatabaei et al[18] (Definition 3) (D) Taghavian and Tavazoei [19] (Defini-tion 4) (E) Almeida et al [12] (Definition 5) and also extendsthe Lyapunov stability theory for nonlinear systems definedby operators introduced in Definitions 5 and 6 Specificallyin (A) the Lyapunov direct method for standard Caputofractional system (13) with 119895 = 1 is proved and the definitionofMittag-Leffler stability is introduced In (B) the same resultfor the case of the modified initialized Caputo fractionalderivative of time-varying order 120572(119905)with 119895 = 2 is proved Forthe case of distributed fractional systems (13) the mentionedresult in (C) for 119895 = 3 is proved In the case of conformalfractional systems in (D) it is shown that (13) is fractionalexponentially stable which implies asymptotic stability for119895 = 4 For the case of Definitions 5 and 6 we show that theproofs are very similar to the one for the case 119895 = 4 Inconsequence Theorems 12 and 16 extend the Lyapunov directmethod for generalized fractional systems defined in (13)

Assumption 11 For 119895 = 2 system (13) is autonomous ie119891(119905 119909(119905)) = 119891(119909(119905))Theorem 12 Consider system (13) with 119895 = 1 2 3 4 5 or 6Let 119881(119905 119909(119905)) be a continuously differentiable function suchthat

1205741 119909 (119905)119897 le 119881 (119905 119909 (119905)) le 1205742 119909 (119905)119897119898 (17)

119879120572119895 119881 (119905 119909 (119905)) le minus1205743 119909 (119905)119897119898 (18)

where 119905 ge 0 120574119894 (119894 = 1 2 3) and 119897 and 119898 are arbitrary positiveconstants If Assumption 11 is fulfilled then the origin of system(13) is asymptotically stable

Proof

(i) For 119895 = 1 (13) is a standard Caputo fractional systemFor this system the proof is the same as the one ofTheorem 51 of Li et al [16] There it is shown that(13) is Mittag-Leffler stable which implies asymptoticstability

(ii) For 119895 = 2 (13) is a fractional system of time-varying order This proof follows from Theorem 1 ofTabatabaei et al [18] That result requires the weakerhypotheses 119881(119905 119909(119905)) ge 0 119881(119905 119909(119905)) = 0 lArrrArr 119909 = 0(which are implied by (17)) and 1198791205722119881(119905 119909(119905)) lt 0 in119863 minus 0 (which is implied by (18))

(iii) For 119895 = 3 (13) is a distributed fractional system Inthis case the proof can be found in Theorem 41 ofTaghavian and Tavazoei [19]

(iv) For 119895 = 4 (13) is a conformable fractional systemThisproof is the same as the one of Theorem 1 of Souahiet al [17] where it is shown that (13) is fractionalexponentially stableThat kind of stability also impliesasymptotic stability

(v) For 119895 = 5 or 6 (13) is a system with local fractionalderivatives or a distributed conformable fractional

system respectively In these instances the proofsare very similar to the one of the previous case(119895 = 4) That proof depends on two facts about theconformable derivative that it satisfies the productrule in the traditional sense and that the sign of1198791205724 119909(119905)determines the monotonicity of 119909(119905) Note from (8)and (12) that these features are also true for theoperators 1198791205725 and 1198791205726

The next result is a partial generalization of Theorem 12being more permissive with the Lyapunov function and itsfractional derivative but requiring a couple of additionalhypotheses

Assumption 13 For 119895 = 3 in (13) the Lyapunov function ofTheorem 16 has a nonzero initial value ie 119881(0 119909(0)) = 0Definition 14 (Teel and Praly [36]) A function ℎ [0infin) 997888rarr[0infin) is said to belong to classK if it is continuous zero atzero and strictly increasing

Assumption 15 For 119895 = 4 5 and 6 in (13) the class Kfunctions ℎ119894 (119894 = 1 2 3) satisfy lim119905997888rarrinfinℎ119894(119905) = infin

Theorem 16 Consider system (13) with 119895 = 1 2 3 4 5 or 6Suppose that there exist classK functions ℎ119894 (119894 = 1 2 3) and acontinuously differentiable function 119881(119905 119909(119905)) such that

ℎ1 (119909 (119905)) le 119881 (119905 119909 (119905)) le ℎ2 (119909 (119905)) (19)

119879120572119895 119881(119905 119909 (119905)) le minusℎ3 (119909 (119905)) (20)

If Assumptions 11 13 and 15 are fulfilled then the origin ofsystem (13) is asymptotically stable

Proof(i) For 119895 = 1 the proof can be found on Theorem 62 of

Li et al [16](ii) For 119895 = 2 the proof is presented in Tabatabaei et al

[18] as explained in item (ii) of Theorem 12 proof(iii) For 119895 = 3 the proof is the same as the one ofTheorem

42 of Taghavian and Tavazoei [19](iv) For 119895 = 4 proof can be found inTheorem 3 of Souahi

et al [17](v) For 119895 = 5 or 6 considering the argument stated in

(v) ofTheorem 12 proof we can readily generalize theresult of 119895 = 4 to cases of the distributed conformableand local fractional derivatives

We now know that Theorem 16 is valid also for Riemann-Liouville-like fractional difference equations (see Theorem36 in Wu Baleanu and Luo [37]) So we conjecture thatTheorem 16 can be valid for a larger family of operators

The following lemma contains a property of the gener-alized fractional differential operator which is useful whenputting into practice the previous Lyapunov Stability Theo-rems

Complexity 5

Lemma 17 Let 119909 R 997888rarr R be a continuous differentiablefunction Then for 119895 = 1 2 3 4 5 or 6 the followingrelationship holds

12119879120572119895 [119909119879 (119905) 119875119909 (119905)] le 119909119879 (119905) 119875119879120572119895 [119909 (119905)] (21)

where 119875 is a Hermitian positive definite matrix

Proof The proof for the cases 119895 = 1 2 4 can be found inAguila-Camacho et al [30] Souahi et al [17] Tabatabaeiet al [18] respectively If 119895 = 3 a proof for when 119875 =119868 is presented in Fernandez-Anaya Nava-Antonio Jamous-Galante Munoz-Vega and Hernandez-Martınez [38] Toobtain the more general version consider inequality (21) with119895 = 1 multiply it by the distribution function 119888(120572) ge 0 andintegrate

int10

12119888 (120572) 1198791205721 [119909119879 (119905) 119875119909 (119905)] 119889120572 = 121198791205723 [119909119879 (119905) 119875119909 (119905)]le

le int10119909119879 (119905) 119875119888 (120572) 1198791205721 [119909 (119905)] 119889120572

= 119909119879 (119905) 1198751198791205723 [119909 (119905)] (22)

We can follow a similar reasoning to prove this lemma for119895 = 5 or 6 by multiplying (21) with 119895 = 4 and 120572 =1 (that is the traditional integer order derivative) by 119896(119905)1minus120572or int10119888(120572)119896(119905)1minus120572119889120572 and using properties (8) or (12) respec-

tively

4 Application to the Consensus of MultiagentSystems of Generalized Fractional Order

In this section we will investigate the problem of consensusfor generalized multiagent systems First we will considersystems with nonlinear dynamics and then we will presentthe linear simplification of that analysis

41 Graph Theory Fundamentals We can describe the inter-action topology of a multagent system with the help of graphtheory A graph G is characterized by its vertices V =V1 V2 V119899 (which represent the agents of the system)and its edges W sube V2 (which correspond to the agentsrsquorelationships) In this paper we will focus on directed graphswhere each edge is an ordered pair (V119894 V119895) this means thatagent 119895 receives information from agent 119894 A graph can berepresented by its adjacency matrix 119860 = [119886119894119895] isin R119899times119899 where119886119894119895 = 1 if (V119894 V119895) isin W and 119886119894119895 = 0 if (V119894 V119895) notin W or by itsLaplacian matrix 119871 = [119897119894119895] isin R119899times119899 where 119897119894119894 = sum119895isin119873119894 119886119894119895 and119897119894119895 = minus119886119894119895 for 119894 = 119895 with 119873119894 the number of connected nodes tonode 119894

The following lemmas will be used in the proofs of ourmain results to gain insight into the graphs associated withthe multiagent systems of our interest

Lemma 18 (W Ren and Cao [24]) If a graph has a directedspanning tree then the Laplacian matrix 119871 has a simple zeroeigenvalue and all its other eigenvalues have positive real partsMoreover all eigenvalues of 119867 = 119871 + 119861 will have positive realparts where 119861 = diag1198871 1198872 119887119899 and 119887119894 ge 0 is not all 0Lemma 19 (Zhang and Tian [39]) Let 119864 = [1119899minus1 minus119868119899minus1] isinR(119899minus1)times119899 and 119865 = ( 0119879119899minus1

minus119868119899minus1) isin R119899times(119899minus1) where 1119899minus1 is the column

vector of ones 119868119899minus1 is the identity matrix and 0119899minus1 is the zerocolumn vector and each of them is of size 119899minus1Then119862 = minus119864119871119865is Hurwitz where 119871 is the Laplacian matrix if and only if theassociated interaction graph has a directed spanning tree

The notion of consensus that will be considered through-out this paper is presented next

Definition 20 A multiagent system accomplishes consensusif it fulfills the following condition

lim119905997888rarrinfin

1003817100381710038171003817119909119894 (119905) minus 119909119896 (119905)1003817100381710038171003817 le 0forall119894 119896 isin 1 2 119899 119894 = 119896 (23)

where 119909119896(119905) is the state of the 119896-th agent

Hereinafter we will suppose for simplicity that all agentsare in a one-dimensional space All our results can be easilygeneralized for 119898 dimensions by means of the Kroneckerproduct Moreover in this work we will consider the matrixnorm

119860 = radic 119899sum119894=1

119898sum119895=1

1198862119894119895 (24)

with 119860 = (119886119894119895) isin R119899times119898 And for any matrix 119876 isin R119899times119899120582max(119876) and 120582min(119876) denote the largest and smallest eigen-values respectively

42 Robust Consensus of Nonlinear Generalized FractionalMultiagent Systems A generalized nonlinear fractionalmulti-agent system can be represented by

119879120572119895 119909119894 (119905) = 119891 (119905 119909119894 (119905)) + 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (25)

where 119895 = 1 2 3 4 5 or 6 and 119909119894(119905)119891(119905 119909119894(119905)) 119906119894(119905) and119908119894(119905)are the state nonlinear dynamics control input and externaldisturbances of the 119894-th agent respectively

As an auxiliary element we will consider a virtual leaderwhich is an isolated agent that designates objectives for thestates of all other agents The behavior of the virtual leader ischaracterized by

119879120572119895 119909119903 (119905) = 119891 (119905 119909119903 (119905)) (26)

6 Complexity

where 119909119903(119905) is the state of the virtual leader To accomplishconsensus in system (25) we will use the following controlinput

119906119894 (119905)= minus120573[ 119899sum

119896=1

119886119894119896 (119909119894 (119905) minus 119909119896 (119905)) + 119887119894 (119909119894 (119905) minus 119909119903 (119905))] (27)

where 119886119894119896 for 119894 119896 isin 1 2 119899 with 119894 = 119896 is the (119894 119896)-thentry of the adjacency matrix 119860 isin R119899times119899 associated with theundirected graph describing the interaction of the agents and120573 ge 0 and 119887119894 for (119894 = 1 2 119899) are positive constants to bechosen as mentioned in Theorem 23

We will require that the following assumptions hold

Assumption 21 The disturbance signal 119908119894(119905) satisfies119908119894(119905) le 119897 lt infin forall119894 isin 1 2 119899Assumption 22 For the multiagent system (25) with 119895 = 3the distribution function 119888(120572) is such that

Lminus1 1119862 (119904) + 120583120582max (119876) ge 0 (28)

where 119862(119904) is defined in terms of 119888(120572) as in (4)

Theorem 23 Consider the generalized fractional nonlinearmultiagent system (25) with the virtual leader (26) and thecontroller (27) Assume that the nonlinear function119891(119905 119909(119905)) isLipschitz (with respect to 119909 and with Lipschitz constant 120579) andthat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 Assumption 11is satisfied and

radic2120573120579 ge 119876 (29)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then robust consensus isachieved

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 Assumptions 21 and 22are satisfied and

120573120579 ge 119876 (30)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then the steady-state errors ofany two agent will converge as 119905 997888rarr infin to the region1198721 where

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic 2119899120582max (119876)120573120583120582min (119876) 119876 119897 (31)

and 120583 = 120573 minus 11987621205792120573

Proof By substituting (27) in system (25) we can write

119879120572119895 119883 (119905) = 119865 (119883 (119905)) minus 120573 [119871119883 (119905) + 119861 (119883 (119905) minus 1199091199031119899)]+ 119882 (119905) (32)

where 119865(119883(119905)) = [119891(1199091(119905)) 119891(119909119899(119905))]119879 Subtracting119879120572119895 [1119899119909119903(119905)] from both sides of (32) and using the change ofvariables 119911119894(119905) = 119909119894(119905) minus 119909119903(119905) 119894 isin 1 2 119899 yields

119879120572119895 119885 (119905) = minus120573119867119885 (119905) + Δ119865 (119885 (119905)) + 119882 (119905) (33)

where119867 is defined as in Lemma 18119885(119905) = [1199111(119905) 119911119899(119905)]119879and Δ119865(119885(119905)) = [119891(1199111(119905) + 119909119903(119905)) minus 119891(119909119903(119905)) 119891(119911119899(119905) +119909119903(119905)) minus 119891(119909119903(119905))]119879 Consider the following Lyapunov candi-date function for system (33)

119881 (119905) = 119885119879119876119885 (119905) (34)

Applying Lemma 17 and substituting (33) we can analyze119879120572119895 119881(119905)119879120572119895 119881 (119905) le 120573119885119879 (119905) [minus119876119867 minus119867119879119876]119885 (119905)

+ 2119885119879 (119905) [119876Δ119865 (119885 (119905)) + 119876119882(119905)] (35)

Using Lemma 18 we can conclude that all the eigenvalues of119867 have positive real parts so that minus119867 is Hurwitz Thus thereexists a matrix 119876 = 119876119879 gt 0 that satisfies minus119867119879119876 minus 119876119867 =minus3119868119899 Applying in (35) this identity along with the property120585119879120577+120577119879120585 le 120581 120585119879120585+(1120581)120577119879120577 which is valid for any 120585 120577 isin R119898we obtain

119879120572119895 119881 (119905) le minus3120573 119885 (119905)2 + 120573 119885 (119905)2 + 1120573 119876119882(119905)2+ 120573 119885 (119905)2 + 1120573 1198762 Δ119865 (119905 119885 (119905))2 (36)

Since 119891(119905 119909(119905)) is Lipschitz with respect to 119909(119905) we cansimplify (36) as follows

119879120572119895 119881 (119905)le minus120573 119885 (119905)2 + 1198991198972 1198762120573

+ 1198762120573119899sum119894=1

(119891 (119905 119911119894 (119905) + 119909119903 (119905)) minus 119891 (119905 119909119903 (119905)))2le minus120573 119885 (119905)2 + 1198991198972 1198762120573 + 1198762 1205792120573 119885 (119905)2le minus120583 119885 (119905)2 + 1198991198972 1198762120573

(37)

where 120583 = 2120573 minus 11987621205792120573 gt 0 by (29)(1) In the following we will use Theorem 12 to prove

that system (33) is asymptotically stable at its origin

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 3

Definition 2 (Tabatabaei et al [18]) The modified initializedCaputo fractional derivative of time-varying order 120572(119905) isdefined as follows

1198791205722 119909 (119905) = 1Γ (1 minus 120572 (119905)) int119905

0

1199091015840 (120577)(119905 minus 120577)120572(119905) 119889120577 + Ψ119909119888 (119905) forall119905 ge 0

(2)

where Ψ119909119888 (119905) = (1Γ(1 minus 120572(119905))) int0minus119888(119905 minus 120577)minus120572(119905)1199091015840(120577)119889120577 captures

the behavior of 119909 before 119905 = 0 assuming that 119909 begins fromminus119888 lt 0 Since we will focus in this paper on systems which areat rest at 119905 lt 0 Ψ119909119888 (119905) = 0Definition 3 (Jiao Chen and Podlubny [31]) The distributedorder fractional derivative in the Caputo sense with respectto the density function 119888 [120583 1] 997888rarr [0 +infin) for some 1 gt120583 gt 0 such that int1

120583119888(120572)119889120572 = 0 is defined as follows

1198791205723 119909 (119905) = int1120583119888 (120572) 1198791205721 119909 (119905) 119889120572 (3)

The Laplace transform of a distributed order derivativewhich will appear in the derivation of our main results is

L 1198791205723 119909 (119905) = 119862 (119904)119883 (119904) minus 1119904 119862 (119904) 119909 (0+) (4)

where 119862(119904) = int119898119898minus1

119888(120572)119904120572119889120572Definition 4 (Souahi et al [17]) The conformable fractionalderivative starting from 119886 of a function 119909 defined on [119886infin) is

1198791205724 119909 (119905) = lim120598997888rarr0

119909 (119905 + 120598 (119905 minus 119886)1minus120572) minus 119909 (119905)120598 (5)

for all 119905 gt 119886 If lim119905997888rarr119886+1198791205724 119909(119905) exists then1198791205724 119909 (119886) = lim

119905997888rarr119886+1198791205724 119909 (119905) (6)

Definition 5 (Almeida et al [12]) Let 119896 [119886 119887] 997888rarr R be acontinuous nonnegative map such that 119896(119905) = 0 whenever119905 gt 119886 and 119909 [119886 119887] 997888rarr R By definition 119909 is 120572-differentiableat 119905 gt 119886 with respect to kernel 119896 if the limit

1198791205725 119909 (119905) = lim120598997888rarr0

119909 (119905 + 120598119896 (119905)1minus120572) minus 119909 (119905)120598 (7)

exists The local fractional derivative at 119905 = 119886 is defined by

1198791205725 119909 (119886) = lim119905997888rarr119886+

1198791205725 119909 (119905) (8)

if the limit exists

Definition 6 Let 119896 and 119909 be functions as in Definition 5and 119888 a density function as in Definition 3 The distributedconformable fractional derivative is defined as

1198791205726 119909 (119905) = int10119888 (120572) 1198791205725 119909 (119905) 119889120572 (9)

Theorem 7 (Almeida et al [12]) A function 119909 [119886 119887] 997888rarr Ris 120572-differentiable at 119905 gt 119886 if and only if it is differentiable at 119905In that case we have the relation

1198791205725 119909 (119905) = 119896 (119905)1minus120572 1199091015840 (119905) 119905 gt 119886 (10)

Notice that Definition 5 is a particular case of Defini-tion 6 Then byTheorem 7 if we take 119896(119905) = 119905 minus 119886 we obtain

1198791205724 119909 (119905) = (119905 minus 119886)1minus120572 1199091015840 (119905) 119905 gt 119886 (11)

In a similar form by substituting (10) in (9) we have

1198791205726 119909 (119905) = 1199091015840 (119905) int10119888 (120572) 119896 (119905)1minus120572 119889120572 (12)

Consider the generalized system of fractional differentialequations of order 120572 isin (0 1)

119879120572119895 119909 = 119891 (119905 119909 (119905)) 119895 = 1 2 3 4 5 or 6 (13)

where 119909 isin R119899 119909(1199050) = 1199090 and 119891 R+ times R119899 997888rarrR119899 is a given nonlinear function is Lipzchitz with respectto the second argument For simplicity and without loss ofgenerality we will consider that the equilibrium points of thesystems analyzed hereafter are at the origin ie 119891(119905 0) = 0forall119905 ge 0

Throughout this paper we will assume that the studiedsystems have unique solutions The existence and uniquenessof the solution of system (13) is discussed in Podlubny [4] Xuand He [32] Ford and Morgado [33] and Bayour and Torres[34] for the cases 119895 = 1 2 3 and 4 respectively The theoryof existence and uniqueness of solutions when 119895 = 5 or 6 canbe easily generalized from Bayour and Torres [34] by takinginto account (11) and (12)

The Final Value Theorem and an important Laplacetransform associated with fractional calculus both used inthe following sections of this article are presented next

Theorem 8 (Duffy [35]) Let 119865(119904) = L119891(119905) If all poles of119904119865(119904) are in the open left-half complex plane then

lim119905997888rarrinfin

119891 (119905) = lim119904997888rarr0

119904119865 (119904) (14)

Definition 9 (Podlubny [4]) A two-parameter function of theMittag-Leffler type is defined by

119864120572120573 (119911) = infinsum119896=0

119911119896Γ (120572119896 + 120573) 120572 gt 0 120573 gt 0 (15)

Lemma 10 (Podlubny [4]) TheMittag-Leffler function of twoparameters satisfies the following relationship

Lminus1 [119904minus(120572minus120573)119904120573 minus 119886 ] = 119905120572minus1119864120573120572 (119886119905120573) 10038161003816100381610038161003816119904120573 minus 11988610038161003816100381610038161003816 lt 1 (16)

3 Lyapunov Stability for GeneralizedFractional Systems

The two Theorems in this section summarize the knownresults for Lyapunov stability theory for nonlinear systems

4 Complexity

of (A) Li et al [16] and Wang and Li [20] (Definition 1)(B) Souahi et al [17] (Definition 2) (C) Tabatabaei et al[18] (Definition 3) (D) Taghavian and Tavazoei [19] (Defini-tion 4) (E) Almeida et al [12] (Definition 5) and also extendsthe Lyapunov stability theory for nonlinear systems definedby operators introduced in Definitions 5 and 6 Specificallyin (A) the Lyapunov direct method for standard Caputofractional system (13) with 119895 = 1 is proved and the definitionofMittag-Leffler stability is introduced In (B) the same resultfor the case of the modified initialized Caputo fractionalderivative of time-varying order 120572(119905)with 119895 = 2 is proved Forthe case of distributed fractional systems (13) the mentionedresult in (C) for 119895 = 3 is proved In the case of conformalfractional systems in (D) it is shown that (13) is fractionalexponentially stable which implies asymptotic stability for119895 = 4 For the case of Definitions 5 and 6 we show that theproofs are very similar to the one for the case 119895 = 4 Inconsequence Theorems 12 and 16 extend the Lyapunov directmethod for generalized fractional systems defined in (13)

Assumption 11 For 119895 = 2 system (13) is autonomous ie119891(119905 119909(119905)) = 119891(119909(119905))Theorem 12 Consider system (13) with 119895 = 1 2 3 4 5 or 6Let 119881(119905 119909(119905)) be a continuously differentiable function suchthat

1205741 119909 (119905)119897 le 119881 (119905 119909 (119905)) le 1205742 119909 (119905)119897119898 (17)

119879120572119895 119881 (119905 119909 (119905)) le minus1205743 119909 (119905)119897119898 (18)

where 119905 ge 0 120574119894 (119894 = 1 2 3) and 119897 and 119898 are arbitrary positiveconstants If Assumption 11 is fulfilled then the origin of system(13) is asymptotically stable

Proof

(i) For 119895 = 1 (13) is a standard Caputo fractional systemFor this system the proof is the same as the one ofTheorem 51 of Li et al [16] There it is shown that(13) is Mittag-Leffler stable which implies asymptoticstability

(ii) For 119895 = 2 (13) is a fractional system of time-varying order This proof follows from Theorem 1 ofTabatabaei et al [18] That result requires the weakerhypotheses 119881(119905 119909(119905)) ge 0 119881(119905 119909(119905)) = 0 lArrrArr 119909 = 0(which are implied by (17)) and 1198791205722119881(119905 119909(119905)) lt 0 in119863 minus 0 (which is implied by (18))

(iii) For 119895 = 3 (13) is a distributed fractional system Inthis case the proof can be found in Theorem 41 ofTaghavian and Tavazoei [19]

(iv) For 119895 = 4 (13) is a conformable fractional systemThisproof is the same as the one of Theorem 1 of Souahiet al [17] where it is shown that (13) is fractionalexponentially stableThat kind of stability also impliesasymptotic stability

(v) For 119895 = 5 or 6 (13) is a system with local fractionalderivatives or a distributed conformable fractional

system respectively In these instances the proofsare very similar to the one of the previous case(119895 = 4) That proof depends on two facts about theconformable derivative that it satisfies the productrule in the traditional sense and that the sign of1198791205724 119909(119905)determines the monotonicity of 119909(119905) Note from (8)and (12) that these features are also true for theoperators 1198791205725 and 1198791205726

The next result is a partial generalization of Theorem 12being more permissive with the Lyapunov function and itsfractional derivative but requiring a couple of additionalhypotheses

Assumption 13 For 119895 = 3 in (13) the Lyapunov function ofTheorem 16 has a nonzero initial value ie 119881(0 119909(0)) = 0Definition 14 (Teel and Praly [36]) A function ℎ [0infin) 997888rarr[0infin) is said to belong to classK if it is continuous zero atzero and strictly increasing

Assumption 15 For 119895 = 4 5 and 6 in (13) the class Kfunctions ℎ119894 (119894 = 1 2 3) satisfy lim119905997888rarrinfinℎ119894(119905) = infin

Theorem 16 Consider system (13) with 119895 = 1 2 3 4 5 or 6Suppose that there exist classK functions ℎ119894 (119894 = 1 2 3) and acontinuously differentiable function 119881(119905 119909(119905)) such that

ℎ1 (119909 (119905)) le 119881 (119905 119909 (119905)) le ℎ2 (119909 (119905)) (19)

119879120572119895 119881(119905 119909 (119905)) le minusℎ3 (119909 (119905)) (20)

If Assumptions 11 13 and 15 are fulfilled then the origin ofsystem (13) is asymptotically stable

Proof(i) For 119895 = 1 the proof can be found on Theorem 62 of

Li et al [16](ii) For 119895 = 2 the proof is presented in Tabatabaei et al

[18] as explained in item (ii) of Theorem 12 proof(iii) For 119895 = 3 the proof is the same as the one ofTheorem

42 of Taghavian and Tavazoei [19](iv) For 119895 = 4 proof can be found inTheorem 3 of Souahi

et al [17](v) For 119895 = 5 or 6 considering the argument stated in

(v) ofTheorem 12 proof we can readily generalize theresult of 119895 = 4 to cases of the distributed conformableand local fractional derivatives

We now know that Theorem 16 is valid also for Riemann-Liouville-like fractional difference equations (see Theorem36 in Wu Baleanu and Luo [37]) So we conjecture thatTheorem 16 can be valid for a larger family of operators

The following lemma contains a property of the gener-alized fractional differential operator which is useful whenputting into practice the previous Lyapunov Stability Theo-rems

Complexity 5

Lemma 17 Let 119909 R 997888rarr R be a continuous differentiablefunction Then for 119895 = 1 2 3 4 5 or 6 the followingrelationship holds

12119879120572119895 [119909119879 (119905) 119875119909 (119905)] le 119909119879 (119905) 119875119879120572119895 [119909 (119905)] (21)

where 119875 is a Hermitian positive definite matrix

Proof The proof for the cases 119895 = 1 2 4 can be found inAguila-Camacho et al [30] Souahi et al [17] Tabatabaeiet al [18] respectively If 119895 = 3 a proof for when 119875 =119868 is presented in Fernandez-Anaya Nava-Antonio Jamous-Galante Munoz-Vega and Hernandez-Martınez [38] Toobtain the more general version consider inequality (21) with119895 = 1 multiply it by the distribution function 119888(120572) ge 0 andintegrate

int10

12119888 (120572) 1198791205721 [119909119879 (119905) 119875119909 (119905)] 119889120572 = 121198791205723 [119909119879 (119905) 119875119909 (119905)]le

le int10119909119879 (119905) 119875119888 (120572) 1198791205721 [119909 (119905)] 119889120572

= 119909119879 (119905) 1198751198791205723 [119909 (119905)] (22)

We can follow a similar reasoning to prove this lemma for119895 = 5 or 6 by multiplying (21) with 119895 = 4 and 120572 =1 (that is the traditional integer order derivative) by 119896(119905)1minus120572or int10119888(120572)119896(119905)1minus120572119889120572 and using properties (8) or (12) respec-

tively

4 Application to the Consensus of MultiagentSystems of Generalized Fractional Order

In this section we will investigate the problem of consensusfor generalized multiagent systems First we will considersystems with nonlinear dynamics and then we will presentthe linear simplification of that analysis

41 Graph Theory Fundamentals We can describe the inter-action topology of a multagent system with the help of graphtheory A graph G is characterized by its vertices V =V1 V2 V119899 (which represent the agents of the system)and its edges W sube V2 (which correspond to the agentsrsquorelationships) In this paper we will focus on directed graphswhere each edge is an ordered pair (V119894 V119895) this means thatagent 119895 receives information from agent 119894 A graph can berepresented by its adjacency matrix 119860 = [119886119894119895] isin R119899times119899 where119886119894119895 = 1 if (V119894 V119895) isin W and 119886119894119895 = 0 if (V119894 V119895) notin W or by itsLaplacian matrix 119871 = [119897119894119895] isin R119899times119899 where 119897119894119894 = sum119895isin119873119894 119886119894119895 and119897119894119895 = minus119886119894119895 for 119894 = 119895 with 119873119894 the number of connected nodes tonode 119894

The following lemmas will be used in the proofs of ourmain results to gain insight into the graphs associated withthe multiagent systems of our interest

Lemma 18 (W Ren and Cao [24]) If a graph has a directedspanning tree then the Laplacian matrix 119871 has a simple zeroeigenvalue and all its other eigenvalues have positive real partsMoreover all eigenvalues of 119867 = 119871 + 119861 will have positive realparts where 119861 = diag1198871 1198872 119887119899 and 119887119894 ge 0 is not all 0Lemma 19 (Zhang and Tian [39]) Let 119864 = [1119899minus1 minus119868119899minus1] isinR(119899minus1)times119899 and 119865 = ( 0119879119899minus1

minus119868119899minus1) isin R119899times(119899minus1) where 1119899minus1 is the column

vector of ones 119868119899minus1 is the identity matrix and 0119899minus1 is the zerocolumn vector and each of them is of size 119899minus1Then119862 = minus119864119871119865is Hurwitz where 119871 is the Laplacian matrix if and only if theassociated interaction graph has a directed spanning tree

The notion of consensus that will be considered through-out this paper is presented next

Definition 20 A multiagent system accomplishes consensusif it fulfills the following condition

lim119905997888rarrinfin

1003817100381710038171003817119909119894 (119905) minus 119909119896 (119905)1003817100381710038171003817 le 0forall119894 119896 isin 1 2 119899 119894 = 119896 (23)

where 119909119896(119905) is the state of the 119896-th agent

Hereinafter we will suppose for simplicity that all agentsare in a one-dimensional space All our results can be easilygeneralized for 119898 dimensions by means of the Kroneckerproduct Moreover in this work we will consider the matrixnorm

119860 = radic 119899sum119894=1

119898sum119895=1

1198862119894119895 (24)

with 119860 = (119886119894119895) isin R119899times119898 And for any matrix 119876 isin R119899times119899120582max(119876) and 120582min(119876) denote the largest and smallest eigen-values respectively

42 Robust Consensus of Nonlinear Generalized FractionalMultiagent Systems A generalized nonlinear fractionalmulti-agent system can be represented by

119879120572119895 119909119894 (119905) = 119891 (119905 119909119894 (119905)) + 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (25)

where 119895 = 1 2 3 4 5 or 6 and 119909119894(119905)119891(119905 119909119894(119905)) 119906119894(119905) and119908119894(119905)are the state nonlinear dynamics control input and externaldisturbances of the 119894-th agent respectively

As an auxiliary element we will consider a virtual leaderwhich is an isolated agent that designates objectives for thestates of all other agents The behavior of the virtual leader ischaracterized by

119879120572119895 119909119903 (119905) = 119891 (119905 119909119903 (119905)) (26)

6 Complexity

where 119909119903(119905) is the state of the virtual leader To accomplishconsensus in system (25) we will use the following controlinput

119906119894 (119905)= minus120573[ 119899sum

119896=1

119886119894119896 (119909119894 (119905) minus 119909119896 (119905)) + 119887119894 (119909119894 (119905) minus 119909119903 (119905))] (27)

where 119886119894119896 for 119894 119896 isin 1 2 119899 with 119894 = 119896 is the (119894 119896)-thentry of the adjacency matrix 119860 isin R119899times119899 associated with theundirected graph describing the interaction of the agents and120573 ge 0 and 119887119894 for (119894 = 1 2 119899) are positive constants to bechosen as mentioned in Theorem 23

We will require that the following assumptions hold

Assumption 21 The disturbance signal 119908119894(119905) satisfies119908119894(119905) le 119897 lt infin forall119894 isin 1 2 119899Assumption 22 For the multiagent system (25) with 119895 = 3the distribution function 119888(120572) is such that

Lminus1 1119862 (119904) + 120583120582max (119876) ge 0 (28)

where 119862(119904) is defined in terms of 119888(120572) as in (4)

Theorem 23 Consider the generalized fractional nonlinearmultiagent system (25) with the virtual leader (26) and thecontroller (27) Assume that the nonlinear function119891(119905 119909(119905)) isLipschitz (with respect to 119909 and with Lipschitz constant 120579) andthat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 Assumption 11is satisfied and

radic2120573120579 ge 119876 (29)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then robust consensus isachieved

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 Assumptions 21 and 22are satisfied and

120573120579 ge 119876 (30)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then the steady-state errors ofany two agent will converge as 119905 997888rarr infin to the region1198721 where

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic 2119899120582max (119876)120573120583120582min (119876) 119876 119897 (31)

and 120583 = 120573 minus 11987621205792120573

Proof By substituting (27) in system (25) we can write

119879120572119895 119883 (119905) = 119865 (119883 (119905)) minus 120573 [119871119883 (119905) + 119861 (119883 (119905) minus 1199091199031119899)]+ 119882 (119905) (32)

where 119865(119883(119905)) = [119891(1199091(119905)) 119891(119909119899(119905))]119879 Subtracting119879120572119895 [1119899119909119903(119905)] from both sides of (32) and using the change ofvariables 119911119894(119905) = 119909119894(119905) minus 119909119903(119905) 119894 isin 1 2 119899 yields

119879120572119895 119885 (119905) = minus120573119867119885 (119905) + Δ119865 (119885 (119905)) + 119882 (119905) (33)

where119867 is defined as in Lemma 18119885(119905) = [1199111(119905) 119911119899(119905)]119879and Δ119865(119885(119905)) = [119891(1199111(119905) + 119909119903(119905)) minus 119891(119909119903(119905)) 119891(119911119899(119905) +119909119903(119905)) minus 119891(119909119903(119905))]119879 Consider the following Lyapunov candi-date function for system (33)

119881 (119905) = 119885119879119876119885 (119905) (34)

Applying Lemma 17 and substituting (33) we can analyze119879120572119895 119881(119905)119879120572119895 119881 (119905) le 120573119885119879 (119905) [minus119876119867 minus119867119879119876]119885 (119905)

+ 2119885119879 (119905) [119876Δ119865 (119885 (119905)) + 119876119882(119905)] (35)

Using Lemma 18 we can conclude that all the eigenvalues of119867 have positive real parts so that minus119867 is Hurwitz Thus thereexists a matrix 119876 = 119876119879 gt 0 that satisfies minus119867119879119876 minus 119876119867 =minus3119868119899 Applying in (35) this identity along with the property120585119879120577+120577119879120585 le 120581 120585119879120585+(1120581)120577119879120577 which is valid for any 120585 120577 isin R119898we obtain

119879120572119895 119881 (119905) le minus3120573 119885 (119905)2 + 120573 119885 (119905)2 + 1120573 119876119882(119905)2+ 120573 119885 (119905)2 + 1120573 1198762 Δ119865 (119905 119885 (119905))2 (36)

Since 119891(119905 119909(119905)) is Lipschitz with respect to 119909(119905) we cansimplify (36) as follows

119879120572119895 119881 (119905)le minus120573 119885 (119905)2 + 1198991198972 1198762120573

+ 1198762120573119899sum119894=1

(119891 (119905 119911119894 (119905) + 119909119903 (119905)) minus 119891 (119905 119909119903 (119905)))2le minus120573 119885 (119905)2 + 1198991198972 1198762120573 + 1198762 1205792120573 119885 (119905)2le minus120583 119885 (119905)2 + 1198991198972 1198762120573

(37)

where 120583 = 2120573 minus 11987621205792120573 gt 0 by (29)(1) In the following we will use Theorem 12 to prove

that system (33) is asymptotically stable at its origin

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

4 Complexity

of (A) Li et al [16] and Wang and Li [20] (Definition 1)(B) Souahi et al [17] (Definition 2) (C) Tabatabaei et al[18] (Definition 3) (D) Taghavian and Tavazoei [19] (Defini-tion 4) (E) Almeida et al [12] (Definition 5) and also extendsthe Lyapunov stability theory for nonlinear systems definedby operators introduced in Definitions 5 and 6 Specificallyin (A) the Lyapunov direct method for standard Caputofractional system (13) with 119895 = 1 is proved and the definitionofMittag-Leffler stability is introduced In (B) the same resultfor the case of the modified initialized Caputo fractionalderivative of time-varying order 120572(119905)with 119895 = 2 is proved Forthe case of distributed fractional systems (13) the mentionedresult in (C) for 119895 = 3 is proved In the case of conformalfractional systems in (D) it is shown that (13) is fractionalexponentially stable which implies asymptotic stability for119895 = 4 For the case of Definitions 5 and 6 we show that theproofs are very similar to the one for the case 119895 = 4 Inconsequence Theorems 12 and 16 extend the Lyapunov directmethod for generalized fractional systems defined in (13)

Assumption 11 For 119895 = 2 system (13) is autonomous ie119891(119905 119909(119905)) = 119891(119909(119905))Theorem 12 Consider system (13) with 119895 = 1 2 3 4 5 or 6Let 119881(119905 119909(119905)) be a continuously differentiable function suchthat

1205741 119909 (119905)119897 le 119881 (119905 119909 (119905)) le 1205742 119909 (119905)119897119898 (17)

119879120572119895 119881 (119905 119909 (119905)) le minus1205743 119909 (119905)119897119898 (18)

where 119905 ge 0 120574119894 (119894 = 1 2 3) and 119897 and 119898 are arbitrary positiveconstants If Assumption 11 is fulfilled then the origin of system(13) is asymptotically stable

Proof

(i) For 119895 = 1 (13) is a standard Caputo fractional systemFor this system the proof is the same as the one ofTheorem 51 of Li et al [16] There it is shown that(13) is Mittag-Leffler stable which implies asymptoticstability

(ii) For 119895 = 2 (13) is a fractional system of time-varying order This proof follows from Theorem 1 ofTabatabaei et al [18] That result requires the weakerhypotheses 119881(119905 119909(119905)) ge 0 119881(119905 119909(119905)) = 0 lArrrArr 119909 = 0(which are implied by (17)) and 1198791205722119881(119905 119909(119905)) lt 0 in119863 minus 0 (which is implied by (18))

(iii) For 119895 = 3 (13) is a distributed fractional system Inthis case the proof can be found in Theorem 41 ofTaghavian and Tavazoei [19]

(iv) For 119895 = 4 (13) is a conformable fractional systemThisproof is the same as the one of Theorem 1 of Souahiet al [17] where it is shown that (13) is fractionalexponentially stableThat kind of stability also impliesasymptotic stability

(v) For 119895 = 5 or 6 (13) is a system with local fractionalderivatives or a distributed conformable fractional

system respectively In these instances the proofsare very similar to the one of the previous case(119895 = 4) That proof depends on two facts about theconformable derivative that it satisfies the productrule in the traditional sense and that the sign of1198791205724 119909(119905)determines the monotonicity of 119909(119905) Note from (8)and (12) that these features are also true for theoperators 1198791205725 and 1198791205726

The next result is a partial generalization of Theorem 12being more permissive with the Lyapunov function and itsfractional derivative but requiring a couple of additionalhypotheses

Assumption 13 For 119895 = 3 in (13) the Lyapunov function ofTheorem 16 has a nonzero initial value ie 119881(0 119909(0)) = 0Definition 14 (Teel and Praly [36]) A function ℎ [0infin) 997888rarr[0infin) is said to belong to classK if it is continuous zero atzero and strictly increasing

Assumption 15 For 119895 = 4 5 and 6 in (13) the class Kfunctions ℎ119894 (119894 = 1 2 3) satisfy lim119905997888rarrinfinℎ119894(119905) = infin

Theorem 16 Consider system (13) with 119895 = 1 2 3 4 5 or 6Suppose that there exist classK functions ℎ119894 (119894 = 1 2 3) and acontinuously differentiable function 119881(119905 119909(119905)) such that

ℎ1 (119909 (119905)) le 119881 (119905 119909 (119905)) le ℎ2 (119909 (119905)) (19)

119879120572119895 119881(119905 119909 (119905)) le minusℎ3 (119909 (119905)) (20)

If Assumptions 11 13 and 15 are fulfilled then the origin ofsystem (13) is asymptotically stable

Proof(i) For 119895 = 1 the proof can be found on Theorem 62 of

Li et al [16](ii) For 119895 = 2 the proof is presented in Tabatabaei et al

[18] as explained in item (ii) of Theorem 12 proof(iii) For 119895 = 3 the proof is the same as the one ofTheorem

42 of Taghavian and Tavazoei [19](iv) For 119895 = 4 proof can be found inTheorem 3 of Souahi

et al [17](v) For 119895 = 5 or 6 considering the argument stated in

(v) ofTheorem 12 proof we can readily generalize theresult of 119895 = 4 to cases of the distributed conformableand local fractional derivatives

We now know that Theorem 16 is valid also for Riemann-Liouville-like fractional difference equations (see Theorem36 in Wu Baleanu and Luo [37]) So we conjecture thatTheorem 16 can be valid for a larger family of operators

The following lemma contains a property of the gener-alized fractional differential operator which is useful whenputting into practice the previous Lyapunov Stability Theo-rems

Complexity 5

Lemma 17 Let 119909 R 997888rarr R be a continuous differentiablefunction Then for 119895 = 1 2 3 4 5 or 6 the followingrelationship holds

12119879120572119895 [119909119879 (119905) 119875119909 (119905)] le 119909119879 (119905) 119875119879120572119895 [119909 (119905)] (21)

where 119875 is a Hermitian positive definite matrix

Proof The proof for the cases 119895 = 1 2 4 can be found inAguila-Camacho et al [30] Souahi et al [17] Tabatabaeiet al [18] respectively If 119895 = 3 a proof for when 119875 =119868 is presented in Fernandez-Anaya Nava-Antonio Jamous-Galante Munoz-Vega and Hernandez-Martınez [38] Toobtain the more general version consider inequality (21) with119895 = 1 multiply it by the distribution function 119888(120572) ge 0 andintegrate

int10

12119888 (120572) 1198791205721 [119909119879 (119905) 119875119909 (119905)] 119889120572 = 121198791205723 [119909119879 (119905) 119875119909 (119905)]le

le int10119909119879 (119905) 119875119888 (120572) 1198791205721 [119909 (119905)] 119889120572

= 119909119879 (119905) 1198751198791205723 [119909 (119905)] (22)

We can follow a similar reasoning to prove this lemma for119895 = 5 or 6 by multiplying (21) with 119895 = 4 and 120572 =1 (that is the traditional integer order derivative) by 119896(119905)1minus120572or int10119888(120572)119896(119905)1minus120572119889120572 and using properties (8) or (12) respec-

tively

4 Application to the Consensus of MultiagentSystems of Generalized Fractional Order

In this section we will investigate the problem of consensusfor generalized multiagent systems First we will considersystems with nonlinear dynamics and then we will presentthe linear simplification of that analysis

41 Graph Theory Fundamentals We can describe the inter-action topology of a multagent system with the help of graphtheory A graph G is characterized by its vertices V =V1 V2 V119899 (which represent the agents of the system)and its edges W sube V2 (which correspond to the agentsrsquorelationships) In this paper we will focus on directed graphswhere each edge is an ordered pair (V119894 V119895) this means thatagent 119895 receives information from agent 119894 A graph can berepresented by its adjacency matrix 119860 = [119886119894119895] isin R119899times119899 where119886119894119895 = 1 if (V119894 V119895) isin W and 119886119894119895 = 0 if (V119894 V119895) notin W or by itsLaplacian matrix 119871 = [119897119894119895] isin R119899times119899 where 119897119894119894 = sum119895isin119873119894 119886119894119895 and119897119894119895 = minus119886119894119895 for 119894 = 119895 with 119873119894 the number of connected nodes tonode 119894

The following lemmas will be used in the proofs of ourmain results to gain insight into the graphs associated withthe multiagent systems of our interest

Lemma 18 (W Ren and Cao [24]) If a graph has a directedspanning tree then the Laplacian matrix 119871 has a simple zeroeigenvalue and all its other eigenvalues have positive real partsMoreover all eigenvalues of 119867 = 119871 + 119861 will have positive realparts where 119861 = diag1198871 1198872 119887119899 and 119887119894 ge 0 is not all 0Lemma 19 (Zhang and Tian [39]) Let 119864 = [1119899minus1 minus119868119899minus1] isinR(119899minus1)times119899 and 119865 = ( 0119879119899minus1

minus119868119899minus1) isin R119899times(119899minus1) where 1119899minus1 is the column

vector of ones 119868119899minus1 is the identity matrix and 0119899minus1 is the zerocolumn vector and each of them is of size 119899minus1Then119862 = minus119864119871119865is Hurwitz where 119871 is the Laplacian matrix if and only if theassociated interaction graph has a directed spanning tree

The notion of consensus that will be considered through-out this paper is presented next

Definition 20 A multiagent system accomplishes consensusif it fulfills the following condition

lim119905997888rarrinfin

1003817100381710038171003817119909119894 (119905) minus 119909119896 (119905)1003817100381710038171003817 le 0forall119894 119896 isin 1 2 119899 119894 = 119896 (23)

where 119909119896(119905) is the state of the 119896-th agent

Hereinafter we will suppose for simplicity that all agentsare in a one-dimensional space All our results can be easilygeneralized for 119898 dimensions by means of the Kroneckerproduct Moreover in this work we will consider the matrixnorm

119860 = radic 119899sum119894=1

119898sum119895=1

1198862119894119895 (24)

with 119860 = (119886119894119895) isin R119899times119898 And for any matrix 119876 isin R119899times119899120582max(119876) and 120582min(119876) denote the largest and smallest eigen-values respectively

42 Robust Consensus of Nonlinear Generalized FractionalMultiagent Systems A generalized nonlinear fractionalmulti-agent system can be represented by

119879120572119895 119909119894 (119905) = 119891 (119905 119909119894 (119905)) + 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (25)

where 119895 = 1 2 3 4 5 or 6 and 119909119894(119905)119891(119905 119909119894(119905)) 119906119894(119905) and119908119894(119905)are the state nonlinear dynamics control input and externaldisturbances of the 119894-th agent respectively

As an auxiliary element we will consider a virtual leaderwhich is an isolated agent that designates objectives for thestates of all other agents The behavior of the virtual leader ischaracterized by

119879120572119895 119909119903 (119905) = 119891 (119905 119909119903 (119905)) (26)

6 Complexity

where 119909119903(119905) is the state of the virtual leader To accomplishconsensus in system (25) we will use the following controlinput

119906119894 (119905)= minus120573[ 119899sum

119896=1

119886119894119896 (119909119894 (119905) minus 119909119896 (119905)) + 119887119894 (119909119894 (119905) minus 119909119903 (119905))] (27)

where 119886119894119896 for 119894 119896 isin 1 2 119899 with 119894 = 119896 is the (119894 119896)-thentry of the adjacency matrix 119860 isin R119899times119899 associated with theundirected graph describing the interaction of the agents and120573 ge 0 and 119887119894 for (119894 = 1 2 119899) are positive constants to bechosen as mentioned in Theorem 23

We will require that the following assumptions hold

Assumption 21 The disturbance signal 119908119894(119905) satisfies119908119894(119905) le 119897 lt infin forall119894 isin 1 2 119899Assumption 22 For the multiagent system (25) with 119895 = 3the distribution function 119888(120572) is such that

Lminus1 1119862 (119904) + 120583120582max (119876) ge 0 (28)

where 119862(119904) is defined in terms of 119888(120572) as in (4)

Theorem 23 Consider the generalized fractional nonlinearmultiagent system (25) with the virtual leader (26) and thecontroller (27) Assume that the nonlinear function119891(119905 119909(119905)) isLipschitz (with respect to 119909 and with Lipschitz constant 120579) andthat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 Assumption 11is satisfied and

radic2120573120579 ge 119876 (29)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then robust consensus isachieved

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 Assumptions 21 and 22are satisfied and

120573120579 ge 119876 (30)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then the steady-state errors ofany two agent will converge as 119905 997888rarr infin to the region1198721 where

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic 2119899120582max (119876)120573120583120582min (119876) 119876 119897 (31)

and 120583 = 120573 minus 11987621205792120573

Proof By substituting (27) in system (25) we can write

119879120572119895 119883 (119905) = 119865 (119883 (119905)) minus 120573 [119871119883 (119905) + 119861 (119883 (119905) minus 1199091199031119899)]+ 119882 (119905) (32)

where 119865(119883(119905)) = [119891(1199091(119905)) 119891(119909119899(119905))]119879 Subtracting119879120572119895 [1119899119909119903(119905)] from both sides of (32) and using the change ofvariables 119911119894(119905) = 119909119894(119905) minus 119909119903(119905) 119894 isin 1 2 119899 yields

119879120572119895 119885 (119905) = minus120573119867119885 (119905) + Δ119865 (119885 (119905)) + 119882 (119905) (33)

where119867 is defined as in Lemma 18119885(119905) = [1199111(119905) 119911119899(119905)]119879and Δ119865(119885(119905)) = [119891(1199111(119905) + 119909119903(119905)) minus 119891(119909119903(119905)) 119891(119911119899(119905) +119909119903(119905)) minus 119891(119909119903(119905))]119879 Consider the following Lyapunov candi-date function for system (33)

119881 (119905) = 119885119879119876119885 (119905) (34)

Applying Lemma 17 and substituting (33) we can analyze119879120572119895 119881(119905)119879120572119895 119881 (119905) le 120573119885119879 (119905) [minus119876119867 minus119867119879119876]119885 (119905)

+ 2119885119879 (119905) [119876Δ119865 (119885 (119905)) + 119876119882(119905)] (35)

Using Lemma 18 we can conclude that all the eigenvalues of119867 have positive real parts so that minus119867 is Hurwitz Thus thereexists a matrix 119876 = 119876119879 gt 0 that satisfies minus119867119879119876 minus 119876119867 =minus3119868119899 Applying in (35) this identity along with the property120585119879120577+120577119879120585 le 120581 120585119879120585+(1120581)120577119879120577 which is valid for any 120585 120577 isin R119898we obtain

119879120572119895 119881 (119905) le minus3120573 119885 (119905)2 + 120573 119885 (119905)2 + 1120573 119876119882(119905)2+ 120573 119885 (119905)2 + 1120573 1198762 Δ119865 (119905 119885 (119905))2 (36)

Since 119891(119905 119909(119905)) is Lipschitz with respect to 119909(119905) we cansimplify (36) as follows

119879120572119895 119881 (119905)le minus120573 119885 (119905)2 + 1198991198972 1198762120573

+ 1198762120573119899sum119894=1

(119891 (119905 119911119894 (119905) + 119909119903 (119905)) minus 119891 (119905 119909119903 (119905)))2le minus120573 119885 (119905)2 + 1198991198972 1198762120573 + 1198762 1205792120573 119885 (119905)2le minus120583 119885 (119905)2 + 1198991198972 1198762120573

(37)

where 120583 = 2120573 minus 11987621205792120573 gt 0 by (29)(1) In the following we will use Theorem 12 to prove

that system (33) is asymptotically stable at its origin

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 5

Lemma 17 Let 119909 R 997888rarr R be a continuous differentiablefunction Then for 119895 = 1 2 3 4 5 or 6 the followingrelationship holds

12119879120572119895 [119909119879 (119905) 119875119909 (119905)] le 119909119879 (119905) 119875119879120572119895 [119909 (119905)] (21)

where 119875 is a Hermitian positive definite matrix

Proof The proof for the cases 119895 = 1 2 4 can be found inAguila-Camacho et al [30] Souahi et al [17] Tabatabaeiet al [18] respectively If 119895 = 3 a proof for when 119875 =119868 is presented in Fernandez-Anaya Nava-Antonio Jamous-Galante Munoz-Vega and Hernandez-Martınez [38] Toobtain the more general version consider inequality (21) with119895 = 1 multiply it by the distribution function 119888(120572) ge 0 andintegrate

int10

12119888 (120572) 1198791205721 [119909119879 (119905) 119875119909 (119905)] 119889120572 = 121198791205723 [119909119879 (119905) 119875119909 (119905)]le

le int10119909119879 (119905) 119875119888 (120572) 1198791205721 [119909 (119905)] 119889120572

= 119909119879 (119905) 1198751198791205723 [119909 (119905)] (22)

We can follow a similar reasoning to prove this lemma for119895 = 5 or 6 by multiplying (21) with 119895 = 4 and 120572 =1 (that is the traditional integer order derivative) by 119896(119905)1minus120572or int10119888(120572)119896(119905)1minus120572119889120572 and using properties (8) or (12) respec-

tively

4 Application to the Consensus of MultiagentSystems of Generalized Fractional Order

In this section we will investigate the problem of consensusfor generalized multiagent systems First we will considersystems with nonlinear dynamics and then we will presentthe linear simplification of that analysis

41 Graph Theory Fundamentals We can describe the inter-action topology of a multagent system with the help of graphtheory A graph G is characterized by its vertices V =V1 V2 V119899 (which represent the agents of the system)and its edges W sube V2 (which correspond to the agentsrsquorelationships) In this paper we will focus on directed graphswhere each edge is an ordered pair (V119894 V119895) this means thatagent 119895 receives information from agent 119894 A graph can berepresented by its adjacency matrix 119860 = [119886119894119895] isin R119899times119899 where119886119894119895 = 1 if (V119894 V119895) isin W and 119886119894119895 = 0 if (V119894 V119895) notin W or by itsLaplacian matrix 119871 = [119897119894119895] isin R119899times119899 where 119897119894119894 = sum119895isin119873119894 119886119894119895 and119897119894119895 = minus119886119894119895 for 119894 = 119895 with 119873119894 the number of connected nodes tonode 119894

The following lemmas will be used in the proofs of ourmain results to gain insight into the graphs associated withthe multiagent systems of our interest

Lemma 18 (W Ren and Cao [24]) If a graph has a directedspanning tree then the Laplacian matrix 119871 has a simple zeroeigenvalue and all its other eigenvalues have positive real partsMoreover all eigenvalues of 119867 = 119871 + 119861 will have positive realparts where 119861 = diag1198871 1198872 119887119899 and 119887119894 ge 0 is not all 0Lemma 19 (Zhang and Tian [39]) Let 119864 = [1119899minus1 minus119868119899minus1] isinR(119899minus1)times119899 and 119865 = ( 0119879119899minus1

minus119868119899minus1) isin R119899times(119899minus1) where 1119899minus1 is the column

vector of ones 119868119899minus1 is the identity matrix and 0119899minus1 is the zerocolumn vector and each of them is of size 119899minus1Then119862 = minus119864119871119865is Hurwitz where 119871 is the Laplacian matrix if and only if theassociated interaction graph has a directed spanning tree

The notion of consensus that will be considered through-out this paper is presented next

Definition 20 A multiagent system accomplishes consensusif it fulfills the following condition

lim119905997888rarrinfin

1003817100381710038171003817119909119894 (119905) minus 119909119896 (119905)1003817100381710038171003817 le 0forall119894 119896 isin 1 2 119899 119894 = 119896 (23)

where 119909119896(119905) is the state of the 119896-th agent

Hereinafter we will suppose for simplicity that all agentsare in a one-dimensional space All our results can be easilygeneralized for 119898 dimensions by means of the Kroneckerproduct Moreover in this work we will consider the matrixnorm

119860 = radic 119899sum119894=1

119898sum119895=1

1198862119894119895 (24)

with 119860 = (119886119894119895) isin R119899times119898 And for any matrix 119876 isin R119899times119899120582max(119876) and 120582min(119876) denote the largest and smallest eigen-values respectively

42 Robust Consensus of Nonlinear Generalized FractionalMultiagent Systems A generalized nonlinear fractionalmulti-agent system can be represented by

119879120572119895 119909119894 (119905) = 119891 (119905 119909119894 (119905)) + 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (25)

where 119895 = 1 2 3 4 5 or 6 and 119909119894(119905)119891(119905 119909119894(119905)) 119906119894(119905) and119908119894(119905)are the state nonlinear dynamics control input and externaldisturbances of the 119894-th agent respectively

As an auxiliary element we will consider a virtual leaderwhich is an isolated agent that designates objectives for thestates of all other agents The behavior of the virtual leader ischaracterized by

119879120572119895 119909119903 (119905) = 119891 (119905 119909119903 (119905)) (26)

6 Complexity

where 119909119903(119905) is the state of the virtual leader To accomplishconsensus in system (25) we will use the following controlinput

119906119894 (119905)= minus120573[ 119899sum

119896=1

119886119894119896 (119909119894 (119905) minus 119909119896 (119905)) + 119887119894 (119909119894 (119905) minus 119909119903 (119905))] (27)

where 119886119894119896 for 119894 119896 isin 1 2 119899 with 119894 = 119896 is the (119894 119896)-thentry of the adjacency matrix 119860 isin R119899times119899 associated with theundirected graph describing the interaction of the agents and120573 ge 0 and 119887119894 for (119894 = 1 2 119899) are positive constants to bechosen as mentioned in Theorem 23

We will require that the following assumptions hold

Assumption 21 The disturbance signal 119908119894(119905) satisfies119908119894(119905) le 119897 lt infin forall119894 isin 1 2 119899Assumption 22 For the multiagent system (25) with 119895 = 3the distribution function 119888(120572) is such that

Lminus1 1119862 (119904) + 120583120582max (119876) ge 0 (28)

where 119862(119904) is defined in terms of 119888(120572) as in (4)

Theorem 23 Consider the generalized fractional nonlinearmultiagent system (25) with the virtual leader (26) and thecontroller (27) Assume that the nonlinear function119891(119905 119909(119905)) isLipschitz (with respect to 119909 and with Lipschitz constant 120579) andthat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 Assumption 11is satisfied and

radic2120573120579 ge 119876 (29)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then robust consensus isachieved

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 Assumptions 21 and 22are satisfied and

120573120579 ge 119876 (30)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then the steady-state errors ofany two agent will converge as 119905 997888rarr infin to the region1198721 where

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic 2119899120582max (119876)120573120583120582min (119876) 119876 119897 (31)

and 120583 = 120573 minus 11987621205792120573

Proof By substituting (27) in system (25) we can write

119879120572119895 119883 (119905) = 119865 (119883 (119905)) minus 120573 [119871119883 (119905) + 119861 (119883 (119905) minus 1199091199031119899)]+ 119882 (119905) (32)

where 119865(119883(119905)) = [119891(1199091(119905)) 119891(119909119899(119905))]119879 Subtracting119879120572119895 [1119899119909119903(119905)] from both sides of (32) and using the change ofvariables 119911119894(119905) = 119909119894(119905) minus 119909119903(119905) 119894 isin 1 2 119899 yields

119879120572119895 119885 (119905) = minus120573119867119885 (119905) + Δ119865 (119885 (119905)) + 119882 (119905) (33)

where119867 is defined as in Lemma 18119885(119905) = [1199111(119905) 119911119899(119905)]119879and Δ119865(119885(119905)) = [119891(1199111(119905) + 119909119903(119905)) minus 119891(119909119903(119905)) 119891(119911119899(119905) +119909119903(119905)) minus 119891(119909119903(119905))]119879 Consider the following Lyapunov candi-date function for system (33)

119881 (119905) = 119885119879119876119885 (119905) (34)

Applying Lemma 17 and substituting (33) we can analyze119879120572119895 119881(119905)119879120572119895 119881 (119905) le 120573119885119879 (119905) [minus119876119867 minus119867119879119876]119885 (119905)

+ 2119885119879 (119905) [119876Δ119865 (119885 (119905)) + 119876119882(119905)] (35)

Using Lemma 18 we can conclude that all the eigenvalues of119867 have positive real parts so that minus119867 is Hurwitz Thus thereexists a matrix 119876 = 119876119879 gt 0 that satisfies minus119867119879119876 minus 119876119867 =minus3119868119899 Applying in (35) this identity along with the property120585119879120577+120577119879120585 le 120581 120585119879120585+(1120581)120577119879120577 which is valid for any 120585 120577 isin R119898we obtain

119879120572119895 119881 (119905) le minus3120573 119885 (119905)2 + 120573 119885 (119905)2 + 1120573 119876119882(119905)2+ 120573 119885 (119905)2 + 1120573 1198762 Δ119865 (119905 119885 (119905))2 (36)

Since 119891(119905 119909(119905)) is Lipschitz with respect to 119909(119905) we cansimplify (36) as follows

119879120572119895 119881 (119905)le minus120573 119885 (119905)2 + 1198991198972 1198762120573

+ 1198762120573119899sum119894=1

(119891 (119905 119911119894 (119905) + 119909119903 (119905)) minus 119891 (119905 119909119903 (119905)))2le minus120573 119885 (119905)2 + 1198991198972 1198762120573 + 1198762 1205792120573 119885 (119905)2le minus120583 119885 (119905)2 + 1198991198972 1198762120573

(37)

where 120583 = 2120573 minus 11987621205792120573 gt 0 by (29)(1) In the following we will use Theorem 12 to prove

that system (33) is asymptotically stable at its origin

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

6 Complexity

where 119909119903(119905) is the state of the virtual leader To accomplishconsensus in system (25) we will use the following controlinput

119906119894 (119905)= minus120573[ 119899sum

119896=1

119886119894119896 (119909119894 (119905) minus 119909119896 (119905)) + 119887119894 (119909119894 (119905) minus 119909119903 (119905))] (27)

where 119886119894119896 for 119894 119896 isin 1 2 119899 with 119894 = 119896 is the (119894 119896)-thentry of the adjacency matrix 119860 isin R119899times119899 associated with theundirected graph describing the interaction of the agents and120573 ge 0 and 119887119894 for (119894 = 1 2 119899) are positive constants to bechosen as mentioned in Theorem 23

We will require that the following assumptions hold

Assumption 21 The disturbance signal 119908119894(119905) satisfies119908119894(119905) le 119897 lt infin forall119894 isin 1 2 119899Assumption 22 For the multiagent system (25) with 119895 = 3the distribution function 119888(120572) is such that

Lminus1 1119862 (119904) + 120583120582max (119876) ge 0 (28)

where 119862(119904) is defined in terms of 119888(120572) as in (4)

Theorem 23 Consider the generalized fractional nonlinearmultiagent system (25) with the virtual leader (26) and thecontroller (27) Assume that the nonlinear function119891(119905 119909(119905)) isLipschitz (with respect to 119909 and with Lipschitz constant 120579) andthat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 Assumption 11is satisfied and

radic2120573120579 ge 119876 (29)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then robust consensus isachieved

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 Assumptions 21 and 22are satisfied and

120573120579 ge 119876 (30)

where 119876 gt 0 is the solution of the Lyapunov equation119867119879119876 + 119876119867 = 3119868119899 and then the steady-state errors ofany two agent will converge as 119905 997888rarr infin to the region1198721 where

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic 2119899120582max (119876)120573120583120582min (119876) 119876 119897 (31)

and 120583 = 120573 minus 11987621205792120573

Proof By substituting (27) in system (25) we can write

119879120572119895 119883 (119905) = 119865 (119883 (119905)) minus 120573 [119871119883 (119905) + 119861 (119883 (119905) minus 1199091199031119899)]+ 119882 (119905) (32)

where 119865(119883(119905)) = [119891(1199091(119905)) 119891(119909119899(119905))]119879 Subtracting119879120572119895 [1119899119909119903(119905)] from both sides of (32) and using the change ofvariables 119911119894(119905) = 119909119894(119905) minus 119909119903(119905) 119894 isin 1 2 119899 yields

119879120572119895 119885 (119905) = minus120573119867119885 (119905) + Δ119865 (119885 (119905)) + 119882 (119905) (33)

where119867 is defined as in Lemma 18119885(119905) = [1199111(119905) 119911119899(119905)]119879and Δ119865(119885(119905)) = [119891(1199111(119905) + 119909119903(119905)) minus 119891(119909119903(119905)) 119891(119911119899(119905) +119909119903(119905)) minus 119891(119909119903(119905))]119879 Consider the following Lyapunov candi-date function for system (33)

119881 (119905) = 119885119879119876119885 (119905) (34)

Applying Lemma 17 and substituting (33) we can analyze119879120572119895 119881(119905)119879120572119895 119881 (119905) le 120573119885119879 (119905) [minus119876119867 minus119867119879119876]119885 (119905)

+ 2119885119879 (119905) [119876Δ119865 (119885 (119905)) + 119876119882(119905)] (35)

Using Lemma 18 we can conclude that all the eigenvalues of119867 have positive real parts so that minus119867 is Hurwitz Thus thereexists a matrix 119876 = 119876119879 gt 0 that satisfies minus119867119879119876 minus 119876119867 =minus3119868119899 Applying in (35) this identity along with the property120585119879120577+120577119879120585 le 120581 120585119879120585+(1120581)120577119879120577 which is valid for any 120585 120577 isin R119898we obtain

119879120572119895 119881 (119905) le minus3120573 119885 (119905)2 + 120573 119885 (119905)2 + 1120573 119876119882(119905)2+ 120573 119885 (119905)2 + 1120573 1198762 Δ119865 (119905 119885 (119905))2 (36)

Since 119891(119905 119909(119905)) is Lipschitz with respect to 119909(119905) we cansimplify (36) as follows

119879120572119895 119881 (119905)le minus120573 119885 (119905)2 + 1198991198972 1198762120573

+ 1198762120573119899sum119894=1

(119891 (119905 119911119894 (119905) + 119909119903 (119905)) minus 119891 (119905 119909119903 (119905)))2le minus120573 119885 (119905)2 + 1198991198972 1198762120573 + 1198762 1205792120573 119885 (119905)2le minus120583 119885 (119905)2 + 1198991198972 1198762120573

(37)

where 120583 = 2120573 minus 11987621205792120573 gt 0 by (29)(1) In the following we will use Theorem 12 to prove

that system (33) is asymptotically stable at its origin

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 7

If 119908119894(119905) = 0 forall119894 then 119897 = 0 As consequence(37) turns into 119879120572119895 119881(119905) le minus120583119885(119905)2 so that (18)is satisfied for 1205723 = 120583 Additionally noting that120582min(119876)119885119879(119905)119885(119905) le 119881(119905) le 120582max(119876)119885119879(119905)119885(119905) itis clear that 119881(119905) satisfies (17) for 1205721 = 120582min(119876)and 1205722 = 120582max(119876) By Theorem 12 we can concludethat system (33) is asymptotically stable at 119884(119905) =0119899minus1 This means according to the definition of 119885(119905)that lim119905997888rarrinfin1199091(119905) minus 119909119894(119905) = 0 forall119894 isin 1 2 119899and hence the multiagent system (25) achieves robustconsensus

(2) Using the inequality 119885119879(119905)119875119885(119905) le 120582max(119876)119885(119905)2 in(34) yields 119881(119905)120582max(119876) le 119885(119905)2 Hence

119879120572119895 119881(119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573 (38)

Let 119906(119905) = 119881(119905) minus 11989911989721198762120582max(119876)120583120573 The generalizedfractional derivative of 119906(119905) can be analyzed as follows

119879120572119895 119906 (119905) le minus 120583120582max (119876)119881 (119905) + 1198991198972 1198762120573le minus 120583120582max (119876)119906 (119905) (39)

There exists a nonnegative function 119898(119905) satisfying119879120572119895 119906 (119905) + 119898 (119905) = minus 120583120582max (119876)119906 (119905) (40)

From this point we will only consider 119895 = 3 and then we willobtain the same result for 119895 = 1 as a particular case Takingthe Laplace transform of (40) produces

119861 (119904) [119880 (119904) minus 119906 (0)119904 ] +119872(119904) = minus 120583120582max (119876)119880 (119904) (41)

where 119861(119904) is defined as in (4) and 119880(119904) and 119872(119904) are theLaplace transforms of 119906(119905) and119898(119905) respectively Solving for119880(119904) we obtain

119880 (119904) = (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) minus 119872 (119904)119861 (119904) + 120583120582max (119876) (42)

Note that the inverse Laplace Transform of the second termof the right-hand side of (42) is nonnegative since 119898(119905)Lminus11(119861(119904) + 120583120582max(119876)) ge 0 Considering this we canturn (42) into

119906 (119905) le Lminus1 (119861 (119904) 119904) 119906 (0)119861 (119904) + 120583120582max (119876) (43)

Substituting the definition of 119906(119905) into (43) yields

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583le Lminus1 119861 (119904) 119906 (0)119904 (119861 (119904) + 120583120582max (119876))

(44)

By usingTheorem 8 we can calculate the limit of (44) as 119905 997888rarrinfin Note that lim119904997888rarr0119861(119904) = 0 Then

lim119905997888rarrinfin

119881 (119905) minus 1198991198972120582max (119876) 1198762120573120583 le lim119904997888rarr0

119861 (119904) 119906 (0)119861 (119904) + 120573120582max (119876) = 0 (45)

Considering that 120582min(119876)119885(119905)2 le 119881(119905) it follows from (45)that

lim119905997888rarrinfin

119885 (119905) le radic119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (46)

According to the definition of 119885(119905) and using inequalityproperties we obtain10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le 1003816100381610038161003816119909119903 (119905) minus 119909119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119909119903 (119905) minus 119909119910 (119905)10038161003816100381610038161003816

le 1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 + 10038161003816100381610038161003816119911119910 (119905)10038161003816100381610038161003816le radic2 (1003816100381610038161003816119911119894 (119905)10038161003816100381610038162 + 10038161003816100381610038161003816119911119910 (119905)100381610038161003816100381610038162)le radic2 119885 (119905)

(47)

forall119894 119910 isin 1 2 119899 Combining (46) and (47) we can analyzethe limit as 119905 997888rarr infin of the difference between any pair ofagents

lim119905997888rarrinfin

10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119876) 120582min (119876) 119876 119897radic120573120583 (48)

forall119894 119910 isin 1 2 119899 which proves that the steady-state errorsbetween the agents converge to1198721

We can prove this theorem with 119895 = 1 by considering thecase 119895 = 3 and setting the distribution function of1198791205723 as 119888(120572) =120575(120572 minus 119886) which turns this operator into the standard Caputofractional derivative of order 119886 Furthermore notice that

Lminus1 1119862 (119904) + 120583120582max (119876)= Lminus1 1119904120573 + 120583120582max (119876)

= 119905120573minus1119864120573120573 (minus 120583120582max (119876) 119905120573) ge 0(49)

where we have used Lemma 10 This means that Assump-tion 22 is satisfied Alternatively the case 119895 = 1 is derivedinTheorem 2 of G Ren and Yu [29]

43 Robust Consensus of Linear of Generalized FractionalMultiagent Systems A linear generalized fractional multia-gent system with external disturbances can be described as aparticular case of (25) with 119891(119905 119909119894) = 0

119879120572119894 119909119894 (119905) = 119906119894 (119905) + 119908119894 (119905) 119894 isin 1 2 119899 (50)

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

8 Complexity

where 119909119894(119905) 119906119894(119905) and 119908119894(119905) are the state control input andexternal disturbances of the 119894th agent respectively

In order to accomplish robust consensus we can use assimpler controller than (27)

119906119894 (119905) = minus120573 119899sum119896=1

119886119894119895 (119909119894 (119905) minus 119909119896 (119905)) (51)

where 120573 ge 0 and 119886119894119896 (119894 119896 = 1 2 119899 119894 = 119896) is the (119894 119896)-th element of the adjacency matrix 119860 isin R119899times119899 associated withthe directed graph describing the interaction of the agents Byfollowing a procedure completely analogous to the one donein the previous section the following theorem can be readilyproved

Theorem 24 Consider the generalized fractional nonlinearmultiagent system (50) with the control input (51) Supposethat the associated fixed directed graph has a directed spanningtree

(1) For 119895 = 1 2 3 4 5 or 6 if 119908119894(119905) = 0 forall119894 then system(50) achieves robust consensus

(2) For 119895 = 1 or 3 if exist 119908119894(119905) = 0 and Assumptions 11 21and 22 are satisfied then the steady-state errors of anytwo agents will converge to the region 1198722 defined as

1198722 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119910 (119905)10038161003816100381610038161003816 le radic2119899120582max (119875) 119875119864 119897120573radic120582min (119875) (52)

where 120582max(119875) and 120582min(119875) are the maximum andminimum eigenvalues of the matrix 119875 gt 0 which is thesolution of the Lyapunov equation119862119879119875+119875119862 = minus2119868119899minus1and 119864 119862 are defined as in Lemma 19

5 Examples

Example 1 Consider a group of 3 undisturbed agentsdescribed by (50) with 119908119894(119905) = 0 forall119894 under the influence of

controller (51) with the interaction graph shown in Figure 1The Laplacian matrix associated with this system is

119871 = [[[2 minus1 minus1minus1 2 minus1minus1 minus1 2

]]] (53)

From Figure 1 it is clear that this graph has a directed span-ning treeTherefore byTheorem24 this system accomplishesconsensus In order to verify our prediction we solved thisproblem for the six types of fractional derivatives addressedin this text To this end we considered the initial conditions1199091(0) = 07996 1199092(0) = 39978 1199093(0) = minus47974 and theparameter 120573 = 1 Additionally we used the differentiationorders given in Table 1

The cases 119895 = 1 and 119895 = 2were analyzed numerically withthe aid of the MATLAB functions developed in Petras [40]and Valerio [41] Valerio Vinagre Domingues and Da Costa[42] Taking advantage of (10) and (12) the cases 119895 = 4 119895 = 5and 119895 = 6 were worked out with MATLABrsquos standard ODESolver Given the limitations of the existing computationalmethods to study fractional distributed order equations wesolved the case 119895 = 3 analytically as it is shown next

We can rewrite the system in vector and obtain

1198791205723119883(119905) = minus120573119871119883 (119905) (54)

By taking the Laplace transform of (54) and solving forX(119904)we get

X (119904) = [119862 (119904) 119868 + 119871]minus1 [119862 (119904)119904 119883 (0)]

= 1119904 (119861 (119904) + 3) [[[119861 (119904) 1199091 (0) + 119902119861 (119904) 1199092 (0) + 119902119861 (119904) 1199093 (0) + 119902

]]] (55)

where 119902 = sum3119894=1 119909119894(0) Substituting 119861(119904) = 119904120573 + 41199041205732decomposing the right hand side of (55) into partial fractionsand taking their inverse Laplace transforms yields

119883(119905) =[[[[[[[[

1199091 (0) + 119902 minus 31199091 (0)2 119905121198641232 (minus11990512) + 31199091 (0) minus 1199022 119905121198641232 (minus311990512)1199092 (0) + 119902 minus 31199092 (0)2 119905121198641232 (minus11990512) + 31199092 (0) minus 1199022 119905121198641232 (minus311990512)1199093 (0) + 119902 minus 31199093 (0)2 119905121198641232 (minus11990512) + 31199093 (0) minus 1199022 119905121198641232 (minus311990512)

]]]]]]]] (56)

which are the expressions shown in Figure 4In Figures 2ndash7 we can see the behavior of the error

between the states of the multiagents In all the cases theseerrors converge to zero as expected and depending on thecharacteristics of the operator 119879120572119895 this rate of convergencevaries

Example 2 Consider again system (54) with the sameinteraction topology as in Example 1 120573 = 1 but this timewith the disturbances 119908119894(119905) = 120574119894 + 119886119894119890minus119888119894119905 where 120574119894 119886119894 119888119894 isin Rforall119894 isin 1 2 3 Let the differentiation orders be 120572 = 05 and119888(120572) = 120575(120572minus23)+4120575(120572minus13) for 119895 = 1 and 119895 = 3 respectivelyAssumption 21 is fulfilled since the external disturbances are

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 9

1

2 3

Figure 1 Interaction graph for the 3 agents of Examples 1 2 and 3

10minus3 10minus2 10minus1 100 101

t

x1 minus x2

x2 minus x3

x2 minus x3

0

2

4

6

8

Figure 2 Linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

101100 102

t

0

2

4

6

8

10

Figure 3 Linear case 119895 = 2

Table 1 Differential orders for simulations

119895 Parameters1 120572 = 052 120572(119905) = 1 minus exp (minus11990550)23 119888 (120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 234 119886 = 0 120572 = 055 119896 (119905) = 1 + 04 log (119905 + 1)6 119888(120572) = 120575 (120572 minus ]) + 4120575 (120572 minus ]2) ] = 23 119896(119905) = 1 + 04 log(119905 + 1)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 4 Linear case 119895 = 3bounded by max119894isin123120574119894 + 119886119894 Hence we only need toshow that Assumption 22 is also satisfied in order to applyTheorem 24 In this specific problem the left-hand side of(28) is

Lminus1 1119862 (119904) + 120573120582max (119875) = L

minus1 111990423 + 411990413 + 3= Lminus1 111990413 + 3 lowastL

minus1 111990413 + 1= [119905minus231198641313 (minus311990513)] lowast [119905minus231198641313 (minus311990513)]= int+infinminusinfin

(119905 minus 120591)minus23 1198641313 (minus3 (119905 minus 120591)13)sdot 119905minus231198641313 (minus311990513) 119889120591

(57)

where we have used Theorems 8 and 16 Considering thatall the factors inside the integral in (57) are nonnegative wecan conclude that Assumption 22 is fulfilled and thereforethe steady-state errors between the agents will convergeasymptotically to 1198721 Solving the equation 119862119879119875 + 119875119862 =

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

10 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 5 Linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 6 Linear case 119895 = 5

minus2119868119899minus1 yields 119875 = (13)119868 so that 120582max(119875) = 120582min(119875) = 13Moreover 119875119864 = max1le119895le3sum2119894=1 |(119875119864)119894119895| = 23 By settingthe parameters 1205741 = minus2 1205742 = 1 1205743 = 2 1198861 = 1 1198862 = 21198863 = minus1 1198881 = 2 1198882 = 15 and 1198883 = 17 one can calculatethat the disturbances are bounded by 119897 = 3 Substituting thesevalues in the definition of1198721 produces

1198721 = 10038161003816100381610038161003816119909119894 (119905) minus 119909119895 (119905)10038161003816100381610038161003816 le 2radic6 asymp 48989 (58)

x1 minus x2

x2 minus x3

x2 minus x3

10minus2 102100

t

0

2

4

6

8

Figure 7 Linear case 119895 = 6

To verify this analysis we solved this system numericallyfor 119895 = 1 (using the MATLAB functions of Petras [40]) andanalytically for 119895 = 3 (since there are no suitable numericalmethods) For the case 119895 = 3 we can take the Laplacetransform of (54) and solve forX(119904)

X (119904) = [119862 (119904) 119868 + 119871]minus1 [W (119904) + 119862 (119904)119904 119883 (0)]= 1119904119862 (119904) (119862 (119904) + 3)

times[[[[[[[[[[

119862 (119904) 1199041199081 (119904) + 1198622 (119904) 1199091 (0) + 119904 3sum119894=1

119908119894 (119904)119862 (119904) 1199041199082 (119904) + 1198622 (119904) 1199092 (0) + 119904 3sum

119894=1

119908119894 (119904)119862 (119904) 1199041199083 (119904) + 1198622 (119904) 1199093 (0) + 119904 3sum

119894=1

119908119894 (119904)

]]]]]]]]]]

(59)

where Lminus1119882(119905) = W(119904) and for simplicity we haveconsidered 1199091(0) + 1199092(0) + 1199093(0) = 0 Substituting 119862(119904) =11990423 + 411990413 and 119908119894(119904) = 120574119894119904 + 119886119894(119904 + 119888119894) one can decomposethe right hand side of (59) into partial fractions and take theirinverse Laplace transforms After extensive calculations weobtain

119883 (119905) = 119866 (119905) + 119867 (119905) + 119891 (119905)119883 (0) (60)

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 11

where 119891(119905) 119866 = [1198921(119905) 1198922(119905) 1198923(119905)]119879 119867 = ℎ(119905)[1 1 1]119879 aredefined as follows

119891 (119905) = 12119905minus139Γ (23) minus 39119905minus2327Γ (13) + 3119905minus231198641313 (minus11990513)2minus 3119905minus231198641313 (minus311990513)54

(61)

119892119894 (119905) = 120574119894 13 minus 4119905minus139Γ (23) + 13119905minus2327Γ (13)+ 1198861198941198882119894 minus 28119888119894 + 27 (9 + 4119888119894) 119890minus119888119894119905+ 13119905minus23119864113 (minus119888119894119905) minus (12 + 119888119894) 119905minus13119864123 (minus119888119894119905)+ 119905minus23 [( 1198861198942 (119888119894 minus 1) minus 1205741198942 )1198641313 (minus11990513)+ ( 12057411989454 minus 1198861198942 (119888119894 minus 27))1198641313 (minus311990513)]

(62)

forall119894 isin 1 2 3 andℎ (119905) = 3sum

119894=1

120574119894 [ 1199051312Γ (43) minus 19144 + 265119905minus131728Γ (23)]+ 119905minus23Γ (13) ( 11988611989412119888119894 minus 335512057411989420736 )+ 119905minus23 [1198641313 (minus11990513) (1205741198946 minus 1198861198946 (119888119894 minus 1))+ 1198641313 (minus311990513) ( 1198861198946 (119888119894 minus 27) minus 120574119894162)+ 1198641313 (minus411990513) ( 120574119894768 minus 11988611989412 (119888119894 minus 64))]+ 119886119894119888119894 (1198883119894 minus 921198882119894 + 1819119888119894 minus 1728) times [228119888119894+ 451198882119894 119890minus119888119894119905 minus (81198882119894 + 265119888119894) 119905minus13119864123 (minus119888119894119905)+ (1198882119894 + 128119888119894 + 144) 119905minus23119864113 (minus119888119894119905)]

(63)

For both cases 119895 = 1 and 119895 = 3 Figure 8 depicts thestates of the agents and Figure 9 the errors between themTo plot our results we used the initial conditions 1199091(0) =minus30 1199092(0) = 10 1199093(0) = 20 From these figures we canconfirm that the steady-state errors of the agents converge tothe calculated region

Example 3 Consider the nonlinear system described by theinteraction graph shown in Figure 1 and (25) and (26) where119891(119905 119909(119905)) = arctan(119909(119905)) for which we can take its Lipschitzconstant as 120579 = 1 For this system one can calculate 119876 =

x1 minus x2

x1 minus x3

x2 minus x3

10minus2 100 102

t

0

10

20

30

40

50

Figure 8 Linear case with perturbation 119895 = 1

x1 minus x2

x1 minus x3

x2 minus x3

104102100 106

t

0

10

20

30

40

Figure 9 Linear case with perturbation 119895 = 3

3radic2623 Setting the parameters of the controller as 120573 = 11198871 = 1 1198872 = 2 and 1198873 = 3 allows us to fulfill inequality(29) and thus according toTheorem 23 this system achievesconsensus

All the simulations start with zero initial conditions andconstant input such that the agents evolve with differenttrajectories at time 119905 = 3 the agents start using the control lawgiven by (27) The simulation for the different operators areshown in Figures 10ndash14 where we plotted the errors betweenstates of the different agents for 119895 = 1 2 4 5 and 6 (see

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

12 Complexity

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

05

1

15

2

25

Figure 10 No-linear case 119895 = 1

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

1

2

3

4

5

Figure 11 No-linear case 119895 = 2

Table 1) with the computational tools already mentionedWe do not present the solution of this system for 119895 = 3since neither the available numerical methods for distributedorder systems nor the Laplace transform technique used inthe previous examples are applicable for the nonlinear case

In all the simulation we can see that while 119905 lt 3 theerror between the agents increases and once the controller isengaged after 119905 ge 3 the errors converge to zero the rate ofconvergence depend on the nature of the operators

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

Figure 12 No-linear case 119895 = 4

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 200t

0

1

2

3

4

5

6

Figure 13 No-linear case 119895 = 5

6 Conclusions

We introduced the distributed conformable derivative whichpreserves the product and chain rules For this and fiveother fractional derivatives we unified the Lyapunov directmethod That result was presented in two theorems the firstbounds the Lyapunov function and its fractional derivative bypowers of the norm of the states and the second by class Kfunctions Moreover we employed this generalized fractionalLyapunov method to prove whether linear and nonlinear

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

Complexity 13

x1 minus x2

x2 minus x3

x2 minus x3

5 10 15 20 25 300t

0

05

1

15

Figure 14 No-linear case 119895 = 6

multiagent systems modeled with different fractional deriva-tives accomplish consensus We found that if the systemis undisturbed the agents converge asymptotically and ifthere are external disturbances the steady-state errors evolvetowards a region which diminishes linearly in size as the gainof the controller is increased It is worth noticing that samecontrol inputs are effective for all the differentiation ordersconsidered in this paper

In the light of these results potential future objectiveswould be to carry out a similar analysis in the presence oftime delays or to study the finite-time consensus problem forfractional multiagent systems possibly employing differentcontrollers

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The financial support for this article is given through Uni-versidad Iberoamericana Campus Ciudad de Mexico andUniversidad Catolica del Uruguay as employers for theauthors

References

[1] G W F Von Leibniz Mathematische Schriften vol 1 Asher1849

[2] A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations New York NYUSA Elsevier 2006

[3] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Science amp Business Media 2011

[4] I Podlubny Fractional Differential Equations Academic PressLondon 1999

[5] S G Samko and B Ross ldquoIntegration and differentiation toa variable fractional orderrdquo Integral Transforms and SpecialFunctions vol 1 no 4 pp 277ndash300 1993

[6] H G Sun W Chen H Wei and Y Q Chen ldquoA comparativestudy of constant-order and variable-order fractional modelsin characterizing memory property of systemsrdquo The EuropeanPhysical Journal Special Topics vol 193 article no 185 no 1 2011

[7] M Caputo Elasticita E Dissipazione Zanichelli Bologna Italy1969

[8] AV Chechkin J Klafter and IM Sokolov ldquoFractional Fokker-Planck equation for ultraslow kineticsrdquo EPL (Europhysics Let-ters) vol 63 no 3 article no 326 2003

[9] MNaber ldquoDistributed order fractional sub-diffusionrdquo Fractalsvol 12 no 1 pp 23ndash32 2004

[10] C F Lorenzo and T T Hartley ldquoVariable order and distributedorder fractional operatorsrdquo Nonlinear Dynamics vol 29 no1ndash4 pp 57ndash98 2002

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] R Almeida M Guzowska and T Odzijewicz ldquoA remarkon local fractional calculus and ordinary derivativesrdquo OpenMathematics vol 14 pp 1122ndash1124 2016

[13] N Laskin Fractional Quantum Mechanics World Scientific2018

[14] D Baleanu J A T Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[15] S S Tabatabaei M J Yazdanpanah S Jafari and J C SprottldquoExtensions in dynamicmodels of happiness Effect ofmemoryrdquoInternational Journal of Happiness and Development vol 1 no4 pp 344ndash356 2014

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems Lyapunov direct method andgeneralized MittagndashLeffler stabilityrdquo Computers ampMathematicswith Applications vol 59 no 5 pp 1810ndash1821 2010

[17] A Souahi A B Makhlouf and M A Hammami ldquoStabilityanalysis of conformable fractional-order nonlinear systemsrdquoIndagationes Mathematicae vol 28 no 6 pp 1265ndash1274 2017

[18] S S Tabatabaei H A Talebi and M Tavakoli ldquoAn adaptiveorderstate estimator for linear systems with non-integer time-varying orderrdquo Automatica vol 84 pp 1ndash9 2017

[19] H Taghavian and M S Tavazoei ldquoStability analysis ofdistributed-order nonlinear dynamic systemsrdquo InternationalJournal of Systems Science vol 49 no 3 pp 523ndash536 2018

[20] YWang and T Li ldquoStability analysis of fractional-order nonlin-ear systems with delayrdquoMathematical Problems in Engineeringvol 2014 Article ID 301235 8 pages 2014

[21] W Ren and R W Beard Distributed Consensus in Multi-VehicleCooperative Control Springer 2008

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

14 Complexity

[22] A Jadbabaie N Motee and M Barahona ldquoOn the stabilityof the Kuramoto model of coupled nonlinear oscillatorsrdquo inProceedings of the American Control Conference (AAC) pp4296ndash4301 IEEE Boston MA USA 2004

[23] R Olfati-Saber and J S Shamma ldquoConsensus filters for sensornetworks and distributed sensor fusionrdquo in Proceedings of the44th IEEE Conference on Decision and Control and the Euro-pean Control Conference (CDC-ECC) pp 6698ndash6703 IEEESeville Spain 2005

[24] W Ren and Y Cao Distributed Coordination of Multi-AgentNetworks Emergent Problems Models And Issues SpringerScience amp Business Media 2010

[25] Z Yu H Jiang C Hu and J Yu ldquoLeader-following consensusof fractional-order multi-agent systems via adaptive pinningcontrolrdquo International Journal of Control vol 88 no 9 pp 1746ndash1756 2015

[26] X Yin D Yue and S Hu ldquoConsensus of fractional-orderheterogeneous multi-agent systemsrdquo IET Control Theory ampApplications vol 7 no 2 pp 314ndash322 2013

[27] C Song J Cao and Y Liu ldquoRobust consensus of fractional-order multi-agent systems with positive real uncertainty viasecond-order neighbors informationrdquo Neurocomputing vol165 pp 293ndash299 2015

[28] G Nava-Antonio G Fernandez-Anaya E G Hernandez-Martinez J Jamous-Galante E D Ferreira-Vazquez and JJ Flores-Godoy ldquoConsensus of multi-agent systems with dis-tributed fractional order dynamicsrdquo in Proceedings of the 14thInternational Workshop on Complex Systems and Networks(IWCSN) pp 190ndash197 IEEE Doha Qatar 2017

[29] G Ren and Y Yu ldquoRobust consensus of fractional multi-agentsystems with external disturbancesrdquo Neurocomputing vol 218pp 339ndash345 2016

[30] N Aguila-Camacho M A Duarte-Mermoud and J A Galle-gos ldquoLyapunov functions for fractional order systemsrdquoCommu-nications in Nonlinear Science andNumerical Simulation vol 19no 9 pp 2951ndash2957 2014

[31] Z Jiao Y Chen and I Podlubny Distributed-Order DynamicSystems Stability Simulation Applications and PerspectivesSpringer Briefs in Electrical and Computer EngineeringSpringer 2012

[32] Y Xu and Z He ldquoExistence and uniqueness results for Cauchyproblem of variable-order fractional differential equationsrdquoJournal of Applied Mathematics and Computing vol 43 no 1-2 pp 295ndash306 2013

[33] N J Ford and M L Morgado ldquoDistributed order equationsas boundary value problemsrdquo Computers amp Mathematics withApplications vol 64 no 10 pp 2973ndash2981 2012

[34] B Bayour and D F M Torres ldquoExistence of solution toa local fractional nonlinear differential equationrdquo Journal ofComputational and Applied Mathematics vol 312 pp 127ndash1332017

[35] D G Duffy Transform Methods for Solving Partial DifferentialEquations Symbolic amp Numeric Computation CRC press 2ndedition 2004

[36] A R Teel and L Praly ldquoA smooth Lyapunov function froma class-KL estimate involving two positive semidefinite func-tionsrdquoESAIM Control Optimisation andCalculus of Variationsvol 5 pp 313ndash367 2000

[37] G-C Wu D Baleanu and W-H Luo ldquoLyapunov functionsfor Riemann-Liouville-like fractional difference equationsrdquoApplied Mathematics and Computation vol 314 pp 228ndash2362017

[38] G Fernandez-Anaya G Nava-Antonio J Jamous-GalanteR Munoz-Vega and E G Hernandez-Martınez ldquoAsymptoticstability of distributed order nonlinear dynamical systemsAsymptotic stability of distributed order nonlinear dynamicalsystemsrdquo Communications in Nonlinear Science and NumericalSimulation48541549 2017

[39] Y Zhang and Y-P Tian ldquoConsentability and protocol designof multi-agent systems with stochastic switching topologyrdquoAutomatica vol 45 no 5 pp 1195ndash1201 2009

[40] I Petras ldquoFractional order chaotic systemsrdquo 2010 httpwwwmathworkscommatlabcentralfileexchange27336-fractional-order-chaotic-systems

[41] DValerio ldquoVariable order derivativesrdquo 2010 httpslamathworkscommatlabcentralfileexchange24444-variable-order-deriva-tives

[42] D Valerio G Vinagre J Domingues and J S Da CostaldquoVariable-order fractional derivatives and their numericalapproximations ImdashReal ordersrdquo In Fractional Signals andSystems 2009

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