# Consensus Tracking of Fractional-Order Multiagent Systems via Fractional-Order Iterative Learning Control

Authors: Shuaishuai Lv; Mian Pan; Xungen Li; Qi Ma; Tianyi Lan; Bingqiang Li; Wenyu Cai
Journal: Complexity
 (2019)

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## Abstract

In this work, the consensus problem of fractional-order multiagent systems with the general linear model of fixed topology is studied. Both distributedPDα-type and Dα-type fractional-order iterative learning control (FOILC) algorithms are proposed. Here, a virtual leader is introduced to generate the desired trajectory, fixed communication topology is considered, and only a subset of followers can access the desired trajectory. The convergence conditions are proved using graph theory, fractional calculus, and λ norm theory. The theoretical analysis shows that the output of each agent completely tracks the expected trajectory in a limited time as the iteration number increases for both PDα-type and Dα-type FOILC algorithms. Extensive numerical simulations are given to demonstrate the feasibility and effectiveness.


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## Body

## 1. Introduction

In recent years, the research on the coordinating control of multiagent systems (MASs) has become a major task in the field of control because of its application in various areas, such as formation control of unmanned aerial vehicles [1], air traffic control [2], and formation of multiple robots [3]. The consensus is an important and fundamental problem for the coordination control of MASs. One object of the consensus problem is that all agents are expected to reach an agreement in some amounts (such as speed, position, phases, and attitudes) by communicating with their local neighbors [4–6].The consensus problem, as an interesting topic for MASs, has achieved many results [5–10]. However, most studies have focused on the consensus of integer-order MASs (IOMASs). In fact, many systems cannot be well described for some complex problems by integer-order models, such as viscoelastic systems, underwater vehicle, and electrical machines [11–13]. Recently, fractional-order calculus is found to be suitable for describing some complex systems, such as macromolecule fluids, porous media, and viscoelastic systems, and have excellent memory and hereditary [14]. Therefore, it is meaningful to study the consensus problem for fractional-order MASs (FOMASs). In particular, integer-order systems are special cases of fractional-order systems, and many researchers have studied the convergence problem and focused on the coordination control of FOMASs. The necessary and sufficient conditions of FOMASs for containment control were presented in [15]. For FOMASs, several consensus problems of FOMASs were studied in [16–19]. Du et al. studied the problem of robust consensus for second-order MASs and proposed a class of novel continuous nonsmooth consensus algorithms [20]. Zhu et al. studied the leader-following consensus problem of FOMASs with general linear and nonlinear models with input time delay and designed a control gain matrix and sufficient condition [21]. In [22], group multiple-lag consensus of nonlinear FOMAS with leader-following was investigated, and two kinds of lag consensus were considered.Notably, the aforementioned studies consider the asymptotic convergence problem of FOMAS, that is, the tracking errors of each agent decreases gradually as time increases. For MAS with repetitive tasks, such as multimechanical arms on industrial production lines, the asymptotic convergence evidently cannot meet the requirements. The complete convergence within a limited time is required [23–25]. Among the control methods, iterative learning control (ILC) can achieve full tracking in a limited time for the repetitive system [26–28]. Therefore, the ILC algorithm is applied for IOMAS to solve the consensus problem recently, especially for complex MASs [29], MASs with switching topologies and communication time delays [30], and nonlinear MASs [31]. The ILC algorithm can achieve high-precision tracking performance and achieve exact tracking within finite time as the iteration number increases. However, most existing works using ILC are suitable only for IOMAS.The consensus problem for the fractional MASs with ILC is seldom discussed. In this study, we consider the leader-following consensus problem of FOMAS with general linear leader and designPDα-type and Dα-type fractional-order ILC (FOILC) controllers. The convergence of the proposed controller is analyzed, and the convergence conditions are derived. The convergence conditions can ensure that the tracking errors of all the agents are gradually decreased to a sufficiently small value as the number of iterations increases. Furthermore, the tracking errors can also tend to zero if the initial states of the agents tend to zero.The rest of the paper is organized as follows. In Section2, graph theory, fractional calculus, and problem formulation are given. The FOILC and the analysis of convergence for FOMAS are discussed in Section 3. In Section 4, the effectiveness of the proposed algorithm is verified by simulation. Section 5 elaborates the conclusions.


## 2. Preliminary Knowledge

In this section, some basic definitions and lemmas are introduced, which will be used in the following sections.


### 2.1. Mathematical Knowledge

The set of real numbers is denoted byℝ, and the set of complex numbers is denoted by ℂ. The set of natural numbers is denoted by ℕ, and i∈ℕ is the iteration number. For a given vector x=x1,x2,…,xnT∈ℝn, x denotes any lp vector norm, where 1≤p≤∞. In particular, x1=∑k=1nxk, x2=xTx, and x∞=maxk=1,…,nxk. For any matrix A∈ℝn×n, A is the induced matrix norm. ρA is the spectral radius. Moreover, ⊗ denotes the Kronecker product, and Im is the m×m identity matrix.Remark 1.
C
m
0
,
T denotes a set consisting of all functions with mth derivatives that are continuous on the finite time interval 0,T.Definition 1.
Continuous vector functionf⋅∈C0,T, where ft=f1t,f2t,…,fntT, is given. The λ norm of the function f is defined as(1)f⋅λ=sup0≤t≤Te−λtmax1≤i≤nfit,where λ is a positive real number.Definition 2 (see [32]).
Theα-order fractional integral of the function ft over the interval t0,t is defined as(2)Dt0t−αft=1Γα∫t0tt−τα−1fτdτ,where α>0 and Γ⋅ is the Gamma function defined as Γα=∫0∞exp−ttα−1dt.
For the functionft, the α-order Caputo derivative over the interval t0,t is defined as(3)Dtαt0Cft=Dt0t−α−α+1dα+1dtα+1ft,where α>0 and α represents the integer part of α.Lemma 1 (see [33, 34]).
The functionfxt,t is assumed continuous. Then, the initial value problem is described as(4)Dtαt0Cxt=fxt,t,xt0=x0,0<α<1.
The function in equation (4) is equivalent to the following nonlinear Volterra integral equation:(5)xt=x0+1Γα∫t0tt−τα−1fxτ,τdτ.


### 2.2. Graph Theory

In this study,N FOMAS and a virtual leader are considered, and vi indicates the ith agent, i=1,2,…,N. The communications among the agents can be described by a directed graph G=V,E,M, where V=v1,v2,…,vn represents the node set, E⊆V×V is the set of edges, and M=almn×n denotes the weighted adjacency matrix, that is, amm=0, alm>0 if l,m∈E and alm=0 otherwise [35]. Nl=mm,l∈E is the set of neighborhood of the lth node. The Laplacian matrix L∈ℝn×n of the graph is defined as(6)L=Din−M,where Din=diagd1in,d2in,…,dnin, diin=∑m∈Nialm.When joining a virtual leader, the relationship between the followers and the leader is expressed by a diagonal matrixD, that is, D=diagd1,d2,…,dn, where dl>0 if the lth agent can receive information from the virtual leader. The trajectory of the leader is only accessible to a small portion of the followers due to communication or sensor limitations. The communication among followers is described by the graph GV,E. If the leader is labeled by vertex 0, then the complete information flow among all the agents can be characterized by a new graph G¯=0∪V,E¯, where E¯ is the new edge set. If the graph is connected or a spanning tree, it can be described as follows.Definition 3 (see [36]).
Among all agents, any agent can receive indirectly or directly the information of the virtual leader in a path. In other words, we do not consider the isolated agent, which means the agent cannot accept any information of the leader, and self-loop agents, which means the agent accepts their own information only.Lemma 2.
IfG¯ is a connected graph, then the matrix L+D associated with the graph G¯ is a positive definite matrix, where L is the Laplacian matrix and D=diagd1,d2,…,dn.
The proof of Lemma 2 is described in [36].


## 2.1. Mathematical Knowledge

The set of real numbers is denoted byℝ, and the set of complex numbers is denoted by ℂ. The set of natural numbers is denoted by ℕ, and i∈ℕ is the iteration number. For a given vector x=x1,x2,…,xnT∈ℝn, x denotes any lp vector norm, where 1≤p≤∞. In particular, x1=∑k=1nxk, x2=xTx, and x∞=maxk=1,…,nxk. For any matrix A∈ℝn×n, A is the induced matrix norm. ρA is the spectral radius. Moreover, ⊗ denotes the Kronecker product, and Im is the m×m identity matrix.Remark 1.
C
m
0
,
T denotes a set consisting of all functions with mth derivatives that are continuous on the finite time interval 0,T.Definition 1.
Continuous vector functionf⋅∈C0,T, where ft=f1t,f2t,…,fntT, is given. The λ norm of the function f is defined as(1)f⋅λ=sup0≤t≤Te−λtmax1≤i≤nfit,where λ is a positive real number.Definition 2 (see [32]).
Theα-order fractional integral of the function ft over the interval t0,t is defined as(2)Dt0t−αft=1Γα∫t0tt−τα−1fτdτ,where α>0 and Γ⋅ is the Gamma function defined as Γα=∫0∞exp−ttα−1dt.
For the functionft, the α-order Caputo derivative over the interval t0,t is defined as(3)Dtαt0Cft=Dt0t−α−α+1dα+1dtα+1ft,where α>0 and α represents the integer part of α.Lemma 1 (see [33, 34]).
The functionfxt,t is assumed continuous. Then, the initial value problem is described as(4)Dtαt0Cxt=fxt,t,xt0=x0,0<α<1.
The function in equation (4) is equivalent to the following nonlinear Volterra integral equation:(5)xt=x0+1Γα∫t0tt−τα−1fxτ,τdτ.


## 2.2. Graph Theory

In this study,N FOMAS and a virtual leader are considered, and vi indicates the ith agent, i=1,2,…,N. The communications among the agents can be described by a directed graph G=V,E,M, where V=v1,v2,…,vn represents the node set, E⊆V×V is the set of edges, and M=almn×n denotes the weighted adjacency matrix, that is, amm=0, alm>0 if l,m∈E and alm=0 otherwise [35]. Nl=mm,l∈E is the set of neighborhood of the lth node. The Laplacian matrix L∈ℝn×n of the graph is defined as(6)L=Din−M,where Din=diagd1in,d2in,…,dnin, diin=∑m∈Nialm.When joining a virtual leader, the relationship between the followers and the leader is expressed by a diagonal matrixD, that is, D=diagd1,d2,…,dn, where dl>0 if the lth agent can receive information from the virtual leader. The trajectory of the leader is only accessible to a small portion of the followers due to communication or sensor limitations. The communication among followers is described by the graph GV,E. If the leader is labeled by vertex 0, then the complete information flow among all the agents can be characterized by a new graph G¯=0∪V,E¯, where E¯ is the new edge set. If the graph is connected or a spanning tree, it can be described as follows.Definition 3 (see [36]).
Among all agents, any agent can receive indirectly or directly the information of the virtual leader in a path. In other words, we do not consider the isolated agent, which means the agent cannot accept any information of the leader, and self-loop agents, which means the agent accepts their own information only.Lemma 2.
IfG¯ is a connected graph, then the matrix L+D associated with the graph G¯ is a positive definite matrix, where L is the Laplacian matrix and D=diagd1,d2,…,dn.
The proof of Lemma 2 is described in [36].


## 3. Problem Description

A FOMAS consists ofN agents indexed by 1,2,…,N and a leader. The dynamic of each follower can be described as follows:(7)Dαxi,jt=Axi,jt+Bui,jt,yi,jt=Cxi,jt,where t∈0,T and i are the time and number of iterations; xi,jt∈ℝm is the state vector of the jth agent; ui,jt∈ℝm1 and yi,jt∈ℝm2 are the input and output vectors, respectively; A,B,C are constant matrices with compatible dimensions; and Dαxi,jt is the α-order Caputo derivative of xi,jt.The dynamic equations described by equation (7) are rewritten in the following compact form:(8)Dαxit=IN⊗Axit+IN⊗Buit,yit=IN⊗Cxit,where xit=xi,1tT,xi,2tT,…,xi,NtTT, uit=ui,1Tt,ui,2Tt,…,ui,NTtT, yit=yi,1Tt,yi,2Tt,…,yi,NTtT, ξit=ξi,1tT,ξi,2tT,…,ξi,NtTT, and ⊗ is the Kronecker product.The expected trajectoryydt is defined on a finite time interval 0,T and is generated by the leader. The dynamics of the leader agent are as follows:(9)Dtαxdt=Axdt+Budt,ydt=Cxdt,where udt is the continuous and unique desired control input.We define thatudt is the continuous and unique desired control input for all followers, that is, udt=u1,d,u2,d,…,uN,dT.This study aims to find the appropriate control input of each agent with ILC to ensure that the trajectories of each agent are exactly the same asydt in a finite time 0,T, that is, finding appropriate ui,kt satisfies(10)limi⟶∞ydt−yi,jt=0,j=1,2,…,N,t∈0,T.In other words, the goal of the function in equation (10) is that each agent in the FOMAS converges to the desired reference trajectory in a finite time with FOILC as the number of iterations increases. The major task is to design a set of distributed FOILC rules for guaranteeing that each individual agent in the network can track the trajectory of the leader under the sparse communication graph G¯.ξ
i
,
j
t denotes the distributed information measured or received by the jth agent at the ith iteration. Specifically,(11)ξi,jt=∑k∈Njaj,kyi,kt−yi,jt+djydt−yi,jt,where aj,k is the element of the weighted adjacency matrix M, that is, aj,k∈M and dj is the element of the diagonal matrix D and describes the relationship between the followers and the leader, that is, dj∈D. We define the tracking error between the j-th and the leader fractional-order agents as follows:(12)ei,jt=ydt−yi,jt.Consequently, equation (15) can be rewritten in a compact form:(13)ξit=L+D⊗Imeit,where eit=ei,1tT,ei,2tT,…,ei,NtTT, ξit=ξi,1tT,ξi,2tT,…,ξi,NtTT, L is the Laplacian matrix of graph G, and D=diagdi,i=1,…,N.


### 3.1. Open-LoopPDα-Type Iterative Learning Control

To solve the consensus tracking problem of FOMASs, the following distributedPDα-type ILC for the jth agent is designed as(14)ui+1,jt=ui,jt+ΓPξi,jt+ΓDξi,jαt.Based on (12), the distributed measurement in equation (11) can be rewritten as(15)ξi,jt=∑k∈Njaj,kydt−yi,jt−ydt+yi,kt+djydt−yi,jt=∑k∈Njaj,kei,jt−ei,kt+djei,jt.By using equation (13), the updating law in equation (14) can be rewritten in terms of the tracking errors for all the followers as(16)ui+1t=uit+L+D⊗ΓPeit+L+D⊗ΓDeiαt.For convenience, we defineγj and j=1,2,…,N as the jth eigenvalue of L+D.To simplify the controller design and convergence analysis, the following assumptions are imposed.Assumption 1.
When the system described by (7) is repeatedly run over the interval 0,T, the initial state of the each subsystem in equation (7) is equal to the expected initial state. In particular, xi,j0=xd0 is satisfied for all j and i.Remark 2.
Assumption1 is the standard condition for ILC to ensure perfect tracking performance. If the assumption is removed, then the tracking performance will be scarified and the extra system information or additional control mechanisms are required. For instance, the initial state learning control is discussed in [35–37]. Notably, perfect tracking can never be achieved without perfect initial condition.Assumption 2.
The communication graphG¯ contains a spanning tree with the leader and the followers being the root.Remark 3.
Assumption 3 is the necessary condition of the consensus tracking problem for the FOMAS. If the graphG¯ does not contain a spanning tree, then an isolated agent exists and cannot follow the trajectory of the leader because it does not know the control objective. However, the graph G does not necessarily contain a spanning tree because the isolated agent can communicate with the selected leader.Theorem 1.
We consider the FOMAS in equation (7) under Assumptions 1 and 2, the communication graph G, and PDα-type updating rule in equation (14). If the learning gains ΓP and ΓD are chosen, then(17)I−L+D⊗ΓDCB+β<1,where I is the identity matrix with the subscript denoting its dimension, L is the Laplacian matrix of G, and D=diagdj,j=1,…,N. The control input ui,jt and output yi,jt converge to udt and ydt, respectively, as the iteration number tends to infinity.Proof.
The convergence analysis is presented as follows. We assume that(18)Δxit=1N⊗xdt−xit,Δuit=1N⊗udt−uit,where β=BL+DΓPC+ΓDCA/λα−A and 1N is a vector in which all entries are 1. From (1), we can obtain(19)eiαt=1N⊗ydαt−yiαt=In⊗Cxdαt−xiαt=In⊗CΔxiαt=In⊗CIn⊗AΔxit+In⊗CIn⊗BΔuit.
Lemma1 yields(20)xit=xi0+1Γα∫0tt−τα−1In⊗Axiτ+In⊗In⊗Buiτdτ.
Accordingly, we can obtain(21)Δxit=1N⊗xdt−xit=1Γα∫0tt−τα−1In⊗AΔxiτ+In⊗BΔuiτdτ.
By taking theλ norm on both sides of (11) and noting Definition 1, we have(22)Δxitλ≤sup0≤t≤Te−λtΔxit≤sup0≤t≤T1Γαe−λt∫0tt−τα−1In⊗AΔxiτ+In⊗BΔuiτdτ≤sup0≤t≤T1Γα∫0tt−τα−1e−λt−τe−λτIn⊗AΔxiτ+In⊗BΔuiτdτ≤sup0≤t≤T1Γα∫0tt−τα−1e−λt−τIn⊗AΔxiτλ+In⊗BΔuiτλdτ≤AΔxitλ+BΔuiτλΓαsup0≤t≤T∫0tt−τα−1e−λt−τdτ.
By contrast, we have(23)∫0tt−τα−1e−λt−τdτ=e−λt∫0tt−τα−1eλτdτ=t−τ=ωe−λt∫0tωα−1eλt−ωdω=λω=se−λtλα∫0tsα−1eλt−sds≤1λα∫0+∞sα−1e−sds=Γαλα.
Substituting equation (23) into equation (22) yields(24)Δxitλ≤AΔxitλ+BΔuitλλα.
Thus, we can find sufficiently largeλ to make(25)λα−A>0.
Therefore, equation (24) is reformulated as(26)Δxitλ≤Bλα−AΔuitλ.
Equations (7) and (18) yield(27)eit=ydt−yit=In⊗CΔxit,(28)Δui+1t=Δuit−L+D⊗ΓPeit−L+D⊗ΓDeiαt.
By substituting (19) and (27) into (28), we can obtain(29)Δui+1t=Δuit−L+D⊗ΓPIn⊗CΔxit−L+D⊗ΓDIn⊗CIn⊗AΔxit−L+D⊗ΓDIn⊗CIn⊗BΔuit=I−L+D⊗ΓDIn⊗CIn⊗BΔuit−L+D⊗ΓPIn⊗C+L+D⊗ΓDIn⊗CIn⊗AΔxit.
By taking theλ norm on both sides of equation (29), we have(30)Δui+1tλ≤I−L+D⊗ΓDIn⊗CIn⊗BΔuitλ+L+D⊗ΓPIn⊗C+L+D⊗ΓDIn⊗CIn⊗AΔxitλ=I−L+D⊗ΓDCBΔuitλ+L+D⊗ΓPC+L+D⊗ΓDCAΔxitλ.
By substituting equation (26) into equation (30), we can obtain(31)Δui+1tλ≤I−L+D⊗ΓDCB+βΔuitλ,where(32)β=BL+DΓPC+ΓDCAλα−A.
Given thatI−L+D⊗ΓDCB<1, sufficiently large λ is obtained, thereby satisfying I−L+D⊗ΓDCB+β≤ρ˜<1. Thus, from equation (31), we have(33)Δui+1tλ≤ρ˜Δuitλ≤ρ˜kΔu1tλ.
When the number of iterations increases, that is,i⟶∞, we have(34)limi⟶∞Δui+1tλ=0,limi⟶∞Δxi+1tλ=0.
Taking theλ norm on both sides yields(35)ei+1tλ≤CΔxi+1tλ.
Accordingly, we can obtain(36)limi⟶∞ei+1tλ=0.
The tracking errors of the agents tend to zero withi⟶∞. Thus, yi,kt converges to ydt. The proof is completed.


### 3.2. Open-LoopDα-Type Iterative Learning Control

For the FOMASs described in (7), if we let ΓP=0 in (14), then the distributed open-loop PDα-type FOILC becomes a distributed open-loop Dα-type FOILC as follows:(37)ui+1,jt=ui,jt+ΓDξi,jαt.Similar to thePDα-type FOILC, the updating law (37) can be rewritten in the vector compact form(38)ui+1t=uit+L+D⊗ΓDeiαt.Then, a corollary can be obtained as follows:Theorem 2.
Consider the FOMAS (7) under the directed graph G¯, suppose Assumption 1 and 2 are satisfied. Let the distributed open-loop Dα-type ILC scheme (38) be applied for the system with learning gain ΓD1 satisfying(39)IN−L+D⊗ΓDCB+β1=ρ1<1,where β1 is an expression about the system parameters and the control parameter as follows: β1=BL+DΓDCA/λα−A, and then limi⟶∞eitλ=0. Namely, for the achievable desired trajectory ydt,t∈0,T, the outputs yit of the FOMASs converge to the desired trajectory ydt when i⟶∞, i.e., limi⟶∞yit=ydt,t∈0,T.Proof.
The proof process of Theorem2 is similar to the proof process of Theorem 1.Remark 4.
In the proof, the constantλ is only a tool for analysis and can be very large, that is, λ⟶∞. The reason is that λ is not used in the implementation and does not affect the actual control performance.Remark 5.
The convergence of FOILC for the FOMASs is uninfluenced by the initial states. We defineM≜I−L+D⊗ΓDCB+β. Suitable ΓP and ΓD can be designed such that ρM<ρ˜ for all t∈0,T. Thus, an appropriate matrix norm can be designed such that M<ρ˜.Remark 6.
According to Lemma2, the matrix L+D is a positive definite matrix when the directed graph G¯ is a connected graph. Then, the matrix −L+D is the Hurwitz stable matrix. Accordingly, the gain matrix ΓD can be found to satisfy the condition in equation (17).Remark 7.
Compared withDα-type FOILC, PDα-type FOILC can reduce the deviation of the limit trajectory from the expected trajectory. So, the increased P-type component reduces the adverse effect of the Dα-type component on the disturbance sensitivity and improves the control performance.


## 3.1. Open-LoopPDα-Type Iterative Learning Control

To solve the consensus tracking problem of FOMASs, the following distributedPDα-type ILC for the jth agent is designed as(14)ui+1,jt=ui,jt+ΓPξi,jt+ΓDξi,jαt.Based on (12), the distributed measurement in equation (11) can be rewritten as(15)ξi,jt=∑k∈Njaj,kydt−yi,jt−ydt+yi,kt+djydt−yi,jt=∑k∈Njaj,kei,jt−ei,kt+djei,jt.By using equation (13), the updating law in equation (14) can be rewritten in terms of the tracking errors for all the followers as(16)ui+1t=uit+L+D⊗ΓPeit+L+D⊗ΓDeiαt.For convenience, we defineγj and j=1,2,…,N as the jth eigenvalue of L+D.To simplify the controller design and convergence analysis, the following assumptions are imposed.Assumption 1.
When the system described by (7) is repeatedly run over the interval 0,T, the initial state of the each subsystem in equation (7) is equal to the expected initial state. In particular, xi,j0=xd0 is satisfied for all j and i.Remark 2.
Assumption1 is the standard condition for ILC to ensure perfect tracking performance. If the assumption is removed, then the tracking performance will be scarified and the extra system information or additional control mechanisms are required. For instance, the initial state learning control is discussed in [35–37]. Notably, perfect tracking can never be achieved without perfect initial condition.Assumption 2.
The communication graphG¯ contains a spanning tree with the leader and the followers being the root.Remark 3.
Assumption 3 is the necessary condition of the consensus tracking problem for the FOMAS. If the graphG¯ does not contain a spanning tree, then an isolated agent exists and cannot follow the trajectory of the leader because it does not know the control objective. However, the graph G does not necessarily contain a spanning tree because the isolated agent can communicate with the selected leader.Theorem 1.
We consider the FOMAS in equation (7) under Assumptions 1 and 2, the communication graph G, and PDα-type updating rule in equation (14). If the learning gains ΓP and ΓD are chosen, then(17)I−L+D⊗ΓDCB+β<1,where I is the identity matrix with the subscript denoting its dimension, L is the Laplacian matrix of G, and D=diagdj,j=1,…,N. The control input ui,jt and output yi,jt converge to udt and ydt, respectively, as the iteration number tends to infinity.Proof.
The convergence analysis is presented as follows. We assume that(18)Δxit=1N⊗xdt−xit,Δuit=1N⊗udt−uit,where β=BL+DΓPC+ΓDCA/λα−A and 1N is a vector in which all entries are 1. From (1), we can obtain(19)eiαt=1N⊗ydαt−yiαt=In⊗Cxdαt−xiαt=In⊗CΔxiαt=In⊗CIn⊗AΔxit+In⊗CIn⊗BΔuit.
Lemma1 yields(20)xit=xi0+1Γα∫0tt−τα−1In⊗Axiτ+In⊗In⊗Buiτdτ.
Accordingly, we can obtain(21)Δxit=1N⊗xdt−xit=1Γα∫0tt−τα−1In⊗AΔxiτ+In⊗BΔuiτdτ.
By taking theλ norm on both sides of (11) and noting Definition 1, we have(22)Δxitλ≤sup0≤t≤Te−λtΔxit≤sup0≤t≤T1Γαe−λt∫0tt−τα−1In⊗AΔxiτ+In⊗BΔuiτdτ≤sup0≤t≤T1Γα∫0tt−τα−1e−λt−τe−λτIn⊗AΔxiτ+In⊗BΔuiτdτ≤sup0≤t≤T1Γα∫0tt−τα−1e−λt−τIn⊗AΔxiτλ+In⊗BΔuiτλdτ≤AΔxitλ+BΔuiτλΓαsup0≤t≤T∫0tt−τα−1e−λt−τdτ.
By contrast, we have(23)∫0tt−τα−1e−λt−τdτ=e−λt∫0tt−τα−1eλτdτ=t−τ=ωe−λt∫0tωα−1eλt−ωdω=λω=se−λtλα∫0tsα−1eλt−sds≤1λα∫0+∞sα−1e−sds=Γαλα.
Substituting equation (23) into equation (22) yields(24)Δxitλ≤AΔxitλ+BΔuitλλα.
Thus, we can find sufficiently largeλ to make(25)λα−A>0.
Therefore, equation (24) is reformulated as(26)Δxitλ≤Bλα−AΔuitλ.
Equations (7) and (18) yield(27)eit=ydt−yit=In⊗CΔxit,(28)Δui+1t=Δuit−L+D⊗ΓPeit−L+D⊗ΓDeiαt.
By substituting (19) and (27) into (28), we can obtain(29)Δui+1t=Δuit−L+D⊗ΓPIn⊗CΔxit−L+D⊗ΓDIn⊗CIn⊗AΔxit−L+D⊗ΓDIn⊗CIn⊗BΔuit=I−L+D⊗ΓDIn⊗CIn⊗BΔuit−L+D⊗ΓPIn⊗C+L+D⊗ΓDIn⊗CIn⊗AΔxit.
By taking theλ norm on both sides of equation (29), we have(30)Δui+1tλ≤I−L+D⊗ΓDIn⊗CIn⊗BΔuitλ+L+D⊗ΓPIn⊗C+L+D⊗ΓDIn⊗CIn⊗AΔxitλ=I−L+D⊗ΓDCBΔuitλ+L+D⊗ΓPC+L+D⊗ΓDCAΔxitλ.
By substituting equation (26) into equation (30), we can obtain(31)Δui+1tλ≤I−L+D⊗ΓDCB+βΔuitλ,where(32)β=BL+DΓPC+ΓDCAλα−A.
Given thatI−L+D⊗ΓDCB<1, sufficiently large λ is obtained, thereby satisfying I−L+D⊗ΓDCB+β≤ρ˜<1. Thus, from equation (31), we have(33)Δui+1tλ≤ρ˜Δuitλ≤ρ˜kΔu1tλ.
When the number of iterations increases, that is,i⟶∞, we have(34)limi⟶∞Δui+1tλ=0,limi⟶∞Δxi+1tλ=0.
Taking theλ norm on both sides yields(35)ei+1tλ≤CΔxi+1tλ.
Accordingly, we can obtain(36)limi⟶∞ei+1tλ=0.
The tracking errors of the agents tend to zero withi⟶∞. Thus, yi,kt converges to ydt. The proof is completed.


## 3.2. Open-LoopDα-Type Iterative Learning Control

For the FOMASs described in (7), if we let ΓP=0 in (14), then the distributed open-loop PDα-type FOILC becomes a distributed open-loop Dα-type FOILC as follows:(37)ui+1,jt=ui,jt+ΓDξi,jαt.Similar to thePDα-type FOILC, the updating law (37) can be rewritten in the vector compact form(38)ui+1t=uit+L+D⊗ΓDeiαt.Then, a corollary can be obtained as follows:Theorem 2.
Consider the FOMAS (7) under the directed graph G¯, suppose Assumption 1 and 2 are satisfied. Let the distributed open-loop Dα-type ILC scheme (38) be applied for the system with learning gain ΓD1 satisfying(39)IN−L+D⊗ΓDCB+β1=ρ1<1,where β1 is an expression about the system parameters and the control parameter as follows: β1=BL+DΓDCA/λα−A, and then limi⟶∞eitλ=0. Namely, for the achievable desired trajectory ydt,t∈0,T, the outputs yit of the FOMASs converge to the desired trajectory ydt when i⟶∞, i.e., limi⟶∞yit=ydt,t∈0,T.Proof.
The proof process of Theorem2 is similar to the proof process of Theorem 1.Remark 4.
In the proof, the constantλ is only a tool for analysis and can be very large, that is, λ⟶∞. The reason is that λ is not used in the implementation and does not affect the actual control performance.Remark 5.
The convergence of FOILC for the FOMASs is uninfluenced by the initial states. We defineM≜I−L+D⊗ΓDCB+β. Suitable ΓP and ΓD can be designed such that ρM<ρ˜ for all t∈0,T. Thus, an appropriate matrix norm can be designed such that M<ρ˜.Remark 6.
According to Lemma2, the matrix L+D is a positive definite matrix when the directed graph G¯ is a connected graph. Then, the matrix −L+D is the Hurwitz stable matrix. Accordingly, the gain matrix ΓD can be found to satisfy the condition in equation (17).Remark 7.
Compared withDα-type FOILC, PDα-type FOILC can reduce the deviation of the limit trajectory from the expected trajectory. So, the increased P-type component reduces the adverse effect of the Dα-type component on the disturbance sensitivity and improves the control performance.


## 4. Simulations

A network consisting of four follower fractional-order agents and a virtual leader fractional-order agent is used to verify the effectiveness of the proposed method. The dynamic function of theith agent is a MIMO system. In equation (7), the matrix A, B, and C and α are given as follows:(40)A=0.10.30.2−0.1,B=0.30.020.010.2,C=0.70.010.010.9,α=0.9,and the desired reference trajectory is(41)yd=t+sint∗2π25t2+sint∗2π,t∈0,1.The virtual leader and the four followers are labeled as 0, 1, 2, 3, and 4. The information topology-directed graph is shown in Figure1.Figure 1
Communication topology among agents in the network.From Figure1, the weighted adjacency matrix M and the in-degree Din can be obtained as follows on the basis of graph theory:(42)M=0000.70.900000.800000.60,Din=diag0.70.90.80.6.The Laplacian matrix and the relationship between the followers and the leader are shown as follows:(43)L=Din−M=0.700−0.7−0.90.9000−0.80.8000−0.60.6,D=diag1.8002.0.


### 4.1. The Results of the Open-LoopPDα-Type FOILC

For the open-loopPDα-type FOILC described in (16), ΓP and ΓD are designed to satisfy the convergence conditions, i.e., ΓP=diag1.11.6 and ΓD=diag1.50.8.By checking the convergence condition and definingλ=100, we obtain that(44)I−L+D⊗ΓDCB=0.9312<1,I−L+D⊗ΓDCB+β=0.9621<1.Therefore, the convergence condition is satisfied.The initial states of all the agents, the initial errors, and the initial inputs are 0, that is,ei0=0. The input signals at the zeroth iteration are set to zero, that is, u0,jt=0, for all agents. Figures 2 and 3 show the tracking performance by the open-loop PDα-type FOILC at 5th, 20th, and 70th iterations. At the 5th iteration, the deviations are very large between the trajectories of the follower and the desired one. As the iteration number increases, the outputs y1(t) and y2(t) of the followers nearly completely track the reference trajectory over the time [0, 1]. By 70 iterations, the maximum tracking errors of y1(t) of the four agents are 0.000326, 0.000843, 0.000916, and 0.000276, and the maximum tracking errors of y2(t) of the four agents are 0.0016, 0.0041, 0.0045, and 0.0014, respectively. And moreover, compared with Agents 2 and 4, both Agents 1 and 3 can get information from the leader directly, so they can converge to the expected trajectory more quickly. We define the maximum errors at the ith iteration as maxj=1,2,3,4yd,1−yi,j,1 and maxj=1,2,3,4yd,2−yi,j,2. Compared with Figure 2(d) and Figure 3(d), the outputs y2(t) converge to the expected trajectory more quickly than y1(t). That is because the trajectory y1d(t) is more complicated than y2d(t) (Figure 4).Figure 2
Results ofy1 with open-loop PDα-type FOILC. (a) Trajectories y1(t) at the 5th iteration. (b) Trajectories y1(t) at the 20th iteration. (c) Trajectories y1(t) at the 70th iteration. (d) Tracking error e1 vs. iteration number.
(a)
(b)
(c)
(d)Figure 3
Results ofy2 with open-loop PDα-type FOILC. (a) Trajectories y2(t) at the 5th iteration. (b) Trajectories y2(t) at the 20th iteration. (c) Trajectories y2(t) at the 70th iteration. (d) Tracking error e2 vs. iteration number.
(a)
(b)
(c)
(d)Figure 4
Results ofy1 with open-loop Dα-type FOILC. (a) Trajectories y1(t) at the 5th iteration. (b) Trajectories y1(t) at the 20th iteration. (c) Trajectories y1(t) at the 70th iteration. (d) Tracking error e1 vs. iteration number.
(a)
(b)
(c)
(d)


### 4.2. The Results of the Open-LoopDα-Type FOILC

In order to compare with the performance of the open-loopPDα-type FOILC, the model and the structure of the FOMASs with open-loop Dα-type FOILC are the same as example 1. The control parameters of the open-loop Dα-type FOILC described in (38) are designed as ΓD=diag1.5,0.8 which are the same as ΓD of the open-loop PDα-type FOILC. Thus, if λ=100 is selected, the convergence condition (39) can be calculated as follows:(45)I−L+D⊗ΓDCB+β1=0.9382<1.Thus, the convergence condition in Theorem1 is satisfied.The simulation results are plotted in (Figure4 and 5). Similar to results of the open-loop PDα-type FOILC, as iteration number increases, the outputs y1(t) and y2(t) of the followers nearly completely track the reference trajectory over the time [0, 1]. But, the maximum tracking errors of y1(t) of the four agents are 0.013, 0.034, 0.037, and 0.011, and the maximum tracking errors of y2(t) of the four agents are 0.1, 0.26, 0.28, and 0.085 by 70 iterations, respectively. Obviously, compared with the open-loop PDα-type FOILC, the convergence speed is slower than the open-loop Dα-type FOILC.Figure 5
Results ofy2 with open-loop Dα-type FOILC. (a) Trajectories y2(t) at the 5th iteration. (b) Trajectories y2(t) at the 20th iteration. (c) Trajectories y2(t) at the 70th iteration. (d) Tracking error e2 vs. iteration number.
(a)
(b)
(c)
(d)


### 4.3. The Results of the Open-LoopPD-Type and D-Type IOILC

In order to reflect the superiority of the fractional controller, we designed an integer-order iterative learning control (IOILC) to apply the FOMASs. ThePD-type and D-type IOILC are as follows:(46)ui+1t=uit+L+D⊗ΓPeit+L+D⊗ΓDdeitdt,ui+1t=uit+L+D⊗ΓDdeitdt.The parameters of the FOMASs are the same as Figure1 and equations (40) and (41). The learning gains of PD-type are ΓP=diag1.11.6, ΓD=diag1.50.8, and the learning gain of D-type is ΓD=diag1.50.8, respectively, which are the same as the parameters of the FOILC proposed in this paper. The results are shown in Figures 6 and 7. It can be seen that the errors of y1(t) and y2(t) decrease first as the iteration number increases when employing the PD-type and D-type IOILC. But, when the iteration number is greater than 60, the error gradually increases. Compared with IOILC, we can see that the FOILC has better performance.Figure 6
Results with open-loopPD-type IOILC. (a) Tracking error e1 vs. iteration number. (b) Tracking error e2 vs. iteration number.
(a)
(b)Figure 7
Results with open-loopD-type FOILC. (a) Tracking error e1 vs. iteration number. (b) Tracking error e2 vs. iteration number.
(a)
(b)


## 4.1. The Results of the Open-LoopPDα-Type FOILC

For the open-loopPDα-type FOILC described in (16), ΓP and ΓD are designed to satisfy the convergence conditions, i.e., ΓP=diag1.11.6 and ΓD=diag1.50.8.By checking the convergence condition and definingλ=100, we obtain that(44)I−L+D⊗ΓDCB=0.9312<1,I−L+D⊗ΓDCB+β=0.9621<1.Therefore, the convergence condition is satisfied.The initial states of all the agents, the initial errors, and the initial inputs are 0, that is,ei0=0. The input signals at the zeroth iteration are set to zero, that is, u0,jt=0, for all agents. Figures 2 and 3 show the tracking performance by the open-loop PDα-type FOILC at 5th, 20th, and 70th iterations. At the 5th iteration, the deviations are very large between the trajectories of the follower and the desired one. As the iteration number increases, the outputs y1(t) and y2(t) of the followers nearly completely track the reference trajectory over the time [0, 1]. By 70 iterations, the maximum tracking errors of y1(t) of the four agents are 0.000326, 0.000843, 0.000916, and 0.000276, and the maximum tracking errors of y2(t) of the four agents are 0.0016, 0.0041, 0.0045, and 0.0014, respectively. And moreover, compared with Agents 2 and 4, both Agents 1 and 3 can get information from the leader directly, so they can converge to the expected trajectory more quickly. We define the maximum errors at the ith iteration as maxj=1,2,3,4yd,1−yi,j,1 and maxj=1,2,3,4yd,2−yi,j,2. Compared with Figure 2(d) and Figure 3(d), the outputs y2(t) converge to the expected trajectory more quickly than y1(t). That is because the trajectory y1d(t) is more complicated than y2d(t) (Figure 4).Figure 2
Results ofy1 with open-loop PDα-type FOILC. (a) Trajectories y1(t) at the 5th iteration. (b) Trajectories y1(t) at the 20th iteration. (c) Trajectories y1(t) at the 70th iteration. (d) Tracking error e1 vs. iteration number.
(a)
(b)
(c)
(d)Figure 3
Results ofy2 with open-loop PDα-type FOILC. (a) Trajectories y2(t) at the 5th iteration. (b) Trajectories y2(t) at the 20th iteration. (c) Trajectories y2(t) at the 70th iteration. (d) Tracking error e2 vs. iteration number.
(a)
(b)
(c)
(d)Figure 4
Results ofy1 with open-loop Dα-type FOILC. (a) Trajectories y1(t) at the 5th iteration. (b) Trajectories y1(t) at the 20th iteration. (c) Trajectories y1(t) at the 70th iteration. (d) Tracking error e1 vs. iteration number.
(a)
(b)
(c)
(d)


## 4.2. The Results of the Open-LoopDα-Type FOILC

In order to compare with the performance of the open-loopPDα-type FOILC, the model and the structure of the FOMASs with open-loop Dα-type FOILC are the same as example 1. The control parameters of the open-loop Dα-type FOILC described in (38) are designed as ΓD=diag1.5,0.8 which are the same as ΓD of the open-loop PDα-type FOILC. Thus, if λ=100 is selected, the convergence condition (39) can be calculated as follows:(45)I−L+D⊗ΓDCB+β1=0.9382<1.Thus, the convergence condition in Theorem1 is satisfied.The simulation results are plotted in (Figure4 and 5). Similar to results of the open-loop PDα-type FOILC, as iteration number increases, the outputs y1(t) and y2(t) of the followers nearly completely track the reference trajectory over the time [0, 1]. But, the maximum tracking errors of y1(t) of the four agents are 0.013, 0.034, 0.037, and 0.011, and the maximum tracking errors of y2(t) of the four agents are 0.1, 0.26, 0.28, and 0.085 by 70 iterations, respectively. Obviously, compared with the open-loop PDα-type FOILC, the convergence speed is slower than the open-loop Dα-type FOILC.Figure 5
Results ofy2 with open-loop Dα-type FOILC. (a) Trajectories y2(t) at the 5th iteration. (b) Trajectories y2(t) at the 20th iteration. (c) Trajectories y2(t) at the 70th iteration. (d) Tracking error e2 vs. iteration number.
(a)
(b)
(c)
(d)


## 4.3. The Results of the Open-LoopPD-Type and D-Type IOILC

In order to reflect the superiority of the fractional controller, we designed an integer-order iterative learning control (IOILC) to apply the FOMASs. ThePD-type and D-type IOILC are as follows:(46)ui+1t=uit+L+D⊗ΓPeit+L+D⊗ΓDdeitdt,ui+1t=uit+L+D⊗ΓDdeitdt.The parameters of the FOMASs are the same as Figure1 and equations (40) and (41). The learning gains of PD-type are ΓP=diag1.11.6, ΓD=diag1.50.8, and the learning gain of D-type is ΓD=diag1.50.8, respectively, which are the same as the parameters of the FOILC proposed in this paper. The results are shown in Figures 6 and 7. It can be seen that the errors of y1(t) and y2(t) decrease first as the iteration number increases when employing the PD-type and D-type IOILC. But, when the iteration number is greater than 60, the error gradually increases. Compared with IOILC, we can see that the FOILC has better performance.Figure 6
Results with open-loopPD-type IOILC. (a) Tracking error e1 vs. iteration number. (b) Tracking error e2 vs. iteration number.
(a)
(b)Figure 7
Results with open-loopD-type FOILC. (a) Tracking error e1 vs. iteration number. (b) Tracking error e2 vs. iteration number.
(a)
(b)


## 5. Conclusion

In this study, a consensus tracking problem of fractional-order multiagent systems with general linear models is formulated. DistributedPDα-type and Dα-type fractional-order ILC control algorithms are proposed to solve the consensus tracking problem of the FOMAS. The convergence is proved, and the convergence condition is presented based on graph theory, fractional calculus, and norm theory. It is shown theoretically, together with numerical simulation, that the proposed method can make the agents reach agreement exactly on a common output trajectory over a finite time interval. Through comparison, we find that the PDα-type FOILC can obtain much better performance in terms of convergence speed. Further research may include the closed-loop PDα-type iterative learning control schemes for fractional-order nonlinear multiagent systems, as well as for experimental verifications.



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