Research ArticleLeader-Following Consensus for Second-Order NonlinearMultiagent Systems with Input Saturation via DistributedAdaptive Neural Network Iterative Learning Control
Xiongfeng Deng 12 Xiuxia Sun 2 Shuguang Liu2 and Boyang Zhang 2
1College of Electrical Engineering Anhui Polytechnic University Wuhu 241000 China2Equipment Management and Unmanned Aerial Vehicle Engineering College Air Force Engineering University Xirsquoan 710051 China
Correspondence should be addressed to Xiongfeng Deng fate2015zero163com
Received 30 December 2018 Revised 19 March 2019 Accepted 30 April 2019 Published 13 May 2019
Academic Editor Mojtaba Ahmadieh Khanesar
Copyright copy 2019 XiongfengDeng et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper the consensus tracking control problem of leader-following nonlinear multiagent systems with iterative learningcontrol is investigatedThe model of each following agent consists of second-order unknown nonlinear dynamics and the externaldisturbance Moreover the input of each following agent is subject to saturation constraint It is assumed that the informationof leader is not available to any following agents and the radial basis function neural network is introduced to approximate thenonlinear dynamics Then a distributed adaptive neural network iterative learning control protocol and the adaptive updating lawsfor the time-varying parameters are proposed respectively A new Lyapunov function is constructed to analyze the validity of thepresented control protocol Finally a numerical example is provided to verify the effectiveness of theoretical results
1 Introduction
In the past few decades cooperative control problems ofmultiagent systems have been paid outstanding attentionowing to their applications in aerospace engineering sensornetworks and power systems [1ndash3] The basic issue forthe cooperative control of multiagent systems is consensuswhich means that the states of a group of agents arrive atagreement under a designed control protocol The consensusproblems of multiagent systems are usually divided into twotypes depending on whether there is a leader in a multiagentsystem namely leaderless consensus problem [4 5] andleader-following consensus problem [6ndash8] For the latter theleader plays the role of a trajectory generator and other agentstry to track the leader
Recently the consensus problems such as the first-ordermultiagent systems [9 10] the second-ordermultiagentsystems [6 7 11ndash15] the high-order multiagent systems [1617] and the fractional-order systems [18 19] have been exten-sively considered Compared with other multiagent systemsthe second-order multiagent systems is more popular forresearchers Many efforts on the consensus of second-order
multiagent systems have been seen in the existing literatureFor examples in [6 7] the leader-following consensus prob-lem with directed communication topology was addressedIn [11] the consensus tracking problem with disturbancesand unmodeled dynamics was studied and the consensusproblem with communication delay was developed in [12]Moreover the formation control problem with time-varyingdelays and the finite-time consensus problem with switchingtopology were discussed as well [13ndash15]
However it should be noted that the consensus problemsmentioned in the above papers do not take into account thecase of input saturation In the practical multiagent systemsinput saturation may exist due to the limitation of sensorsor actuators The occurrence of input saturation may reducethe performance of a system cause oscillations and evenresult in instability Some papers have explored the consensusproblems of multiagent systems with input saturation Theconsensus control problems of first-order and second-ordermultiagent systems with input saturation were considered in[20ndash23] The finite-time consensus control problem for thesecond-order linear multiagent systems with bounded inputand without velocity measurements was developed in [24]
HindawiComplexityVolume 2019 Article ID 9858504 13 pageshttpsdoiorg10115520199858504
2 Complexity
while the coordinated tracking problem of a class of linearmultiagent systems with actuator magnitude constraint wasstudied in [25] However the related results achieved in [22ndash24] are mainly based on the linear multiagent systems withinput saturation Hence the first motivation of this paperis to discuss the consensus problem of nonlinear multiagentsystems with input saturation
It is worth pointing out that the consensus problemsmentioned in the above literatures are only obtained in thetime domain Based on the prior knowledge of the system theiterative learning control can repeatedly perform tasks withina finite time interval and increase the tracking accuracy asthe number of repetitions increases [26]Themain differencefrom the traditional control methods is that the iterativelearning control can achieve the tracking problem from theperspective of time domain and iterative domain Currentlythe method has been used to achieve the consensus problemsofmultiagent systems In [27] the consensus problem ofmul-tiagent systems with sliding mode iterative learning controlwas investigated In [28 29] the tracking problems of multi-agent systems were addressed by using the designed iterativelearning control method In [30] the formation trackingcontrol problemwith distributed formation iterative learningapproach was discussed while the nonrepetitive formationtracking problem of multiagent systems with line-of-sightand angle constraints under a novel iterative learning controlmethodwas discussed in [31]Moreover the iterative learningcontrol for the high-order and heterogeneous multiagentsystems were studied in [32 33] respectively Different from[27ndash33] the iterative learning control method was appliedto deal with the input saturation problem of robotic armsystems in [34] and the iterative learning control protocolfor the nonrepetitive trajectory tracking of mobile robotswith fault-tolerant and output constraints was presented in[35] However to the best of our knowledge there are fewpapers that discussed the issue of the iterative learning controlfor the consensus problem of nonlinear multiagent systemswith external disturbances and input saturation which is thesecond motivation of this paper
Inspired by the above analysis the iterative learningcontrol for the nonlinear multiagent systems with externaldisturbance and input saturation is discussed in this workThe main contributions are summarized as follows
(i) A class of leader-following second-order nonlinearmultiagent systems with external disturbance and input satu-ration is considered The nonlinear dynamics of each follow-ing agent is unknown Compared with [22ndash24] the controlprotocol design will be more complicated
(ii) Motivated by [28 36] the radial basis function (RBF)neural network is adopted to approximate the unknown non-linear terms of all following agents in this paper Also it issupposed that the information of leader is not available to anyfollowing agents
(iii) Based on the RBF neural network and iterative learn-ing control approach a distributed adaptive neural networkiterative learning control protocol is proposed and the adap-tive updating laws for time-varying parameters are presentedrespectively Then the effectiveness of the designed controlprotocol is checked by simulation example
The rest of this paper is planned as follows In Section 2graph theory RBF neural network and some useful defini-tions and lemmas are introduced In Section 3 the consen-sus problem formulation the control protocol design andconvergence analysis are described Finally the simulationanalysis and conclusions are provided in Sections 4 and 5respectively
2 Preliminaries
In this section the preliminaries on the graph theory neuralnetwork approximation and some useful definitions andlemmas are introduced for the discussion and analysis below
21 Graph Theory Let an undirected graph G = (VEA)consist of 119899 nodes where the set of nodes isV = V1 sdot sdot sdot V119899and the set of edges is E sube V times V The weighted adjacencymatrix is defined asA = [119886119894119895] isin 119877119899times119899 in which 119886119894119895 = 119886119895119894 gt 0if (119894 119895) isin E and 119886119894119895 = 119886119895119894 = 0 otherwise It is assumed that119886119894119894 = 0 The set of neighbors of node 119894 is defined by N119894 =V119895 (V119895 V119894) isin E The Laplacian matrix of G is defined asL = DminusA whereD = diag1198891 sdot sdot sdot 119889119899 with 119889119894 = sum119899119895=1 119886119894119895
In this paper an augmented graph G with 119899 followingagents whose information topology graph isG and one leaderagent is considered Let 119887119894 be the connection matrix betweenagent 119894 and the leader If agent 119894 gets the information of leaderthen 119887119894 gt 0 otherwise 119887119894 = 0 Hence the connection matrixbetween the leader and following agents is defined as B =diag1198871 sdot sdot sdot 119887119899 Also it is obtained that H = L + B is amatrix associated with G
Lemma 1 (see [37]) If the graph G is connected then thesymmetric matrixH associated withG is positive definite
22 Neural Network Approximation As amethod of process-ing nonlinear dynamics the neural network is mostly usedbecause of its universal approximation capabilities In thispaper the RBF neural network is considered to approximatethe unknown nonlinear dynamics of agents Consider acontinuous function 119910(119909) 119877119899 997888rarr 119877 which can beapproximated by the RBF neural network as
119910 (119909) = 119882T120593 (119909) (1)
where 119909 isin Ω119909sub 119877119899 is the input vector of neural network119882 = [1199081 sdot sdot sdot 119908119871]T isin 119877119871 is the weight matrix of output layer120593(119909) = [1205931(119909) sdot sdot sdot 120593119871(119909)]T isin 119877119871 is the basis function vector
and the basis function is considered to be Gaussian functionas 120593119894(119909) = exp(minus(119909 minus 120585119894)T(119909 minus 120585119894)21205792119894 ) for 119894 = 1 sdot sdot sdot 119871 where120585119894 = [1205851198941 sdot sdot sdot 120585119894119899]T isin 119877119899 is the center vector and 120579119894 is the widthof the Gaussian function 119871 is the node number of hiddenlayer
where119882lowast is the optimal constant weight vector and 119900(119909) isin 119877is the approximation error which satisfies 119900(119909) le 119900lowast with119900lowast being an unknown positive constant
Complexity 3
It should be highlighted that the optimal weight vector119882lowast is only used for analytical purpose The optimal weightvector 119882lowast is defined so that 119900(119909) is minimized for all 119909 isinΩ119909sub 119877119899 that is
119882lowast = argmin119882isin119877119871
sup119909isinΩ119909
10038161003816100381610038161003816119910119900119901 (119909) minus 119882T120593 (119909)10038161003816100381610038161003816 (3)
Remark 2 The approximation ability of a neural networkrelies on the number of hidden layer nodes 119871 The larger thenumber of 119871 the better the approximation effect Howeverthere is no goodway to select119871 in the existing literature It canbe roughly estimated according to the control requirementsIn addition the Gaussian function for 120593119894(119909) is considered inthis paper and it can be replaced by other basis functionssuch as the spline function the sigmoid function and thehyperbolic tangent function as long as they satisfy the natureof the basis function
Some useful definitions and lemmas are given as follows
Definition 3 (see [38]) A convergent series sequence Δ 119896 isdenoted byΔ 119896 = 119888119896119898 where 119896 isin 119885+ 119888 gt 0 and119898(isin 119885+) ge 2are the parameters to be designed
Lemma 4 (see [38]) For a given sequence 119888119896119898 where119896 isin 119885+ 119888 gt 0 and 119898(isin 119885+) ge 2 it is held thatlim119896997888rarrinfinsum119896119895=1(119888119895119898) le 2119888Lemma 5 (see [39]) For any 119887(isin 119877) gt 0 and 120577 gt 0 thehyperbolic tangent function satisfies 0 le |119887|minus119887 tanh(119887120577) le 119902120577where 119902 = 02785Lemma 6 Let 119886 isin 119877119899times1 119887 isin 119877119899times1 and 119862 isin 119877119899times119899 then it canbe obtained that 119886T119862119887 = tr119862119887119886T where tr(sdot) represents thetrace operation
3 Main Results
In this section the tracking problem of nonlinear multiagentsystems with input saturation is discussed Based on theneural network approximation technique and the iterativelearning control approach the distributed adaptive controlprotocol and the adaptive updating laws are presentedrespectively Then the convergence of proposed controlprotocol is illustrated by a designed Lyapunov function
31 Problem Formulation Consider a class of leader-following second-order nonlinear multiagent systems withthe external disturbance and input saturation the dynamicsof the 119894th following agent at 119896th iteration are described asfollows
where 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 and 119906119896119894 (119905) isin 119877 are theposition velocity and control input of the 119894th following
agent respectively 119891(119909119896119894 (119905) V119896119894 (119905)) represents the unknownnonlinear function 119889119896119894 (119905) is unknown but bounded externaldisturbance that is there exists 119889119896119894 (119905) le 119889lowast119894 with 119889lowast119894 beingan unknown positive constant and 119896 denotes the iterationnumber and 119905 isin [0 119879] sat(119906119896119894 (119905)) is the saturation functionwhich is defined as
sat (119906119896119894 (119905)) =119906 119906119896119894 (119905) gt 119906119906119896119894 (119905) minus119906 le 119906119896119894 (119905) le 119906minus119906 119906119896119894 (119905) lt minus119906
(5)
where 119906 gt 0 is the upper bound of saturation function andprespecified
where 1199090(119905) isin 119877 V0(119905) isin 119877 and 1199060(119905) isin 119877 are the positionvelocity and input of leader respectively and 119891(1199090(119905) V0(119905))represents the unknown nonlinear function Referring to theliterature [40] it is also assumed that the control input ofleader is nonzero but bounded that is there exists 1199060(119905) le119906lowast0 with 119906lowast0 being a positive constant
According to the multiagent systems (4) and (7) thetracking errors of position and velocity are defined as
119890119896119909119894 (119905) = 1199090 (119905) minus 119909119896119894 (119905) (8)
119890119896V119894 (119905) = V0 (119905) minus V119896119894 (119905) (9)
Let 119890119896119909(119905) = [1198901198961199091(119905) sdot sdot sdot 119890119896119909119899(119905)]T and 119890119896V(119905) = [119890119896V1(119905) sdot sdot sdot 119890119896V119899(119905)]T then119890119896119909 (119905) = 11198991199090 (119905) minus 119909119896 (119905) (10)
119890119896V (119905) = 1119899V0 (119905) minus V119896 (119905) (11)
where 1119899 = [1 sdot sdot sdot 1]TAssumption 7 Theunknown nonlinear item 119891(1199090(119905) V0(119905)) isbounded namely there exists 119891(1199090(119905) V0(119905)) le 119891lowast0 where119891lowast0 is an unknown constant
Assumption 8 The alignment initial conditions that is119909119896119894 (0) = 119909119896minus1119894 (119879) and V119896119894 (0) = V119896minus1119894 (119879) for each following agentare satisfied Also it is assumed that the trajectory of leader isspatially closed that is 1199090(0) = 1199090(119879) and V0(0) = V0(119879)
4 Complexity
According to Assumption 8 hence it can be gotten that119890119896119909119894(0) = 119890119896minus1119909119894 (119879) and 119890119896V119894(0) = 119890119896minus1V119894 (119879) for each following agentDefinition 9 For any initial condition the consensus track-ing problem of leader-following second-order nonlinearmultiagent systems with input saturation is achieved iflim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 =1 sdot sdot sdot 119899 over the interval [0 119879] are satisfied
The control objective of this paper is to design theappropriate control scheme 119906119896119894 (119905) for 119894 = 1 sdot sdot sdot 119899 and theadaptive updating laws such that the states of all the followingagents can track the trajectory of leader over the interval[0 119879] as the iteration number 119896 tends to infinity32 Control Protocol Design According to the multiagentsystems (4) and (7) the consensus tracking errors are definedas
119886119894119895 (V119896119895 (119905) minus V119896119894 (119905)) + 119887119894 (V0 (119905) minus V119896119894 (119905)) (13)
And directly from (12) and (13) we get
120576119896119909 (119905) = H (11198991199090 (119905) minus 119909119896 (119905)) = H119890119896119909 (119905) (14)
120576119896V (119905) = H (1119899V0 (119905) minus V119896 (119905)) = H119890119896V (119905) (15)
where 120576119896119909(119905) = [1205761198961199091(119905) sdot sdot sdot 120576119896119909119899(119905)]T and 120576119896V (119905) =[120576119896V1(119905) sdot sdot sdot 120576119896V119899(119905)]TRemark 10 In this paper we only discuss the states of eachagent as 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 1199090(119905) isin 119877 and V0(119905) isin 119877 Forthe case of 119909119896119894 (119905) isin 119877119901 V119896119894 (119905) isin 119877119901 1199090(119905) isin 119877119901 and V0(119905) isin 119877119901we have
where otimes is the Kronecker product 119868119901 is the unit matrix with119901 dimension and all of the related results can be changed byapplying the Kronecker product operation
Considering 120576119896119909(119905) and 120576119896V (119905) a sliding mode function isdesigned as
where 120572 gt 0 is a positive constant and 119904119896(119905) =[1199041198961(119905) sdot sdot sdot 119904119896119899(119905)]TSo the derivative of 119904119896(119905) is
119904119896 (119905) = 120576119896V (119905) + 120572 120576119896119909 (119905) = H (1119899V0 (119905) minus V119896 (119905))+ 120572H119890119896V (119905) = H (1119899119891 (1199090 (119905) V0 (119905))minus 119891 (119909119896 (119905) V119896 (119905)) + 11198991199060 (119905) minus sat (119906119896 (119905))minus 119889119896 (119905)) + 120572H119890119896V (119905)
(19)
For the unknown nonlinear parts 119891(119909119896(119905) V119896(119905)) weintroduce the RBF neural network to approximate them Inview of the approximation properties of RBF neural network119891(119909119896119894 (119905) V119896119894 (119905)) can be described as
where 120573 gt 0 120599119894 gt 0 and 120574119894 gt 0 are constants to bedesigned and 120575119906119896(119905) = 119906119896(119905) minus sat(119906119896(119905)) with 119906119896(119905) =[1199061198961(119905) sdot sdot sdot 119906119896119899(119905)]T and 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]T
The vector form of control protocol (24) can be written as
where 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]Tand 120592119896(119905) = [1205921198961(119905) sdot sdot sdot 120592119896119899(119905)]T with 120592119896119894 (119905) = 120578119896119894 (119905) tanh(120578119896119894 (119905)119904119896119894 (119905)Δ 119896)Remark 11 In the control protocol (24) the time-varyingparameters 120590119896119894 (119905) and 120578119896119894 (119905) are introduced The purposeof designing 120590119896119894 (119905) is to compensate the saturation error120575119906119896119894 (119905) and the purpose of designing 120578119896119894 (119905) is to eliminate theinfluence of approximation error 119900(119909119896119894 (119905) V119896119894 (119905)) and externaldisturbance 119889119896119894 (119905) In other words the objective of designingadaptive updating laws is to seek the distributed adaptive iter-ative learning control protocol for time-varying parameterssuch that the tracking problem can be solved over the interval[0 119879]33 Convergence Analysis In what follows the main result ofthis paper is given inTheorem 12
Theorem 12 Consider the leader-following second-order non-linear multiagent systems with input saturation (4) and (7)and suppose that Assumptions 7 and 8 are held and thecommunication topology G is connected Let the distributedadaptive neural network iterative learning control protocol (24)and the adaptive updating laws (25) (26) and (27) be appliedthen all the following agents can track the trajectory of leadernamely lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for119894 = 1 sdot sdot sdot 119899 over the interval [0 119879]
Proof Design the following Lyapunov function candidate
where 119896(119905) = 119882lowast minus 119896(119905) Γ = diag1205741119868119871 sdot sdot sdot 120574119899119868119871 and1205780 gt 0 is a constant to be determined laterConsider the difference between 119881119896(119905) and 119881119896minus1(119905) that
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
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2 Complexity
while the coordinated tracking problem of a class of linearmultiagent systems with actuator magnitude constraint wasstudied in [25] However the related results achieved in [22ndash24] are mainly based on the linear multiagent systems withinput saturation Hence the first motivation of this paperis to discuss the consensus problem of nonlinear multiagentsystems with input saturation
It is worth pointing out that the consensus problemsmentioned in the above literatures are only obtained in thetime domain Based on the prior knowledge of the system theiterative learning control can repeatedly perform tasks withina finite time interval and increase the tracking accuracy asthe number of repetitions increases [26]Themain differencefrom the traditional control methods is that the iterativelearning control can achieve the tracking problem from theperspective of time domain and iterative domain Currentlythe method has been used to achieve the consensus problemsofmultiagent systems In [27] the consensus problem ofmul-tiagent systems with sliding mode iterative learning controlwas investigated In [28 29] the tracking problems of multi-agent systems were addressed by using the designed iterativelearning control method In [30] the formation trackingcontrol problemwith distributed formation iterative learningapproach was discussed while the nonrepetitive formationtracking problem of multiagent systems with line-of-sightand angle constraints under a novel iterative learning controlmethodwas discussed in [31]Moreover the iterative learningcontrol for the high-order and heterogeneous multiagentsystems were studied in [32 33] respectively Different from[27ndash33] the iterative learning control method was appliedto deal with the input saturation problem of robotic armsystems in [34] and the iterative learning control protocolfor the nonrepetitive trajectory tracking of mobile robotswith fault-tolerant and output constraints was presented in[35] However to the best of our knowledge there are fewpapers that discussed the issue of the iterative learning controlfor the consensus problem of nonlinear multiagent systemswith external disturbances and input saturation which is thesecond motivation of this paper
Inspired by the above analysis the iterative learningcontrol for the nonlinear multiagent systems with externaldisturbance and input saturation is discussed in this workThe main contributions are summarized as follows
(i) A class of leader-following second-order nonlinearmultiagent systems with external disturbance and input satu-ration is considered The nonlinear dynamics of each follow-ing agent is unknown Compared with [22ndash24] the controlprotocol design will be more complicated
(ii) Motivated by [28 36] the radial basis function (RBF)neural network is adopted to approximate the unknown non-linear terms of all following agents in this paper Also it issupposed that the information of leader is not available to anyfollowing agents
(iii) Based on the RBF neural network and iterative learn-ing control approach a distributed adaptive neural networkiterative learning control protocol is proposed and the adap-tive updating laws for time-varying parameters are presentedrespectively Then the effectiveness of the designed controlprotocol is checked by simulation example
The rest of this paper is planned as follows In Section 2graph theory RBF neural network and some useful defini-tions and lemmas are introduced In Section 3 the consen-sus problem formulation the control protocol design andconvergence analysis are described Finally the simulationanalysis and conclusions are provided in Sections 4 and 5respectively
2 Preliminaries
In this section the preliminaries on the graph theory neuralnetwork approximation and some useful definitions andlemmas are introduced for the discussion and analysis below
21 Graph Theory Let an undirected graph G = (VEA)consist of 119899 nodes where the set of nodes isV = V1 sdot sdot sdot V119899and the set of edges is E sube V times V The weighted adjacencymatrix is defined asA = [119886119894119895] isin 119877119899times119899 in which 119886119894119895 = 119886119895119894 gt 0if (119894 119895) isin E and 119886119894119895 = 119886119895119894 = 0 otherwise It is assumed that119886119894119894 = 0 The set of neighbors of node 119894 is defined by N119894 =V119895 (V119895 V119894) isin E The Laplacian matrix of G is defined asL = DminusA whereD = diag1198891 sdot sdot sdot 119889119899 with 119889119894 = sum119899119895=1 119886119894119895
In this paper an augmented graph G with 119899 followingagents whose information topology graph isG and one leaderagent is considered Let 119887119894 be the connection matrix betweenagent 119894 and the leader If agent 119894 gets the information of leaderthen 119887119894 gt 0 otherwise 119887119894 = 0 Hence the connection matrixbetween the leader and following agents is defined as B =diag1198871 sdot sdot sdot 119887119899 Also it is obtained that H = L + B is amatrix associated with G
Lemma 1 (see [37]) If the graph G is connected then thesymmetric matrixH associated withG is positive definite
22 Neural Network Approximation As amethod of process-ing nonlinear dynamics the neural network is mostly usedbecause of its universal approximation capabilities In thispaper the RBF neural network is considered to approximatethe unknown nonlinear dynamics of agents Consider acontinuous function 119910(119909) 119877119899 997888rarr 119877 which can beapproximated by the RBF neural network as
119910 (119909) = 119882T120593 (119909) (1)
where 119909 isin Ω119909sub 119877119899 is the input vector of neural network119882 = [1199081 sdot sdot sdot 119908119871]T isin 119877119871 is the weight matrix of output layer120593(119909) = [1205931(119909) sdot sdot sdot 120593119871(119909)]T isin 119877119871 is the basis function vector
and the basis function is considered to be Gaussian functionas 120593119894(119909) = exp(minus(119909 minus 120585119894)T(119909 minus 120585119894)21205792119894 ) for 119894 = 1 sdot sdot sdot 119871 where120585119894 = [1205851198941 sdot sdot sdot 120585119894119899]T isin 119877119899 is the center vector and 120579119894 is the widthof the Gaussian function 119871 is the node number of hiddenlayer
where119882lowast is the optimal constant weight vector and 119900(119909) isin 119877is the approximation error which satisfies 119900(119909) le 119900lowast with119900lowast being an unknown positive constant
Complexity 3
It should be highlighted that the optimal weight vector119882lowast is only used for analytical purpose The optimal weightvector 119882lowast is defined so that 119900(119909) is minimized for all 119909 isinΩ119909sub 119877119899 that is
119882lowast = argmin119882isin119877119871
sup119909isinΩ119909
10038161003816100381610038161003816119910119900119901 (119909) minus 119882T120593 (119909)10038161003816100381610038161003816 (3)
Remark 2 The approximation ability of a neural networkrelies on the number of hidden layer nodes 119871 The larger thenumber of 119871 the better the approximation effect Howeverthere is no goodway to select119871 in the existing literature It canbe roughly estimated according to the control requirementsIn addition the Gaussian function for 120593119894(119909) is considered inthis paper and it can be replaced by other basis functionssuch as the spline function the sigmoid function and thehyperbolic tangent function as long as they satisfy the natureof the basis function
Some useful definitions and lemmas are given as follows
Definition 3 (see [38]) A convergent series sequence Δ 119896 isdenoted byΔ 119896 = 119888119896119898 where 119896 isin 119885+ 119888 gt 0 and119898(isin 119885+) ge 2are the parameters to be designed
Lemma 4 (see [38]) For a given sequence 119888119896119898 where119896 isin 119885+ 119888 gt 0 and 119898(isin 119885+) ge 2 it is held thatlim119896997888rarrinfinsum119896119895=1(119888119895119898) le 2119888Lemma 5 (see [39]) For any 119887(isin 119877) gt 0 and 120577 gt 0 thehyperbolic tangent function satisfies 0 le |119887|minus119887 tanh(119887120577) le 119902120577where 119902 = 02785Lemma 6 Let 119886 isin 119877119899times1 119887 isin 119877119899times1 and 119862 isin 119877119899times119899 then it canbe obtained that 119886T119862119887 = tr119862119887119886T where tr(sdot) represents thetrace operation
3 Main Results
In this section the tracking problem of nonlinear multiagentsystems with input saturation is discussed Based on theneural network approximation technique and the iterativelearning control approach the distributed adaptive controlprotocol and the adaptive updating laws are presentedrespectively Then the convergence of proposed controlprotocol is illustrated by a designed Lyapunov function
31 Problem Formulation Consider a class of leader-following second-order nonlinear multiagent systems withthe external disturbance and input saturation the dynamicsof the 119894th following agent at 119896th iteration are described asfollows
where 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 and 119906119896119894 (119905) isin 119877 are theposition velocity and control input of the 119894th following
agent respectively 119891(119909119896119894 (119905) V119896119894 (119905)) represents the unknownnonlinear function 119889119896119894 (119905) is unknown but bounded externaldisturbance that is there exists 119889119896119894 (119905) le 119889lowast119894 with 119889lowast119894 beingan unknown positive constant and 119896 denotes the iterationnumber and 119905 isin [0 119879] sat(119906119896119894 (119905)) is the saturation functionwhich is defined as
sat (119906119896119894 (119905)) =119906 119906119896119894 (119905) gt 119906119906119896119894 (119905) minus119906 le 119906119896119894 (119905) le 119906minus119906 119906119896119894 (119905) lt minus119906
(5)
where 119906 gt 0 is the upper bound of saturation function andprespecified
where 1199090(119905) isin 119877 V0(119905) isin 119877 and 1199060(119905) isin 119877 are the positionvelocity and input of leader respectively and 119891(1199090(119905) V0(119905))represents the unknown nonlinear function Referring to theliterature [40] it is also assumed that the control input ofleader is nonzero but bounded that is there exists 1199060(119905) le119906lowast0 with 119906lowast0 being a positive constant
According to the multiagent systems (4) and (7) thetracking errors of position and velocity are defined as
119890119896119909119894 (119905) = 1199090 (119905) minus 119909119896119894 (119905) (8)
119890119896V119894 (119905) = V0 (119905) minus V119896119894 (119905) (9)
Let 119890119896119909(119905) = [1198901198961199091(119905) sdot sdot sdot 119890119896119909119899(119905)]T and 119890119896V(119905) = [119890119896V1(119905) sdot sdot sdot 119890119896V119899(119905)]T then119890119896119909 (119905) = 11198991199090 (119905) minus 119909119896 (119905) (10)
119890119896V (119905) = 1119899V0 (119905) minus V119896 (119905) (11)
where 1119899 = [1 sdot sdot sdot 1]TAssumption 7 Theunknown nonlinear item 119891(1199090(119905) V0(119905)) isbounded namely there exists 119891(1199090(119905) V0(119905)) le 119891lowast0 where119891lowast0 is an unknown constant
Assumption 8 The alignment initial conditions that is119909119896119894 (0) = 119909119896minus1119894 (119879) and V119896119894 (0) = V119896minus1119894 (119879) for each following agentare satisfied Also it is assumed that the trajectory of leader isspatially closed that is 1199090(0) = 1199090(119879) and V0(0) = V0(119879)
4 Complexity
According to Assumption 8 hence it can be gotten that119890119896119909119894(0) = 119890119896minus1119909119894 (119879) and 119890119896V119894(0) = 119890119896minus1V119894 (119879) for each following agentDefinition 9 For any initial condition the consensus track-ing problem of leader-following second-order nonlinearmultiagent systems with input saturation is achieved iflim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 =1 sdot sdot sdot 119899 over the interval [0 119879] are satisfied
The control objective of this paper is to design theappropriate control scheme 119906119896119894 (119905) for 119894 = 1 sdot sdot sdot 119899 and theadaptive updating laws such that the states of all the followingagents can track the trajectory of leader over the interval[0 119879] as the iteration number 119896 tends to infinity32 Control Protocol Design According to the multiagentsystems (4) and (7) the consensus tracking errors are definedas
119886119894119895 (V119896119895 (119905) minus V119896119894 (119905)) + 119887119894 (V0 (119905) minus V119896119894 (119905)) (13)
And directly from (12) and (13) we get
120576119896119909 (119905) = H (11198991199090 (119905) minus 119909119896 (119905)) = H119890119896119909 (119905) (14)
120576119896V (119905) = H (1119899V0 (119905) minus V119896 (119905)) = H119890119896V (119905) (15)
where 120576119896119909(119905) = [1205761198961199091(119905) sdot sdot sdot 120576119896119909119899(119905)]T and 120576119896V (119905) =[120576119896V1(119905) sdot sdot sdot 120576119896V119899(119905)]TRemark 10 In this paper we only discuss the states of eachagent as 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 1199090(119905) isin 119877 and V0(119905) isin 119877 Forthe case of 119909119896119894 (119905) isin 119877119901 V119896119894 (119905) isin 119877119901 1199090(119905) isin 119877119901 and V0(119905) isin 119877119901we have
where otimes is the Kronecker product 119868119901 is the unit matrix with119901 dimension and all of the related results can be changed byapplying the Kronecker product operation
Considering 120576119896119909(119905) and 120576119896V (119905) a sliding mode function isdesigned as
where 120572 gt 0 is a positive constant and 119904119896(119905) =[1199041198961(119905) sdot sdot sdot 119904119896119899(119905)]TSo the derivative of 119904119896(119905) is
119904119896 (119905) = 120576119896V (119905) + 120572 120576119896119909 (119905) = H (1119899V0 (119905) minus V119896 (119905))+ 120572H119890119896V (119905) = H (1119899119891 (1199090 (119905) V0 (119905))minus 119891 (119909119896 (119905) V119896 (119905)) + 11198991199060 (119905) minus sat (119906119896 (119905))minus 119889119896 (119905)) + 120572H119890119896V (119905)
(19)
For the unknown nonlinear parts 119891(119909119896(119905) V119896(119905)) weintroduce the RBF neural network to approximate them Inview of the approximation properties of RBF neural network119891(119909119896119894 (119905) V119896119894 (119905)) can be described as
where 120573 gt 0 120599119894 gt 0 and 120574119894 gt 0 are constants to bedesigned and 120575119906119896(119905) = 119906119896(119905) minus sat(119906119896(119905)) with 119906119896(119905) =[1199061198961(119905) sdot sdot sdot 119906119896119899(119905)]T and 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]T
The vector form of control protocol (24) can be written as
where 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]Tand 120592119896(119905) = [1205921198961(119905) sdot sdot sdot 120592119896119899(119905)]T with 120592119896119894 (119905) = 120578119896119894 (119905) tanh(120578119896119894 (119905)119904119896119894 (119905)Δ 119896)Remark 11 In the control protocol (24) the time-varyingparameters 120590119896119894 (119905) and 120578119896119894 (119905) are introduced The purposeof designing 120590119896119894 (119905) is to compensate the saturation error120575119906119896119894 (119905) and the purpose of designing 120578119896119894 (119905) is to eliminate theinfluence of approximation error 119900(119909119896119894 (119905) V119896119894 (119905)) and externaldisturbance 119889119896119894 (119905) In other words the objective of designingadaptive updating laws is to seek the distributed adaptive iter-ative learning control protocol for time-varying parameterssuch that the tracking problem can be solved over the interval[0 119879]33 Convergence Analysis In what follows the main result ofthis paper is given inTheorem 12
Theorem 12 Consider the leader-following second-order non-linear multiagent systems with input saturation (4) and (7)and suppose that Assumptions 7 and 8 are held and thecommunication topology G is connected Let the distributedadaptive neural network iterative learning control protocol (24)and the adaptive updating laws (25) (26) and (27) be appliedthen all the following agents can track the trajectory of leadernamely lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for119894 = 1 sdot sdot sdot 119899 over the interval [0 119879]
Proof Design the following Lyapunov function candidate
where 119896(119905) = 119882lowast minus 119896(119905) Γ = diag1205741119868119871 sdot sdot sdot 120574119899119868119871 and1205780 gt 0 is a constant to be determined laterConsider the difference between 119881119896(119905) and 119881119896minus1(119905) that
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
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Complexity 3
It should be highlighted that the optimal weight vector119882lowast is only used for analytical purpose The optimal weightvector 119882lowast is defined so that 119900(119909) is minimized for all 119909 isinΩ119909sub 119877119899 that is
119882lowast = argmin119882isin119877119871
sup119909isinΩ119909
10038161003816100381610038161003816119910119900119901 (119909) minus 119882T120593 (119909)10038161003816100381610038161003816 (3)
Remark 2 The approximation ability of a neural networkrelies on the number of hidden layer nodes 119871 The larger thenumber of 119871 the better the approximation effect Howeverthere is no goodway to select119871 in the existing literature It canbe roughly estimated according to the control requirementsIn addition the Gaussian function for 120593119894(119909) is considered inthis paper and it can be replaced by other basis functionssuch as the spline function the sigmoid function and thehyperbolic tangent function as long as they satisfy the natureof the basis function
Some useful definitions and lemmas are given as follows
Definition 3 (see [38]) A convergent series sequence Δ 119896 isdenoted byΔ 119896 = 119888119896119898 where 119896 isin 119885+ 119888 gt 0 and119898(isin 119885+) ge 2are the parameters to be designed
Lemma 4 (see [38]) For a given sequence 119888119896119898 where119896 isin 119885+ 119888 gt 0 and 119898(isin 119885+) ge 2 it is held thatlim119896997888rarrinfinsum119896119895=1(119888119895119898) le 2119888Lemma 5 (see [39]) For any 119887(isin 119877) gt 0 and 120577 gt 0 thehyperbolic tangent function satisfies 0 le |119887|minus119887 tanh(119887120577) le 119902120577where 119902 = 02785Lemma 6 Let 119886 isin 119877119899times1 119887 isin 119877119899times1 and 119862 isin 119877119899times119899 then it canbe obtained that 119886T119862119887 = tr119862119887119886T where tr(sdot) represents thetrace operation
3 Main Results
In this section the tracking problem of nonlinear multiagentsystems with input saturation is discussed Based on theneural network approximation technique and the iterativelearning control approach the distributed adaptive controlprotocol and the adaptive updating laws are presentedrespectively Then the convergence of proposed controlprotocol is illustrated by a designed Lyapunov function
31 Problem Formulation Consider a class of leader-following second-order nonlinear multiagent systems withthe external disturbance and input saturation the dynamicsof the 119894th following agent at 119896th iteration are described asfollows
where 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 and 119906119896119894 (119905) isin 119877 are theposition velocity and control input of the 119894th following
agent respectively 119891(119909119896119894 (119905) V119896119894 (119905)) represents the unknownnonlinear function 119889119896119894 (119905) is unknown but bounded externaldisturbance that is there exists 119889119896119894 (119905) le 119889lowast119894 with 119889lowast119894 beingan unknown positive constant and 119896 denotes the iterationnumber and 119905 isin [0 119879] sat(119906119896119894 (119905)) is the saturation functionwhich is defined as
sat (119906119896119894 (119905)) =119906 119906119896119894 (119905) gt 119906119906119896119894 (119905) minus119906 le 119906119896119894 (119905) le 119906minus119906 119906119896119894 (119905) lt minus119906
(5)
where 119906 gt 0 is the upper bound of saturation function andprespecified
where 1199090(119905) isin 119877 V0(119905) isin 119877 and 1199060(119905) isin 119877 are the positionvelocity and input of leader respectively and 119891(1199090(119905) V0(119905))represents the unknown nonlinear function Referring to theliterature [40] it is also assumed that the control input ofleader is nonzero but bounded that is there exists 1199060(119905) le119906lowast0 with 119906lowast0 being a positive constant
According to the multiagent systems (4) and (7) thetracking errors of position and velocity are defined as
119890119896119909119894 (119905) = 1199090 (119905) minus 119909119896119894 (119905) (8)
119890119896V119894 (119905) = V0 (119905) minus V119896119894 (119905) (9)
Let 119890119896119909(119905) = [1198901198961199091(119905) sdot sdot sdot 119890119896119909119899(119905)]T and 119890119896V(119905) = [119890119896V1(119905) sdot sdot sdot 119890119896V119899(119905)]T then119890119896119909 (119905) = 11198991199090 (119905) minus 119909119896 (119905) (10)
119890119896V (119905) = 1119899V0 (119905) minus V119896 (119905) (11)
where 1119899 = [1 sdot sdot sdot 1]TAssumption 7 Theunknown nonlinear item 119891(1199090(119905) V0(119905)) isbounded namely there exists 119891(1199090(119905) V0(119905)) le 119891lowast0 where119891lowast0 is an unknown constant
Assumption 8 The alignment initial conditions that is119909119896119894 (0) = 119909119896minus1119894 (119879) and V119896119894 (0) = V119896minus1119894 (119879) for each following agentare satisfied Also it is assumed that the trajectory of leader isspatially closed that is 1199090(0) = 1199090(119879) and V0(0) = V0(119879)
4 Complexity
According to Assumption 8 hence it can be gotten that119890119896119909119894(0) = 119890119896minus1119909119894 (119879) and 119890119896V119894(0) = 119890119896minus1V119894 (119879) for each following agentDefinition 9 For any initial condition the consensus track-ing problem of leader-following second-order nonlinearmultiagent systems with input saturation is achieved iflim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 =1 sdot sdot sdot 119899 over the interval [0 119879] are satisfied
The control objective of this paper is to design theappropriate control scheme 119906119896119894 (119905) for 119894 = 1 sdot sdot sdot 119899 and theadaptive updating laws such that the states of all the followingagents can track the trajectory of leader over the interval[0 119879] as the iteration number 119896 tends to infinity32 Control Protocol Design According to the multiagentsystems (4) and (7) the consensus tracking errors are definedas
119886119894119895 (V119896119895 (119905) minus V119896119894 (119905)) + 119887119894 (V0 (119905) minus V119896119894 (119905)) (13)
And directly from (12) and (13) we get
120576119896119909 (119905) = H (11198991199090 (119905) minus 119909119896 (119905)) = H119890119896119909 (119905) (14)
120576119896V (119905) = H (1119899V0 (119905) minus V119896 (119905)) = H119890119896V (119905) (15)
where 120576119896119909(119905) = [1205761198961199091(119905) sdot sdot sdot 120576119896119909119899(119905)]T and 120576119896V (119905) =[120576119896V1(119905) sdot sdot sdot 120576119896V119899(119905)]TRemark 10 In this paper we only discuss the states of eachagent as 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 1199090(119905) isin 119877 and V0(119905) isin 119877 Forthe case of 119909119896119894 (119905) isin 119877119901 V119896119894 (119905) isin 119877119901 1199090(119905) isin 119877119901 and V0(119905) isin 119877119901we have
where otimes is the Kronecker product 119868119901 is the unit matrix with119901 dimension and all of the related results can be changed byapplying the Kronecker product operation
Considering 120576119896119909(119905) and 120576119896V (119905) a sliding mode function isdesigned as
where 120572 gt 0 is a positive constant and 119904119896(119905) =[1199041198961(119905) sdot sdot sdot 119904119896119899(119905)]TSo the derivative of 119904119896(119905) is
119904119896 (119905) = 120576119896V (119905) + 120572 120576119896119909 (119905) = H (1119899V0 (119905) minus V119896 (119905))+ 120572H119890119896V (119905) = H (1119899119891 (1199090 (119905) V0 (119905))minus 119891 (119909119896 (119905) V119896 (119905)) + 11198991199060 (119905) minus sat (119906119896 (119905))minus 119889119896 (119905)) + 120572H119890119896V (119905)
(19)
For the unknown nonlinear parts 119891(119909119896(119905) V119896(119905)) weintroduce the RBF neural network to approximate them Inview of the approximation properties of RBF neural network119891(119909119896119894 (119905) V119896119894 (119905)) can be described as
where 120573 gt 0 120599119894 gt 0 and 120574119894 gt 0 are constants to bedesigned and 120575119906119896(119905) = 119906119896(119905) minus sat(119906119896(119905)) with 119906119896(119905) =[1199061198961(119905) sdot sdot sdot 119906119896119899(119905)]T and 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]T
The vector form of control protocol (24) can be written as
where 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]Tand 120592119896(119905) = [1205921198961(119905) sdot sdot sdot 120592119896119899(119905)]T with 120592119896119894 (119905) = 120578119896119894 (119905) tanh(120578119896119894 (119905)119904119896119894 (119905)Δ 119896)Remark 11 In the control protocol (24) the time-varyingparameters 120590119896119894 (119905) and 120578119896119894 (119905) are introduced The purposeof designing 120590119896119894 (119905) is to compensate the saturation error120575119906119896119894 (119905) and the purpose of designing 120578119896119894 (119905) is to eliminate theinfluence of approximation error 119900(119909119896119894 (119905) V119896119894 (119905)) and externaldisturbance 119889119896119894 (119905) In other words the objective of designingadaptive updating laws is to seek the distributed adaptive iter-ative learning control protocol for time-varying parameterssuch that the tracking problem can be solved over the interval[0 119879]33 Convergence Analysis In what follows the main result ofthis paper is given inTheorem 12
Theorem 12 Consider the leader-following second-order non-linear multiagent systems with input saturation (4) and (7)and suppose that Assumptions 7 and 8 are held and thecommunication topology G is connected Let the distributedadaptive neural network iterative learning control protocol (24)and the adaptive updating laws (25) (26) and (27) be appliedthen all the following agents can track the trajectory of leadernamely lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for119894 = 1 sdot sdot sdot 119899 over the interval [0 119879]
Proof Design the following Lyapunov function candidate
where 119896(119905) = 119882lowast minus 119896(119905) Γ = diag1205741119868119871 sdot sdot sdot 120574119899119868119871 and1205780 gt 0 is a constant to be determined laterConsider the difference between 119881119896(119905) and 119881119896minus1(119905) that
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
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4 Complexity
According to Assumption 8 hence it can be gotten that119890119896119909119894(0) = 119890119896minus1119909119894 (119879) and 119890119896V119894(0) = 119890119896minus1V119894 (119879) for each following agentDefinition 9 For any initial condition the consensus track-ing problem of leader-following second-order nonlinearmultiagent systems with input saturation is achieved iflim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 =1 sdot sdot sdot 119899 over the interval [0 119879] are satisfied
The control objective of this paper is to design theappropriate control scheme 119906119896119894 (119905) for 119894 = 1 sdot sdot sdot 119899 and theadaptive updating laws such that the states of all the followingagents can track the trajectory of leader over the interval[0 119879] as the iteration number 119896 tends to infinity32 Control Protocol Design According to the multiagentsystems (4) and (7) the consensus tracking errors are definedas
119886119894119895 (V119896119895 (119905) minus V119896119894 (119905)) + 119887119894 (V0 (119905) minus V119896119894 (119905)) (13)
And directly from (12) and (13) we get
120576119896119909 (119905) = H (11198991199090 (119905) minus 119909119896 (119905)) = H119890119896119909 (119905) (14)
120576119896V (119905) = H (1119899V0 (119905) minus V119896 (119905)) = H119890119896V (119905) (15)
where 120576119896119909(119905) = [1205761198961199091(119905) sdot sdot sdot 120576119896119909119899(119905)]T and 120576119896V (119905) =[120576119896V1(119905) sdot sdot sdot 120576119896V119899(119905)]TRemark 10 In this paper we only discuss the states of eachagent as 119909119896119894 (119905) isin 119877 V119896119894 (119905) isin 119877 1199090(119905) isin 119877 and V0(119905) isin 119877 Forthe case of 119909119896119894 (119905) isin 119877119901 V119896119894 (119905) isin 119877119901 1199090(119905) isin 119877119901 and V0(119905) isin 119877119901we have
where otimes is the Kronecker product 119868119901 is the unit matrix with119901 dimension and all of the related results can be changed byapplying the Kronecker product operation
Considering 120576119896119909(119905) and 120576119896V (119905) a sliding mode function isdesigned as
where 120572 gt 0 is a positive constant and 119904119896(119905) =[1199041198961(119905) sdot sdot sdot 119904119896119899(119905)]TSo the derivative of 119904119896(119905) is
119904119896 (119905) = 120576119896V (119905) + 120572 120576119896119909 (119905) = H (1119899V0 (119905) minus V119896 (119905))+ 120572H119890119896V (119905) = H (1119899119891 (1199090 (119905) V0 (119905))minus 119891 (119909119896 (119905) V119896 (119905)) + 11198991199060 (119905) minus sat (119906119896 (119905))minus 119889119896 (119905)) + 120572H119890119896V (119905)
(19)
For the unknown nonlinear parts 119891(119909119896(119905) V119896(119905)) weintroduce the RBF neural network to approximate them Inview of the approximation properties of RBF neural network119891(119909119896119894 (119905) V119896119894 (119905)) can be described as
where 120573 gt 0 120599119894 gt 0 and 120574119894 gt 0 are constants to bedesigned and 120575119906119896(119905) = 119906119896(119905) minus sat(119906119896(119905)) with 119906119896(119905) =[1199061198961(119905) sdot sdot sdot 119906119896119899(119905)]T and 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]T
The vector form of control protocol (24) can be written as
where 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]Tand 120592119896(119905) = [1205921198961(119905) sdot sdot sdot 120592119896119899(119905)]T with 120592119896119894 (119905) = 120578119896119894 (119905) tanh(120578119896119894 (119905)119904119896119894 (119905)Δ 119896)Remark 11 In the control protocol (24) the time-varyingparameters 120590119896119894 (119905) and 120578119896119894 (119905) are introduced The purposeof designing 120590119896119894 (119905) is to compensate the saturation error120575119906119896119894 (119905) and the purpose of designing 120578119896119894 (119905) is to eliminate theinfluence of approximation error 119900(119909119896119894 (119905) V119896119894 (119905)) and externaldisturbance 119889119896119894 (119905) In other words the objective of designingadaptive updating laws is to seek the distributed adaptive iter-ative learning control protocol for time-varying parameterssuch that the tracking problem can be solved over the interval[0 119879]33 Convergence Analysis In what follows the main result ofthis paper is given inTheorem 12
Theorem 12 Consider the leader-following second-order non-linear multiagent systems with input saturation (4) and (7)and suppose that Assumptions 7 and 8 are held and thecommunication topology G is connected Let the distributedadaptive neural network iterative learning control protocol (24)and the adaptive updating laws (25) (26) and (27) be appliedthen all the following agents can track the trajectory of leadernamely lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for119894 = 1 sdot sdot sdot 119899 over the interval [0 119879]
Proof Design the following Lyapunov function candidate
where 119896(119905) = 119882lowast minus 119896(119905) Γ = diag1205741119868119871 sdot sdot sdot 120574119899119868119871 and1205780 gt 0 is a constant to be determined laterConsider the difference between 119881119896(119905) and 119881119896minus1(119905) that
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 120573 gt 0 120599119894 gt 0 and 120574119894 gt 0 are constants to bedesigned and 120575119906119896(119905) = 119906119896(119905) minus sat(119906119896(119905)) with 119906119896(119905) =[1199061198961(119905) sdot sdot sdot 119906119896119899(119905)]T and 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]T
The vector form of control protocol (24) can be written as
where 120590119896(119905) = [1205901198961 (119905) sdot sdot sdot 120590119896119899(119905)]Tand 120592119896(119905) = [1205921198961(119905) sdot sdot sdot 120592119896119899(119905)]T with 120592119896119894 (119905) = 120578119896119894 (119905) tanh(120578119896119894 (119905)119904119896119894 (119905)Δ 119896)Remark 11 In the control protocol (24) the time-varyingparameters 120590119896119894 (119905) and 120578119896119894 (119905) are introduced The purposeof designing 120590119896119894 (119905) is to compensate the saturation error120575119906119896119894 (119905) and the purpose of designing 120578119896119894 (119905) is to eliminate theinfluence of approximation error 119900(119909119896119894 (119905) V119896119894 (119905)) and externaldisturbance 119889119896119894 (119905) In other words the objective of designingadaptive updating laws is to seek the distributed adaptive iter-ative learning control protocol for time-varying parameterssuch that the tracking problem can be solved over the interval[0 119879]33 Convergence Analysis In what follows the main result ofthis paper is given inTheorem 12
Theorem 12 Consider the leader-following second-order non-linear multiagent systems with input saturation (4) and (7)and suppose that Assumptions 7 and 8 are held and thecommunication topology G is connected Let the distributedadaptive neural network iterative learning control protocol (24)and the adaptive updating laws (25) (26) and (27) be appliedthen all the following agents can track the trajectory of leadernamely lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) and lim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for119894 = 1 sdot sdot sdot 119899 over the interval [0 119879]
Proof Design the following Lyapunov function candidate
where 119896(119905) = 119882lowast minus 119896(119905) Γ = diag1205741119868119871 sdot sdot sdot 120574119899119868119871 and1205780 gt 0 is a constant to be determined laterConsider the difference between 119881119896(119905) and 119881119896minus1(119905) that
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 119882119896(119905) = lowast minus 119882119896(119905) = minus 119882119896(119905) is considered in (40)and the adaptive updating laws 119896(119905) 120578119896119894 (119905) and 119882119896(119905) areapplied
Substituting (38)-(41) into (30) it can be obtained that
where 119884119896(119905) = [(119904119896(119905))T (119890119896V(119905))T]T and Φ = (12)[(2120573 minus1)119868119899 minus120572119868119899 minus120572119868119899 119868119899]It is clear thatΦ is the positive-definite matrix if it satisfies120573 gt (1205722 + 1)2 In addition we have 120578119896119894 (119905) gt 0 from the
adaptive updating law (26) then it can be obtained fromLemma 5 that119899sum119894=1
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
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
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Complexity
Considering Assumption 8 we have 120576119896119909(0) = 120576119896minus1119909 (119879)and 120576119896V (0) = 120576119896minus1V (119879) then 119904119896(0) = 119904119896minus1(119879) is easilyobtained Moreover we have 120590119896119894 (0) = 120590119896minus1119894 (119879) from (25)120578119896119894 (0) = 120578119896minus1119894 (119879) from (26) and 119896119894 (0) = 119896minus1119894 (119879) from (27)Consequently we get from (47)
Δ 2 le 1198811 (119879) + 2119899119879119902119888(50)
Obviously it can be derived that the boundedness of119881119896(119879) is guaranteed for any iteration provided 1198811(119879) isbounded In the Appendix the boundedness of 1198811(119905) isproved
The boundedness of 1198811(119905) indicates the boundedness of1198811(119879) Hence 119881119896(119879) is bounded from (50) for all 119896 isin 119885+From (48) it is gotten that 119881119896(119905) is uniformly bounded overthe interval [0 119879]
According to (50) we have
119881119896 (119879) le 1198811 (119879) minus 120582min (Φ) 119896sum119895=2
Owing to the boundedness of 1198811(119879) and the pos-itiveness of 119881119896(119879) we obtain that the series
1 0
4 53
2
Figure 1 Communication topology
sum119896119895=2 int1198790 (119884119895(120591))T119884119895(120591)119889120591 is convergent Furthermore it
is easy to get that lim119896997888rarrinfin int1198790(119904119896(120591))T119904119896(120591)119889120591 = 0 and
lim119896997888rarrinfin int1198790(119890119896V(120591))T119890119896V(120591)119889120591 = 0 According to (14) and
(18) we have lim119896997888rarrinfin int1198790(119890119896119909(120591))T119890119896119909(120591)119889120591 = 0 Consider the
Barbalat-like Lemma [41] we obtain lim119896997888rarrinfin119890119896119909(119905) = 0 andlim119896997888rarrinfin119890119896V (119905) = 0 uniformly over the interval [0 119879] Thenit follows from (10) and (11) that lim119896997888rarrinfin119909119896119894 (119905) = 1199090(119905) andlim119896997888rarrinfinV
119896119894 (119905) = V0(119905) for 119894 = 1 sdot sdot sdot 119899 which implies that all
the following agents can track the leader uniformly over theinterval [0 119879] The proof is completed
4 Simulation Analysis
In this section a numerical example is provided to check thevalidity of the proposed distributed adaptive neural networkiterative learning control protocol (24) The undirected com-munication topology consists of five following agents and oneleader agent (labelled as 0) is given in Figure 1
The weighted adjacency matrices from Figure 1 are
A =[[[[[[[[[
0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 01 0 0 0 0
]]]]]]]]]
B = diag 1 0 0 0 0
(52)
The dynamics of five following agents are described as
119896119894 (119905) = V119896119894 (119905)V119896119894 (119905) = 119909119896119894 (119905) cos (V119896119894 (119905)) + sat (119906119896119894 (119905)) + 119889119896119894 (119905)
119894 = 1 2 3 4 5(53)
The disturbance of the 119894th following agent is 119889119896119894 (119905) =1199111sin(1199081119905)+1199112sin(1199082119905) where 119911119894 and119908119894 (119894 = 1 2) are arbitraryreal numbers 119911119894 isin [0 1] and 119908119894 isin [1 2]
The dynamics of leader are given as
0 (119905) = V0 (119905)V0 (119905) = (1199090 (119905))2 sin (V0 (119905)) minus cos (2120587119905) (54)
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
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
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 9
0 02 04 06 08 1 12 14 16 18 2
004
006
008
01
Time
x0 x1 x2 x3 x4 x5
0 02 04 06 08 1 12 14 16 18 2minus02minus01
00102
Time
v0 v1 v2 v3 v4 v5
Ｒ 0Ｒ
ＣＰ 0
ＰＣ
Figure 2 Tracking results of position and velocity
0 5 10 15 20 25 30 35 40 45 500
05
1
Iteration number
0 5 10 15 20 25 30 35 40 45 500
2
4
6
Iteration number
Ｒ1
Ｒ2
Ｒ3
Ｒ4
Ｒ5
Ｐ1
Ｐ2
Ｐ3
Ｐ4
Ｐ5
Nor
m o
fＲＣ
Nor
m o
fＰＣ
Figure 3 Error norms of position and velocity
The initial states of five following agents and the leaderare set as 119909(0) = [minus05 05 09 03 minus02]T V(0) =[02 06 minus04 minus08 10]T 1199090(0) = 01 and V0(0) = 0 Thesimulation time 119905 isin [0 2] and the iterationnumber 119896max = 50
The RBF neural network for 119891(119909119896119894 (119905) V119896119894 (119905))) contains 7nodes with the centers 120585119894 evenly spaced in the range [minus3 3]and the widths 120579119894 = 20 for 119894 = 1 sdot sdot sdot 5 The initial valuesof 120590119894(0) 120578119894(0) and 119882119894(0) are 1205901(0) = 01 1205902(0) = 0051205903(0) = 015 1205904(0) = 01 and 1205905(0) = 005 1205781(0) = 051205782(0) = 15 1205783(0) = 20 1205784(0) = 15 1205785(0) = 05 and119882119894(0) = [1 1 1 1 1 1 1]T (119894 = 1 sdot sdot sdot 5) Other parameters
By applying the control protocol (24) and the adaptiveupdating laws (25)-(27) the simulation results for 50 itera-tions are shown in Figures 2 3 4 5 6 and 7
The tracking results of five following agents at the 50thiteration are shown in Figure 2 which implies that the con-sensus tracking problem of leader-following second-ordernonlinear multiagent systems with input saturation can besolved by adopting the proposed control protocol (24) Due
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
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
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Complexity
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
4
45
5
55
Iteration number
Nor
m o
f sat
urat
ed in
put u
i
ＭＮ(Ｏ1)
ＭＮ(Ｏ2)
ＭＮ(Ｏ3)
ＭＮ(Ｏ4)
ＭＮ(Ｏ5)
Figure 4 Norm of saturated inputs
0 5 10 15 20 25 30 35 40 45 50
07
08
09
1
11
12
13
14
15
Iteration number
1
2
3
4
5
Nor
m o
fＣ
Figure 5 Response of 120590119896119894 (119905)to the application of alignment initial condition the finaltrajectories of five following agents can be synchronized withthe leader The error curves of position and velocity at 50iterations are shown in Figure 3
Figure 4 gives the saturated input results at 50 iterationsAlthough the control inputs are constrained the tracking
0 5 10 15 20 25 30 35 40 45 500
05
1
15
2
25
3
35
Iteration number
Nor
m o
fＣ
1
2
3
4
5
Figure 6 Response of 120578119896119894 (119905)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Iteration number
Nor
m o
f esti
mat
ed W
i
Norm of estimated W1
Norm of estimated W2
Norm of estimated W3
Norm of estimated W4
Norm of estimated W5
Figure 7 Response of estimated 119896119894 (119905)
problem with the designed distributed adaptive neural net-work iterative learning control protocol can be achievedvery well It means that the proposed control protocol iseffective from another perspective In addition the responses
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
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
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 11
of adaptive updating laws 120590119896119894 (119905) 120578119896119894 (119905) and 119896119894 (119905) at 50iterations are given in Figures 5 6 and 7 respectively
5 Conclusions
In this paper the consensus tracking problem of the leader-following nonlinear multiagent systems was addressed TheRBF neural network was adopted to approximate theunknown nonlinear terms of all following agents The dis-tributed adaptive neural network iterative learning controlprotocol was designed and the adaptive updating laws fortime-varying parameters were proposed respectively Thenthe convergence of proposed control protocol was analyzedby a designed Lyapunov function It was proved that whenthere exists the input saturation the tracking control problemwas solved under the designed control protocol Finally forthe validity of the theoretical analysis a simulation examplewas verified by the simulation example
Appendix
The Proof of the Boundedness of 1198811(119905)From the definition of 119881119896(119905) we have
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Thus the boundedness of 1198811(119905) is obtained The proof iscompleted
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Z Liang J Y Wang and Z W Sun ldquoRobust decentralizedcoordinated attitude control of spacecraft formationrdquo ActaAstronautica vol 69 no 5-6 pp 280ndash288 2011
[2] J Wu S Yuan S Ji G Zhou Y Wang and Z Wang ldquoMulti-agent system design and evaluation for collaborative wirelesssensor network in large structure health monitoringrdquo ExpertSystems with Applications vol 37 no 3 pp 2028ndash2036 2010
[3] M da Rosa A Leite da Silva and V Miranda ldquoMulti-agentsystems applied to reliability assessment of power systemsrdquogtInternational Journal of Electrical Power amp Energy Systems vol42 no 1 pp 367ndash374 2012
[4] A Sakaguchi and T Ushio ldquoDynamic pinning consensuscontrol of multi-agent systemsrdquo IEEE Control Systems Lettersvol 1 no 2 pp 340ndash345 2017
[5] Y Xie and Z Lin ldquoGlobal optimal consensus for multi-agentsystems with bounded controlsrdquo Systems amp Control Letters vol102 pp 104ndash111 2017
[6] W Guo ldquoLeader-following consensus of the second-ordermulti-agent systems under directed topologyrdquo ISA Transac-tions vol 65 pp 116ndash124 2016
[7] C Wang X Wang and H Ji ldquoLeader-following consensus fora class of second-order nonlinear multi-agent systemsrdquo Systemsamp Control Letters vol 89 pp 61ndash65 2016
[8] X Jin ldquoFault tolerant finite-time leader-follower formation con-trol for autonomous surface vessels with LOS range and angleconstraintsrdquo Automatica vol 68 pp 228ndash236 2016
[9] YGao B Liu J Yu JMa andT Jiang ldquoConsensus of first-ordermulti-agent systems with intermittent interactionrdquo Neurocom-puting vol 129 pp 273ndash278 2014
[10] J Feng and G-X Wen ldquoAdaptive NN consensus trackingcontrol of a class of nonlinear multi-agent systemsrdquo Neurocom-puting vol 151 no 1 pp 288ndash295 2015
[11] G Hu ldquoRobust consensus tracking of a class of second-ordermulti-agent dynamic systemsrdquo Systems amp Control Letters vol61 no 1 pp 134ndash142 2012
[13] L Han X Dong Q Li and Z Ren ldquoFormation tracking con-trol for time-delayed multi-agent systems with second-orderdynamicsrdquo Chinese Journal of Aeronautics vol 30 no 1 pp348ndash357 2017
[14] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012
[15] F Wang X Chen Y He and M Wu ldquoFinite-time consensusproblem for second-ordermulti-agent systems under switchingtopologiesrdquo Asian Journal of Control vol 19 no 5 pp 1756ndash1766 2017
[16] M H Rezaei and M B Menhaj ldquoStationary average consensusfor high-order multi-agent systemsrdquo IET Control Theory ampApplications vol 11 no 5 pp 723ndash731 2017
[17] C Sun G Hu and L Xie ldquoRobust consensus tracking for aclass of high-order multi-agent systemsrdquo International Journalof Robust and Nonlinear Control vol 26 no 3 pp 578ndash5982016
[18] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[19] H Yang F Wang and F Han ldquoContainment control of frac-tional order multi-agent systems with time delaysrdquo IEEECAAJournal of Automatica Sinica vol 5 no 3 pp 727ndash732 2018
[20] X Yi T Yang J Wu and K Johansson ldquoDistributed event-triggered control for global consensus of multi-agent systemswith input saturationrdquo Automatica vol 100 pp 1ndash9 2019
[21] Y Li J Xiang andWWei ldquoConsensus problems for linear time-invariant multi-agent systems with saturation constraintsrdquo IETControl Theory amp Applications vol 5 no 6 pp 823ndash829 2011
[22] J Yan X-P Guan X-Y Luo and X Yang ldquoConsensus andtrajectory planning with input constraints for multi-agent sys-temsrdquoActaAutomatica Sinica vol 38 no 7 pp 1074ndash1082 2012
[23] J Lyu J Qin D Gao and Q Liu ldquoConsensus for constrainedmulti-agent systems with input saturationrdquo International Jour-nal of Robust and Nonlinear Control vol 26 no 14 pp 2977ndash2993 2016
[24] B Zhang Y Jia and F Matsuno ldquoFinite-time observers formulti-agent systems without velocity measurements and withinput saturationsrdquo Systems amp Control Letters vol 68 no 1 pp86ndash94 2014
[25] H S Su M Z Q Chen and G R Chen ldquoRobust semi-global coordinated tracking of linear multi-agent systems withinput saturationrdquo International Journal of Robust and NonlinearControl vol 25 no 14 pp 2375ndash2390 2015
[26] D A Bristow M Tharayil and A G Alleyne ldquoA survey ofiterative learning control a learning-based method for high-performance tracking controlrdquo IEEE Control SystemsMagazinevol 26 no 3 pp 96ndash114 2006
[27] X F Deng X X Sun and R Liu ldquoQuantized consensus controlfor second-order nonlinear multi-agent systems with slidingmode iterative learning approachrdquo International Journal ofAeronautical and Space Sciences vol 19 no 2 pp 518ndash533 2018
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
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
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 13
[28] X Deng X Sun R Liu and S Liu ldquoConsensus controlof leader-following nonlinear multi-agent systems with dis-tributed adaptive iterative learning controlrdquo International Jour-nal of Systems Science vol 49 no 16 pp 3247ndash3260 2018
[29] S-P Yang J-X Xu D-QHuang and Y Tan ldquoOptimal iterativelearning control design for multi-agent systems consensustrackingrdquo Systems amp Control Letters vol 69 pp 80ndash89 2014
[30] D Meng and Y Jia ldquoFormation control for multi-agent systemsthrough an iterative learning design approachrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 2 pp 340ndash361 2014
[31] X Jin ldquoNonrepetitive leaderndashfollower formation tracking formultiagent systems with LOS range and angle constraints usingiterative learning controlrdquo IEEE Transactions on Cyberneticsvol 49 no 5 pp 1748ndash1758 2019
[32] X Jin ldquoAdaptive iterative learning control for high-order non-linear multi-agent systems consensus trackingrdquo Systems ampControl Letters vol 89 pp 16ndash23 2016
[33] J Li and J Li ldquoIterative learning control approach for a kind ofheterogeneousmulti-agent systems with distributed initial statelearningrdquo Applied Mathematics and Computation vol 265 pp1044ndash1057 2015
[34] T Meng and W He ldquoIterative Learning Control of a RoboticArm Experiment Platform with Input Constraintrdquo IEEE Trans-actions on Industrial Electronics vol 65 no 1 pp 664ndash672 2018
[35] X Jin ldquoFault-tolerant iterative learning control for mobilerobots non-repetitive trajectory tracking with output con-straintsrdquo Automatica vol 94 pp 63ndash71 2018
[36] M Lv Y Wang S Baldi Z Liu and Z Wang ldquoA DSC methodfor strict-feedback nonlinear systems with possibly unboundedcontrol gain functionsrdquo Neurocomputing vol 275 pp 1383ndash1392 2018
[37] Y Hong J Hu and L Gao ldquoTracking control for multi-agent consensus with an active leader and variable topologyrdquoAutomatica vol 42 no 7 pp 1177ndash1182 2006
[38] S Zhu M X Sun and X X He ldquoIterative learning control ofstrict-feedback nonlinear time-varying systemsrdquoActaAutomat-ica Sinica vol 36 no 3 pp 454ndash458 2010
[39] M M Polycarpou and P A Ioannou ldquoA robust adaptivenonlinear control designrdquo Automatica vol 32 no 3 pp 423ndash427 1996
[40] Z Li X Liu W Ren and L Xie ldquoDistributed tracking con-trol for linear multiagent systems with a leader of boundedunknown inputrdquo IEEE Transactions on Automatic Control vol58 no 2 pp 518ndash523 2013
[41] M Sun ldquoA Barbalat-like lemma with its application to learningcontrolrdquo IEEE Transactions on Automatic Control vol 54 no 9pp 2222ndash2225 2009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![MANAGING DESIGN PROCESS COMPLEXITY: A VALUE-OF-INFORMATION … · analysis tasks that generate information for decisions [10-13]. Hence, the complexity of design processes is a result](https://static.documents.pub/doc/80x56/5f47c7e47bf58702224c72cf/managing-design-process-complexity-a-value-of-information-analysis-tasks-that-generate.jpg)
