# Fractional-Order Iterative Learning Control with Initial State Learning for a Class of Multiagent Systems

Authors: Xungen Li; Shuaishuai Lv; Mian Pan; Qi Ma; Wenyu Cai
Journal: Complexity
 (2020)

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## Abstract

To solve the consensus problem of fractional-order multiagent systems with nonzero initial states, both open- and closed-loop PDα-type fractional-order iterative learning control are presented. Considering the nonzero states, an initial state learning mechanism is designed. The finite time convergences of the proposed methods are discussed in detail and strictly proved by using Lebesgue-p norm theory and fractional-order calculus. The convergence conditions of the proposed algorithms are presented. Finally, some simulations are applied to verify the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Fractional-order multiagent systems (FOMASs) is composed of multiple agents, which can coordinate with each other to perceive the external environment, and apply fractional-order calculus principle. Due to the autonomy, fault tolerance, flexibility, scalability, and collaboration capabilities of the FOMASs, it can be applied to the intelligent environment perception and intelligent operation, such as air formation control, traffic vehicle control, data convergence, sensor networks, and so on [1–4]. In order to realize the wide application of FOMASs, it is necessary to design the coordinated control effectively, including consensus control, formation control, coalescence control, and rendezvous control. And the consensus problem is the basic problem in FOMASs distributed coordination control. Its purpose is to design an appropriate distributed consensus control protocol based on the neighbor states of the agent and its own state information, so that the states of all the agents converge to the same value at a specific position or a certain moment.The consensus problem of FOMASs was studied in [5] for the first time, in which the relationship between the consensus problem of FOMAS and the number of agents and fractional orders was discussed, and some control strategies were given to improve the convergence speed of the FOMASs. In the same year, Cao and Ren [6] also applied the consensus theory to the formation control problem of FOMASs. Since then, the research and application of FOMASs consensus problems have been emerging, including linear fractional-order multiagents [7–10] and nonlinear fractional-order multiagents [11–14]. Song and Cao [7] used the stability theory of FOSs and linear matrix inequality to study the consensus problem of linear FOMASs. And then they further considered the robust consensus problem of linear FOMASs when the fractional order satisfies α∈0,2[8]. Yu et al. [9] used the algebraic graph theory tool and the Lyapunov method to study the consensus problem of nonlinear FOMASs with a leader-following structure. Similarly, in [10, 11], the adaptive control and the sampling data control were designed to solve the consensus problem of nonlinear and linear FOMASs with and without leader-following structure, and some sufficient and necessary conditions related to fractional order, coupling gain, and Laplacian matrix spectrum were obtained to ensure that the system can achieve consensus. For the study of nonlinear FOMASs, there are also literatures [12–14].However, most of the research just consider the asymptotic convergence problem of FOMASs, which means the tracking errors of the fractional-order agents gradually converge to zero as time increases. On some special occasions, such as industrial automatic production lines, the asymptotic convergence cannot meet the actual demands. As we all know, fractional-order iterative learning control (FOILC) methods for repetitive running systems can achieve complete tracking problems in finite time [15,16]. In [17,18], both distributed Dα- and PDα-type FOILC were proposed and applied to linear FOMASs with fixed topology. Furthermore, for the linear time-varying integer-order system, Luo et al. proposed a FOILC framework with initial state learning and presented sufficient and necessary conditions for open-loop and closed-loop Dα-type FOILC. But for FOMASs, it has not been researched using open and closed FOILC.In the literature [17], the consensus problem of FOMASs is discussed using FOLIC. However, the authors just considered the zero initial states of FOMASs, which must ensure the strict positioning of the initial state during the iteration process. In this paper, for linear time-varying FOMASs with fixing the initial states over the directed graph, we design several fractional iterative learning controllers with the initial states learning algorithms. The contributions are summarized as follows. First, considering the nonzero initial state of FOMASs, we propose three different forms of fractional-order iterative learning updating laws. Second, an initial state learning algorithm together with the FOILC updating laws is designed. Finally, the convergences of the proposed algorithm are discussed and the convergence conditions are presented. The theoretical analysis and simulation experiments verify the effectiveness of the proposed method. The results show that both the tracking errors and the nonzero initial states can tend to zero in finite time as the iterative number increases.The remainder of this paper is organized as follows. Section2 overviews the related theories related to this article, including the graph theory, the definition of fractional calculus, and the problem formulation. The algorithm design and analysis employing FOILC with initial learning are discussed in Section 3. Section 4 demonstrates the simulation results to verify the effectiveness of the proposed methods. And briefly, conclusions are presented in Section 5.


## 2. Preliminaries

In this part, first, we introduce some basic definitions, lemmas, and properties, which will be used in the following sections.


### 2.1. Graph Theory

ConsiderN multiagents with the same dynamic. The direct graph G=V,E,M is used to describe the information transfer between multiagents, where V=v1,…,vN is the node set, E⊆V×V is the edge set, and M=aikN×N is the adjacency matrix of the direct graph. k,i∈E⊆V×V is a direct edge of the agents k and i. The set of neighbors of the ith agent is denoted by ℕi=k∈V:k,i∈E. The matrix element aik>0 represents node k passing information to node i; otherwise, aik=0. Here, the communication topology graph has no self-loop phenomenon, namely, ai,i=0. D=diagdi,i∈SN is defined as the degree matrix, where di=∑k=1Nai,k, and L=D−M is the Laplacian matrix of the direct graph.


### 2.2. The Norm

In this paper, the vector Euclidian norm and its induced matrix norm is defined as⋅. Im∈ℝm×m is the identity matrix. ℂm0,T is defined as a function set and the mth derivative of ℂm0,T∧ # is continuous over a finite time interval 0,T. ℝ and ℕ are the sets of real and natural numbers. SN=0,1,…,N. Denote the Kronecker product by ⊗, for some matrices A,B, C, and D, the following properties will be satisfied such that(1)kA⊗B=kA⊗B=A⊗kB,(2)A+B⊗C=A⊗C+B⊗C,(3)A⊗BC⊗D=AC⊗BD,(4)A⊗B=A⋅B.Definition 1.
Assuming the continuous vector functionf:0,T⟶ℝnft=f1t,f2t,…,fntT, the Lebesgue-p norm of ft is defined as(5)ftp=∫0Tmax1≤i≤n|fit|pdt1/p,1≤p<∞.Lemma 1 (see [19]).
Assuming the functionsgt∈Lq0,T and ht∈Lp0,T, then the convolution generalized Yong inequality of the functions gt and ht is(6)g∗htr≤gtqhtp,where 1≤p,q,r≤∞, 1/r=1/p+1/q−1, and g∗ht=∫0tgt−τhτdτ is the convolution integral of gt and ht. In particular, if r=p, the inequality is converted to g∗htp≤gt1htp.


### 2.3. Fractional Calculus

Definition 2 (see [21]).
The [21, 22] Riemann–Liouville fractional integrals of ft with order α∈0,1 are defined as(7)Dt0t−αft=1Γα∫t0tt−τα−1fτdτ,t>t0,Dtt−αft=1Γα∫tTτ−tα−1fτdτ,t<T,where Γ⋅ is gamma function. The left- and right-sided Caputo derivatives are(8)t0CDtαft=t0Dt-α-α+1dα+1dtα+1ft,t>t0,DtCTαft=tDT-α-α+1dα+1dtα+1ft,t<T,where α∈R+ and α means the integral part of α.Lemma 2 (see [20]).
Suppose the functionsft,gt are continuous in 0,T, and DtCTαft, D0CTαgtt∈0,T exist, then the fractional integration by parts is(9)∫0TDtCTαftgtdt=∫0TftD0CTαgtdt.Definition 3.
(see [20, 23]). The Mittag–Leffler function can be described as(10)Eα,βz=∑k=0∞zkΓαk+βα>0,β>0,z∈Cn×n.
Particularly, whenβ=1, we can obtain(11)Eα,1z=Eαz=∑k=0∞zkΓαk+1,α>0,z∈Cn×n.Lemma 3 (see [20]).
LetΦα,βA,t=tβ−1Eα,βAtαt∈0,+∞α>0,β>0,z∈Cn×n; then we have(12)DτCt1−αftΦα,1A,t−τ=Φα,αA,t−τ,0<α<1,ddτΦα,1A,t−τ=−Φα,αA,t−τA,α>0,A∈Cn×n.Lemma 4 (see [23]).
For the initial value problem(13)t0CDtαxt=Axt+But,xt0=x0.,A∈Cn×n,B∈Cn×p,0<α<1.The Volterra-type nonlinear integral equation can be obtained as(14)xt=Φα,1A,tx0+∫t0tΦα,αA,t−τBuτdτ.Property 1.
Ifft∈ℂt0,∞, then D1−αDαft=f1t,α∈0,1, where f1t=d/dtft.


## 2.1. Graph Theory

ConsiderN multiagents with the same dynamic. The direct graph G=V,E,M is used to describe the information transfer between multiagents, where V=v1,…,vN is the node set, E⊆V×V is the edge set, and M=aikN×N is the adjacency matrix of the direct graph. k,i∈E⊆V×V is a direct edge of the agents k and i. The set of neighbors of the ith agent is denoted by ℕi=k∈V:k,i∈E. The matrix element aik>0 represents node k passing information to node i; otherwise, aik=0. Here, the communication topology graph has no self-loop phenomenon, namely, ai,i=0. D=diagdi,i∈SN is defined as the degree matrix, where di=∑k=1Nai,k, and L=D−M is the Laplacian matrix of the direct graph.


## 2.2. The Norm

In this paper, the vector Euclidian norm and its induced matrix norm is defined as⋅. Im∈ℝm×m is the identity matrix. ℂm0,T is defined as a function set and the mth derivative of ℂm0,T∧ # is continuous over a finite time interval 0,T. ℝ and ℕ are the sets of real and natural numbers. SN=0,1,…,N. Denote the Kronecker product by ⊗, for some matrices A,B, C, and D, the following properties will be satisfied such that(1)kA⊗B=kA⊗B=A⊗kB,(2)A+B⊗C=A⊗C+B⊗C,(3)A⊗BC⊗D=AC⊗BD,(4)A⊗B=A⋅B.Definition 1.
Assuming the continuous vector functionf:0,T⟶ℝnft=f1t,f2t,…,fntT, the Lebesgue-p norm of ft is defined as(5)ftp=∫0Tmax1≤i≤n|fit|pdt1/p,1≤p<∞.Lemma 1 (see [19]).
Assuming the functionsgt∈Lq0,T and ht∈Lp0,T, then the convolution generalized Yong inequality of the functions gt and ht is(6)g∗htr≤gtqhtp,where 1≤p,q,r≤∞, 1/r=1/p+1/q−1, and g∗ht=∫0tgt−τhτdτ is the convolution integral of gt and ht. In particular, if r=p, the inequality is converted to g∗htp≤gt1htp.


## 2.3. Fractional Calculus

Definition 2 (see [21]).
The [21, 22] Riemann–Liouville fractional integrals of ft with order α∈0,1 are defined as(7)Dt0t−αft=1Γα∫t0tt−τα−1fτdτ,t>t0,Dtt−αft=1Γα∫tTτ−tα−1fτdτ,t<T,where Γ⋅ is gamma function. The left- and right-sided Caputo derivatives are(8)t0CDtαft=t0Dt-α-α+1dα+1dtα+1ft,t>t0,DtCTαft=tDT-α-α+1dα+1dtα+1ft,t<T,where α∈R+ and α means the integral part of α.Lemma 2 (see [20]).
Suppose the functionsft,gt are continuous in 0,T, and DtCTαft, D0CTαgtt∈0,T exist, then the fractional integration by parts is(9)∫0TDtCTαftgtdt=∫0TftD0CTαgtdt.Definition 3.
(see [20, 23]). The Mittag–Leffler function can be described as(10)Eα,βz=∑k=0∞zkΓαk+βα>0,β>0,z∈Cn×n.
Particularly, whenβ=1, we can obtain(11)Eα,1z=Eαz=∑k=0∞zkΓαk+1,α>0,z∈Cn×n.Lemma 3 (see [20]).
LetΦα,βA,t=tβ−1Eα,βAtαt∈0,+∞α>0,β>0,z∈Cn×n; then we have(12)DτCt1−αftΦα,1A,t−τ=Φα,αA,t−τ,0<α<1,ddτΦα,1A,t−τ=−Φα,αA,t−τA,α>0,A∈Cn×n.Lemma 4 (see [23]).
For the initial value problem(13)t0CDtαxt=Axt+But,xt0=x0.,A∈Cn×n,B∈Cn×p,0<α<1.The Volterra-type nonlinear integral equation can be obtained as(14)xt=Φα,1A,tx0+∫t0tΦα,αA,t−τBuτdτ.Property 1.
Ifft∈ℂt0,∞, then D1−αDαft=f1t,α∈0,1, where f1t=d/dtft.


## 3. Problem Description

ConsideringN homogeneous fractional-order linear time-delay MASs, it is assumed that each agent is completely nonregular and has repeated operational characteristics in a finite time interval. At the ith iteration, the dynamics of the jth agent can be described as follows:(15)D0Ctαxi,jt=Axi,jt+Bui,jt,yi,jt=Cxi,jt,where t∈0,T, D0Ctαxi,jt is the left-sided α-order derivative of xi,jt, α∈0,1. xi,jt∈ℝm is the state vectors, ui,jt∈ℝm1 and yi,jt∈ℝm2 are the input and output vectors, respectively, and A,B,C are constant matrices with m×m, m×m1, and m2×m.The expected trajectoryydt on the finite-time interval 0,T is generated by the virtual leader and it is described as(16)D0Ctαxdt=Axdt+Budt,ydt=Cxdt,where udt is the desired control input, and it is continuous and unique control input.If the virtual leader is the agent 0, the new graph can be expressed asG¯=0∪V,E¯,M¯, where E¯ and M¯ are the new edge set and the new adjacency matrix of G¯. The purpose is to design appropriate FOILC algorithms that enable each agent in the network topology to track the leader’s trajectory over a finite time interval.ξi,jt is defined as the distributed information of the jth agent, which is measured or received from other agents at the ith iteration. Consider(17)ξi,jt=∑k∈Njaj,kyi,kt−yi,jt+sjydt−yi,jt,where aj,k is the entry of adjacency matrix M, sj=1 if the jth agent can obtain the desired trajectory, and sj=0 otherwise.The tracking error of thejth agent is defined as ei,jt=ydt−yi,jt. Then, equation (17) can be reorganized as(18)ξi,jt=∑k∈Njaj,kei,jt−ei,kt+sjei,jt.Define column stack vectors in theith iteration(19)xit=xi,1tT,xi,2tT,…,xi,NtTT,eit=ei,1tT,ei,2tT,…,ei,NtTT,uit=ui,1tT,ui,2tT,…,ui,NtTT,ξit=ξi,1tT,ξi,2tT,…,ξi,NtTT.According to (19), (18) can be reorganized in a compact form(20)ξit=L+S⊗Imeit,where L is the Laplacian matrix of graph G, Im is unit matrix, and S=diagsj,j∈SN.Similarly, equation (15) can be rearranged as(21)D0Ctαxit=IN⊗Axit+IN⊗Buit.yit=IN⊗Cxit.


### 3.1. Open-LoopPDα-type FOILC

For FOMASs described by (15), considering the nonzero initial state, the open-loop PDα-type FOILC algorithm with initial state learning is proposed as follows:(22)ui+1,jt=ui,jt+ΓP1ξi,jt+ΓD1D0Ctαξi,jt,xi+1,j0=xi,j0+BΓD1ξi,j0.Similar to (20), the updating law (22) can be rewritten as(23)ui+1t=uit+L+S⊗ΓP1eit+L+S⊗ΓD1D0Ctαeit,xi+10=xi0+L+S⊗BΓD1ei0.In order to facilitate the convergence analysis of the proposed methods, the following assumptions hold.Assumption 1.
CB is of full column rank.Remark 1.
In order to guarantee the flawless tracking performance, a typical supposition, i.e., identical initialization condition, is needed to be made in the ILC design. Remember that accurate tracking can only be accomplished with perfect initial conditions.Assumption 2.
(see [17]). The graph G¯ contains a spanning tree with the leader being the root.Remark 2.
This supposition is a prerequisite for the FOMASs consensus tracking problem, which means all followers can receive the leader’s information directly or indirectly. Otherwise, due to the absence of data to make their control inputs accurate, the isolated agents cannot keep track of the leader’s trajectory.Theorem 1.
Consider the FOMASs (15) and under the communication graph G¯, if Assumption 1 and 2 are satisfied. Distributed PDα-type updating rule (23) is applied to the FOMASs (15). If the matrices A,B,C and the learning gains ΓP1 and ΓD1 satisfy the following condition:(24)ρ1=I−L+S⊗CBΓD1+β<1,where β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1, then limi⟶∞ei+1tp=0. Namely, the output yit converges uniformly to the desired trajectory ydt as i⟶∞.Proof.
The convergence discussed is as follows.
Based on Lemma4, we can write the FOMASs (15) as follows:(25)xit=Φα,1IN⊗A,txi0+∫0tΦα,αIN⊗A,t−τIN⊗Buiτdτ.
According to equalities (21), (23), and (25), we can obtain(26)ei+1t=1N⊗ydt−yi+1t=1N⊗ydt−yit−yi+1t−yit=eit−IN⊗Cxi+1t−xit=eit−IN⊗CΦα,1IN⊗A,txi+10−xi0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗Bui+1τ−uiτdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD1ei0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗BL+S⊗ΓP1eiτdτ−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗BL+S⊗ΓD1D0CTαeitdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD1ei0−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1eiτdτ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓD1D0CTαeiτdτwhere 1⋅ is a vector in which all entries are 1.
From Lemma2 and 3, we can see that(27)∫0tΦα,αIN⊗A,t−τL+S⊗BΓD1D0Ctαeitdτ=∫0tDτCt1−αΦα,1IN⊗A,t−τL+S⊗BΓD1D0Ctαeitdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1D0Ct1−αD0Ctαeitdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1ei′τdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1deiτ=Φα,1IN⊗A,t−τL+S⊗BΓD1eiτ|0t−∫0tddτΦα,1IN⊗A,t−τL+S⊗BΓD1eiτdτ=Φα,1IN⊗A,0L+S⊗BΓD1eit−Φα,1IN⊗A,tL+S⊗BΓD1ei0+∫0tΦα,1IN⊗A,t−τIN⊗AIN⊗BL+S⊗ΓD1eiτdτ=L+S⊗BΓD1eit−Φα,1IN⊗A,tL+S⊗BΓD1ei0+∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD1eiτdτ.
Taking (27) into (26), we further get(28)ei+1t=I−IN⊗CL+S⊗BΓD1eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1eiτdτ−IN⊗C∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD1eiτdτ=I−L+S⊗CBΓD1eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1+ABΓD1eiτdτ.
According to Lemma1, taking Lebesgue-p norm on both sides of (28), we achieve(29)ei+1tp≤I−L+S⊗CBΓD1+βeitp=ρ1eitp,where(30)β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1.
Recalling the condition ofρ<1, it deduces that(31)ei+1tp≤ρ1eitp≤ρ1ie1tp.
So, as the iterations number increases, i.e.,i⟶∞, we obtain(32)limi⟶∞ei+1tp=0.
It shows that the tracking errors of all the agents tend to reach zero in finite time wheni⟶∞. The proof is completed. When ΓP1=0, the open-loop PDα-type fractional-order algorithm degenerates into the Dα-type fractional-order algorithm, which has the following form:(33)ui+1t=uit+L+S⊗ΓD1eiαt.
Thus, the following corollary can be obtained.Corollary 1.
Consider the FOMASs (15) and under the communication graph G¯, if Assumptions 1 and 2 are satisfied. Distributed Dα-type updating rule (33) is applied to the FOMASs (15). Assuming that(34)I−L+S⊗CBΓD1+β0<1
holds for all [0,T], where β0=IN⊗CL+S⊗ABΓD1Φα,αIN⊗A,t1, then limi⟶∞ei+1tp=0. Namely, the output yit converges uniformly to the desired trajectory ydt as i⟶∞.Proof.
The process of proof is similar to Theorem1.


### 3.2. Closed-LoopPDα -Type FOILC

The closed-loopPDα -type FOILC updating law for the FOMASs (15) is designed as follows:(35)ui+1,jt=ui,jt+ΓP2ξi+1,jt+ΓD2D0Ctαξi+1,jt,xi+1,j0=xi,j0+BΓD2ξi+1,j0.Similar to (23), the updating law (35) can be rewritten by the Kronecker product as(36)ui+1t=uit+L+S⊗ΓP2ei+1t+L+S⊗ΓD2D0Ctαei+1t,xi+10=xi0+L+S⊗BΓD2ei+10,where L and S are the same as defined in (20).Theorem 2.
Consider the FOMASs (15) under a directed graph G¯, if Assumptions 1 and 2 hold. The closed-loop PDα-type FOILC described in (36) is applied for the system (15). If learning gains ΓP2 and ΓD2 satisfy(37)0<ρ2=1I+L+S⊗CBΓD2−1−γ−1<1,where(38)γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1,then limi⟶∞ei+1tp=0. Hence, the system outputs yit can fully track the desired trajectory ydt in a finite time when i⟶∞ for all t∈0,T; that is, limi⟶∞yit=ydt,t∈0,T.Proof.
From (15) and (36), we can get(39)ei+1t=eit−IN⊗Cxi+1t−xit=eit−IN⊗CΦα,1IN⊗A,txi+10−xi0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗Bui+1τ−uiτdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD2ei+10−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2ei+1τdτ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓD2D0Ctαei+1tdτ,where 1⋅ is a vector in which all entries are 1.
Similar to the derivation of (27), one can conclude that(40)∫0tΦα,αIN⊗A,t−τL+S⊗BΓD2D0Ctαei+1tdτ=L+S⊗BΓD2ei+1t−Φα,1IN⊗A,tL+S⊗BΓD2ei+10+∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD2ei+1τdτ.
Substituting (40) into (39), it yields(41)ei+1t=eit−IN⊗CL+S⊗BΓD2ei+1t−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2ei+1τdτ−IN⊗C∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD2ei+1τdτ=eit−L+S⊗CBΓD2ei+1τ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2+ABΓD2ei+1τdτ.
Therefore,(42)I+L+S⊗CBΓD2ei+1t=eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2+ABΓD2ei+1τdτ.
According to Assumption1, one can find a feedback gain matrix of differentiation ΓD2 such that I+L+S⊗CBΓD2 is a nonsingular matrix. Therefore, premultiplying by I+L+S⊗CBΓD2−1 on both sides of (42), taking Lebesgue-p norm, and adopting the generalized Young inequality of convolution integral, it can be concluded that(43)ei+1tp≤I+L+S⊗CBΓD2−1eitp+λei+1tp,where(44)γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1.
Further(45)ei+1tp≤1I+L+S⊗CBΓD2−1−γ−1eitp=ρ2eitp.
Recalling the condition ofρ2<1, according to inequality (43), it is deduced that(46)ei+1tp≤ρ2eitp≤ρ2ie1tp.
From (45), when the number of iterations is large enough, i.e., i⟶∞, we obtain(47)limi⟶∞ei+1tp⟶0.
So, it can be proved that the errors of all the fractional-order agents tend to zero asi⟶∞. For the FOMASs (15), ifΓP2=0 in (37), then the PDα -type FOILC will become Dα-type FOILC.(48)ui+1t=uit+L+S⊗ΓD2ei+1αt.
Thus, according to Theorem2, we can obtain a corollary as follows.Corollary 2.
For the FOMASs (15) under a directed graph G¯, suppose Assumptions 1 and 2 hold. If the learning gain ΓD2 in (48) is chosen such that(49)1I+L+S⊗CBΓD2−1−γ0−1<1,where(50)γ0=IN⊗CL+S⊗ABΓD2Φα,αIN⊗A,t1.
Then the tracking error satisfieslimi⟶∞ei+1tp=0. Namely, the outputs yit of the FOMASs (15) converge to the desired trajectory ydt uniformly in a finite time when i⟶∞, i.e., limi⟶∞yit=ydt,t∈0,T.Proof.
The proof process of the corollary is similar to Theorem2.


### 3.3. Open-Closed-LoopPDα-Type FOILC

Considering the FOMASs (15), an open-closed-loop PDα-type FOILC is designed as(51)ui+1,jt=ui,jt+ΓP1ξi,jt+ΓD1D0Ctαξi,jt+ΓP2ξi+1,jt+ΓD2D0Ctαξi+1,jt,xi+1,j0=xi,j0+BΓD1ξi,j0+ΓD2ξi+1,j0.Similar to (25), the updating law (51) can be rewritten by the Kronecker product as(52)ui+1t=uit+L+S⊗ΓP1eit+L+S⊗ΓD1D0Ctαeit+L+S⊗ΓP2ei+1t+L+S⊗ΓD2D0Ctαei+1t,xi+10=xi0+L+S⊗BΓD1ei0+ΓD2ei+10,where L and S are the same as defined in (20) and (36).Theorem 3.
Consider the FOMASs (15) under a directed graph G¯, if Assumptions 1 and 2 hold. Let the distributed closed-loop PDα-type FOILC described in (52) be applied for the system with learning gains ΓP1ΓP2,ΓD1, and ΓD2 satisfying(53)ρ2ρ1<1,where(54)ρ1=I−L+S⊗CBΓD1+β,ρ2=1I+L+S⊗CBΓD2−1−γ−1>0,β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1.Then limi⟶∞ei+1tp=0. Thus, the system outputs yit of the fractional-order agents converge to ydt when i⟶∞ for all t∈0,T; that is, limi⟶∞yit=ydt,t∈0,T.Remark 3.
According to the conditions of Theorems1 and 2, in the sense of Lebesgue-p norm, the convergence conditions of the proposed algorithms are determined by the learning gain and the properties of the system.


## 3.1. Open-LoopPDα-type FOILC

For FOMASs described by (15), considering the nonzero initial state, the open-loop PDα-type FOILC algorithm with initial state learning is proposed as follows:(22)ui+1,jt=ui,jt+ΓP1ξi,jt+ΓD1D0Ctαξi,jt,xi+1,j0=xi,j0+BΓD1ξi,j0.Similar to (20), the updating law (22) can be rewritten as(23)ui+1t=uit+L+S⊗ΓP1eit+L+S⊗ΓD1D0Ctαeit,xi+10=xi0+L+S⊗BΓD1ei0.In order to facilitate the convergence analysis of the proposed methods, the following assumptions hold.Assumption 1.
CB is of full column rank.Remark 1.
In order to guarantee the flawless tracking performance, a typical supposition, i.e., identical initialization condition, is needed to be made in the ILC design. Remember that accurate tracking can only be accomplished with perfect initial conditions.Assumption 2.
(see [17]). The graph G¯ contains a spanning tree with the leader being the root.Remark 2.
This supposition is a prerequisite for the FOMASs consensus tracking problem, which means all followers can receive the leader’s information directly or indirectly. Otherwise, due to the absence of data to make their control inputs accurate, the isolated agents cannot keep track of the leader’s trajectory.Theorem 1.
Consider the FOMASs (15) and under the communication graph G¯, if Assumption 1 and 2 are satisfied. Distributed PDα-type updating rule (23) is applied to the FOMASs (15). If the matrices A,B,C and the learning gains ΓP1 and ΓD1 satisfy the following condition:(24)ρ1=I−L+S⊗CBΓD1+β<1,where β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1, then limi⟶∞ei+1tp=0. Namely, the output yit converges uniformly to the desired trajectory ydt as i⟶∞.Proof.
The convergence discussed is as follows.
Based on Lemma4, we can write the FOMASs (15) as follows:(25)xit=Φα,1IN⊗A,txi0+∫0tΦα,αIN⊗A,t−τIN⊗Buiτdτ.
According to equalities (21), (23), and (25), we can obtain(26)ei+1t=1N⊗ydt−yi+1t=1N⊗ydt−yit−yi+1t−yit=eit−IN⊗Cxi+1t−xit=eit−IN⊗CΦα,1IN⊗A,txi+10−xi0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗Bui+1τ−uiτdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD1ei0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗BL+S⊗ΓP1eiτdτ−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗BL+S⊗ΓD1D0CTαeitdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD1ei0−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1eiτdτ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓD1D0CTαeiτdτwhere 1⋅ is a vector in which all entries are 1.
From Lemma2 and 3, we can see that(27)∫0tΦα,αIN⊗A,t−τL+S⊗BΓD1D0Ctαeitdτ=∫0tDτCt1−αΦα,1IN⊗A,t−τL+S⊗BΓD1D0Ctαeitdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1D0Ct1−αD0Ctαeitdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1ei′τdτ=∫0tΦα,1IN⊗A,t−τL+S⊗BΓD1deiτ=Φα,1IN⊗A,t−τL+S⊗BΓD1eiτ|0t−∫0tddτΦα,1IN⊗A,t−τL+S⊗BΓD1eiτdτ=Φα,1IN⊗A,0L+S⊗BΓD1eit−Φα,1IN⊗A,tL+S⊗BΓD1ei0+∫0tΦα,1IN⊗A,t−τIN⊗AIN⊗BL+S⊗ΓD1eiτdτ=L+S⊗BΓD1eit−Φα,1IN⊗A,tL+S⊗BΓD1ei0+∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD1eiτdτ.
Taking (27) into (26), we further get(28)ei+1t=I−IN⊗CL+S⊗BΓD1eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1eiτdτ−IN⊗C∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD1eiτdτ=I−L+S⊗CBΓD1eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP1+ABΓD1eiτdτ.
According to Lemma1, taking Lebesgue-p norm on both sides of (28), we achieve(29)ei+1tp≤I−L+S⊗CBΓD1+βeitp=ρ1eitp,where(30)β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1.
Recalling the condition ofρ<1, it deduces that(31)ei+1tp≤ρ1eitp≤ρ1ie1tp.
So, as the iterations number increases, i.e.,i⟶∞, we obtain(32)limi⟶∞ei+1tp=0.
It shows that the tracking errors of all the agents tend to reach zero in finite time wheni⟶∞. The proof is completed. When ΓP1=0, the open-loop PDα-type fractional-order algorithm degenerates into the Dα-type fractional-order algorithm, which has the following form:(33)ui+1t=uit+L+S⊗ΓD1eiαt.
Thus, the following corollary can be obtained.Corollary 1.
Consider the FOMASs (15) and under the communication graph G¯, if Assumptions 1 and 2 are satisfied. Distributed Dα-type updating rule (33) is applied to the FOMASs (15). Assuming that(34)I−L+S⊗CBΓD1+β0<1
holds for all [0,T], where β0=IN⊗CL+S⊗ABΓD1Φα,αIN⊗A,t1, then limi⟶∞ei+1tp=0. Namely, the output yit converges uniformly to the desired trajectory ydt as i⟶∞.Proof.
The process of proof is similar to Theorem1.


## 3.2. Closed-LoopPDα -Type FOILC

The closed-loopPDα -type FOILC updating law for the FOMASs (15) is designed as follows:(35)ui+1,jt=ui,jt+ΓP2ξi+1,jt+ΓD2D0Ctαξi+1,jt,xi+1,j0=xi,j0+BΓD2ξi+1,j0.Similar to (23), the updating law (35) can be rewritten by the Kronecker product as(36)ui+1t=uit+L+S⊗ΓP2ei+1t+L+S⊗ΓD2D0Ctαei+1t,xi+10=xi0+L+S⊗BΓD2ei+10,where L and S are the same as defined in (20).Theorem 2.
Consider the FOMASs (15) under a directed graph G¯, if Assumptions 1 and 2 hold. The closed-loop PDα-type FOILC described in (36) is applied for the system (15). If learning gains ΓP2 and ΓD2 satisfy(37)0<ρ2=1I+L+S⊗CBΓD2−1−γ−1<1,where(38)γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1,then limi⟶∞ei+1tp=0. Hence, the system outputs yit can fully track the desired trajectory ydt in a finite time when i⟶∞ for all t∈0,T; that is, limi⟶∞yit=ydt,t∈0,T.Proof.
From (15) and (36), we can get(39)ei+1t=eit−IN⊗Cxi+1t−xit=eit−IN⊗CΦα,1IN⊗A,txi+10−xi0−IN⊗C∫0tΦα,αIN⊗A,t−τIN⊗Bui+1τ−uiτdτ=eit−IN⊗CΦα,1IN⊗A,tL+S⊗BΓD2ei+10−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2ei+1τdτ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓD2D0Ctαei+1tdτ,where 1⋅ is a vector in which all entries are 1.
Similar to the derivation of (27), one can conclude that(40)∫0tΦα,αIN⊗A,t−τL+S⊗BΓD2D0Ctαei+1tdτ=L+S⊗BΓD2ei+1t−Φα,1IN⊗A,tL+S⊗BΓD2ei+10+∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD2ei+1τdτ.
Substituting (40) into (39), it yields(41)ei+1t=eit−IN⊗CL+S⊗BΓD2ei+1t−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2ei+1τdτ−IN⊗C∫0tΦα,1IN⊗A,t−τL+S⊗ABΓD2ei+1τdτ=eit−L+S⊗CBΓD2ei+1τ−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2+ABΓD2ei+1τdτ.
Therefore,(42)I+L+S⊗CBΓD2ei+1t=eit−IN⊗C∫0tΦα,αIN⊗A,t−τL+S⊗BΓP2+ABΓD2ei+1τdτ.
According to Assumption1, one can find a feedback gain matrix of differentiation ΓD2 such that I+L+S⊗CBΓD2 is a nonsingular matrix. Therefore, premultiplying by I+L+S⊗CBΓD2−1 on both sides of (42), taking Lebesgue-p norm, and adopting the generalized Young inequality of convolution integral, it can be concluded that(43)ei+1tp≤I+L+S⊗CBΓD2−1eitp+λei+1tp,where(44)γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1.
Further(45)ei+1tp≤1I+L+S⊗CBΓD2−1−γ−1eitp=ρ2eitp.
Recalling the condition ofρ2<1, according to inequality (43), it is deduced that(46)ei+1tp≤ρ2eitp≤ρ2ie1tp.
From (45), when the number of iterations is large enough, i.e., i⟶∞, we obtain(47)limi⟶∞ei+1tp⟶0.
So, it can be proved that the errors of all the fractional-order agents tend to zero asi⟶∞. For the FOMASs (15), ifΓP2=0 in (37), then the PDα -type FOILC will become Dα-type FOILC.(48)ui+1t=uit+L+S⊗ΓD2ei+1αt.
Thus, according to Theorem2, we can obtain a corollary as follows.Corollary 2.
For the FOMASs (15) under a directed graph G¯, suppose Assumptions 1 and 2 hold. If the learning gain ΓD2 in (48) is chosen such that(49)1I+L+S⊗CBΓD2−1−γ0−1<1,where(50)γ0=IN⊗CL+S⊗ABΓD2Φα,αIN⊗A,t1.
Then the tracking error satisfieslimi⟶∞ei+1tp=0. Namely, the outputs yit of the FOMASs (15) converge to the desired trajectory ydt uniformly in a finite time when i⟶∞, i.e., limi⟶∞yit=ydt,t∈0,T.Proof.
The proof process of the corollary is similar to Theorem2.


## 3.3. Open-Closed-LoopPDα-Type FOILC

Considering the FOMASs (15), an open-closed-loop PDα-type FOILC is designed as(51)ui+1,jt=ui,jt+ΓP1ξi,jt+ΓD1D0Ctαξi,jt+ΓP2ξi+1,jt+ΓD2D0Ctαξi+1,jt,xi+1,j0=xi,j0+BΓD1ξi,j0+ΓD2ξi+1,j0.Similar to (25), the updating law (51) can be rewritten by the Kronecker product as(52)ui+1t=uit+L+S⊗ΓP1eit+L+S⊗ΓD1D0Ctαeit+L+S⊗ΓP2ei+1t+L+S⊗ΓD2D0Ctαei+1t,xi+10=xi0+L+S⊗BΓD1ei0+ΓD2ei+10,where L and S are the same as defined in (20) and (36).Theorem 3.
Consider the FOMASs (15) under a directed graph G¯, if Assumptions 1 and 2 hold. Let the distributed closed-loop PDα-type FOILC described in (52) be applied for the system with learning gains ΓP1ΓP2,ΓD1, and ΓD2 satisfying(53)ρ2ρ1<1,where(54)ρ1=I−L+S⊗CBΓD1+β,ρ2=1I+L+S⊗CBΓD2−1−γ−1>0,β=IN⊗CL+S⊗BΓP1+ABΓD1Φα,αIN⊗A,t1γ=IN⊗CL+S⊗BΓP2+ABΓD2Φα,αIN⊗A,t1.Then limi⟶∞ei+1tp=0. Thus, the system outputs yit of the fractional-order agents converge to ydt when i⟶∞ for all t∈0,T; that is, limi⟶∞yit=ydt,t∈0,T.Remark 3.
According to the conditions of Theorems1 and 2, in the sense of Lebesgue-p norm, the convergence conditions of the proposed algorithms are determined by the learning gain and the properties of the system.


## 4. Simulation

In this section, five fractional-order agents are considered, including a virtual leader and four followers. The directed fixed communication topology among agents is shown in Figure1, where the fractional-order agents are labeled with 0, 1, 2, 3, and 4, respectively. The virtual leader has directed edges to agents 1 and 3.Figure 1
Communication graph among agents in the network.From Figure1, the Laplacian matrix L and the information transfer matrix S of the leader to the followers can be obtained as follows:(55)L=1−100−120−1001−10−1−12,S=diag1,0,1,0.The dynamic model of thejth agent is described as(56)Dαxjt=0.425−6xjt+1001ujt,yjt=0.85001xjt,Here, t∈0,1, α=0.75.Let the virtual leader be the given expected reference trajectory(57)yd1=t2+sin2πt,t∈0,1,yd2=sin2πt,t∈0,1.In the following simulations, the initial states of the followers at first iteration are set asx0,1=0.10.3T, x0,2=−0.5−0.7T , x0,3=0.20.4T , and x0,4=−0.60.8T . The control objective of the initial state is xd=00T and the initial control is set as u0,jt=0,j=1,2,3,4 for all agents.Case 1.
Open-loop PDα-type: the open-loop PDα-type is applied to the multiagent system (1). Based on Theorem 1, the gains are selected as ΓP1=0.4000.6,ΓD1=0.2000.3. Thus, we can calculate ρ1=I−L+D⊗ΓD1CB+β1=0.9421<1, which satisfies the convergence condition (26).
The simulation results are shown in Figures2–4. The initial states of the followers at the first iteration are x0,1=0.10.3T, x0,2=−0.5−0.7T, x0,3=0.20.4T, and x0,4=−0.60.8T. And the desired initial states of the four followers are zero; that is x0,j=00T for j = 1, 2, 3, 4. Figure 2 shows the initial state learning process. It can be seen that the initial states x1 and x2 of the multiagent at time zero have a large error from the desired state at the beginning of the iteration, because the initial control is set as u0,jt=0,j=1,2,3,4 for all agents. But as the number of iterations increases, the errors of the initial states gradually decrease. When the number of iterations reaches the 40th iteration, the initial state of x2 also converges to the desired initial state. And when the number of iterations reaches the 60th iteration, the initial state of x1 converges to the desired initial state. Figure 3 shows the output tracking results of y1 and y2. It can be seen that each subsystem does not track the desired trajectory at the 5th iteration. With the increase of the number of iterations, when it reaches the 100th iteration, both the outputs y1 and y2 of all the agents fully track the desired trajectory over the time period [0, 1]. We define the errors in the 2-norm sense at the ith iteration as yd,1−y1i,j2 and yd,2−y2i,j for j=1,2,3,4. Figure 4 depicts the tracking errors in each iteration; it shows that the tracking errors converge to zero as the iteration number increases. By the 60th iteration, the tracking errors of y1 of the four followers in the 2-norm sense are 0.000456, 0.000862, 0.000351, and 0.000785, respectively. By the 80th iteration, the tracking errors of y2 of the four followers in the 2-norm sense are 0.000648, 0.000978, 0.000596, and 0.000895, respectively.Figure 2
Initial state profile vs. iteration number by open-loop PDα-type. (a) Initial state learning of x1. (b) Initial state learning of x2.
(a)(b)Figure 3
The tracking results of all agents at different iterations by open-loop PDα-type. (a) Output y1 at the 5th iteration. (b) Output y1 at the 100th iteration. (c) Output y2 at the 5th iteration. (d) Output y2 at the 100th iteration.
(a)(b)(c)(d)Figure 4
The 2-norm of tracking errors for all agents in each interaction by open-loop PDα-type. (a) Tracking errors of y1 with iterations. (b) Tracking errors of y2 with iterations.
(a)(b)Case 2.
Closed-loop PDα-type: the initial inputs and initial state of the multiagents are the same as Case 1. Based on Theorem 2, we select the learning gains as ΓP2=0.504000.396,ΓD2=6007. Clearly, ρ2=1/I+H⊗ΓD2CB=0.2146<1; thus, the convergence condition can be satisfied.
Figures5–7 show the trajectory tracking performances employing the closed-loop PDα-type ILC scheme. As it can be seen from Figure 5, similar to the simulation results of Case 1, the initial state of the agents tends to reach the desired initial state as the iteration number increases. Figure 6 shows the outputs y1 and y2 with closed-loop PDα-type ILC at the 5th and 30th iterations. From Figure 6, the trajectories y1 and y2 of the followers can track the desired trajectory generated by the leader as the iteration number increases over the time period [0, 1]. Figure 7 shows the tracking errors of y1 and y2 of the four followers in 2-norm sense with the number of iterations. It can be seen that the errors gradually decrease and approach zero as the number of iteration increases. By the 30th iteration, the tracking errors of y1 of the four followers in 2-norm sense are 0.000279, 0.000648, 0.000324, and 0.000472. The tracking errors of y2 of the four followers in 2-norm sense are, respectively, 0.000187, 0.000547, 0.000298, and 0.000385. Besides, compared with the open-loop PDα-type, the closed-loop FOILC performs better and has faster convergence speed than the open-loop one.Figure 5
Initial state profile vs. iteration number by closed-loop PDα-type. (a) Initial state learning of x1. (b) Initial state learning of x2.
(a)(b)Figure 6
The tracking results of all agents at different iterations by closed-loop PDα-type. (a) Trajectories of y1 at the 5th iteration. (b) Trajectories of y1 at the 30th iteration. (c) Trajectories of y2 at the 5th iteration. (d) Trajectories of y2 at the 30th iteration.
(a)(b)(c)(d)Figure 7
The 2-norm of tracking errors for all agents in each interaction by closed-loop PDα-type. (a) Tracking errors of y1 with iterations. (b) Tracking errors of y2 with iterations.
(a)(b)Case 3.
Open-closed-loopPDα-type
In this simulation, the initial states and inputs are the same as Case1 and Case 2. According to Theorem 3, the learning gain matrix can be obtained as follows:(58)ΓP1=0.616000.484,ΓD1=4.5001.8,ΓP2=0.5000.4,ΓD2=6007.Clearly, ρ2ρ1=0.245<1; thus, the convergence condition in Theorem 3 can be satisfied.
The simulation results with open-closed-loop PDα-type FOILC are presented in Figures 8–10. The results are similar to those of the open-loop and closed-loop PDα-type FOILC. From the results, both the initial states and the outputs can converge to the desired values. And we can conclude that the proposed FOILC scheme with initial state learning works well as the iteration number increases. Figure 9 shows the output tracking results of y1 and y2. It can be seen that the followers can fully track the desired trajectory as the iteration increases over the time period [0, 1].In addition, compared with open-loop PDα-type FOILC and closed-loop PDα-type, applying open-closed-loop PDα-type FOILC has better performance in the initial state and for the outputs.Figure 8
Initial state profile vs. iteration number by open-closed-loop PDα-type. (a) Initial state learning of x1. (b) Initial state learning of x2.
(a)(b)Figure 9
The tracking results of all agents at different iterations by open-closed-loop PDα-type. (a) Trajectories of y1 at the 5th iteration. (b) Trajectories of y1 at the 30th iteration. (c) Trajectories of y2 at the 5th iteration. (d) Trajectories of y2 at the 30th iteration.
(a)(b)(c)(d)Figure 10
The 2-norm of tracking errors for all agents in each interaction by open-closed-loop PDα-type. (a) Tracking errors of y1 with iterations. (b) Tracking errors of y2 with iterations.
(a)(b)


## 5. Conclusion

In this paper, we have discussed the consensus problem with fixed communication graph, which has been addressed for fractional-order multiagent systems with initial state shift. Considering the initial state learning mechanism, open-loop PDα type, closed-loop PDα type, and open-closed-loop PDα type FOILC are proposed. The theoretical convergence of the proposed algorithm is analyzed and sufficient conditions are presented. Theoretical analysis shows that the proposed algorithms can guarantee the tracking errors of all the agents and the errors in the initial state tend to be zero in a finite time as the number of iterations increases. Finally, some simulation examples are used to validate the effectiveness. As a recommendation for the future, the convergence and robustness of fractional-order nonlinear systems can be studied by using the proposed method of this paper.



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