# Performance Index Based Observer-Type Iterative Learning Control for Consensus Tracking of Uncertain Nonlinear Fractional-Order Multiagent Systems

Authors: Liming Wang; Guoshan Zhang
Journal: Complexity
 (2019)

---

## Abstract

This paper is devoted to the perfect tracking problem of consensus and the monotonic convergence problem of input errors for the uncertain nonlinear fractional-order multiagent systems (FOMASs), where there exist the linear coupling relations between the fractional order, the perturbations of the system matrix, and input matrix. For the FOMASs including one leader agent and multiple follower agents, an observer-type fractional-order iterative learning consensus protocol is proposed. Based on the two-dimensional analysis for the FOMASs, a novel performance index, which can exhibit the monotonic convergence of the input errors, is constructed by using the definition of fractional integral. A Lyapunov-like method is applied to derive the sufficient conditions in terms of linear matrix inequalities, which can guarantee the perfect tracking of consensus and the monotonic convergence of input errors. Finally, the numerical simulation results including comparisons to traditional two-dimensional analysis are presented to demonstrate the effectiveness of the proposed methods.


---

## Body

## 1. Introduction

Over the past decades, due to the potential applications of the consensus for multiagent systems (MASs), their studies have attracted more and more attention [1–6]. Among most of the existing results, the MASs are assumed to have the integer-order dynamics, such as first-order dynamics [7, 8], second-order dynamics [9, 10], and higher-order dynamics [11, 12]. Recently, it has been found that many practical coordinate behaviors of agents in the complex environment, including vehicles moving on the top of macromolecule fluids and porous media [13] and aircrafts traveling at the high speeds in dust storm, rain/snow environment [14], often exhibit the fractional-order dynamics. Since the fractional-order multiagent systems (FOMASs) include the integer-order multiagent systems (IOMASs) as their special cases, it is significant to consider the consensus problem of FOMASs. The consensus problem of FOMASs has been widely investigated from various perspectives, such as heterogeneities [15], uncertainties [16] and information transmission delays [17] or information processing delays [18]. Furthermore, some control methods, including sliding mode control [19], event-triggered control [20], and impulsive control [21], are also developed for the consensus of FOMASs.In the above literatures of FOMASs, the consensus is achieved in the one-dimensional system framework evolving along the time axis, and these studies are not suitable for arbitrary high precision tracking tasks of the FOMASs. The iterative learning control (ILC) method can create a two-dimensional process with time axis and iteration axis as two independent directions [22]. Thus, by iteratively adjusting the input signal from one iteration to the next, the ILC method can achieve the perfect consensus. Here the perfect consensus refers to the phenomena that the states or outputs of all follower agents converge to that of the leader agent in a finite time interval as the iteration step goes to infinity. The ILC method has been used for the perfect consensus of the IOMASs. [23, 24] have proposed the ILC protocols for the perfect consensus and the formation control of IOMASs, respectively. It is indicated in [25] that the iterative learning convergence rate of consensus can be enhanced by input sharing among agents with integer-order dynamics. Further, the perfect consensus problem of IOMASs with output saturation is solved by a distributed ILC algorithm [26]. Recently, to obtain the better performance of perfect consensus, the ILC scheme is combined with event-triggered control method [27].In [23–27], the time-weighted norm based contraction mapping (CM) method is used to analyze the convergence of learning process. The method ignores the dynamics of agents and cannot derive the consensus convergence conditions guaranteeing the good transients of learning process. On the other hand, the bad learning transients, especially unacceptable overshoots, will be the obstacles to applying the ILC method into the practical problems, and should try to be overcome. The composite energy function (CEF) method is one of the remedies for the bad transient phenomena, because the CEF method is defined in L2 norm and can lead to point-wise convergence [28]. There have existed some instructive results of the CEF method in the perfect consensus problem for the linear IOMASs, including first-order agents [29], second-order agents [30] and high-order agents [31]. The similar methods are applied to the perfect consensus problems of the nonlinear IOMASs with unknown control direction [32]. In [33], the CEF method is used to design the ILC protocol for the perfect consensus of continuous and discrete IOMASs.The above-mentioned literatures about the ILC consensus assume that the states of agents are measurable, while this is not always true in practice. There have been some literatures about the observer-based perfect consensus of IOMASs [34, 35] and the observer-based asymptotic consensus of the FOMASs [36–39]. It should be mentioned that the models in [34–39] are free from the perturbations, and the constructed observers are applicable to the systems without any uncertainty. Uncertainties are inevitable in the real physical systems, thus it should pay attention to the observer-based control for the consensus of uncertain FOMASs. There have been a few literatures about the observer-based control for the uncertain systems merely including one node [40–44]. From the perspectives of relations between the parameters of observers and those of corresponding systems, the existing results about observer-based control for the uncertain system can be generally classified into the following two cases. In the first case (see [40, 41]), the observers are constructed by using the parameters of nominal model of the corresponding systems, and the constructed observers are systems without any uncertainty. In the second case (see [42–44]), the time-varying structural uncertainties are assumed to be measurable, and the observers are constructed by using the uncertain time-varying system matrices same as the corresponding systems. Thus the constructed observers in fact are the time-varying uncertain systems. To the best of authors’ knowledge, little work has been done about the observer-based control for consensus of uncertain MASs, especially uncertain FOMASs. Due to the invalidation of the well-known Leibniz rule in the fractional derivatives and the particularities of the perfect consensus problem, it is still a great challenge to design an observer-type ILC protocol for the good transients of learning process and the perfect tracking of consensus for the FOMASs.Inspired by the above work, in this paper, we investigate the consensus perfect tracking problem with good transients for uncertain nonlinear FOMASs, where there exist the linear coupling relations among the fractional order, the perturbations of the system matrix and input matrix. Firstly, the perfect tracking problem of consensus and the monotonic convergence problem of input errors are formulated and are transformed into the control problem of the tracking error system, and a performance index reflecting the monotonic convergence of the input errors is constructed. Secondly, an observer-type fractional-order ILC consensus protocol is designed, and the consensus sufficient conditions are derived for the nonlinear FOMASs without any uncertainty. Then, the method is further developed for the consensus of the uncertain nonlinear FOMASs, and the related consensus criterions are derived. Finally, two numerical examples are presented to validate the proposed methods. The main contributions of this paper can be summarized as follows:(1) Investigate the perfect tracking problem of consensus and the monotonic convergence problem of ILC law for a class of general models of uncertain nonlinear FOMASs with the parameter coupling. The proposed models can include the following existing models as their special cases: (a) linear IOMASs with (or without) uncertainties and parameter coupling [34, 35]; (b) nonlinear IOMASs with (or without) uncertainties and parameter coupling [25]; (c) linear FOMAS with (or without) uncertainties and parameter coupling [16, 20]. (d) Nonlinear FOMASs with (or without) uncertainties and parameter coupling [21]. In the existing studies about FOMASs, the asymptotic consensus or the finite time consensus has been considered, while there has not been any report about the perfect consensus of FOMASs. Moreover, in the existing literatures about the perfect consensus of IOMASs, the monotonic convergence problem of the learning law is seldom considered.(2) Based on the observed information of agents’ neighbors, an observer-type fractional-order ILC consensus protocol is designed. Based on the definition of the fractional integral, a novel performance index, which can reflect the monotonic convergence of the input errors, is constructed after the two-dimensional (2D) analysis for the nonlinear FOMASs. The stability and the monotonic convergence of iterative learning process are analyzed by a Lyapunov-like method, thereby the sufficient conditions are derived for the perfect consensus of the nonlinear FOMASs with or without uncertainties, respectively.(3) By solving the linear matrix inequalities (LMIs) in the sufficient conditions, the appropriate learning gain matrices are obtained directly. Under the proposed consensus protocol with the obtained learning gain matrices, the perfect tracking of consensus and the monotonic convergence of the input errors are achieved simultaneously. That is, the consensus perfect tracking with good transients is achieved.The rest of this paper is organized as follows. In Section2, notations, basic terminologies in graph theory, and some existing results about fractional order calculus are introduced. Both the perfect tracking problem of consensus and the monotonic convergence problem of input errors for the uncertain nonlinear FOMASs with the parameter coupling relations are formulated. Subsequently, a Lyapunov-like method is applied to deal with the formulated problems and the related sufficient conditions in terms of LMIs are derived for the nonlinear FOMASs without any uncertainty, then the reasonable extension to the uncertain nonlinear FOMASs are presented in Section 3. Two numerical illustrative examples are presented in Section 4. Finally, the conclusions are drawn in Section 5.


## 2. Background and Preliminaries

In this section, some basic notations, useful lemmas and the algebraic graph theory as well as the model of FOMASs are presented.


### 2.1. Notations

LetR, Rm×n, and Z+ be the set of real numbers, the set of m×n real matrices and the set of nonnegative integers, respectively. The superscript T denotes the transposition of vector or matrix, and Im (0m) denotes the m×m identity (zero) matrix; sym{M} indicates the expression M+MT, and diagc1,c2,…,cn denotes the block diagonal matrix with ci,i=1,2,…,n as its diagonal elements; M>0 (M<0) implies a symmetric positive (negative) definite matrix, and the notation ∗ denotes the entries determined by symmetry; · refers to the Euclidean norm for vectors and to the spectrum norm for matrices, and the notation ⊗ denotes the Kronecker product of two matrices. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. Moreover, G=(V,E,A) denotes a weighted directed graph with the set of vertices V=1,2,…,N, the set of edges E∈V×V, and the adjacency matrix A=akj∈RN×N. In particular, akk=0, and akj=1 if j,k∈E, and akj=0 otherwise. The detailed description about graph theory can refer to [8, 23].


### 2.2. Some Definitions and Lemmas

The following definitions and lemmas will be used for the convergence analysis in the main results.Definition 1 (see [45]).
The definition of fractional integral is described byDt-αt0f(t)=1/Γα∫t0t(t-τ)α-1f(τ)dτ,α>0, where Γ(α) is the well-known Gamma function.Definition 2 (see [45]).
The Caputo derivative is defined asDtαf(t)t0C=Dtα-mt0Dtmf(t),α∈(m-1,m), where Dtm(·) denotes the classical m-order derivative, and m is a positive integer.Note. In the following, Dtαft0C is denoted as Dtαft for simplicity.Property 3 (see [45, 46]).
Letα,β>0. If f1(t) and f2(t) are continuous on the interval t∈[0,T], then the following relations hold:(1)
DtαDt-βf1(t)=Dtα-βf1(t);(2)
Dtα(f1(t)±f2(t))=Dtαf1(t)±Dtαf2(t);(3)
DtαDt-αf1(t)=f1(t);(4)
Dt-αDtαf1(t)=f1(t)-f1(0).Property 4 (see [47]).
Letx(t)∈Rn be a vector of differentiable functions. Then, for any time instant, the following relation holds: DtαxT(t)PxT(t)≤xT(t)PDtαx(t)+DtαxT(t)Px(t),∀α∈(0,1],∀t≥0, where the equal sign corresponds to the case of α=1, and P∈Rn×n is a constant, square, symmetric and positive definite matrix.Lemma 5 (see [48]).
Given matricesQ=QT, M, N of appropriate dimensions. The inequality Q+M∑(t)N+[M∑(t)N]T<0 holds for any ∑(t) satisfying ∑tT∑(t)≤I if and only if there exists a scalar constant ε>0 such that Q+εNTN+ε-1MMT<0.


### 2.3. Model Description and Problem Formulation

Consider the uncertain nonlinear FOMASs consisting of one leader agent andN follower agents, which work in a repeatable control environment. The leader agent, labeled as d, has nonlinear dynamics [49, 50] as(1)Dtαxdt=Atxdt+ft,xdt+Btudt,ydt=Cxdt,where α∈(0,1) and t∈0,T. xd(t)∈Rn, yd(t)∈Rm and ud(t)∈Rp are the state, output and control input of the leader agent, and f(·)∈Rn is a nonlinear function satisfying Assumption 6. A(t)∈Rn×n, B(t)∈Rn×p and C∈Rm×n are the system matrix, input matrix and output matrix, and A(t) and B(t) satisfy Assumption 8.Assumption 6 (see [51]).
Assume thatf(·) is a Lipschitz continuous nonlinear function, that is, there exists a positive constant l such that for any z1(t),z2(t)∈Rn, f(t,z1(t))-f(t,z2(t))≤lz1(t)-z2(t), where l is called as the Lipschitz constant.Remark 7.
Assumption6 is a classical Lipschitz condition for a continuous nonlinear function. It guarantees the existence and uniqueness of the solution of system (1) and will be used to prove Lemma 9 and to derive the consensus convergence conditions (See Theorems 20 and 23).Assumption 8.
Assume that there exist the simple linear coupling relations among the fractional orderα, the perturbation ΔA(t) of system matrix A(t) and the perturbation ΔB(t) of input matrix B(t) [16]. More specifically, the linear coupling relations are described by(2)At=A0+μαΔAt,Bt=B0+μαΔBt,where μ>0 is a scalar constant, and A0 and B0 are the constant matrices, and A0 is Hurwitz and (A0,B0,C) is stabilizable and detectable. The time-varying matrices ΔA(t) and ΔB(t) are measurable and satisfy [42–44](3)ΔAt=E∑tF1,ΔBt=E∑tF2,where E, F1 and F2 are the known real constant matrices of appropriate dimensions, while ∑(t) is a time-varying matrix satisfying ∑tT∑(t)≤I.The follower agents are labeled byj,j∈1,2,…,N, and the interaction topology among all the follower agents is described by the fixed graph G=(V,E,A), where V=1,2,…,N, E and A are the edge set and the weighted adjacency matrix of G, respectively. Thus, the interaction topology among all agents can be described by the graph G-=V-,E-,A-, where V-=0,1,2,…,N, E- and A- are the edge set and the weighted adjacency matrix of G-, respectively.At thekth iteration, the dynamics of the follower agent j take the following form(4)Dtαxk,jt=Atxk,jt+ft,xk,jt+Btuk,jt,yk,jt=Cxk,jt,where k∈Z+ denotes the iteration step. xk,jt∈Rn, yk,j(t)∈Rm and uk,j(t)∈Rp are the state, output and control input of the system (4), and the other notations are the same as those in (1). Here the system matrix and input matrix of all agents are assumed to be identical. As pointed out in [52], this case is reasonable, because it can represent birds and school of fishes.Lemma 9.
Let the scalar constantε1>0, the positive definite matrix Q1=Q1T∈Rn×n, and assume that Assumptions 6 and 8 hold. If A0 is Hurwitz, and there exist a positive definite matrix P1=P1T satisfying(5)A0TP1+P1A0+l2In+ε1μαF1TF1+Q1P1P1EP1-In0∗∗-ε1μ-1α-1In<0,then the fractional-order nonlinear uncertain agent (4) under no control is stable.
Lemma9 provides the sufficient condition guaranteeing that the fractional-order nonlinear uncertain agent (4) under no control is asymptotically stable, and its detailed proof is given in the Appendix.Remark 10.
(a) Ifμ=0 and ΔA(t)=ΔB(t)=0, the model will degenerate into the nonlinear FOMASs without any uncertainty and parameter coupling relation. (b) If f(·)=0, the model will degenerate into the uncertain linear FOMASs, where there exist linear coupling relations among α, ΔA(t), and ΔB(t). (c) If μ=1/α, the model will degenerate into the special uncertain nonlinear FOMASs, where there is no coupling relation among α, ΔA(t), and ΔB(t), but A(t)=A0+ΔA(t) and B(t)=B0+ΔB(t). (d) If μ≠1/α, ΔA(t)≠0, ΔB(t)≠0 and f(·)≠0, the model will become the more general uncertain nonlinear FOMASs, where there exist linear coupling relations among α, ΔA(t) and ΔB(t). The asymptotic consensus of FOMASs in Case (a) and (b) has been studied [16, 20, 21], while there is no report about the perfect consensus of FOMASs in all of the above cases.
Assume that the state of each follower agent is not measurable. To estimate the states of the follower agentj, its state observer is designed as(6)Dtαx^k,jt=Atx^k,jt+ft,x^k,jt+Btuk,jt+Lyk,jt-Cx^k,jt,where x^k,j(t)∈Rn denotes the observed state vector, L∈Rn×m is the gain matrix of observer, and the other notations are the same as those in (1).Remark 11.
Similar to [42–44], this paper assumes that the time-varying structural uncertainties ΔA(t) and ΔB(t) are measurable. This assumption holds in many applications. For example, the dynamics of an aero-engine can be described as a linear system subject to the time-varying measurable uncertainties [53]. By using the uncertain time-varying system matrices same as the corresponding agents, we construct the state observers. Thus the constructed observers in fact are the time-varying uncertain systems [42–44], which are different from the observer constructed by the nominal parameters [40, 41].Let the observation errorx~k,j(t)=xk,j(t)-x^k,j(t). From (4) and (6), we have(7)Dtαx~k,jt=At-LCx~k,jt+ft,xk,jt-ft,x^k,jt.For the observation error system (7), we can obtain the following Lemma:Lemma 12.
Let the scalar constantε2>0, the positive definite matrix Q2=Q2T∈Rn×n, and assume that Assumptions 6 and 8 hold. If L can be chosen such that A0-LC is Hurwitz, and there exist a positive definite matrix P2=P2T satisfying(8)A0-LCTP2+P2A0-LC+l2In+ε2μαF1TF1+Q2P2P2EP2-In0∗∗-ε2μ-1α-1In<0,then the observation error x~k,j(t) is bounded and converges asymptotically to zero as time goes to infinity.Lemma12 provides the sufficient condition guaranteeing the boundedness and asymptotic convergence of the observation errors, and its proof is similar to that of Lemma 9, thus is omitted.Remark 13.
In this paper, the system matrixA0 is assumed to be Hurwitz (i.e., the agent is assumed to be stable). It is necessary, because it ensures that (5) has positive definite solutions P1, which will be needed in the proofs of Theorems 20 and 23. Meanwhile, it should be pointed out that it is meaningful to consider the observer-based perfect consensus of the FOMASs consisting of the stable agents. On the one hand, for the leader-following FOMASs such as (1) and (4), even though the system matrix A0 is Hurwitz, when the state of agent is not measurable, to ensure the boundedness and convergence of the observation errors, the state observer and its gain matrices have to be designed (See Lemma 12 and Corollary 19). On the other hand, this paper focuses on the perfect consensus tracking problem. For the MASs consisting of the stable agents, the asymptotic consensus can be achieved without requiring any control. However, if no control is applied, even for the MASs consisting of the stable agents, the perfect consensus cannot be achieved, and especially the good transients such as the monotonic convergence of input errors cannot be achieved.Letζk,j(t) denote the information received by the follower agent j at the kth iteration. More specifically,(9)ζk,jt=∑i∈Njajix^k,it-x^k,jt+djxdt-x^k,jt,where ajk is the j,kth entry in the adjacency matrix A, and Nj is the neighborhood set of the follower agent j. dj=1 if the follower agent j can receive directly the information of the leader agent, and dj=0 otherwise.Let the consensus tracking errorek,j(t)=xd(t)-x^k,j(t). The compact format of (9) is rewritten as(10)ζkt=H⊗Inekt,where ζk(t)=[ζk,1T(t),ζk,2T(t),…,ζk,NT(t)]T, ek(t)=[ek,1T(t),ek,2T(t),…,ek,NT(t)]T, H=G+D, G is the Laplacian matrix of graph G, and D=diag{d1,d2,…,dN}.Based on (10), an observer-type ILC consensus protocol is designed as(11)uk+1,jt=uk,jt+Kζk,jt+ΓDtαζk,jt,where both K∈Rp×n and Γ∈Rp×n are the learning gain matrices to be designed later.By (10), the compact format of (11) is rewritten as(12)uk+1t=ukt+H⊗Kekt+H⊗ΓDtαekt,where uk(t)=(uk,1T(t),uk,2T(t),…,uk,NT(t))T.Letδuk,j(t)=ud(t)-uk,j(t). From (1) and (6), we have(13)Dtαek,jt=Atek,jt+Btδuk,jt-LCx~k,jt+ft,xdt-ft,x^k,jt.Letϕk,j(t)=f(t,xd(t))-f(t,x^k,j(t)) and φk,j(t)=f(t,xk,j(t))-f(t,x^k,j(t)). The compact formats of (13) and (7) are rewritten as(14)Dtαekt=IN⊗Atekt+IN⊗Btδukt-IN⊗LCx~kt+ϕkt,(15)Dtαx~kt=IN⊗At-LCx~kt+φkt,where δuk(t)=(δuk,1T(t),δuk,2T(t),…,δuk,NT(t))T, ϕk(t)=(ϕk,1T(t),ϕk,2T(t),…,ϕk,NT(t))T,x~k(t)=(x~k,1T(t),x~k,2T(t),…,x~k,NT(t))T and φk(t)=(φk,1T(t),φk,2T(t),…,φk,NT(t))T.Combining (14) and (15) results in a system(16)Dtαθkt=R1θkt+R2δukt+Ψkt,where θk(t)=[ekT(t),x~k(t)T]T,Ψk(t)=[ϕkT(t),φkT(t)]T, R1=IN⊗A(t)-IN⊗(LC)0IN⊗(A(t)-LC) and R2=IN⊗B(t)0.By (12), we have(17)δuk+1t=1Nudt-uk+1t=δukt-H⊗Kekt-H⊗ΓDtαekt=δukt-H⊗Kekt-H⊗ΓIN⊗Atekt+IN⊗Btδukt-IN⊗LCx~kt+ϕkt=R3θkt+R4δukt-H⊗Γϕkt,where R3=-H⊗(K+ΓA(t))H⊗(ΓLC) and R4=I(Np)-H⊗(ΓB(t)).Combining (16) and (17) results in a 2D Roesser system [54](18)Dtαθktδuk+1t=R1R2R3R4θktδukt+Ψkt-H⊗Γϕkt.Based on Assumption6, we obtain that ϕk,j(t)2≤lek,j(t)2 and φk,j(t)2≤lx~k,j(t)2. Thus, according to the 2D system theory [54], it is not difficult to know from (18) that limk→+∞⁡[θkT(t),δukT(t)]T=0 holds if the condition R42<1 is satisfied. However, the condition R42<1 only indicates the asymptotic convergence of input errors, but cannot guarantee that the control input errors converge monotonically to zero as a function of iteration steps. Based on the observations, this paper mainly focuses on the following problems:Problem Statement. Given the nonlinear FOMAS (1) and (4), let the observer (6) and the ILC consensus protocol (11) be applied. This paper will design the learning gain matrices K and Γ such that:(i) The monotonic convergence of the input errors is achieved, i.e.,(19)δuk+1t2<γδukt2holds for any t∈[0,T], k∈Z+, where γ∈(0,1) is a prescribed scalar constant indicating the convergence rate of consensus.(ii) The outputs of all follower agents converge to that of the leader agent in the finite time intervalt∈[0,T] as the iteration step k goes to infinity, that is,(20)limk→+∞⁡ydt-yk,jt2=0,j=1,2,…,N,t∈0,T.To deal with the problem shown in (19), we define a novel performance index as(21)Jk,γ=1Γα∫0TT-sα-1δuk+1Tsδuk+1s-γ2δukTsδuksds.Since the inequality(T-s)α-1>0 holds for all s∈[0,T] when α∈(0,1), it is not difficult to see from (21) that the problem shown in (19) will be solved if J(k,γ)<0 is satisfied. Moreover, the problem shown in (20) will be solved if limk→+∞⁡θk(t)=0 is satisfied for all t∈[0,T]. Thus, for two problems shown in (19) and (20), the control objectives in this paper can be further summarized as follows:(22)Jk,γ<0,γ∈0,1,k∈Z+;(23)limk→+∞⁡x~kt=0,limk→+∞⁡ekt=0,k∈Z+,t∈0,T.Remark 14.
In the exiting literatures [55, 56], the performance indexes are expressed in terms of integer-order integral, and the optimal controllers and the consensus conditions are obtained by the inverse optimal method. Since each agent in this paper is described by the fractional-order differential equation, it is difficult to extend the performance indexes in [55, 56] to the FOMASs in this paper. Moreover, due to the existence of structural uncertainties, the inverse optimal method is not applicable. Thus, in this paper, the performance index is designed in terms of fraction-order integral of linear quadratic forms of input errors, and the Lyapunov-like method instead of the inverse optimal method is applied to determine the learning gains and to derive the consensus convergence conditions (See Theorems 20 and 23).Remark 15.
If the fractional orderα=1, then the performance index (21) will degenerate into(24)Jk,γ=∫0Tδuk+1Tsδuk+1s-γ2δukTsδuksdt,which has been used to solve the robust ILC problem of uncertain time-delay integer-order systems [57]. Thus, the performance index (21) includes (24) as its special case.To simplify the analysis, the following assumptions are needed.Assumption 16.
The initial states of each follower agent and its observer are reset to the same as that of the leader agent after each execution.
To better focus on the main problem, the strictly identical initial conditions [28] are assumed in Assumption 16. On the other hand, if Assumption 16 is not satisfied, the initial state learning law for each follower agent and its observer can be designed by the method in [29] before the methods in this paper are used.Assumption 17.
The communication graphG- at least contains a spanning tree with the leader agent being the root.


## 2.1. Notations

LetR, Rm×n, and Z+ be the set of real numbers, the set of m×n real matrices and the set of nonnegative integers, respectively. The superscript T denotes the transposition of vector or matrix, and Im (0m) denotes the m×m identity (zero) matrix; sym{M} indicates the expression M+MT, and diagc1,c2,…,cn denotes the block diagonal matrix with ci,i=1,2,…,n as its diagonal elements; M>0 (M<0) implies a symmetric positive (negative) definite matrix, and the notation ∗ denotes the entries determined by symmetry; · refers to the Euclidean norm for vectors and to the spectrum norm for matrices, and the notation ⊗ denotes the Kronecker product of two matrices. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. Moreover, G=(V,E,A) denotes a weighted directed graph with the set of vertices V=1,2,…,N, the set of edges E∈V×V, and the adjacency matrix A=akj∈RN×N. In particular, akk=0, and akj=1 if j,k∈E, and akj=0 otherwise. The detailed description about graph theory can refer to [8, 23].


## 2.2. Some Definitions and Lemmas

The following definitions and lemmas will be used for the convergence analysis in the main results.Definition 1 (see [45]).
The definition of fractional integral is described byDt-αt0f(t)=1/Γα∫t0t(t-τ)α-1f(τ)dτ,α>0, where Γ(α) is the well-known Gamma function.Definition 2 (see [45]).
The Caputo derivative is defined asDtαf(t)t0C=Dtα-mt0Dtmf(t),α∈(m-1,m), where Dtm(·) denotes the classical m-order derivative, and m is a positive integer.Note. In the following, Dtαft0C is denoted as Dtαft for simplicity.Property 3 (see [45, 46]).
Letα,β>0. If f1(t) and f2(t) are continuous on the interval t∈[0,T], then the following relations hold:(1)
DtαDt-βf1(t)=Dtα-βf1(t);(2)
Dtα(f1(t)±f2(t))=Dtαf1(t)±Dtαf2(t);(3)
DtαDt-αf1(t)=f1(t);(4)
Dt-αDtαf1(t)=f1(t)-f1(0).Property 4 (see [47]).
Letx(t)∈Rn be a vector of differentiable functions. Then, for any time instant, the following relation holds: DtαxT(t)PxT(t)≤xT(t)PDtαx(t)+DtαxT(t)Px(t),∀α∈(0,1],∀t≥0, where the equal sign corresponds to the case of α=1, and P∈Rn×n is a constant, square, symmetric and positive definite matrix.Lemma 5 (see [48]).
Given matricesQ=QT, M, N of appropriate dimensions. The inequality Q+M∑(t)N+[M∑(t)N]T<0 holds for any ∑(t) satisfying ∑tT∑(t)≤I if and only if there exists a scalar constant ε>0 such that Q+εNTN+ε-1MMT<0.


## 2.3. Model Description and Problem Formulation

Consider the uncertain nonlinear FOMASs consisting of one leader agent andN follower agents, which work in a repeatable control environment. The leader agent, labeled as d, has nonlinear dynamics [49, 50] as(1)Dtαxdt=Atxdt+ft,xdt+Btudt,ydt=Cxdt,where α∈(0,1) and t∈0,T. xd(t)∈Rn, yd(t)∈Rm and ud(t)∈Rp are the state, output and control input of the leader agent, and f(·)∈Rn is a nonlinear function satisfying Assumption 6. A(t)∈Rn×n, B(t)∈Rn×p and C∈Rm×n are the system matrix, input matrix and output matrix, and A(t) and B(t) satisfy Assumption 8.Assumption 6 (see [51]).
Assume thatf(·) is a Lipschitz continuous nonlinear function, that is, there exists a positive constant l such that for any z1(t),z2(t)∈Rn, f(t,z1(t))-f(t,z2(t))≤lz1(t)-z2(t), where l is called as the Lipschitz constant.Remark 7.
Assumption6 is a classical Lipschitz condition for a continuous nonlinear function. It guarantees the existence and uniqueness of the solution of system (1) and will be used to prove Lemma 9 and to derive the consensus convergence conditions (See Theorems 20 and 23).Assumption 8.
Assume that there exist the simple linear coupling relations among the fractional orderα, the perturbation ΔA(t) of system matrix A(t) and the perturbation ΔB(t) of input matrix B(t) [16]. More specifically, the linear coupling relations are described by(2)At=A0+μαΔAt,Bt=B0+μαΔBt,where μ>0 is a scalar constant, and A0 and B0 are the constant matrices, and A0 is Hurwitz and (A0,B0,C) is stabilizable and detectable. The time-varying matrices ΔA(t) and ΔB(t) are measurable and satisfy [42–44](3)ΔAt=E∑tF1,ΔBt=E∑tF2,where E, F1 and F2 are the known real constant matrices of appropriate dimensions, while ∑(t) is a time-varying matrix satisfying ∑tT∑(t)≤I.The follower agents are labeled byj,j∈1,2,…,N, and the interaction topology among all the follower agents is described by the fixed graph G=(V,E,A), where V=1,2,…,N, E and A are the edge set and the weighted adjacency matrix of G, respectively. Thus, the interaction topology among all agents can be described by the graph G-=V-,E-,A-, where V-=0,1,2,…,N, E- and A- are the edge set and the weighted adjacency matrix of G-, respectively.At thekth iteration, the dynamics of the follower agent j take the following form(4)Dtαxk,jt=Atxk,jt+ft,xk,jt+Btuk,jt,yk,jt=Cxk,jt,where k∈Z+ denotes the iteration step. xk,jt∈Rn, yk,j(t)∈Rm and uk,j(t)∈Rp are the state, output and control input of the system (4), and the other notations are the same as those in (1). Here the system matrix and input matrix of all agents are assumed to be identical. As pointed out in [52], this case is reasonable, because it can represent birds and school of fishes.Lemma 9.
Let the scalar constantε1>0, the positive definite matrix Q1=Q1T∈Rn×n, and assume that Assumptions 6 and 8 hold. If A0 is Hurwitz, and there exist a positive definite matrix P1=P1T satisfying(5)A0TP1+P1A0+l2In+ε1μαF1TF1+Q1P1P1EP1-In0∗∗-ε1μ-1α-1In<0,then the fractional-order nonlinear uncertain agent (4) under no control is stable.
Lemma9 provides the sufficient condition guaranteeing that the fractional-order nonlinear uncertain agent (4) under no control is asymptotically stable, and its detailed proof is given in the Appendix.Remark 10.
(a) Ifμ=0 and ΔA(t)=ΔB(t)=0, the model will degenerate into the nonlinear FOMASs without any uncertainty and parameter coupling relation. (b) If f(·)=0, the model will degenerate into the uncertain linear FOMASs, where there exist linear coupling relations among α, ΔA(t), and ΔB(t). (c) If μ=1/α, the model will degenerate into the special uncertain nonlinear FOMASs, where there is no coupling relation among α, ΔA(t), and ΔB(t), but A(t)=A0+ΔA(t) and B(t)=B0+ΔB(t). (d) If μ≠1/α, ΔA(t)≠0, ΔB(t)≠0 and f(·)≠0, the model will become the more general uncertain nonlinear FOMASs, where there exist linear coupling relations among α, ΔA(t) and ΔB(t). The asymptotic consensus of FOMASs in Case (a) and (b) has been studied [16, 20, 21], while there is no report about the perfect consensus of FOMASs in all of the above cases.
Assume that the state of each follower agent is not measurable. To estimate the states of the follower agentj, its state observer is designed as(6)Dtαx^k,jt=Atx^k,jt+ft,x^k,jt+Btuk,jt+Lyk,jt-Cx^k,jt,where x^k,j(t)∈Rn denotes the observed state vector, L∈Rn×m is the gain matrix of observer, and the other notations are the same as those in (1).Remark 11.
Similar to [42–44], this paper assumes that the time-varying structural uncertainties ΔA(t) and ΔB(t) are measurable. This assumption holds in many applications. For example, the dynamics of an aero-engine can be described as a linear system subject to the time-varying measurable uncertainties [53]. By using the uncertain time-varying system matrices same as the corresponding agents, we construct the state observers. Thus the constructed observers in fact are the time-varying uncertain systems [42–44], which are different from the observer constructed by the nominal parameters [40, 41].Let the observation errorx~k,j(t)=xk,j(t)-x^k,j(t). From (4) and (6), we have(7)Dtαx~k,jt=At-LCx~k,jt+ft,xk,jt-ft,x^k,jt.For the observation error system (7), we can obtain the following Lemma:Lemma 12.
Let the scalar constantε2>0, the positive definite matrix Q2=Q2T∈Rn×n, and assume that Assumptions 6 and 8 hold. If L can be chosen such that A0-LC is Hurwitz, and there exist a positive definite matrix P2=P2T satisfying(8)A0-LCTP2+P2A0-LC+l2In+ε2μαF1TF1+Q2P2P2EP2-In0∗∗-ε2μ-1α-1In<0,then the observation error x~k,j(t) is bounded and converges asymptotically to zero as time goes to infinity.Lemma12 provides the sufficient condition guaranteeing the boundedness and asymptotic convergence of the observation errors, and its proof is similar to that of Lemma 9, thus is omitted.Remark 13.
In this paper, the system matrixA0 is assumed to be Hurwitz (i.e., the agent is assumed to be stable). It is necessary, because it ensures that (5) has positive definite solutions P1, which will be needed in the proofs of Theorems 20 and 23. Meanwhile, it should be pointed out that it is meaningful to consider the observer-based perfect consensus of the FOMASs consisting of the stable agents. On the one hand, for the leader-following FOMASs such as (1) and (4), even though the system matrix A0 is Hurwitz, when the state of agent is not measurable, to ensure the boundedness and convergence of the observation errors, the state observer and its gain matrices have to be designed (See Lemma 12 and Corollary 19). On the other hand, this paper focuses on the perfect consensus tracking problem. For the MASs consisting of the stable agents, the asymptotic consensus can be achieved without requiring any control. However, if no control is applied, even for the MASs consisting of the stable agents, the perfect consensus cannot be achieved, and especially the good transients such as the monotonic convergence of input errors cannot be achieved.Letζk,j(t) denote the information received by the follower agent j at the kth iteration. More specifically,(9)ζk,jt=∑i∈Njajix^k,it-x^k,jt+djxdt-x^k,jt,where ajk is the j,kth entry in the adjacency matrix A, and Nj is the neighborhood set of the follower agent j. dj=1 if the follower agent j can receive directly the information of the leader agent, and dj=0 otherwise.Let the consensus tracking errorek,j(t)=xd(t)-x^k,j(t). The compact format of (9) is rewritten as(10)ζkt=H⊗Inekt,where ζk(t)=[ζk,1T(t),ζk,2T(t),…,ζk,NT(t)]T, ek(t)=[ek,1T(t),ek,2T(t),…,ek,NT(t)]T, H=G+D, G is the Laplacian matrix of graph G, and D=diag{d1,d2,…,dN}.Based on (10), an observer-type ILC consensus protocol is designed as(11)uk+1,jt=uk,jt+Kζk,jt+ΓDtαζk,jt,where both K∈Rp×n and Γ∈Rp×n are the learning gain matrices to be designed later.By (10), the compact format of (11) is rewritten as(12)uk+1t=ukt+H⊗Kekt+H⊗ΓDtαekt,where uk(t)=(uk,1T(t),uk,2T(t),…,uk,NT(t))T.Letδuk,j(t)=ud(t)-uk,j(t). From (1) and (6), we have(13)Dtαek,jt=Atek,jt+Btδuk,jt-LCx~k,jt+ft,xdt-ft,x^k,jt.Letϕk,j(t)=f(t,xd(t))-f(t,x^k,j(t)) and φk,j(t)=f(t,xk,j(t))-f(t,x^k,j(t)). The compact formats of (13) and (7) are rewritten as(14)Dtαekt=IN⊗Atekt+IN⊗Btδukt-IN⊗LCx~kt+ϕkt,(15)Dtαx~kt=IN⊗At-LCx~kt+φkt,where δuk(t)=(δuk,1T(t),δuk,2T(t),…,δuk,NT(t))T, ϕk(t)=(ϕk,1T(t),ϕk,2T(t),…,ϕk,NT(t))T,x~k(t)=(x~k,1T(t),x~k,2T(t),…,x~k,NT(t))T and φk(t)=(φk,1T(t),φk,2T(t),…,φk,NT(t))T.Combining (14) and (15) results in a system(16)Dtαθkt=R1θkt+R2δukt+Ψkt,where θk(t)=[ekT(t),x~k(t)T]T,Ψk(t)=[ϕkT(t),φkT(t)]T, R1=IN⊗A(t)-IN⊗(LC)0IN⊗(A(t)-LC) and R2=IN⊗B(t)0.By (12), we have(17)δuk+1t=1Nudt-uk+1t=δukt-H⊗Kekt-H⊗ΓDtαekt=δukt-H⊗Kekt-H⊗ΓIN⊗Atekt+IN⊗Btδukt-IN⊗LCx~kt+ϕkt=R3θkt+R4δukt-H⊗Γϕkt,where R3=-H⊗(K+ΓA(t))H⊗(ΓLC) and R4=I(Np)-H⊗(ΓB(t)).Combining (16) and (17) results in a 2D Roesser system [54](18)Dtαθktδuk+1t=R1R2R3R4θktδukt+Ψkt-H⊗Γϕkt.Based on Assumption6, we obtain that ϕk,j(t)2≤lek,j(t)2 and φk,j(t)2≤lx~k,j(t)2. Thus, according to the 2D system theory [54], it is not difficult to know from (18) that limk→+∞⁡[θkT(t),δukT(t)]T=0 holds if the condition R42<1 is satisfied. However, the condition R42<1 only indicates the asymptotic convergence of input errors, but cannot guarantee that the control input errors converge monotonically to zero as a function of iteration steps. Based on the observations, this paper mainly focuses on the following problems:Problem Statement. Given the nonlinear FOMAS (1) and (4), let the observer (6) and the ILC consensus protocol (11) be applied. This paper will design the learning gain matrices K and Γ such that:(i) The monotonic convergence of the input errors is achieved, i.e.,(19)δuk+1t2<γδukt2holds for any t∈[0,T], k∈Z+, where γ∈(0,1) is a prescribed scalar constant indicating the convergence rate of consensus.(ii) The outputs of all follower agents converge to that of the leader agent in the finite time intervalt∈[0,T] as the iteration step k goes to infinity, that is,(20)limk→+∞⁡ydt-yk,jt2=0,j=1,2,…,N,t∈0,T.To deal with the problem shown in (19), we define a novel performance index as(21)Jk,γ=1Γα∫0TT-sα-1δuk+1Tsδuk+1s-γ2δukTsδuksds.Since the inequality(T-s)α-1>0 holds for all s∈[0,T] when α∈(0,1), it is not difficult to see from (21) that the problem shown in (19) will be solved if J(k,γ)<0 is satisfied. Moreover, the problem shown in (20) will be solved if limk→+∞⁡θk(t)=0 is satisfied for all t∈[0,T]. Thus, for two problems shown in (19) and (20), the control objectives in this paper can be further summarized as follows:(22)Jk,γ<0,γ∈0,1,k∈Z+;(23)limk→+∞⁡x~kt=0,limk→+∞⁡ekt=0,k∈Z+,t∈0,T.Remark 14.
In the exiting literatures [55, 56], the performance indexes are expressed in terms of integer-order integral, and the optimal controllers and the consensus conditions are obtained by the inverse optimal method. Since each agent in this paper is described by the fractional-order differential equation, it is difficult to extend the performance indexes in [55, 56] to the FOMASs in this paper. Moreover, due to the existence of structural uncertainties, the inverse optimal method is not applicable. Thus, in this paper, the performance index is designed in terms of fraction-order integral of linear quadratic forms of input errors, and the Lyapunov-like method instead of the inverse optimal method is applied to determine the learning gains and to derive the consensus convergence conditions (See Theorems 20 and 23).Remark 15.
If the fractional orderα=1, then the performance index (21) will degenerate into(24)Jk,γ=∫0Tδuk+1Tsδuk+1s-γ2δukTsδuksdt,which has been used to solve the robust ILC problem of uncertain time-delay integer-order systems [57]. Thus, the performance index (21) includes (24) as its special case.To simplify the analysis, the following assumptions are needed.Assumption 16.
The initial states of each follower agent and its observer are reset to the same as that of the leader agent after each execution.
To better focus on the main problem, the strictly identical initial conditions [28] are assumed in Assumption 16. On the other hand, if Assumption 16 is not satisfied, the initial state learning law for each follower agent and its observer can be designed by the method in [29] before the methods in this paper are used.Assumption 17.
The communication graphG- at least contains a spanning tree with the leader agent being the root.


## 3. Main Results

In this section, we will consider the perfect tracking problem of consensus for the nonlinear FOMASs (1) and (4) by using Lyapunov-like method, and will present the sufficient conditions guaranteeing that two problems shown in (19) and (20) are solved simultaneously. First, the case of nonlinear FOMASs (1) and (4) without any uncertainty is considered.


### 3.1. Nonlinear FOMASs without Any Uncertainty

In this subsection, bothA(t) and B(t) are assumed to be the known matrices without any uncertainty, that is, A(t)=A0 and B(t)=B0. In this case, we can state the following two corollaries:Corollary 18.
Let the positive definite matrixQ1=Q1T∈Rn×n, and assume that Assumption 6 holds. If A0 is Hurwitz, and there exist a positive definite matrix P1=P1T satisfying(25)A0TP1+P1A0+l2In+Q1P1P1-In<0.then the fractional-order nonlinear agent (4) without any uncertainty under no control is stable.Corollary 19.
Let the positive definite matrixQ2=Q2T∈Rn×n, and assume that Assumption 6 holds. If the observer gain matrix L is chosen such that A0-LC is Hurwitz, and there exists a positive definite matrix P2=P2T∈Rn×n satisfying(26)A0-LCTP2+P2A0-LC+l2In+Q2P2P2-In<0,then the observation error is bounded, and converges asymptotically to zero as time goes to infinity.Corollaries18 and 19 can be directly derived from Lemmas 9 and 12, respectively. Thus their proofs are omitted.Based on the previous preparation, we can state the following results.Theorem 20.
Assume that Assumptions6, 16, and 17 and the conditions in Corollaries 18 and 19 are satisfied for the nonlinear FOMASs (1) and (4) without any uncertainty, and let the observer (6) and the consensus protocol (11) are applied. If, for a scalar γ∈(0,1), there exist matrices Xi,i=1,2, satisfying(27)Ξ1=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗IN⊗symA0TP1+IN⊗Q1+IN⊗P1P1+l2INn-IN⊗P1LCIN⊗P1B0∗∗∗∗-IN⊗CTLTP1IN⊗symA0-LCTP2+IN⊗Q2+IN⊗P2P2+l2INn0∗∗∗∗∗∗-γ2INp<0,where P1 and Q1 (P2 and Q2) are given in Corollary 18 (Corollary 19). Then both the input errors and the output tracking errors converge to zero for all t∈[0,T] as the iteration step goes to infinity and the monotonic convergence of input errors is obtained for all k∈Z+. Moreover, if (27) holds, then the learning gain matrices K and Γ are determined by(28)K=X1,Γ=X2.Proof.
(a) Proof of (19)
Let(29)P=diagIN⊗P1,IN⊗P2,Q=diagIN⊗Q1,IN⊗Q2.
Note that bothP=PT and Q=QT are positive definite, we can define a Lyapunov function as(30)Vθkt=θkTtPθkt+1Γα∫0tt-sα-1θkTsQθksds.It is easy to know that V(θk(t)) is positive semidefinite. Moreover, according to Assumption 16, we obtain that V(θk(0))=0.
Letξk(t)=θkT(t)δukT(t)T. By Definition 2 and Properties 3 and 4, we have(31)DtαVθkt≤DtαθktTPθkt+θkTtPDtαθkt+θkTtQθkt=R1θkt+R2δukt+ΨktTPθkt+θkTtPR1θkt+R2δukt+Ψkt+θkTtQθkt=θkTtR1TP+PR1+Qθkt+θkTtPR2δukt+δukTtR2TPθkt+2θkTtPΨkt≤θktδuktTR1TP+PR1+QPR2∗0θktδukt+θkTtPPθkt+ϕktφktTϕktφkt≤θktδuktTR1TP+PR1+Q+PPPR2∗0θktδukt+l2ekTtekt+l2x~ktTx~kt≤θktδuktTR1TP+PR1+Q+PP+l2I2NnPR2∗0θktδukt=ξkTtΠ1ξkt,where(32)Π1=R1TP+PR1+Q+PP+l2I2NnPR2∗0,R1=IN⊗A0-IN⊗LC0IN⊗A0-LC,R2=IN⊗B00.
Based on (17), we have(33)δuk+1Ttδuk+1t=R3θkt+R4δukt-H⊗ΓϕktTR3θkt+R4δukt-H⊗Γϕkt=θkTtR3TR3θkt+δukTtR4TR3θkt-ϕkTtHT⊗ΓTR3θkt+θkTtR3TR4δukt+δukTtR4TR4δukt-ϕkTtHT⊗ΓTR4δukt-θkTtR3TH⊗Γϕkt-δukTtR4TH⊗Γϕkt+ϕkTtHT⊗ΓTH⊗Γϕkt=ξkTtR3TR4TR3R4ξkt-2θkTtR3TH⊗Γϕkt-2δukTtR4TH⊗Γϕkt+ϕkTtHT⊗ΓTH⊗Γϕkt≤ξkTtR3TR4TR3R4ξkt+θkTtR3TR3θkt+δukTtR4TR4δukt+3ϕkTtHT⊗ΓTH⊗Γϕkt=ξkTtΩ2T0R3TR3T0R4T0R4TΩ200R4R30R3R4ξkt,where(34)Ω2=3H⊗Γ0,R3=-H⊗K+ΓA0H⊗ΓLC,R4=INp-H⊗ΓB0.
By Property3, (31) and (33), the index J(k,γ) can be rewritten as(35)Jk,γ=Jk,γ+1Γα∫0TT-sα-1DsαVθksds-VθkT+Vθk0≤1Γα∫0TT-sα-1δuk+1Tsδuk+1s-γ2δukTsδuks+DsαVθksds=1Γα∫0TT-sα-1ξkTsΠ2ξksds,where the matrix Π2 satisfies(36)Π2=R1TP+PR1+Q+PP+l2I2NnPR2∗-γ2INp-Ω2T0R3TR3T0R4T0R4T-I4NpΩ200R4R30R3R4.
By (29), (27) is rewritten as(37)Ξ1=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗symIN⊗P1A0-IN⊗P1LC0IN⊗P2A0-P2LC+Q+IN⊗P1P100IN⊗P2P2+l2I2NnIN⊗P1B00∗∗∗∗∗-γ2INp<0.
When (27) is satisfied, by selecting the learning gain matrices in (28) and substituting them into (37), we have(38)Ξ1=-INp000Ω20∗-INp000R4∗∗-INp0R30∗∗∗-INpR3R4∗∗∗∗R1TP+PR1+Q+PP+l2I2NnPR2∗∗∗∗∗-γ2INp<0.
Based on the Schur complement formula, it is easy to know from (38) that Π2<0 is satisfied, which indicates that J(k,γ)<0 in (22) holds for all non-zero vectors ξk(t). That is, the monotonic convergence of input errors shown in (19) is achieved.
(b) The proof of (20)
LetΠ3=R1TP+PR1+Q+PP+l2I(2Nn)PR2∗-γ2I(Np). Applying the Schur complement formula to (38), we obtain that Π3<0. Thus, based on (31) and (33), we have(39)DtαVθkt-γ2δukTtδukt≤ξkTtΠ1ξkt+ξkTtdiag0,-γ2INpξkt=ξkTtΠ3ξkt≤0.
From (30), (31), and (39), we have(40)θkTtPθkt≤Vθkt≤1Γα∫0tγ2t-sα-1δukTsδuksds+Vθk0=1Γα∫0tγ2t-sα-1δukTsδuksds.
When (27) is satisfied, it is easy to know that limk→+∞⁡δuk(t)2=0 holds. Thus, taking the limit of k→+∞ on both sides of (40), we obtain that limk→+∞⁡θk(t)2=0, which indicates that limk→+∞⁡yd(t)-yk,j(t)2=0 holds. That is, the perfect tracking of consensus shown in (20) is achieved.
This ends the proof of Theorem20.Remark 21.
Theorem20 provides the sufficient condition in terms of LMIs, which belongs to the category of ILC where a consensus convergence condition is related to the learning gain matrices, the known plant knowledge and the structure of communication graph [33]. However, compared to [33], the obtained LMI condition not only provides a design method of learning gain matrices to achieve the perfect tracking of consensus, but also guarantees the monotonic convergence of the input errors and achieves the good learning transients.Remark 22.
Among the existing results about the consensus of FOMASs [15–21, 36–38], the controllers are merely related to time t, and the asymptotic consensus problem or the finite time consensus problem is addressed. In contrast, this paper has the following two distinct differences: (a) The controllers (11) is related to both time t and iterative learning step k. (b) The perfect tracking of consensus is obtained. That is, the consensus will be maintained perfectly in the finite time interval t∈[0,T] as the iteration step k goes to infinity.


### 3.2. Extension to Uncertain Nonlinear FOMASs with Parameter Coupling Relations

By using the knowledge of plant as the priors, Theorem20 provides an LMI condition guaranteeing the perfect tracking of consensus and the monotonic convergence of the input errors. However, if the parameters of nonlinear FOMASs are subject to the perturbation, Theorem 20 is no longer suitable, thus should be further improved. Next, by studying the uncertain nonlinear FOMASs satisfying Assumptions 6, 8, 16, and 17, we obtain Theorem 23.Theorem 23.
Assume that Assumptions6, 8, 16, and 17 and the conditions in Lemmas 9 and 12 are satisfied for the nonlinear FOMASs (1) and (4), and let the observer (6) and the consensus protocol (11) are applied. If, for a scalar γ∈(0,1), there exist matrices Xi,i=1,2 satisfying(41)-I6NnWWTΠ4<0,where W and Π4 are given in (42) and (43), ε is a known positive scalar constant, P1 and Q1 (P2 and Q2) are given in Lemma 9 (Lemma 12). Then both the input errors and the output tracking errors converge to zero for all t∈[0,T] as the iteration step goes to infinity and the monotonic convergence of input errors is obtained for all k∈Z+. Moreover, if (41) holds, then the learning gain matrices K and Γ are determined by (28).(42)W=00-εHT⊗ETX2T-εHT⊗ETX2TεIN⊗ETP10000000εIN⊗ETP200-εHT⊗ETX2T0-εHT⊗ETX2TεIN⊗ETP1000000μαεIN⊗F10000000μαεIN⊗F10000000μαεIN⊗F2,(43)Π4=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗IN⊗symA0TP1+IN⊗Q1+IN⊗P1P1+l2INn-IN⊗P1LCIN⊗P1B0∗∗∗∗-IN⊗CTLTP1IN⊗symA0-LCTP2+IN⊗Q2+IN⊗P2P2+l2INn0∗∗∗∗∗∗-γ2INp,Proof.
According to Theorem20, we know that the proof of Theorem 23 will be accomplished if Ξ1<0 is satisfied when A0 and B0 in (27) are replaced with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), respectively. Thus, we should prove that after A0 and B0 in (27) are respectively replaced with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), Ξ1<0 holds for all ∑(t) satisfying ∑tT∑(t)≤I if (41) is satisfied.
By replacing respectivelyA0 and B0 in (27) with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), we have(44)Ξ1=Π4+YΨtZ+YΨtZT,where Π4 is given in (43), and Y, Z, and Ψ(t) satisfy(45)Y=μα000000000000IN⊗F1T000IN⊗F1T000IN⊗F2T,(46)Z=00-HT⊗ETX2T-HT⊗ETX2TIN⊗ETP10000000IN⊗ETP200-HT⊗ETX2T0-HT⊗ETX2TIN⊗ETP100,(47)Ψt=IN⊗∑Tt000IN⊗∑Tt000IN⊗∑Tt.
SinceΠ4 is symmetric, by using Lemma 5, we know that Ξ1<0 holds for any ∑(t) satisfying ∑tT∑(t)≤I if there exists a scalar constant ε>0 such that(48)Π4+εZTZ+ε-1YYT<0.
By using (45) and (46), W in (42) can be rewritten as W=εZYT/ε. Thus, the inequality (48) is equivalent to(49)Π4+WTW<0.
By using the Schur complement formula, (49) can be rewritten as (41). Thus, based on Theorem 20, for the uncertain nonlinear FOMASs (1) and (4) satisfying Assumptions 6, 8, 16, and 17 and the conditions in Lemmas 9 and 12, two problems shown in (19) and (20) can be solved simultaneously if (41) is satisfied.
This ends the proof of Theorem23.Remark 24.
[57] investigates the performance index based perfect synchronization of integer-order linear uncertain systems, where there is only a leader and a follower, and there is no coupling among parameters. [35] proposes the observer-type ILC protocol for the perfect consensus of linear IOMASs without any uncertainty and any parameter coupling. Compared to [35, 57], the results in this paper have the following distinct differences: (a) Each agent is described by the uncertain fractional-order nonlinear differential equation with linear coupling relation among parameters. (b) A novel performance index is designed as the form of the fractional integral of control input errors, and is used together with the Lyapunov-like method to derive the consensus sufficient conditions that can guarantee the perfect tracking of consensus and the better learning transients of ILC law. Moreover, if α=1, ΔA(t)=0, ΔB(t)=0 and f(·)=0, then the model in this paper will degenerate into the model of linear IOMAS in [35], Theorem 20 with α=1 and l=0 can solve the consensus problem this kind of special IOMASs.Remark 25.
Ifμ=1/α, then the FOMASs (1) and (4) will degenerate into the special uncertain nonlinear FOMASs without any parameter coupling. If α=1, then the FOMASs (1) and (4) will degenerate into the uncertain nonlinear IOMASs. Theorem 23 presents the sufficient conditions to guarantee that above two kinds of special uncertain nonlinear MASs can achieve the perfect tracking of consensus and the monotonic convergence of input errors simultaneously. The results show that Theorems 23 has broader applicability in the perfect consensus control of MASs. The FOMASs in this paper consist of N autonomous agents with the nonlinear dynamics satisfying the global Lipschitz condition. The consensus perfect tracking problem of T-S fuzzy systems [58] should be a further investigated topic, but it is still a challenging problem to design the distributed ILC protocol for the perfect consensus of MASs consisting of T-S fuzzy subsystems.Remark 26.
[40, 41, 59] design the observer-based repetitive-control systems for the systems with periodic time-varying uncertainties, with unknown aperiodic disturbances and without any uncertainties, respectively. The repetitive control (RC) process is also an iterative learning process (ILP). However, there exist the following differences between the ILPs of [40, 41, 59] and ILP of this paper. The ILPs of [40, 41, 59] are operating on an infinite time horizon, the state of the RC progresses continuously from one period to the next period (i.e., the state at the beginning of a period is the same as the final state of the system in the previous period). Thus the ILPs of [40, 41, 59] do not have time resetting or any iteration axis, and the RC controllers in [40, 41, 59] are merely related to time t. In contrast, in the ILP of this paper, the operation time is finite, that is t∈[0,T]. The identical initial condition (i.e., the initial state tracking error is zero in each iteration) is required (See Assumption 16). This requirement indicates that both time and state need to be reset after each iteration. Thus the ILP of this paper has a time axis and an iteration axis, and the controller is related to both time t and iterative learning step k.


## 3.1. Nonlinear FOMASs without Any Uncertainty

In this subsection, bothA(t) and B(t) are assumed to be the known matrices without any uncertainty, that is, A(t)=A0 and B(t)=B0. In this case, we can state the following two corollaries:Corollary 18.
Let the positive definite matrixQ1=Q1T∈Rn×n, and assume that Assumption 6 holds. If A0 is Hurwitz, and there exist a positive definite matrix P1=P1T satisfying(25)A0TP1+P1A0+l2In+Q1P1P1-In<0.then the fractional-order nonlinear agent (4) without any uncertainty under no control is stable.Corollary 19.
Let the positive definite matrixQ2=Q2T∈Rn×n, and assume that Assumption 6 holds. If the observer gain matrix L is chosen such that A0-LC is Hurwitz, and there exists a positive definite matrix P2=P2T∈Rn×n satisfying(26)A0-LCTP2+P2A0-LC+l2In+Q2P2P2-In<0,then the observation error is bounded, and converges asymptotically to zero as time goes to infinity.Corollaries18 and 19 can be directly derived from Lemmas 9 and 12, respectively. Thus their proofs are omitted.Based on the previous preparation, we can state the following results.Theorem 20.
Assume that Assumptions6, 16, and 17 and the conditions in Corollaries 18 and 19 are satisfied for the nonlinear FOMASs (1) and (4) without any uncertainty, and let the observer (6) and the consensus protocol (11) are applied. If, for a scalar γ∈(0,1), there exist matrices Xi,i=1,2, satisfying(27)Ξ1=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗IN⊗symA0TP1+IN⊗Q1+IN⊗P1P1+l2INn-IN⊗P1LCIN⊗P1B0∗∗∗∗-IN⊗CTLTP1IN⊗symA0-LCTP2+IN⊗Q2+IN⊗P2P2+l2INn0∗∗∗∗∗∗-γ2INp<0,where P1 and Q1 (P2 and Q2) are given in Corollary 18 (Corollary 19). Then both the input errors and the output tracking errors converge to zero for all t∈[0,T] as the iteration step goes to infinity and the monotonic convergence of input errors is obtained for all k∈Z+. Moreover, if (27) holds, then the learning gain matrices K and Γ are determined by(28)K=X1,Γ=X2.Proof.
(a) Proof of (19)
Let(29)P=diagIN⊗P1,IN⊗P2,Q=diagIN⊗Q1,IN⊗Q2.
Note that bothP=PT and Q=QT are positive definite, we can define a Lyapunov function as(30)Vθkt=θkTtPθkt+1Γα∫0tt-sα-1θkTsQθksds.It is easy to know that V(θk(t)) is positive semidefinite. Moreover, according to Assumption 16, we obtain that V(θk(0))=0.
Letξk(t)=θkT(t)δukT(t)T. By Definition 2 and Properties 3 and 4, we have(31)DtαVθkt≤DtαθktTPθkt+θkTtPDtαθkt+θkTtQθkt=R1θkt+R2δukt+ΨktTPθkt+θkTtPR1θkt+R2δukt+Ψkt+θkTtQθkt=θkTtR1TP+PR1+Qθkt+θkTtPR2δukt+δukTtR2TPθkt+2θkTtPΨkt≤θktδuktTR1TP+PR1+QPR2∗0θktδukt+θkTtPPθkt+ϕktφktTϕktφkt≤θktδuktTR1TP+PR1+Q+PPPR2∗0θktδukt+l2ekTtekt+l2x~ktTx~kt≤θktδuktTR1TP+PR1+Q+PP+l2I2NnPR2∗0θktδukt=ξkTtΠ1ξkt,where(32)Π1=R1TP+PR1+Q+PP+l2I2NnPR2∗0,R1=IN⊗A0-IN⊗LC0IN⊗A0-LC,R2=IN⊗B00.
Based on (17), we have(33)δuk+1Ttδuk+1t=R3θkt+R4δukt-H⊗ΓϕktTR3θkt+R4δukt-H⊗Γϕkt=θkTtR3TR3θkt+δukTtR4TR3θkt-ϕkTtHT⊗ΓTR3θkt+θkTtR3TR4δukt+δukTtR4TR4δukt-ϕkTtHT⊗ΓTR4δukt-θkTtR3TH⊗Γϕkt-δukTtR4TH⊗Γϕkt+ϕkTtHT⊗ΓTH⊗Γϕkt=ξkTtR3TR4TR3R4ξkt-2θkTtR3TH⊗Γϕkt-2δukTtR4TH⊗Γϕkt+ϕkTtHT⊗ΓTH⊗Γϕkt≤ξkTtR3TR4TR3R4ξkt+θkTtR3TR3θkt+δukTtR4TR4δukt+3ϕkTtHT⊗ΓTH⊗Γϕkt=ξkTtΩ2T0R3TR3T0R4T0R4TΩ200R4R30R3R4ξkt,where(34)Ω2=3H⊗Γ0,R3=-H⊗K+ΓA0H⊗ΓLC,R4=INp-H⊗ΓB0.
By Property3, (31) and (33), the index J(k,γ) can be rewritten as(35)Jk,γ=Jk,γ+1Γα∫0TT-sα-1DsαVθksds-VθkT+Vθk0≤1Γα∫0TT-sα-1δuk+1Tsδuk+1s-γ2δukTsδuks+DsαVθksds=1Γα∫0TT-sα-1ξkTsΠ2ξksds,where the matrix Π2 satisfies(36)Π2=R1TP+PR1+Q+PP+l2I2NnPR2∗-γ2INp-Ω2T0R3TR3T0R4T0R4T-I4NpΩ200R4R30R3R4.
By (29), (27) is rewritten as(37)Ξ1=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗symIN⊗P1A0-IN⊗P1LC0IN⊗P2A0-P2LC+Q+IN⊗P1P100IN⊗P2P2+l2I2NnIN⊗P1B00∗∗∗∗∗-γ2INp<0.
When (27) is satisfied, by selecting the learning gain matrices in (28) and substituting them into (37), we have(38)Ξ1=-INp000Ω20∗-INp000R4∗∗-INp0R30∗∗∗-INpR3R4∗∗∗∗R1TP+PR1+Q+PP+l2I2NnPR2∗∗∗∗∗-γ2INp<0.
Based on the Schur complement formula, it is easy to know from (38) that Π2<0 is satisfied, which indicates that J(k,γ)<0 in (22) holds for all non-zero vectors ξk(t). That is, the monotonic convergence of input errors shown in (19) is achieved.
(b) The proof of (20)
LetΠ3=R1TP+PR1+Q+PP+l2I(2Nn)PR2∗-γ2I(Np). Applying the Schur complement formula to (38), we obtain that Π3<0. Thus, based on (31) and (33), we have(39)DtαVθkt-γ2δukTtδukt≤ξkTtΠ1ξkt+ξkTtdiag0,-γ2INpξkt=ξkTtΠ3ξkt≤0.
From (30), (31), and (39), we have(40)θkTtPθkt≤Vθkt≤1Γα∫0tγ2t-sα-1δukTsδuksds+Vθk0=1Γα∫0tγ2t-sα-1δukTsδuksds.
When (27) is satisfied, it is easy to know that limk→+∞⁡δuk(t)2=0 holds. Thus, taking the limit of k→+∞ on both sides of (40), we obtain that limk→+∞⁡θk(t)2=0, which indicates that limk→+∞⁡yd(t)-yk,j(t)2=0 holds. That is, the perfect tracking of consensus shown in (20) is achieved.
This ends the proof of Theorem20.Remark 21.
Theorem20 provides the sufficient condition in terms of LMIs, which belongs to the category of ILC where a consensus convergence condition is related to the learning gain matrices, the known plant knowledge and the structure of communication graph [33]. However, compared to [33], the obtained LMI condition not only provides a design method of learning gain matrices to achieve the perfect tracking of consensus, but also guarantees the monotonic convergence of the input errors and achieves the good learning transients.Remark 22.
Among the existing results about the consensus of FOMASs [15–21, 36–38], the controllers are merely related to time t, and the asymptotic consensus problem or the finite time consensus problem is addressed. In contrast, this paper has the following two distinct differences: (a) The controllers (11) is related to both time t and iterative learning step k. (b) The perfect tracking of consensus is obtained. That is, the consensus will be maintained perfectly in the finite time interval t∈[0,T] as the iteration step k goes to infinity.


## 3.2. Extension to Uncertain Nonlinear FOMASs with Parameter Coupling Relations

By using the knowledge of plant as the priors, Theorem20 provides an LMI condition guaranteeing the perfect tracking of consensus and the monotonic convergence of the input errors. However, if the parameters of nonlinear FOMASs are subject to the perturbation, Theorem 20 is no longer suitable, thus should be further improved. Next, by studying the uncertain nonlinear FOMASs satisfying Assumptions 6, 8, 16, and 17, we obtain Theorem 23.Theorem 23.
Assume that Assumptions6, 8, 16, and 17 and the conditions in Lemmas 9 and 12 are satisfied for the nonlinear FOMASs (1) and (4), and let the observer (6) and the consensus protocol (11) are applied. If, for a scalar γ∈(0,1), there exist matrices Xi,i=1,2 satisfying(41)-I6NnWWTΠ4<0,where W and Π4 are given in (42) and (43), ε is a known positive scalar constant, P1 and Q1 (P2 and Q2) are given in Lemma 9 (Lemma 12). Then both the input errors and the output tracking errors converge to zero for all t∈[0,T] as the iteration step goes to infinity and the monotonic convergence of input errors is obtained for all k∈Z+. Moreover, if (41) holds, then the learning gain matrices K and Γ are determined by (28).(42)W=00-εHT⊗ETX2T-εHT⊗ETX2TεIN⊗ETP10000000εIN⊗ETP200-εHT⊗ETX2T0-εHT⊗ETX2TεIN⊗ETP1000000μαεIN⊗F10000000μαεIN⊗F10000000μαεIN⊗F2,(43)Π4=-INp0003H⊗X200∗-INp0000INp-H⊗X2B0∗∗-INp0-H⊗X1+X2A0H⊗X2LC0∗∗∗-INp-H⊗X1+X2A0H⊗X2LCINp-H⊗X2B0∗∗∗∗IN⊗symA0TP1+IN⊗Q1+IN⊗P1P1+l2INn-IN⊗P1LCIN⊗P1B0∗∗∗∗-IN⊗CTLTP1IN⊗symA0-LCTP2+IN⊗Q2+IN⊗P2P2+l2INn0∗∗∗∗∗∗-γ2INp,Proof.
According to Theorem20, we know that the proof of Theorem 23 will be accomplished if Ξ1<0 is satisfied when A0 and B0 in (27) are replaced with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), respectively. Thus, we should prove that after A0 and B0 in (27) are respectively replaced with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), Ξ1<0 holds for all ∑(t) satisfying ∑tT∑(t)≤I if (41) is satisfied.
By replacing respectivelyA0 and B0 in (27) with A(t)=A0+μαΔA(t) and B(t)=B0+μαΔB(t), we have(44)Ξ1=Π4+YΨtZ+YΨtZT,where Π4 is given in (43), and Y, Z, and Ψ(t) satisfy(45)Y=μα000000000000IN⊗F1T000IN⊗F1T000IN⊗F2T,(46)Z=00-HT⊗ETX2T-HT⊗ETX2TIN⊗ETP10000000IN⊗ETP200-HT⊗ETX2T0-HT⊗ETX2TIN⊗ETP100,(47)Ψt=IN⊗∑Tt000IN⊗∑Tt000IN⊗∑Tt.
SinceΠ4 is symmetric, by using Lemma 5, we know that Ξ1<0 holds for any ∑(t) satisfying ∑tT∑(t)≤I if there exists a scalar constant ε>0 such that(48)Π4+εZTZ+ε-1YYT<0.
By using (45) and (46), W in (42) can be rewritten as W=εZYT/ε. Thus, the inequality (48) is equivalent to(49)Π4+WTW<0.
By using the Schur complement formula, (49) can be rewritten as (41). Thus, based on Theorem 20, for the uncertain nonlinear FOMASs (1) and (4) satisfying Assumptions 6, 8, 16, and 17 and the conditions in Lemmas 9 and 12, two problems shown in (19) and (20) can be solved simultaneously if (41) is satisfied.
This ends the proof of Theorem23.Remark 24.
[57] investigates the performance index based perfect synchronization of integer-order linear uncertain systems, where there is only a leader and a follower, and there is no coupling among parameters. [35] proposes the observer-type ILC protocol for the perfect consensus of linear IOMASs without any uncertainty and any parameter coupling. Compared to [35, 57], the results in this paper have the following distinct differences: (a) Each agent is described by the uncertain fractional-order nonlinear differential equation with linear coupling relation among parameters. (b) A novel performance index is designed as the form of the fractional integral of control input errors, and is used together with the Lyapunov-like method to derive the consensus sufficient conditions that can guarantee the perfect tracking of consensus and the better learning transients of ILC law. Moreover, if α=1, ΔA(t)=0, ΔB(t)=0 and f(·)=0, then the model in this paper will degenerate into the model of linear IOMAS in [35], Theorem 20 with α=1 and l=0 can solve the consensus problem this kind of special IOMASs.Remark 25.
Ifμ=1/α, then the FOMASs (1) and (4) will degenerate into the special uncertain nonlinear FOMASs without any parameter coupling. If α=1, then the FOMASs (1) and (4) will degenerate into the uncertain nonlinear IOMASs. Theorem 23 presents the sufficient conditions to guarantee that above two kinds of special uncertain nonlinear MASs can achieve the perfect tracking of consensus and the monotonic convergence of input errors simultaneously. The results show that Theorems 23 has broader applicability in the perfect consensus control of MASs. The FOMASs in this paper consist of N autonomous agents with the nonlinear dynamics satisfying the global Lipschitz condition. The consensus perfect tracking problem of T-S fuzzy systems [58] should be a further investigated topic, but it is still a challenging problem to design the distributed ILC protocol for the perfect consensus of MASs consisting of T-S fuzzy subsystems.Remark 26.
[40, 41, 59] design the observer-based repetitive-control systems for the systems with periodic time-varying uncertainties, with unknown aperiodic disturbances and without any uncertainties, respectively. The repetitive control (RC) process is also an iterative learning process (ILP). However, there exist the following differences between the ILPs of [40, 41, 59] and ILP of this paper. The ILPs of [40, 41, 59] are operating on an infinite time horizon, the state of the RC progresses continuously from one period to the next period (i.e., the state at the beginning of a period is the same as the final state of the system in the previous period). Thus the ILPs of [40, 41, 59] do not have time resetting or any iteration axis, and the RC controllers in [40, 41, 59] are merely related to time t. In contrast, in the ILP of this paper, the operation time is finite, that is t∈[0,T]. The identical initial condition (i.e., the initial state tracking error is zero in each iteration) is required (See Assumption 16). This requirement indicates that both time and state need to be reset after each iteration. Thus the ILP of this paper has a time axis and an iteration axis, and the controller is related to both time t and iterative learning step k.


## 4. Simulation Results

In this section, to illustrate the effectiveness of the proposed methods, we present two numerical simulation examples.


### 4.1. Example 1

Consider the leader-following FOMASs, which are described by (1) and (4) and consist of a leader agent and three follower agents over the fixed communication graph shown in Figure 1. It is easy to know that the communication graph in Figure 1 satisfies Assumption 17, and H=2-1001-1-1-12. Moreover, the parameters in (1), (4) and (6) satisfy(50)A0=01-0.5-0.6,B0=01,ft,xt=0.1sin⁡x1t,0.1sin⁡x2tT,C=1,1,α=0.96,μ=0.05,and the uncertain matrices are expressed by ∑(t)=diag{1-2exp⁡(-3t),cos(5t)} and(51)E=010.80.8,F1=1.2001.2,F2=0.50.5.Thus the Lipschitz constant l=0.1. Take L=[0.09,0.19]T, it is easily check that both the system matrix A0 and the observer matrix A0-LC are Hurwitz. By solving the inequality (5) with Q1=0.1000.1 and the inequality (8) with Q2=0.3000.3, we obtain that P1=0.0468-0.0284-0.02840.2213 and P2=0.62830.24650.24650.9018. To perform the simulation, the initial states of each follower agent and its observer are set the same as that of leader agent, the control input of leader agent is set as ud(t)=sin(0.2πt), while the zero initial control input for each follower agent is adopted. Five scalars are prescribed as T=5, γ=0.1, ε1=1.5, ε2=2 and ε=4.Figure 1
Communication graph among all agents. Leader: agent 0; Followers: agents 1, 2, and 3 (color online).By using the Matlab LMI toolbox to solve the LMI (41) with Q1=0.1000.1, Q2=0.3000.3, P1=0.0468-0.0284-0.02840.2213 and P2=0.62830.24650.24650.9018, we obtain K=[0.0748,0.0931] and Γ=[-0.0034,0.1496], which indicates that the LMI (41) has feasible solutions. Thus, according to Theorem 23, we can conclude that under the observer (6) and the controller (11) with the obtained learning gain matrices, the perfect tracking of consensus and the monotonic convergence of input errors can be achieved simultaneously for uncertain nonlinear FOMASs (1) and (4) with the parameters satisfying (50) and (51). Next, we will further confirm the conclusion. By (2) and (3) satisfying (50) and (51), we randomly generate the system matrix A(t) and input matrix B(t), then obtain the numerical simulation results as shown in Figure 2. In Figure 2, both the output tracking errors and the control input errors converge monotonically to zero as the iteration step goes to infinity, which further confirms the previous conclusion.Figure 2
Under the observer (6) and the controller (11) with K=[0.0748,0.0931] and Γ=[-0.0034,0.1496]: (a) Output tracking errors versus iteration numbers; (b) input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)The superiority of proposed method becomes clear in the following comparison. Figure3 shows that the simulation results under the observer (6) and the controller (11) with the learning gain matrices K=[0,0] and Γ=[-0.6908,0.3945]. It is easy to verify that the condition I-H⊗(ΓB)2<1 holds. According to [54], the observer-type ILC law (11) is stable, which also can be confirmed by Figure 3. However, this ILC process is not monotonously convergent but is asymptotically stable, and the high-overshoots (as much as 94) are generated even though the input errors approach zero as the iteration step tends to infinity. Thus, these results show that merely the 2D analysis of ILC cannot present the sufficient conditions guaranteeing the monotonic convergence of input errors.Figure 3
Under the observer (6) and the controller (11) with K=[0,0] and Γ=[-0.6908,0.3945]: (a) output tracking errors versus iteration numbers; (b) Input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)By comparing Figures2 and 3, it is easily to know that the proposed method can achieve the perfect tracking of consensus and the good learning transients of control inputs simultaneously.


### 4.2. Example 2

Consider the leader-following multiple Newcastle robot systems [60] consisting of a leader robot and six follower robots. The communication graph among all robots is shown in Figure 4. It is easy to know that the communication graph in Figure 4 satisfies Assumption 17, and H=2-10000-110000002-1-100002-1-10-10010000-101. The motion equations of leader robot and follower robots are respectively described by(52)mq¨dt+ctq˙dt+κtqdt=βtτdt,(53)mq¨k,jt+ctq˙k,jt+κtqk,jt=βtτk,jt,where k∈Z+,j=1,2,…,6. The mass m=2500kg, the viscous damping factor c(t)=c0+Δc(t)Ns/mm, the linear stiffness of the force sensor κ(t)=κ0+0.5Δκ(t)N/mm, and the coefficient of control inputs β(t)=β0+Δβ(t). c0=7Ns/mm, κ0=44.5N/mm and β0=12500 (maxt∈[0,T]⁡|Δc(t)|=0.8Ns/mm, maxt∈[0,T]⁡|Δκ(t)|=5.5Ns/mm and maxt∈[0,T]⁡|Δβ(t)|=100) are the nominal values (the perturbation scopes) of uncertain viscous damping factor, uncertain linear stiffness and uncertain coefficient of control inputs, respectively. The control inputs of leader robot τd(t)=1000sin⁡(0.5πt)N. Let xd(t)=[qdT(t),q˙dT(t)]T and xk,j(t)=[qk,jT(t),q˙k,jT(t)]T. Then, (52) and (53) can be rewritten respectively as (1) and (4), where(54)A0=01-17.8-2.8,B0=05,C=10,ft,xt=0,α=1,μ=1,E=0010,F1=2.20.3200,F2=00.04,∑t=diagsin0.1t,-cos⁡0.5t.Thus the Lipschitz constant l=0. Take L=[0.3,0.2]T, it is easily check that both the system matrix A0 and the observer matrix A0-LC are Hurwitz. By solving the inequality (5) with Q1=0.2000.2 and the inequality (8) with Q2=0.4000.4, we obtain that P1=2.53020.22000.22000.1524 and P2=3.13230.24910.24910.1842. To perform the simulation, the initial states of each follower robot and its observer are set the same as that of leader robot, the zero initial control input for each follower robot is adopted. Five scalars are prescribed as T=3, γ=0.01, ε1=0.1, ε2=0.15 and ε=0.01.Figure 4
Communication graph among all robots. Leader: robot 0; Followers: robots 1, 2,…,6 (color online).By using the Matlab LMI toolbox to solve the LMI (41) with Q1=0.2000.2, Q2=0.4000.4, P1=2.53020.22000.22000.1524 and P2=3.13230.24910.24910.1842, we obtain K=[0.3113,0.2105] and Γ=[-0.0032,0.0549], which indicates that the LMI (41) has feasible solutions. Under the observer (6) and the controller (11) with the obtained learning gain matrices, we obtain the simulation results as shown in Figure 5. The results illustrate that the proposed method is still suitable for the perfect tracking problem of consensus and the monotonic convergence problem of input errors for uncertain linear IOMASs.Figure 5
Under the observer (6) and the controller (11) with K=[0.3113,0.2105] and Γ=[-0.0032,0.0549]: (a) output tracking errors versus iteration numbers; (b) input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)


## 4.1. Example 1

Consider the leader-following FOMASs, which are described by (1) and (4) and consist of a leader agent and three follower agents over the fixed communication graph shown in Figure 1. It is easy to know that the communication graph in Figure 1 satisfies Assumption 17, and H=2-1001-1-1-12. Moreover, the parameters in (1), (4) and (6) satisfy(50)A0=01-0.5-0.6,B0=01,ft,xt=0.1sin⁡x1t,0.1sin⁡x2tT,C=1,1,α=0.96,μ=0.05,and the uncertain matrices are expressed by ∑(t)=diag{1-2exp⁡(-3t),cos(5t)} and(51)E=010.80.8,F1=1.2001.2,F2=0.50.5.Thus the Lipschitz constant l=0.1. Take L=[0.09,0.19]T, it is easily check that both the system matrix A0 and the observer matrix A0-LC are Hurwitz. By solving the inequality (5) with Q1=0.1000.1 and the inequality (8) with Q2=0.3000.3, we obtain that P1=0.0468-0.0284-0.02840.2213 and P2=0.62830.24650.24650.9018. To perform the simulation, the initial states of each follower agent and its observer are set the same as that of leader agent, the control input of leader agent is set as ud(t)=sin(0.2πt), while the zero initial control input for each follower agent is adopted. Five scalars are prescribed as T=5, γ=0.1, ε1=1.5, ε2=2 and ε=4.Figure 1
Communication graph among all agents. Leader: agent 0; Followers: agents 1, 2, and 3 (color online).By using the Matlab LMI toolbox to solve the LMI (41) with Q1=0.1000.1, Q2=0.3000.3, P1=0.0468-0.0284-0.02840.2213 and P2=0.62830.24650.24650.9018, we obtain K=[0.0748,0.0931] and Γ=[-0.0034,0.1496], which indicates that the LMI (41) has feasible solutions. Thus, according to Theorem 23, we can conclude that under the observer (6) and the controller (11) with the obtained learning gain matrices, the perfect tracking of consensus and the monotonic convergence of input errors can be achieved simultaneously for uncertain nonlinear FOMASs (1) and (4) with the parameters satisfying (50) and (51). Next, we will further confirm the conclusion. By (2) and (3) satisfying (50) and (51), we randomly generate the system matrix A(t) and input matrix B(t), then obtain the numerical simulation results as shown in Figure 2. In Figure 2, both the output tracking errors and the control input errors converge monotonically to zero as the iteration step goes to infinity, which further confirms the previous conclusion.Figure 2
Under the observer (6) and the controller (11) with K=[0.0748,0.0931] and Γ=[-0.0034,0.1496]: (a) Output tracking errors versus iteration numbers; (b) input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)The superiority of proposed method becomes clear in the following comparison. Figure3 shows that the simulation results under the observer (6) and the controller (11) with the learning gain matrices K=[0,0] and Γ=[-0.6908,0.3945]. It is easy to verify that the condition I-H⊗(ΓB)2<1 holds. According to [54], the observer-type ILC law (11) is stable, which also can be confirmed by Figure 3. However, this ILC process is not monotonously convergent but is asymptotically stable, and the high-overshoots (as much as 94) are generated even though the input errors approach zero as the iteration step tends to infinity. Thus, these results show that merely the 2D analysis of ILC cannot present the sufficient conditions guaranteeing the monotonic convergence of input errors.Figure 3
Under the observer (6) and the controller (11) with K=[0,0] and Γ=[-0.6908,0.3945]: (a) output tracking errors versus iteration numbers; (b) Input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)By comparing Figures2 and 3, it is easily to know that the proposed method can achieve the perfect tracking of consensus and the good learning transients of control inputs simultaneously.


## 4.2. Example 2

Consider the leader-following multiple Newcastle robot systems [60] consisting of a leader robot and six follower robots. The communication graph among all robots is shown in Figure 4. It is easy to know that the communication graph in Figure 4 satisfies Assumption 17, and H=2-10000-110000002-1-100002-1-10-10010000-101. The motion equations of leader robot and follower robots are respectively described by(52)mq¨dt+ctq˙dt+κtqdt=βtτdt,(53)mq¨k,jt+ctq˙k,jt+κtqk,jt=βtτk,jt,where k∈Z+,j=1,2,…,6. The mass m=2500kg, the viscous damping factor c(t)=c0+Δc(t)Ns/mm, the linear stiffness of the force sensor κ(t)=κ0+0.5Δκ(t)N/mm, and the coefficient of control inputs β(t)=β0+Δβ(t). c0=7Ns/mm, κ0=44.5N/mm and β0=12500 (maxt∈[0,T]⁡|Δc(t)|=0.8Ns/mm, maxt∈[0,T]⁡|Δκ(t)|=5.5Ns/mm and maxt∈[0,T]⁡|Δβ(t)|=100) are the nominal values (the perturbation scopes) of uncertain viscous damping factor, uncertain linear stiffness and uncertain coefficient of control inputs, respectively. The control inputs of leader robot τd(t)=1000sin⁡(0.5πt)N. Let xd(t)=[qdT(t),q˙dT(t)]T and xk,j(t)=[qk,jT(t),q˙k,jT(t)]T. Then, (52) and (53) can be rewritten respectively as (1) and (4), where(54)A0=01-17.8-2.8,B0=05,C=10,ft,xt=0,α=1,μ=1,E=0010,F1=2.20.3200,F2=00.04,∑t=diagsin0.1t,-cos⁡0.5t.Thus the Lipschitz constant l=0. Take L=[0.3,0.2]T, it is easily check that both the system matrix A0 and the observer matrix A0-LC are Hurwitz. By solving the inequality (5) with Q1=0.2000.2 and the inequality (8) with Q2=0.4000.4, we obtain that P1=2.53020.22000.22000.1524 and P2=3.13230.24910.24910.1842. To perform the simulation, the initial states of each follower robot and its observer are set the same as that of leader robot, the zero initial control input for each follower robot is adopted. Five scalars are prescribed as T=3, γ=0.01, ε1=0.1, ε2=0.15 and ε=0.01.Figure 4
Communication graph among all robots. Leader: robot 0; Followers: robots 1, 2,…,6 (color online).By using the Matlab LMI toolbox to solve the LMI (41) with Q1=0.2000.2, Q2=0.4000.4, P1=2.53020.22000.22000.1524 and P2=3.13230.24910.24910.1842, we obtain K=[0.3113,0.2105] and Γ=[-0.0032,0.0549], which indicates that the LMI (41) has feasible solutions. Under the observer (6) and the controller (11) with the obtained learning gain matrices, we obtain the simulation results as shown in Figure 5. The results illustrate that the proposed method is still suitable for the perfect tracking problem of consensus and the monotonic convergence problem of input errors for uncertain linear IOMASs.Figure 5
Under the observer (6) and the controller (11) with K=[0.3113,0.2105] and Γ=[-0.0032,0.0549]: (a) output tracking errors versus iteration numbers; (b) input errors δuk(t)2 versus iteration numbers (color online).
(a) (b)


## 5. Conclusions

In this paper, an observer-type fractional-order ILC protocol is designed to solve the perfect tracking problem of consensus and the monotonic convergence problem of input errors for the uncertain nonlinear fractional-order multiagent systems (FOMASs), where there exist the linear coupling relations among the fractional order, and the perturbation of the systems matrix and input matrix. By applying the 2D analysis method to analyzing the iterative learning process of such kind of FOMASs, the design problem of a robust monotonically convergent iterative learning law can be transformed into a robust control problem of the tracking error system. Based on the constructed performance index in terms of the fractional integral, the Lyapunov-like method is applied to derive the LMI conditions guaranteeing simultaneously the perfect tracking of consensus and the monotonic convergence of the input errors. The effectiveness of the proposed method is illustrated in the numerical simulation results including comparison to those by the traditional ILC method, which shows the superiority of the novel method over the traditional ILC method in terms of achieving good transients of the iterative learning process. A future work may be to consider an extension of the proposed methods to the heterogeneous FOMASs.



---

*Source: 9734094-2019-11-07.xml*