predefined-time consensus of nonlinear first-order systems using...

Research ArticlePredefined-Time Consensus of Nonlinear First-OrderSystems Using a Time Base Generator

J Armando Colunga 1 Carlos R Vaacutezquez 2

Heacutector M Becerra 1 and David Goacutemez-Gutieacuterrez 23

1Centro de Investigacion en Matematicas (CIMAT) AC Computer Science Department GTO Mexico2Tecnologico de Monterrey Escuela de Ingenierıa y Ciencias Zapopan Mexico3Intel Tecnologıa de Mexico Multi-Agent Autonomous Systems Lab Intel Labs Zapopan Mexico

Correspondence should be addressed to Hector M Becerra hectorbecerracimatmx

Received 16 June 2018 Revised 11 September 2018 Accepted 17 September 2018 Published 9 October 2018

Academic Editor Ju H Park

Copyright copy 2018 J Armando Colunga 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

This paper proposes a couple of consensus algorithms for multiagent systems in which agents have first-order nonlinear dynamicsreaching the consensus state at a predefined time independently of the initial conditions The proposed consensus protocols arebased on the so-called time base generators (TBGs) which are time-dependent functions used to build time-varying controllaws Predefined-time convergence to the consensus is proved for connected undirected communication topologies and directedtopologies having a spanning tree Furthermore one of the proposed protocols is based on the super-twisting controller providingrobustness against disturbances while maintaining the predefined-time convergence property The performance of the proposedmethods is illustrated in simulations and it is compared with finite-time fixed-time and predefined-time consensus protocols Itis shown that the proposed TBG protocols represent an advantage not only in the possibility to define a settling time but also inproviding smoother and smaller control actions than existing finite-time fixed-time and predefined-time consensus

1 Introduction

Recent years have seen an increasing interest in algorithmsallowing a group of systems to reach a common value forits internal state through local interaction This problem hasbeen addressed from different viewpoints in the consensusof multiagent systems (MASs) [1 2] and in the synchroniza-tion of complex dynamical networks [3ndash5] On the one handconsensus protocols have been applied to flocking [6] forma-tion control [7 8] and distributed resource allocation [9 10]On the other hand the results on synchronization of complexdynamical networks have been applied to neuroscience [3]power-grids [3] and the chaotic synchronization for securecommunication in swarms [11]

Several works have been published proposing consensusand synchronization algorithms for different types of systemsconsidering static [12ndash14] and dynamic networks [4 5]Regarding first-order agents the standard protocol (the input

of an agent is a linear combination of the errors between theagentrsquos state and those of its neighbors) achieves consensusif the graph topology is strongly connected [15 16] Thisalgorithm achieves consensus also for dynamic topologiesswitching among connected graphs [15 17] For directedgraphs topologies a common requirement is that the graphcontains a spanning-tree (eg [18]) All these algorithmsare based on linear protocols and achieve consensus in anasymptotic fashion where convergence rate is a function ofthe algebraic connectivity of the graph

With the aim of developing consensus protocols satisfy-ing real-time constraints finite-time consensus has receiveda great deal of attentionThe main focus has been in definingnonlinear protocols evaluated either on each of the neighborsrsquoerrors or on the sum of the neighborsrsquo errors For finite-time convergence binary protocols based on the sign(∙)function have been proposed to achieve consensus to theaverage value [19 20] the average-min-max value [21] the

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1957070 11 pageshttpsdoiorg10115520181957070

2 Mathematical Problems in Engineering

median value [22] and the maximum or minimum value[23] of the agentrsquos initial conditions Continuous finite-timeprotocols based on the | ∙ |120572sign(∙) function with 0 lt 120572 lt1 have been introduced in [12 24] Some algorithms havebeen extended to achieve finite-time consensus for stronglyconnected dynamic topologies [20 25] Fixed-time protocolshave also been proposed in [26 27] In these there exists abound for the convergence time that is independent of theinitial conditions [28 29] The protocols are mainly basedon functions of the form | ∙ |119901sign(∙) + | ∙ |119902sign(∙) with0 lt 119902 lt 1 lt 119901

The methods mentioned above cannot be used in appli-cations where real-time constraints have to be accomplishedsince in both finite-time and fixed-time consensus therelationship between the protocol parameters and the conver-gence time is not straightforwardThus it is difficult to definethe convergence time bound as a function of the protocolparameters Moreover the convergence time bound is oftenoverestimated

To address the consensus design problem with real-timeconstraints predefined-time convergence has to be investi-gated In this the time at which consensus is achieved ispredefined a priori as a parameter of the consensus protocolbeing independent of the initial conditions Few works haveaddressed the predefined-time consensus problem In [3031] linear protocols that use time-varying control gainswere proposed for reaching network consensus at a presettime By using a motion planning strategy predefined-time consensus protocols based on a time-varying samplingsequence convergent to a specified settling time are proposedin [32 33] Preliminary versions of a couple of predefined-time protocols were proposed in [34] taking advantage oftime base generators (TBGs) which are parametric timesignals that converge to zero in a specified time [35] and canbe tracked using feedback controllers

In this work a couple of consensus algorithms based onTBGs are proposed as an extension to [34] One algorithmis based on a linear-feedback and the other is based onthe super-twisting controller for the tracking of the TBGsignal The proposed approach can be applied to systems inwhich agents belong to the class of first-order controllablelinear systems and nonlinear systems that can be linearizedby state feedback [36] Comparisons between the proposedprotocols and finite-time fixed-time and predefined-timeprotocols are provided through simulations showing theadvantages of the proposed approach In contrast with workson predefined-time consensus [30ndash32] the contribution ofthis paper is threefold First the proposed controllers yieldsmoother auxiliary control signals (without considering thelinearization terms) with smaller magnitudes Second thereare no parameters in the proposed consensus protocolsdepending on the algebraic connectivity of the graph con-sidered Third a super-twisting controller is presented todeal with perturbed dynamics in the networkrsquos agents Withrespect to our preliminary conference version [34] two mainextensions are reported herein (1) the new linear-feedbackprotocol drives the agents towards the initial conditionsaverage (for undirected graphs) or the weighted sum ofthe initial conditions (for directed graphs) whereas in ourprevious work the consensus value was artificially specified

(2) convergence to the consensus value in predefined timeis demonstrated even if uncertainty in the knowledge of theinitial state is introduced in the controller at difference of ourprevious results where it was assumed that the initial statewas perfectly known

It is known that second-order systems can representa broader class of real systems than first-order systemsHowever the last class of systems has also broad applicabilityfor instance in robotics for kinematic control of holonomicrobotsWe are very interested in generalizing our results evenfor systems of a higher order than two taking into accountexisting results like the ones in [37 38]

The rest of this paper is organized as follows Section 2introduces some background in algebraic graph theorydefines the class of systems for which the proposed methodis applicable and formulates the problem to be solved in thispaper Section 3 presents the proposed consensus protocolsa linear-feedback and a super-twisting controller for thetracking of the TBG signal Section 4 presents simulationresults and a comparison of the proposed controllers withexisting approaches Section 5 provides conclusions

2 Preliminaries and Problem Definition

21 Algebraic Graph Theory Agentsrsquo communication is rep-resented by a graph G = (119881119864119860) which consists of a setof vertices (nodes or agents in this work) 119881 = 1 119873 aset of edges 119864 sube 119881 times 119881 and a weighted adjacency matrix119860 = [119886119894119895]with nonnegative adjacency elements 119886119894119895 ie 119886119894119895 gt 0iff (119894 119895) isin 119864 The set of neighbors of agent 119894 is denoted by119873119894 = 119895 isin 119881 (119895 119894) isin 119864Definition 1 (see [39 40]) A graphG is called directed if thereis an entry of 119860 that fulfills 119886119894119895 gt 0 ie (119894 119895) isin 119864 representsthat agent 119895 receives the information of the state value119909119894 of theagent 119894 otherwise 119886119894119895 = 0 ie (119895 119894) notin 119864 A graphG is calledundirected if the pairs of nodes are not ordered 119886119894119895 = 119886119895119894 gt 0otherwise 119886119894119895 = 119886119895119894 = 0 ie (119894 119895) (119895 119894) notin 119864Definition 2 (see [39 40]) A directed path (respectivelyundirected path) from nodes V119895 to V119894 is a sequence of distinctedges of the form (V119894 V1198941) (V1198941 V1198942) (V119894119897 V119895) in a directedgraph (respectively undirected graph) A graph is stronglyconnected (respectively connected) if there exists a directedpath (respectively undirected path) between any two distinctvertices V119894 and V119895 inG

Definition 3 (see [16 18]) A directed graphG has a directedspanning tree if there exists a node V119894 (a root) such that allother nodes can be linked to V119894 via a directed path

If an undirected graph has a directed spanning tree thenthe graph is connected A strongly connected directed graphcontains at least one directed spanning tree [16 18]

Definition 4 (see [16]) LetG be a graph on119873 vertices labelledas 1 119873The Laplacianmatrix ofG is defined as the119873times119873matrix 119871 = [119897119894119895] where

119897119894119895 =

minus119886119894119895 if 119894 = 119895119873sum119896=1119896 =119894

119886119894119896 if 119894 = 119895 (1)

Mathematical Problems in Engineering 3

In this work we will propose and analyze consensusprotocols for two kinds of networks connected undirectedgraphs and directed graphs with a directed spanning tree[2 16 18]

Lemma 5 ([2 16 18]) 119871 has at most one zero eigenvalue1205821(119871) with 1119873 as the corresponding right eigenvector where1119873 denotes a column vector with all entries equal to one ie1205821(119871) = 0 Re(120582119894(119871)) gt 0 forall119894 = 2 119873 and 1198711119873 = 0119873Furthermore if the graph is undirected and connected 119871 has aleft eigenvector 120574 that satisfies 120574119879119871 = 0119879119873 and 120574 = 1119873 where0119873 represents a column vector of zeros If the graph is directedand has a spanning tree then 120574119879119871 = 0119879119873 and 120574

1198791119873 = 1Finally let us introduce a matrix transformation that will

be useful later on Let 119869 isin C119873times119873 be the Jordan formassociatedwith 119871 Then there exists a nonsingular matrix 119878 isin C119873times119873

such that

119878minus1119871119878 = 119869 = [ 0 0119879119873minus10119873minus1

] (2)

where isin C(119873minus1)times(119873minus1) is the Jordan matrix associated withthe nonzero eigenvalues of 119871 [39 41] and 0119873minus1 represents acolumn vector of zeros of size119873 minus 122 Problem Definition In this paper we will consider a setof119873 agents connected through a network which aremodeledas first-order nonlinear systems as follows

119894 (119905) = 119891119894 (119909119894) + 119892119894 (119909119894) 119906119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (3)

Where for the agent 119894 119909119894(119905) isin R is the systemrsquos state 119906119894(119905) isinR is the control input 119891119894(119909119894) and 119892119894(119909119894) are smooth nonlinearfunctions and 120588119894(119905) isin R is a bounded disturbance In orderto represent the multiagent system (MAS) dynamics denote119909 = [1199091 119909119873]119879 119906 = [1199061 119906119873]119879 119891 = [1198911 119891119873]119879119892 = [1198921 119892119873]119879 and 120588 = [1205881 120588119873]119879 isin R119873 Then thecomplete dynamics (3) can be written as

(119905) = 119891 (119909) + 119892 (119909) 119906 (119905) + 120588 (119905) (4)

Let us propose a linearizing control law of the form 119906119894 =(minus119891119894(119909119894)+V119894)119892119894(119909119894) for each agent (3) where V119894 is an auxiliarycontrol input and it is assumed that119892119894(119909119894) = 0 ie the relativedegree is well defined [36] and the agent is controllable Theclosed-loop dynamics for the 119894-th agent are now linear andgiven by

119894 (119905) = V119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (5)

The complete closed-loop dynamics for all the agents canbe written as

(119905) = 119907 (119905) + 120588 (119905) (6)

where 119907 = [V1 V119873]119879 isin R119873In a MAS the consensus error of agent 119894 with respect to

its neighbors is defined as [2]

119890119894 (119905) = sum119895isin119873119894

119886119894119895 (119909119895 (119905) minus 119909119894 (119905)) 119894 isin 1 119873 (7)

The consensus error for the complete MAS can beexpressed in a compact form as [2]

119890 (119905) = [1198901 (119905) 119890119873 (119905)]119879 = minus119871119909 (119905) (8)

Problem statement given a MAS with agents modeled asfirst-order nonlinear systems (3) andwith an associated graphG design a consensus protocol for each agent in the form119906119894 = 120574(119890119894 119909119894 119905) such that the state of all the agents reachesa consensus value 119909lowast in a predefined time 119905119891 from any initialstate 119909(0) ie 119909119894(119905) 997888rarr 119909lowast forall119894 = 1 119873 as 119905 997888rarr 119905119891Therefore the consensus error (8) converges to zero at time119905119891Remark 6 The consensus problem in MASs is closely relatedto the synchronization problem in complex dynamical net-works but they have been addressed from different view-points [42] In fact the consensus problem can be posedas a complete-synchronization problem In the former theanalysis is often based on the matrix Laplacian approach[2] and the agents have simple dynamics such as singleintegrator dynamics a chain of integrators or linear systemsThe recent focus has been on defining finite-time fixed-timeand predefined-time protocols for fixed and dynamic net-works In the synchronization problem in complex networksthe analysis is often based in the Master stability functionformalism [3 42] and addresses interconnected systems withcomplex dynamics for instance chaotic attractors Althoughrecently some works have addressed synchronization indynamic networks eg Markov jump topologies [4 5] mostof the work has been focused on static topologies For furtherdetails on the synchronization problem we refer the readerto [3]

23 Time Base Generator (TBG) Time base generators areparametric functions of time particularly designed to sta-bilize a system in such a way that its state describes aconvenient transient profile TBGs have been previously usedto achieve predefined-time convergence of first and higherorder dynamics for single systems in [35] For the case ofsingle first-order systems TBGs have been used for thecontrol of robots at kinematic level [43]

A time base generator (TBG) is a continuous and differen-tiable time-dependent polynomial function ℎ(119905) described as[35]

ℎ (119905) = 120591 (119905) sdotC if 119905 isin [0 119905119891]0 otherwise

(9)

where 120591(119905) = [119905119903 119905119903minus1 119905 1] is the time basis vector andC is a vector of coefficients of proper dimensions For first-order systems it is required to compute a TBG fulfilling thefollowing conditions at initial time and final time 119905119891

ℎ (119905) = 1 if 119905 = 00 if 119905 ge 119905119891

ℎ (119905) = 0 if 119905 = 0 or 119905 ge 119905119891(10)


In [35] a couple of procedures for calculating the coef-ficients C fulfilling the required constraints have beenintroduced In particular when a high degree polynomial isused an optimization procedure can be applied to find thebest coefficients

3 Consensus Protocols Using the TBG

In this section we present consensus protocols that takeadvantage of the TBG to enforce convergence of a MASin predefined time In the first two subsections we presentresults for unperturbed systems ie for 120588 = 0119873 while inSection 33 we consider perturbed systems

31 Previous Results In [34] a consensus protocol wasproposed for the MAS (4) by introducing a time-varyingcontrol gain in order to drive the system to a neighborhoodabout a given consensus value 119909lowast in a predefined time 119905119891from any initial state 119909(0) Consequently the consensus error(8) converges to a neighborhood about the origin at 119905119891 Suchconsensus protocol is similar in nature to those in [30 31]

In the same preliminary work [34] we have proposed thefollowing open-loop controller to enforce predefined-timeconvergence to a desired consensus value 119909 provided that theMAS (4) is not perturbed and the initial state is known

119906119894 (119905) = 1119892119894 (119909119894) (minus119891119894 (119909119894) + V119894) 119894 isin 1 119873V119894 = ℎ (119905) (119909119894 (0) minus 119909)

(11)

where ℎ(119905) fulfills (10) In that work it was demonstratedthat consensus to 119909 is achieved in time 119905119891 and the resultingconsensus error trajectory in the interval 119905 isin [0 119905119891] is givenby

(119905) = ℎ (119905) 1198900 = minusℎ (119905)1198711199090 (12)

where 119890(0) = 1198900 = [1198901(0) 119890119873(0)]119879 is the consensuserror vector (8) at the initial condition 119909(0) = 1199090 =[1199091(0) 119909119873(0)]119879

To provide closed-loop stability of the tracking errora different protocol was proposed in [34] to address thepredefined-time consensus problem as a trajectory trackingproblem where the reference trajectory depends on the TBGSuch protocol is described as follows

119906119894 (119905) = 1119892119894 (119909119894) (minus119891119894 (119909119894) + V119894) 119894 isin 1 119873V119894 = ℎ (119905) (119909119894 (0) minus 119909) + 119896119891 (119890119894 (119905) minus ℎ (119905) 119890119894 (0))

(13)

This protocol guarantees stability during the transientbehavior and also after the settling time ie for any 119905 ge 119905119891The controller for each agent in (13) consists of two termsthe first one is a feed-forward term that yields the evolutionof each agentrsquos error from the initial value 119890119894(0) to the originat time 119905119891 following a transient profile defined by the TBGas 119890119894(119905) = ℎ(119905)119890119894(0) the second term is a feedback term thatcorrects any deviation of 119890119894(119905) from the transient profile 119890119894(119905)providing global stability of the error dynamics

32 Consensus Based on the Error Feedback In the confer-ence paper [34] whose main results were summarized aboveconsensus protocols were proposed achieving predefined-time convergence to a consensus value 119909 which must bespecified in the control law Moreover it was assumed thatthe initial state is accurately known since this value is used inthe protocols

Nevertheless in most of the protocols proposed in theliterature the consensus value is a quantity that inherentlyresults from the initial states and the connectivity of the graphmodel [15] The consensus value may represent importantinformation to be recovered as in a network of sensorsFor this reason in this paper we propose a new consensusprotocol that yields predefined-time convergence of theunperturbed MAS (4) without the need of specifying aconsensus value Moreover we consider uncertainties in theknowledge of the initial conditions

In the sequel consider that the consensus protocolhas an estimate of the initial state denoted as 0 =[1199091(0) 119909119873(0)]119879 instead of the real initial state 1199090 In thisway

0 = 1199090 + 119889 (14)

where 119889 = [1198891 119889119873]119879 is a vector of uncertainty in theinitial state From (14) the uncertain initial consensus errorcan be computed as

0 = [1198901 (119905) 119890119873 (119905)]119879 = minus1198710 (15)

The following theorem introduces one of the main resultsof this paper a feedback-based consensus protocol ableto achieve consensus without the need of specifying theconsensus value and able to track the trajectories given bythe TBG when there is uncertainty in the initial state 0introduced in the controller

Theorem 7 Consider a MAS with a connected undirectedcommunication topology or directed topology having a span-ning tree modeled as in (4) Let 119896119891 isin R+ be a constant state-feedback gain Then the time-varying feedback control law

119906119894 (119905) = 1119892119894 (119909119894) (minus119891119894 (119909119894) + V119894) 119894 isin 1 119873V119894 = minusℎ (119905) 119890119894 (0) + 119896119891 (119890119894 (119905) minus ℎ (119905) 119890119894 (0))

(16)

with ℎ(119905) as in (10) and 119890119894(0) computed from the initial stateestimate 0 provides global asymptotic stability of the trackingerror 120585(119905) = 119890(119905) minus ℎ(119905)0 Therefore assuming 120588(119905) = 0119873predefined-time convergence of the MAS (4) to a consensusvalue 119909lowast is achieved at 119905119891 from any initial state 1199090

Proof First let us demonstrate the stability of the trackingerrorThedynamics (3)with120588119894(119905) = 0 forall119894 = 1 119873 underthe control action (16) can be written in vector notation as

(119905) = minusℎ (119905) 0 + 119896119891 (119890 (119905) minus ℎ (119905) 0) (17)


Computing (119905) from (8) and using the linearized dynam-ics (17) we have

(119905) = (119905) minus ℎ (119905) 0 = minus119871 (119905) minus ℎ (119905) 0= minus119871 (119905) + ℎ (119905)1198710= minusℎ (119905)11987120 minus 119896119891119871120585 (119905) + ℎ (119905)1198710= minus119896119891119871120585 (119905) + ℎ (119905)119871 (0 minus 1198710)

(18)

By using ℎ(0) = 1 the initial tracking error is computedas

120585 (0) = 1198900 minus ℎ (0) 0 = minus1198711199090 + 119871 (1199090 + 119889) = 119871119889= 119878119869119878minus1119889 (19)

with 119871 = 119878119869119878minus1 where 119869 isin C119873times119873 is the Jordan formassociated with 119871 and 119878 isin C119873times119873 is a nonsingular matrix [39]

Let 120577(119905) = 119878minus1120577(119905) then 120577(0) = 119869119878minus1119889 and according to(2) it follows that the first entry of 120577(0) is null ie 1205771(0) = 0Now taking the time derivative of 120577(119905) and using (18) we have (119905) ≜ [ 1205771 (119905) 120577119873 (119905)]119879 = 119878minus1 (119905)

= 119878minus1 (minus119896119891119871120585 (119905) + ℎ (119905)119871 (0 minus 1198710))= minus119896119891119878minus1 (119878119869119878minus1) 119878120577 (119905)

+ ℎ (119905) 119878minus1 (119878119869119878minus1) (0 minus (119878119869119878minus1) 0)= minus119896119891119869120577 (119905) + ℎ (119905) 119869 (119878minus1 minus 119869119878minus1) 0

(20)

Let 120601(119905) = [1205772(119905) 120577119873(119905)]119879 and 119878minus1 = [ 119903119884 ] where 119903 isinR1times119873 and 119884 isin C(119873minus1)times119873 Then we can verify that (20) can bewritten as

1205771 (119905) = 0 (119905) = minus119896119891120601 (119905) + 119865 (119905) (21)

where 119865(119905) = ℎ(119905)(119884 minus 119884)(0) and isin C(119873minus1)times(119873minus1) is theJordanmatrix associated with 119871 for 1205822 120582119873 Re(120582119894(119871)) gt0 forall119894 = 2 119873

It can be shown that the solutions of the differentialequations (21) are the following [44]

1205771 (119905) = 1205771 (0) = 0120601 (119905) = 119890minus119896119891119905 (120601 (0) + int119905

0119890119896119891119904119865 (119904) 119889119904)

119905 isin [0 119905119891](22)

where 119890minus119896119891119905 isin C(119873minus1)times(119873minus1) is the exponential matrix Thenit follows that

lim119905997888rarr+infin120601 (119905) = lim

119905997888rarr+infin[1205772 (119905) 120577119873 (119905)]119879 = 0119873minus1 (23)

since the integral is a finite value due to ℎ(119905) = 0 forall119905 ge 119905119891 andminus119896119891 has eigenvalues with negative real parts [18 45] Noticethat the convergence speed is determined by the dominanteigenvalue 1205822 of 119871 ([15]) and the control gain 119896119891

Since lim119905997888rarr+infin120577119894(119905) = 0 forall119894 isin 1 119873 thenlim119905997888rarr+infin120585 (119905) = 119878 lim

119905997888rarr+infin120577 (119905)

= 119878 lim119905997888rarr+infin

[1205771 (119905) 120577119873 (119905)]119879= 119878 [0 0]119879 = 0119873

(24)

Therefore the tracking error converges to the origin Infact from (21) it can be seen that the tracking error is globallyasymptotically stable since the real part of the eigenvaluesof minus119896119891 is negative 1205771(119905) = 0 and 119865(119905) can be seen as adisturbance that becomes null for 119905 ge 119905119891 The stability ofthe tracking error implies that 119890(119905) follows ℎ(119905)0 thus 119890(119905)converges to the origin at the predefined-time 119905119891 Any smallfinal error in time 119905 ge 119905119891 is corrected by the control input119906119894 = (1119892119894(119909119894))(minus119891119894(119909119894) + 119896119891119890119894(119905)) which is applied to eachagent for 119905 ge 119905119891

By using the property 1198711119873 = 0119873 (in accordance toLemma 5) the converge of the consensus error (8) to theorigin implies

119909 (119905) 997888rarr 119909 (119905119891) = 119909lowast1119873 as 119905 997888rarr 119905119891 (25)

ie

119909119894 (119905) 997888rarr 119909lowast as 119905 997888rarr 119905119891 forall119894 = 1 119873 (26)

That is the system (4) achieves predefined-time conver-gence to a consensus value 119909lowast at 119905119891 using the consensusprotocol (16)

Now let us compute the final consensus value Firstdefine 119910(119905) = 120574119879119909(119905) where 120574 is defined in Lemma 5 For thetwo types of considered graphs 119910(119905) is an invariant quantitysince

119910 (119905) = 120574119879 (119905) = 120574119879 (minusℎ (119905) 0 + 119896119891 (119890 (119905) minus ℎ (119905) 0))= 120574119879119871 (ℎ (119905) 0 minus 119896119891 (119909 (119905) minus ℎ (119905) 0)) = 0

forall119909 (119905)(27)

where 120574119879119871 = 0119879119873 by Lemma 5 Thus

lim119905997888rarr119905119891

119910 (119905) = 119910 (0) = 120574119879119909 (0) (28)

Moreover from (25) it follows that

lim119905997888rarr119905119891

119910 (119905) = 120574119879 lim119905997888rarr119905119891119909 (119905) = 120574119879 (119909lowast1119873) (29)

Then for a connected undirected graph and by Lemma 5(120574 = 1119873) it is obtained that

120574119879119909lowast1119873 = 1205741198791199090119909lowast (1119873)119879 1119873 = (1119873)119879 1199090

(30)


and solving for 119909lowast one has119909lowast = sum119873119894=1 119909119894 (0)119873 = 119860V119890 (1199090) (31)

Hence the consensus value 119909lowast is the average of the initialstate 1199090 for connected undirected graphs Now for a directedgraphwith a directed spanning tree and by Lemma 5 (1205741198791119873 =1) it results in

119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)

Therefore the consensus value is given by (32) for adirected graph with a directed spanning tree

Thus system (4) with120588(119905) = 0119873 achieves predefined-timeconvergence to a consensus value 119909lowast at 119905119891 in the presence ofuncertainty in the initial state1199090 using the consensus protocol(16)

Remark 8 It is worth noting that in contrast to the use ofthe consensus protocol (13) the feed-forward term of V119894 in(16) is not sufficient to drive the consensus error trajectory tothe TBG reference trajectory (119905) = ℎ(119905)0 even for 119889 = 0119873consequently system (4) under the first term of (16) does notreach predefined-time convergence to a consensus value 119909lowastat 119905119891 In such a case the final state is given by 119909(119905119891) = (119868119873 minus119871)1199090 minus 119871119889 where 119868119873 denotes the identity matrix of size 119873Therefore the use of the feedback term in (16) is required inany case

It can be seen that the consensus protocol (16) cansolve the problem of predefined-time consensus stated inSection 22 by tracking the trajectories given by the TBG evenwhen considering uncertainty 119889 in the initial state In thissense the compensation term for tracking must guaranteereaching the TBG trajectories before the desired convergencetime 119905119891 This means that the tracking control gain 119896119891 mustbe higher for a larger uncertainty in the initial state Thenthere is a limit on the values of 119889 that can be managed by theconsensus protocol

Remark 9 Theorem 7 represents a novel contribution onconsensus for nonlinear MAS First the predefined-timeconsensus is solved for nonlinear first-order systems Secondthe auxiliary control inputs V119894(119905) in (16) are smooth signalseven at 119905 = 0 which is not the case of other approachesreported in the literature Third the proposed consensusprotocols do not have parameters depending on the algebraicconnectivity of the graph considered Finally the stabilityanalysis is performed by using standard techniques for linearsystems (upon feedback linearization of the original systems)while most of the works regarding finite-time and fixed-time and even the few existing works on predefined-timeconvergence adopt more sophisticated techniques such ashomogeneity theory and Lyapunov functions

Remark 10 The result introduced by Theorem 7 comparedto our previous conference work [34] is twofold (1) a newconsensus protocol is proposed achieving predefined-time

convergence of the unperturbed MAS (4) without the needof specifying a consensus value (2) the effect of consideringuncertainty in the knowledge of the initial conditions isanalyzed demonstrating that convergence to a consensusvalue can still be achieved

33 Robust Predefined-Time Consensus In the previous con-sensus protocol we have considered unperturbed dynamics(ie 120588(119905) = 0119873) To effectively deal with disturbances arobust protocol is introduced in this section by combiningthe super-twisting controller (STC) which can compensatefor matched uncertaintiesdisturbances [46] with the TBGleading to a robust predefined-time controller

In this scheme it is assumed that there exits a leader119909119897(119905) isin R in the network system whose dynamics can bemodeled as in [47] by

119897 (119905) = 119906119897 (119905) (33)

where 119906119897(119905) isin R is the leaderrsquos control input In this case119873 agents are now the followers with dynamics (3) Theconsensus error for each follower is defined as [47]

119890119891119894 (119905) = sum119895isin119873119894

119886119894119895 (119909119895 (119905) minus 119909119894 (119905)) minus 119887119894 (119909119894 (119905) minus 119909119897 (119905)) (34)

where 119887119894 isin R represents the leaderrsquos adjacency with 119887119894 gt 0if agent 119894 is a neighbor of the leader otherwise 119887119894 = 0 forall119894 =1 119873

The tracking error is given by

120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)

where ℎ(119905)119890119894(0) is the TBG reference trajectory for the 119894-thand 119890119894(0) is computed from the initial state estimate 0 as in(15)The tracking errorwill be used as the sliding surface thus

119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)

Theorem 11 Consider a MAS modeled as in (4) with a leader(33) For each agent 119894 define 120573119894 = (sum119895isin119873119894 119886119894119895+119887119894) and the slidingsurface (36) Consider a function ℎ(119905) as in (10) There existgains 1198961 gt 0 and 1198962 gt 984858(119872) for some function 984858(∙) and aconstant 119872 gt 0 such that the following nonlinear controllerdefined for each agent

119906119894 (119905) = (120573119894119892119894)minus1 (V119894 + V119894) 119894 isin 1 119873V119894 = 119887119894119906119897 + sum

119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) minus 120573119894119891119894 minus ℎ (119905) 119890119894 (0)V119894 = 1198961 1003816100381610038161003816119904119894100381610038161003816100381612 sign (119904119894) minus 119908119894119894 = minus1198962 sign (119904119894)

(37)

achieves predefined-time convergence to the leaderrsquos state 119909119897 at119905119891 allowing 120588119894(119905) = 0 from any initial state 119909119894(0)


Proof First taking the time derivative of the tracking error(35) andusing the dynamics of the followers (3) and the leader(33) it is obtained that

120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894

119886119894119895 (119895 (119905) minus 119894 (119905)) minus 119887119894 (119894 (119905) minus 119897 (119905))minus ℎ (119905) 119890119894 (0)

= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)

Now using (38) and plugging the control law (37) in thetime derivative of the sliding surface (36) it results in

119904119894 = 120585119894 (119905) = minus1198961 1003816100381610038161003816119904119894100381610038161003816100381612 sign (119904119894) + 119908119894 + 120588119889119894119894 = minus1198962 sign (119904119894) (39)

where 120588119889119894 = sum119895isin119873119894 119886119894119895120588119895 minus 120573119894120588119894 Let 119911119894 = 119908119894 + 120588119889119894 then theprevious expression can be rewritten as

119904119894 = minus1198961 1003816100381610038161003816119904119894100381610038161003816100381612 sign (119904119894) + 119911119894119894 = minus1198962 sign (119904119894) + 120588119889119894

(40)

It has been proven in [46] that for a bounded con-tinuously differentiable disturbance ie if |120588119889119894 | lt 119871 and| 120588119889119894 | lt 119872 for some constants 119871 gt 0 119872 gt 0 thesecond-order dynamics (40) converge globally to the origin(119904119894 = 0 119911119894 = 0) in finite time in spite of the disturbanceif adequate positive control gains 1198961 gt 0 and 1198962 gt 984858(119872)are used Moreover the remaining dynamics of the trackingerror system are constrained to the sliding surface such that119904119894 = 119904119894 = 0 meaning that each agent 119890119891119894 (119905) follows ℎ(119905)119890119894(0)Consequently the consensus error converges to zero in thepredefined-time 119905119891 and the consensus value is given by theleaderrsquos state 119909119897 [47]Remark 12 To the best of our knowledge the only worksthat propose predefined-time consensus protocols by otherauthors are [30ndash32] The consensus control law (37) hasthe advantage over those protocols of providing robustnessagainst large matched perturbations

Some important issues to be considered during real-world applications of the proposed distributed control pro-tocols are threefold First all the clocks of the agents inthe network must be synchronized to achieve predefined-time convergence Second physical constraints of the systemsmust be taken into account to set 119905119891 considering that a small119905119891 will result in large control inputs Thus the maximumallowable input of each agent must be taken into consider-ation to set 119905119891 Third we are currently assuming no timedelay in the agentrsquos communication however timedelaysmayexist in some systems and may affect the convergence to theconsensus

1 2

3

4

56

7

8

1 2

3

4

56

7

8

Figure 1 Communication graphs Left undirected graphG1 Rightdirected graphG2

4 Simulation Results

In this section simulation results are shown to illustrate theadvantages of the proposed protocols All the simulationsconsider a MAS of 8 agents described by (3) with 119891119894(119909119894) = 1199092119894for agents 1 to 4 119891119894(119909119894) = sin(119909119894) for agents 5 to 8 and119892119894(119909119894) = 5 for all the 8 agents The convergence time ispreset to 119905119891 = 5 seconds For this the TBG is described bythe function ℎ(119905) = 2(119905119905119891)3 minus 3(119905119905119891)2 + 1 which fulfills(9) ie ℎ(0) = 1 ℎ(5) = 0 and ℎ(0) = ℎ(5) = 0 Bothproposed distributed control laws are implemented as definedin (16) and (37) The implementation essentially requires thecomputation of the TGB reference and its time derivativewhich means to evaluate the functions ℎ(119905) and ℎ(119905) at eachiteration and compute the consensus error for each agentdepending on the neighbors state as given by (7)

We use the communication topologies shown in Figure 1(obtained from [30]) denoted as G1 and G2 where G1is a connected undirected graph and G2 is a directedgraph having a directed spanning tree The simulations wereimplemented inMATLABusing the Euler forwardmethod toapproximate the time derivatives with a time step of 01ms

The initial states of the eight agents are 1199090 = [minus061 352minus107 073 148 019 197 072]119879 The uncertainties in theinitial states are set to be 119889 = [022 015 minus039 027 034021 minus036 minus009]11987941 Simulation of the Consensus Protocol Based on the ErrorFeedback This subsection is devoted to showing the perfor-mance of the TBG protocol (16) Figures 2 and 3 show thetransient behavior of the MAS under (16) with 119896119891 = 100for G1 and G2 respectively For the case shown in Figure 2perfect knowledge of the initial conditions is assumed whilein Figure 3 there exists uncertainty in the initial conditionknowledge given by 119889 In both cases the agentrsquos state (topgraphs in the figures) reaches consensus at the predefined 5seconds

It can be seen in Figure 2 (middle) that the auxiliarycontrol inputs V119894(119905) generated by the control protocol (16) aresmooth signals even at 119905 = 0 This is due to the properties ofthe TBG which yields that each V119894 starts at zero and returnsto zero at time 119905119891This is not the case for the control inputs 119906119894since they include the nonlinearities that must be cancelled


Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1

Figure 2 Predefined-time TBG-tracking controller (16) for G1The convergence time is preset to 5 sec and the consensus value issum119873119894=1 119909119894(0)119873 = 08676

Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2

Figure 3 Predefined-time TBG-tracking controller (16) for G2and uncertainty in the initial state In the error trajectories thecontinuous lines represent the evolution of the errors whereas theTBG reference is drawn with dash line The convergence time ispreset to 5 sec and the consensus value is 1205741198791199090 = 00946

In the case where the uncertainty 119889 in the initial state isconsidered (Figure 3) the second term of V119894 in (16) is notinitially null to compensate the deviation from the referencegiven by the TBG It can be seen in Figure 3 (middle andbottom) that the consensus errors do not initiate over thereferences and consequently the auxiliary controls are notnull at 119905 = 042 Comparison with Others Approaches In this subsectionfirst we compare the proposed control law (16) with otherpredefined-time control protocols [30ndash32] As described inSection 1 to the authorsrsquo knowledge these are the existingapproaches that solve the problemof predefined-time conver-gence Each one of these works refers to their proposal witha different adjective preset-time consensus [30] prescribed-time consensus [31] and specified-time consensus [32] Wecan see in Figures 4ndash6 that consensus is achieved by usingany of these approaches We have to say that the controlprotocols of [30 31] are very similar both of them are basedon the use of a time-varying control gain Between theirsimilarities we can see in Figures 4 and 5 the large initialvalue of the auxiliary control inputs This is also the case forthe specified-time consensus [32] of Figure 6 Notice that theauxiliary control inputs of the TBG-based controller shown

Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1

Figure 4 Preset-time controller [30] forG1 The convergence timeis preset to 5 sec and the consensus value issum119873119894=1 119909119894(0)119873 = 08676

Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2

Figure 5 Prescribed-time controller [31] for G2 The convergencetime is present to 5 sec and the consensus value is 00946

Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1

Figure 6 Specified-time controller [32] for G1 The convergencetime is present to 5 sec and the consensus value is 05196

in Figure 2 started in zero and were smoother and of lowermagnitude compared to the large initial auxiliary controleffort of the other predefined-time consensus approaches

Additionally we can see that the specified-time consensusprotocol in Figure 6 generates discontinuities in the auxiliaryinputs along the time according to its planned switchingstrategy Moreover although this consensus approach wastested on the undirected graphG1 the consensus value is notthe average of the initial state as one could expect

Following the comparison we evaluate the TBGcontroller (16) and different finite-time fixed-time andpredefined-time consensus protocols reported in theliterature by applying the protocols to the MAS with thecommunication topologies G1 and G2 In particular theprotocols in [20 48] ensure finite-time convergence and theone proposed in [27] guarantees fixed-time convergenceThe predefined-time protocols [30ndash32] and the proposed

Mathematical Problems in Engineering 9Se

ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20

Prescribed-Time [31]

Specified-time [32]

Predefined-time with TBG

Fixed-time [27]Finite-time [48]

Finite-time [20]

Preset-time [30]

minus50 minus40 minus30 minus20 minus10 0 10 20 30 40 50

Mean on initial state (x0) within 1

Figure 7 Comparison of the TBG-tracking controller (16) withother approaches forG1 Top settling time as a function of the initialcondition 1199090 Bottommaximum value of the auxiliary control inputas a function of the initial condition 1199090

max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]


TBG-tracking controller (16) achieve predefined-timeconvergence The finite-time and fixed-time controllers weretuned to achieve a similar convergence time around the 5seconds of predefined time from the initial condition 1199090Then several simulations were performed for different initialstates of the form 1205721199090 where 120572 was varying in such a waythat themean of 1205721199090 ranges from minus50 to 50The same controlgains were used in all the experiments Finally for everyexperiment the convergence time of the system defined asthe time at which 119890 lt 1 times 10minus4 was measured similarlythe maximum absolute value of the auxiliary control input 119907was recorded

The results are shown in Figures 7 and 8 where itcan be observed that the proposed TBG-based control (16)and the other predefined-time controllers [30ndash32] are ableto maintain the same convergence time for all the initialconditions Furthermore note that the maximum auxiliarycontrol efforts were lower for both the TBG-based controland the finite-time control [20] The last control law is of theform 119896 sign(∙) In particular the consensus protocols [30 31]explicitly include a parameter that must be set accordinglyto an eigenvalue of the Laplacian matrix which depends onthe connectivity of the graph considered This is not the case

Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7

Figure 9 Predefined-time TBG-tracking controller (16) which isnot robust for perturbed dynamics andG2

Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7

Figure 10 Robust predefined-time TBG controller (37) for per-turbed systems andG1 In the error trajectories the continuous linesrepresent the evolution of the errors whereas the TBG reference isdrawn with dash line The convergence time is preset to 5 sec

for the proposed control protocolsTherefore the simulationsshow that the proposed TBG protocol (16) represents anadvantage not only in the possibility to define a settling timebut also in providing smooth and efficient control actions

43 Robust Predefined-Time Consensus All the previous sim-ulations considered ideal dynamics in (3) ie 120588119894(119905) = 0 for allthe agents In fact the TBG-tracking controller (16) is not ableto deal with matched bounded disturbances For instanceFigure 9 shows the MAS with the topology G2 under thisprotocol andmatching perturbations 120588119894(119905) = 120572119894(1+1 sin(5119905))with 120572119894 randomly selected in (0 05) It can be observed thatthe state of the MAS does not establish in a consensus valuedue to the perturbation A similar behavior can be obtainedwith the other existing predefined-time consensus protocols[30ndash32]

In order to deal with the described matching perturba-tions 120588119894(119905) = 120572119894(1+1 sin(5119905)) the proposed robust predefined-time TBG controller (37) has to be used For this a leaderis defined with dynamics (33) having communication onlywith the first follower agent ie 1198871 = 1 and 119887119894 = 0 forall119894 =2 8 The gains of the robust controller (37) were set as1198961 = 5 and 1198962 = 5 and the leaderrsquos state was maintainedconstant and equal to 119909119897(119905) = minus5 by setting its control inputas 119906119897 = 0 Figures 10 and 11 show the results obtainedwhen the


Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7


communication between the followers is described byG1 andG2 respectively Notice that the error trajectories are drivento zero in the predefined time 119905119891 = 5 while the control inputkeeps oscillating after 119905119891 in order to reject the disturbance120588(119905)5 Conclusions

In this paper a couple of predefined-time consensus proto-cols for first-order nonlinearMASunder both undirected anddirected communication topologies have been proposed Inthese protocols the convergence time to achieve consensuscan be set by the user and it is independent of the initialstate conditions For this the TBG (time base generator)is combined with feedback controllers to achieve closed-loop stability and robustness In particular one protocol usesthe super-twisting controller to deal with perturbations Theperformance of the proposed controllers has been comparedwith existing finite-time fixed-time and predefined-timecontrollers through simulations The results have shown thatthe proposed protocols achieve consensus in the predefinedtime independently of the initial conditions and exhibitclosed-loop stability Moreover a benefit of the proposedcontrollers is that they yield smoother control signals withsmaller magnitude than the existing approaches reportedin the literature Furthermore the proposed schemes canreach predefined-time consensus even when there existsuncertainty in the knowledge of the initial conditions and thesuper-twisting protocol is robust against matching perturba-tions

Future works will focus on extending the results pre-sented in this paper to the case of high-order nonlinearMAS as well as in considering the predefined-time consensusproblem in Markovian jump topologies [4 5]

Data Availability

The paper is a theoretical result on consensus withpredefined-time convergence All proofs are presentedand simulations are provided to illustrate the result In the

authorsrsquo opinion there is no need to provide the numericalexperiments simulations can be provided andmade availableunder request

Conflicts of Interest

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

Acknowledgments

The first two authors were supported in part by Cona-cyt [grant No 220796] The authors would also like toacknowledge the partial support of Intel Corporation for thedevelopment of this work

References

[1] F Jiang and L Wang ldquoFinite-time information consensusfor multi-agent systems with fixed and switching topologiesrdquoPhysicaDNonlinear Phenomena vol 238 no 16 pp 1550ndash15602009

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

[3] A Arenas A D Guilera J Kurths Y Moreno and C ZhouldquoSynchronization in complex networksrdquo Physics Reports vol469 no 3 pp 93ndash153 2008

[4] H Shen J H Park Z-G Wu and Z Zhang ldquoFinite-time Hinfinsynchronization for complex networks with semi-Markov jumptopologyrdquo Communications in Nonlinear Science and NumericalSimulation vol 24 no 1ndash3 pp 40ndash51 2015

[5] H Shen S Huo J Cao and T Huang ldquoGeneralized stateestimation for Markovian coupled networks under round-robin protocol and redundant channelsrdquo IEEE Transactions onCybernetics pp 1ndash10 2018

[6] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[7] K-K Oh M-C Park and H-S Ahn ldquoA survey of multi-agentformation controlrdquo Automatica vol 53 pp 424ndash440 2015

[8] W Ren ldquoDistributed attitude alignment in spacecraft formationflyingrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 2-3 pp 95ndash113 2007

[9] Y Xu G Yan K Cai and Z Lin ldquoFast centralized integerresource allocation algorithm and its distributed extension overdigraphsrdquo Neurocomputing vol 270 pp 91ndash100 2017

[10] Y Xu T Han K Cai Z Lin G Yan and M Fu ldquoA distributedalgorithm for resource allocation over dynamic digraphsrdquo IEEETransactions on Signal Processing vol 65 no 10 pp 2600ndash26122017

[11] D Gomez-Gutierrez and R De La Guardia ldquoChaotic-basedsynchronization for secure network communications USPatent 9438422B2rdquo 2016

[12] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[13] Z Zuo and L Tie ldquoA new class of finite-time nonlinearconsensus protocols for multi-agent systemsrdquo InternationalJournal of Control vol 87 no 2 pp 363ndash370 2014


[14] D Gomez-Gutierrez J Ruiz-Leon S Celikovsky and J DSanchez-Torres ldquoA finite-time consensus algorithmwith simplestructure for fixed networksrdquo Computacion y Sistemas vol 22no 2 2018

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

[16] W Ren and R W BeardDistributed Consensus in Multi-VehicleCooperative Control Springer 2008

[17] K Cai and H Ishii ldquoAverage consensus on arbitrary stronglyconnected digraphs with time-varying topologiesrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 59 no 4 pp 1066ndash1071 2014

[18] Z Li and Z Duan Cooperative Control of Multi-Agent SystemsA Consensus Region Approach CRC Press Boca Raton FlaUSA 2014

[19] H Sayyaadi and M R Doostmohammadian ldquoFinite-timeconsensus in directed switching network topologies and time-delayed communicationsrdquo Scientia Iranica vol 18 no 1 pp 75ndash85 2011

[20] M Franceschelli A Giua A Pisano and E Usai ldquoFinite-timeconsensus for switching network topologies with disturbancesrdquoNonlinear Analysis Hybrid Systems vol 10 pp 83ndash93 2013

[21] J Cortes ldquoFinite-time convergent gradient flows with applica-tions to network consensusrdquo Automatica vol 42 no 11 pp1993ndash2000 2006

[22] M Franceschelli A Giua and A Pisano ldquoFinite-time consen-sus on the median value with robustness propertiesrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 4 pp 1652ndash1667 2017

[23] B Liu W Lu and T Chen ldquoConsensus in continuous-timemultiagent systems under discontinuous nonlinear protocolsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 2 pp 290ndash301 2015

[24] Y-K Zhu X-P Guan and X-Y Luo ldquoFinite-time consensusfor multi-agent systems via nonlinear control protocolsrdquo Inter-national Journal of Automation and Computing vol 10 no 5pp 455ndash462 2013

[25] Y Shang ldquoFinite-time consensus for multi-agent systems withfixed topologiesrdquo International Journal of Systems Science vol43 no 3 pp 499ndash506 2012

[26] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013

[27] Z Zuo Q L Han B Ning X Ge and X M Zhang ldquoAnoverview of recent advances in fixed-time cooperative controlof multiagent systemsrdquo IEEE Transactions on Industrial Infor-matics vol 14 no 6 pp 2322ndash2334 2018

[28] E Cruz-Zavala J A Moreno and L Fridman ldquoUniformSecond-Order SlidingMode Observer for mechanical systemsrdquoin Proceedings of the 2010 11th International Workshop onVariable Structure Systems VSS 2010 pp 14ndash19 Mexico June2010

[29] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012

[30] C Yong X Guangming and L Huiyang ldquoReaching consensusat a preset time single-integrator dynamics caserdquo in Proceedingsof the 31st Chinese Control Conference (CCC) pp 6220ndash6225IEEE 2012

[31] Y Wang Y Song D J Hill and M Krstic ldquoPrescribed-timeconsensus and containment control of networked multiagentsystemsrdquo IEEE Transactions on Cybernetics 2018

[32] Y Zhao and Y Liu ldquoSpecified-time consensus for multi-agent systemsrdquo in Proceedings of the 2017 Chinese AutomationCongress (CAC) pp 732ndash737 Jinan October 2017

[33] Y Liu Y Zhao W Ren and G Chen ldquoAppointed-time con-sensus accurate and practical designsrdquo Automatica vol 89 pp425ndash429 2018

[34] J A Colunga C R Vazquez H M Becerra and D Gomez-Gutierrez ldquoPredefined-time consensus using a time base gen-erator (TBG)rdquo in Proceedings of the IFAC Second Conferenceon Modelling Identification and Control of Nonlinear Systems(MICNON) vol 51 pp 246ndash253 2018

[35] H M Becerra C R Vazquez G Arechavaleta and J DelfinldquoPredefined-time convergence control for high-order integratorsystems using time base generatorsrdquo IEEE Transactions onControl Systems Technology vol 26 no 5 pp 1866ndash1873 2018

[36] H K Khalil and JW GrizzleNonlinear Systems vol 3 PrenticeHall New Jersey NJ USA 2002

[37] Q Ma S Xu F L Lewis B Zhang and Y Zou ldquoCooper-ative output regulation of singular heterogeneous multiagentsystemsrdquo IEEE Transactions on Cybernetics vol 46 no 6 pp1471ndash1475 2016

[38] Q Ma ldquoCooperative control of multi-agent systems withunknown control directionsrdquo Applied Mathematics and Com-putation vol 292 pp 240ndash252 2017

[39] A H Roger and R J Charles Matrix Analysis CambridgeUniversity Press New York NY USA 2nd edition 2013

[40] W Yu G Wen G Chen and J Cao Distributed CooperativeControl of Multi-agent Systems John Wiley and Sons 2016

[41] Z Li and Z Duan ldquoDistributed consensus protocol designfor general linear multi-agent systems a consensus regionapproachrdquo IET Control Theory amp Applications vol 8 no 18 pp2145ndash2161 2014

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

[43] G Jarquın G Arechavaleta and V Parra-Vega ldquoTimeparametrization of prioritized inverse kinematics based onterminal attractorsrdquo in Proceedings of the 2011 IEEERSJInternational Conference on Intelligent Robots and SystemsCelebrating 50 Years of Robotics IROSrsquo11 pp 1612ndash1617 USASeptember 2011

[44] M Braun Differential Equations and Their Applications AnIntroduction to Applied Mathematics vol 11 Springer Scienceand Business Media Third edition 1983

[45] H K Khalil and J Grizzle Nonlinear systems vol 3 Prenticehall Upper Saddle River New Jersey NJ USA 2002

[46] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

[47] S Mondal R Su and L Xie ldquoHeterogeneous consensus ofhigher-order multi-agent systems with mismatched uncertain-ties using sliding mode controlrdquo International Journal of Robustand Nonlinear Control vol 27 no 13 pp 2303ndash2320 2017

[48] Y Cao and W Ren ldquoFinite-time consensus for multi-agentnetworks with unknown inherent nonlinear dynamicsrdquo Auto-matica vol 50 no 10 pp 2648ndash2656 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


median value [22] and the maximum or minimum value[23] of the agentrsquos initial conditions Continuous finite-timeprotocols based on the | ∙ |120572sign(∙) function with 0 lt 120572 lt1 have been introduced in [12 24] Some algorithms havebeen extended to achieve finite-time consensus for stronglyconnected dynamic topologies [20 25] Fixed-time protocolshave also been proposed in [26 27] In these there exists abound for the convergence time that is independent of theinitial conditions [28 29] The protocols are mainly basedon functions of the form | ∙ |119901sign(∙) + | ∙ |119902sign(∙) with0 lt 119902 lt 1 lt 119901

The methods mentioned above cannot be used in appli-cations where real-time constraints have to be accomplishedsince in both finite-time and fixed-time consensus therelationship between the protocol parameters and the conver-gence time is not straightforwardThus it is difficult to definethe convergence time bound as a function of the protocolparameters Moreover the convergence time bound is oftenoverestimated

To address the consensus design problem with real-timeconstraints predefined-time convergence has to be investi-gated In this the time at which consensus is achieved ispredefined a priori as a parameter of the consensus protocolbeing independent of the initial conditions Few works haveaddressed the predefined-time consensus problem In [3031] linear protocols that use time-varying control gainswere proposed for reaching network consensus at a presettime By using a motion planning strategy predefined-time consensus protocols based on a time-varying samplingsequence convergent to a specified settling time are proposedin [32 33] Preliminary versions of a couple of predefined-time protocols were proposed in [34] taking advantage oftime base generators (TBGs) which are parametric timesignals that converge to zero in a specified time [35] and canbe tracked using feedback controllers

In this work a couple of consensus algorithms based onTBGs are proposed as an extension to [34] One algorithmis based on a linear-feedback and the other is based onthe super-twisting controller for the tracking of the TBGsignal The proposed approach can be applied to systems inwhich agents belong to the class of first-order controllablelinear systems and nonlinear systems that can be linearizedby state feedback [36] Comparisons between the proposedprotocols and finite-time fixed-time and predefined-timeprotocols are provided through simulations showing theadvantages of the proposed approach In contrast with workson predefined-time consensus [30ndash32] the contribution ofthis paper is threefold First the proposed controllers yieldsmoother auxiliary control signals (without considering thelinearization terms) with smaller magnitudes Second thereare no parameters in the proposed consensus protocolsdepending on the algebraic connectivity of the graph con-sidered Third a super-twisting controller is presented todeal with perturbed dynamics in the networkrsquos agents Withrespect to our preliminary conference version [34] two mainextensions are reported herein (1) the new linear-feedbackprotocol drives the agents towards the initial conditionsaverage (for undirected graphs) or the weighted sum ofthe initial conditions (for directed graphs) whereas in ourprevious work the consensus value was artificially specified

(2) convergence to the consensus value in predefined timeis demonstrated even if uncertainty in the knowledge of theinitial state is introduced in the controller at difference of ourprevious results where it was assumed that the initial statewas perfectly known

It is known that second-order systems can representa broader class of real systems than first-order systemsHowever the last class of systems has also broad applicabilityfor instance in robotics for kinematic control of holonomicrobotsWe are very interested in generalizing our results evenfor systems of a higher order than two taking into accountexisting results like the ones in [37 38]

The rest of this paper is organized as follows Section 2introduces some background in algebraic graph theorydefines the class of systems for which the proposed methodis applicable and formulates the problem to be solved in thispaper Section 3 presents the proposed consensus protocolsa linear-feedback and a super-twisting controller for thetracking of the TBG signal Section 4 presents simulationresults and a comparison of the proposed controllers withexisting approaches Section 5 provides conclusions

2 Preliminaries and Problem Definition

21 Algebraic Graph Theory Agentsrsquo communication is rep-resented by a graph G = (119881119864119860) which consists of a setof vertices (nodes or agents in this work) 119881 = 1 119873 aset of edges 119864 sube 119881 times 119881 and a weighted adjacency matrix119860 = [119886119894119895]with nonnegative adjacency elements 119886119894119895 ie 119886119894119895 gt 0iff (119894 119895) isin 119864 The set of neighbors of agent 119894 is denoted by119873119894 = 119895 isin 119881 (119895 119894) isin 119864Definition 1 (see [39 40]) A graphG is called directed if thereis an entry of 119860 that fulfills 119886119894119895 gt 0 ie (119894 119895) isin 119864 representsthat agent 119895 receives the information of the state value119909119894 of theagent 119894 otherwise 119886119894119895 = 0 ie (119895 119894) notin 119864 A graphG is calledundirected if the pairs of nodes are not ordered 119886119894119895 = 119886119895119894 gt 0otherwise 119886119894119895 = 119886119895119894 = 0 ie (119894 119895) (119895 119894) notin 119864Definition 2 (see [39 40]) A directed path (respectivelyundirected path) from nodes V119895 to V119894 is a sequence of distinctedges of the form (V119894 V1198941) (V1198941 V1198942) (V119894119897 V119895) in a directedgraph (respectively undirected graph) A graph is stronglyconnected (respectively connected) if there exists a directedpath (respectively undirected path) between any two distinctvertices V119894 and V119895 inG

Definition 3 (see [16 18]) A directed graphG has a directedspanning tree if there exists a node V119894 (a root) such that allother nodes can be linked to V119894 via a directed path

If an undirected graph has a directed spanning tree thenthe graph is connected A strongly connected directed graphcontains at least one directed spanning tree [16 18]

Definition 4 (see [16]) LetG be a graph on119873 vertices labelledas 1 119873The Laplacianmatrix ofG is defined as the119873times119873matrix 119871 = [119897119894119895] where

119897119894119895 =

minus119886119894119895 if 119894 = 119895119873sum119896=1119896 =119894

119886119894119896 if 119894 = 119895 (1)






such that


] (2)


119894 (119905) = 119891119894 (119909119894) + 119892119894 (119909119894) 119906119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (3)


(119905) = 119891 (119909) + 119892 (119909) 119906 (119905) + 120588 (119905) (4)


119894 (119905) = V119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (5)


(119905) = 119907 (119905) + 120588 (119905) (6)



119890119894 (119905) = sum119895isin119873119894

119886119894119895 (119909119895 (119905) minus 119909119894 (119905)) 119894 isin 1 119873 (7)


119890 (119905) = [1198901 (119905) 119890119873 (119905)]119879 = minus119871119909 (119905) (8)





(9)


ℎ (119905) = 1 if 119905 = 00 if 119905 ge 119905119891

ℎ (119905) = 0 if 119905 = 0 or 119905 ge 119905119891(10)








(11)


(119905) = ℎ (119905) 1198900 = minusℎ (119905)1198711199090 (12)




(13)





0 = 1199090 + 119889 (14)


0 = [1198901 (119905) 119890119873 (119905)]119879 = minus1198710 (15)




(16)



(119905) = minusℎ (119905) 0 + 119896119891 (119890 (119905) minus ℎ (119905) 0) (17)




(18)







(20)


1205771 (119905) = 0 (119905) = minus119896119891120601 (119905) + 119865 (119905) (21)



1205771 (119905) = 1205771 (0) = 0120601 (119905) = 119890minus119896119891119905 (120601 (0) + int119905

0119890119896119891119904119865 (119904) 119889119904)

119905 isin [0 119905119891](22)



119905997888rarr+infin[1205772 (119905) 120577119873 (119905)]119879 = 0119873minus1 (23)



119905997888rarr+infin120577 (119905)

= 119878 lim119905997888rarr+infin

[1205771 (119905) 120577119873 (119905)]119879= 119878 [0 0]119879 = 0119873

(24)




ie





forall119909 (119905)(27)


lim119905997888rarr119905119891

119910 (119905) = 119910 (0) = 120574119879119909 (0) (28)


lim119905997888rarr119905119891

119910 (119905) = 120574119879 lim119905997888rarr119905119891119909 (119905) = 120574119879 (119909lowast1119873) (29)


120574119879119909lowast1119873 = 1205741198791199090119909lowast (1119873)119879 1119873 = (1119873)119879 1199090

(30)




119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)










119897 (119905) = 119906119897 (119905) (33)


119890119891119894 (119905) = sum119895isin119873119894




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)


119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)



119895isin119873119894


(37)




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018













such that


] (2)


119894 (119905) = 119891119894 (119909119894) + 119892119894 (119909119894) 119906119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (3)


(119905) = 119891 (119909) + 119892 (119909) 119906 (119905) + 120588 (119905) (4)


119894 (119905) = V119894 (119905) + 120588119894 (119905) 119894 isin 1 119873 (5)


(119905) = 119907 (119905) + 120588 (119905) (6)



119890119894 (119905) = sum119895isin119873119894

119886119894119895 (119909119895 (119905) minus 119909119894 (119905)) 119894 isin 1 119873 (7)


119890 (119905) = [1198901 (119905) 119890119873 (119905)]119879 = minus119871119909 (119905) (8)





(9)


ℎ (119905) = 1 if 119905 = 00 if 119905 ge 119905119891

ℎ (119905) = 0 if 119905 = 0 or 119905 ge 119905119891(10)








(11)


(119905) = ℎ (119905) 1198900 = minusℎ (119905)1198711199090 (12)




(13)





0 = 1199090 + 119889 (14)


0 = [1198901 (119905) 119890119873 (119905)]119879 = minus1198710 (15)




(16)



(119905) = minusℎ (119905) 0 + 119896119891 (119890 (119905) minus ℎ (119905) 0) (17)




(18)







(20)


1205771 (119905) = 0 (119905) = minus119896119891120601 (119905) + 119865 (119905) (21)



1205771 (119905) = 1205771 (0) = 0120601 (119905) = 119890minus119896119891119905 (120601 (0) + int119905

0119890119896119891119904119865 (119904) 119889119904)

119905 isin [0 119905119891](22)



119905997888rarr+infin[1205772 (119905) 120577119873 (119905)]119879 = 0119873minus1 (23)



119905997888rarr+infin120577 (119905)

= 119878 lim119905997888rarr+infin

[1205771 (119905) 120577119873 (119905)]119879= 119878 [0 0]119879 = 0119873

(24)




ie





forall119909 (119905)(27)


lim119905997888rarr119905119891

119910 (119905) = 119910 (0) = 120574119879119909 (0) (28)


lim119905997888rarr119905119891

119910 (119905) = 120574119879 lim119905997888rarr119905119891119909 (119905) = 120574119879 (119909lowast1119873) (29)


120574119879119909lowast1119873 = 1205741198791199090119909lowast (1119873)119879 1119873 = (1119873)119879 1199090

(30)




119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)










119897 (119905) = 119906119897 (119905) (33)


119890119891119894 (119905) = sum119895isin119873119894




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)


119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)



119895isin119873119894


(37)




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018















(11)


(119905) = ℎ (119905) 1198900 = minusℎ (119905)1198711199090 (12)




(13)





0 = 1199090 + 119889 (14)


0 = [1198901 (119905) 119890119873 (119905)]119879 = minus1198710 (15)




(16)



(119905) = minusℎ (119905) 0 + 119896119891 (119890 (119905) minus ℎ (119905) 0) (17)




(18)







(20)


1205771 (119905) = 0 (119905) = minus119896119891120601 (119905) + 119865 (119905) (21)



1205771 (119905) = 1205771 (0) = 0120601 (119905) = 119890minus119896119891119905 (120601 (0) + int119905

0119890119896119891119904119865 (119904) 119889119904)

119905 isin [0 119905119891](22)



119905997888rarr+infin[1205772 (119905) 120577119873 (119905)]119879 = 0119873minus1 (23)



119905997888rarr+infin120577 (119905)

= 119878 lim119905997888rarr+infin

[1205771 (119905) 120577119873 (119905)]119879= 119878 [0 0]119879 = 0119873

(24)




ie





forall119909 (119905)(27)


lim119905997888rarr119905119891

119910 (119905) = 119910 (0) = 120574119879119909 (0) (28)


lim119905997888rarr119905119891

119910 (119905) = 120574119879 lim119905997888rarr119905119891119909 (119905) = 120574119879 (119909lowast1119873) (29)


120574119879119909lowast1119873 = 1205741198791199090119909lowast (1119873)119879 1119873 = (1119873)119879 1199090

(30)




119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)










119897 (119905) = 119906119897 (119905) (33)


119890119891119894 (119905) = sum119895isin119873119894




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)


119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)



119895isin119873119894


(37)




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018











(18)







(20)


1205771 (119905) = 0 (119905) = minus119896119891120601 (119905) + 119865 (119905) (21)



1205771 (119905) = 1205771 (0) = 0120601 (119905) = 119890minus119896119891119905 (120601 (0) + int119905

0119890119896119891119904119865 (119904) 119889119904)

119905 isin [0 119905119891](22)



119905997888rarr+infin[1205772 (119905) 120577119873 (119905)]119879 = 0119873minus1 (23)



119905997888rarr+infin120577 (119905)

= 119878 lim119905997888rarr+infin

[1205771 (119905) 120577119873 (119905)]119879= 119878 [0 0]119879 = 0119873

(24)




ie





forall119909 (119905)(27)


lim119905997888rarr119905119891

119910 (119905) = 119910 (0) = 120574119879119909 (0) (28)


lim119905997888rarr119905119891

119910 (119905) = 120574119879 lim119905997888rarr119905119891119909 (119905) = 120574119879 (119909lowast1119873) (29)


120574119879119909lowast1119873 = 1205741198791199090119909lowast (1119873)119879 1119873 = (1119873)119879 1199090

(30)




119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)










119897 (119905) = 119906119897 (119905) (33)


119890119891119894 (119905) = sum119895isin119873119894




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)


119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)



119895isin119873119894


(37)




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018











119909lowast = 12057411987911990901205741198791119873

= 1205741198791199090 (32)










119897 (119905) = 119906119897 (119905) (33)


119890119891119894 (119905) = sum119895isin119873119894




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (35)


119904119894 (119905) = 120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0) (36)



119895isin119873119894


(37)




120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018










120585119894 (119905) = 119890119891119894 (119905) minus ℎ (119905) 119890119894 (0)= sum119895isin119873119894


= minus120573119894 (119891119894 + 119892119894119906119894) + sum119895isin119873119894

119886119894119895 (119891119895 + 119892119895119906119895) + 119887119894119906119897minus 120573119894120588119894 + sum

119895isin119873119894

119886119894119895120588119895 minus ℎ (119905) 119890119894 (0)

(38)





(40)



1 2

3

4

56

7

8

1 2

3

4

56

7

8








Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









Stat

es x

minus5

0

5

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

0 1 2 3 4 5 6 7

Con

trol

inpu

ts u

minus2minus1

0

Time (sec) within 1


Stat

es x

minus2024

Con

sens

user

ror e

minus10minus5

05

Auxi

liary

cont

rol i

nput

s v

minus5

0

5

0 1 2 3 4 5 6 7Time (sec) within 2



Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s v

minus2024

0 1 2 3 4 5 6 7Time (sec) within 1


Stat

es x

minus2024

Auxi

liary

cont

rol i

nput

s vminus4minus2

02

0 1 2 3 4 5 6 7Time (sec) within 2


Stat

es x

minus2024

0 1 2 3 4 5 6 7

Auxi

liary

cont

rol i

nput

s v

minus2minus1

01

Time (sec) within 1






ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









ttlin

gtim

e (se

c)

468

1012

max

(v)

0

10

20


Specified-time [32]



Finite-time [20]

Preset-time [30]




max

(v)

Settl

ing

time (

sec)

510152025

02468




Specified-time [32]



Finite-time [20]

Preset-time [30]




Stat

es x

minus5

0

5

Con

sens

user

ror e

minus10

0

10

Auxi

liary

cont

rol i

nput

s v

minus1minus05

005

Time (sec) within 2

0 1 2 3 4 5 6 7


Stat

es x

minus505

Con

sens

user

ror e

minus100

10

Auxi

liary

cont

rol i

nput

s v

minus10minus5

05

Con

trol

inpu

ts u

minus6minus4minus2

0

Time (sec) within 1

0 1 2 3 4 5 6 7






Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018









Stat

es x

Con

sens

user

ror e

Auxi

liary

cont

rol i

nput

s vC

ontro

lin

puts

u

minus505

minus10minus5

05

minus10minus5

0

minus6minus4minus2

0

Time (sec) within 2

0 1 2 3 4 5 6 7





Data Availability





Acknowledgments


References

























































Journal of












Journal of







Volume 2018






Volume 2018



















































Journal of












Journal of







Volume 2018






Volume 2018








predefined-time consensus of nonlinear first-order systems using...

Documents