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

To solve the consensus problem of fractional-order multiagent systems with nonzero initial states, both open- and closed-loop PD α -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.


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][2][3][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. e 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][8][9][10] and nonlinear fractional-order multiagents [11][12][13][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 leaderfollowing 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][13][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 D α -and PD α -type FOILC were proposed and applied to linear FOMASs with fixed topology. Furthermore, for the linear time-varying integer-order system, Luo  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. e 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. e theoretical analysis and simulation experiments verify the effectiveness of the proposed method. e results show that both the tracking errors and the nonzero initial states can tend to zero in finite time as the iterative number increases. e remainder of this paper is organized as follows. Section 2 overviews the related theories related to this article, including the graph theory, the definition of fractional calculus, and the problem formulation. e 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.

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

Graph eory.
Consider N multiagents with the same dynamic. e direct graph G � V, E, M { } is used to describe the information transfer between multiagents, where V � v 1 , . . . , v N is the node set, E⊆V × V is the edge set, and M � (a ik ) N×N is the adjacency matrix of the direct graph. (k, i) ∈ E⊆V × V is a direct edge of the agents k and i. e set of neighbors of the ith agent is denoted by N i � k ∈ V: (k, { i) ∈ E}. e matrix element a ik > 0 represents node k passing information to node i; otherwise, a ik � 0. Here, the communication topology graph has no self-loop phenomenon, namely, a i,i � 0. D � diag d i , i ∈ S N is defined as the degree matrix, where d i � N k�1 a i,k , and L � D − M is the Laplacian matrix of the direct graph.

e Norm.
In this paper, the vector Euclidian norm and its induced matrix norm is defined as ‖ · ‖. I m ∈ R m×m is the identity matrix. Denote the Kronecker product by ⊗ , for some matrices A,B, C, and D, the following properties will be satisfied such that ‖A ⊗ B‖ � ‖A‖ · ‖B‖.
Lemma 4 (see [23]). For the initial value problem e Volterra-type nonlinear integral equation can be obtained as

Problem Description
Considering N homogeneous fractional-order linear timedelay 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: are the input and output vectors, respectively, and A, B, C are constant matrices with m × m, m × m1, and m2 × m. e expected trajectory y d (t) on the finite-time interval [0, T] is generated by the virtual leader and it is described as where u d (t) 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 as G � 0 ∪ V, E, M , where E and M are the new edge set and the new adjacency matrix of G. e 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. ξ i,j (t) is defined as the distributed information of the jth agent, which is measured or received from other agents at the ith iteration. Consider where a j,k is the entry of adjacency matrix M, s j � 1 if the jth agent can obtain the desired trajectory, and s j � 0 otherwise. e tracking error of the jth agent is defined as en, equation (17) can be reorganized as Define column stack vectors in the ith iteration According to (19), (18) can be reorganized in a compact form where L is the Laplacian matrix of graph G, I m is unit matrix, and S � diag s j , j ∈ S N .

Complexity 3
Similarly, equation (15) can be rearranged as 3.1. Open-Loop PD α -type FOILC. For FOMASs described by (15), considering the nonzero initial state, the open-loop PD α -type FOILC algorithm with initial state learning is proposed as follows: Similar to (20), the updating law (22) can be rewritten as 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]). e graph G contains a spanning tree with the leader being the root.

Remark 2.
is 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. (15) and under the communication graph G, if Assumption 1 and 2 are satisfied. Distributed PD α -type updating rule (23) is applied to the FOMASs (15). If the matrices A, B, C and the learning gains Γ P1 and Γ D1 satisfy the following condition:

Theorem 1. Consider the FOMASs
Proof. e convergence discussed is as follows. Based on Lemma 4, we can write the FOMASs (15) as follows: According to equalities (21), (23), and (25), we can obtain Complexity where 1 (·) is a vector in which all entries are 1.
From Lemma 2 and 3, we can see that Taking (27) into (26), we further get According to Lemma 1, taking Lebesgue-p norm on both sides of (28), we achieve where Recalling the condition of ρ < 1, it deduces that So, as the iterations number increases, i.e., i ⟶ ∞, we obtain It shows that the tracking errors of all the agents tend to reach zero in finite time when i ⟶ ∞. e proof is completed. When Γ P1 � 0, the open-loop PD α -type fractional-order algorithm degenerates into the D α -type fractional-order algorithm, which has the following form:  (15). Assuming that Namely, the output y i (t) converges uniformly to the desired trajectory y d (t) as i ⟶ ∞.
Proof. e process of proof is similar to eorem 1.

Closed-Loop PD α -Type FOILC.
e closed-loop PD α -type FOILC updating law for the FOMASs (15) is designed as follows: Similar to (23), the updating law (35) can be rewritten by the Kronecker product as where L and S are the same as defined in (20). (36) is applied for the system (15). If learning gains Γ P2 and Γ D2 satisfy

Theorem 2. Consider the FOMASs (15) under a directed graph G, if Assumptions 1 and 2 hold. e closed-loop PD α -type FOILC described in
then lim i⟶∞ ‖e i+1 (t)‖ p � 0. Hence, the system outputs y i (t) can fully track the desired trajectory y d (t) in a finite time Proof. From (15) and (36), we can get where 1 (·) is a vector in which all entries are 1. Similar to the derivation of (27), one can conclude that erefore, According to Assumption 1, one can find a feedback gain matrix of differentiation Γ D2 such that I + (L + S) ⊗ CBΓ D2 is a nonsingular matrix. erefore, premultiplying by (I + (L+ S) ⊗ CBΓ D2 ) − 1 on both sides of (42), taking Lebesgue-p norm, and adopting the generalized Young inequality of convolution integral, it can be concluded that where Recalling the condition of ρ 2 < 1, according to inequality (43), it is deduced that From (45), when the number of iterations is large enough, i.e., i ⟶ ∞, we obtain So, it can be proved that the errors of all the fractionalorder agents tend to zero as i ⟶ ∞. For the FOMASs (15), ifΓ P2 � 0 in (37), then the PD α -type FOILC will become D α -type FOILC.
us, according to eorem 2, 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 Γ D2 in (48) is chosen such that
where en the tracking error satisfies lim i⟶∞ ‖e i+1 (t)‖ p � 0. Namely, the outputs y i (t) of the FOMASs (15) converge to the desired trajectory y d (t) uniformly in a finite time when i ⟶ ∞, i.e., lim i⟶∞ y i (t) � y d (t), (t ∈ [0, T]).

Proof.
e proof process of the corollary is similar to eorem 2.

Complexity 7
Similar to (25), the updating law (51) can be rewritten by the Kronecker product as where L and S are the same as defined in (20) where en lim i⟶∞ ‖e i+1 (t)‖ p � 0. us, the system outputs y i (t) of the fractional-order agents converge to y d (t) when i ⟶ ∞ for all t ∈ [0, T]; that is, lim i⟶∞ y i (t) � y d (t), (t ∈ [0, T]).

Remark 3.
According to the conditions of eorems 1 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.

Simulation
In this section, five fractional-order agents are considered, including a virtual leader and four followers. e directed fixed communication topology among agents is shown in Figure 1, where the fractional-order agents are labeled with 0, 1, 2, 3, and 4, respectively. e virtual leader has directed edges to agents 1 and 3.
From Figure 1, the Laplacian matrix L and the information transfer matrix S of the leader to the followers can be obtained as follows: 0, 1, 0).
e dynamic model of the jth agent is described as Here, t ∈ [0, 1], α � 0.75. Let the virtual leader be the given expected reference trajectory In the following simulations, the initial states of the followers at first iteration are set as e control objective of the initial state is x d � 0 0 T and the initial control is set as u 0,j (t) � 0, j � 1, 2, 3, 4 for all agents. And the desired initial states of the four followers are zero; that is x 0,j � 0 0 T 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 u 0,j (t) � 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 8 Complexity   CB‖) � 0.2146 < 1; thus, the convergence condition can be satisfied. Figures 5-7 show the trajectory tracking performances employing the closed-loop PD α -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 PD α -type ILC at the 5th and 30th iterations.

Case 3. Open-closed-loop PD α -type
In this simulation, the initial states and inputs are the same as Case 1 and Case 2. According to eorem 3, the learning gain matrix can be obtained as follows:     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

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 PD α type, closed-loop PD α type, and open-closed-loop PD α type FOILC are proposed. e theoretical convergence of the proposed algorithm is analyzed and sufficient conditions are presented. eoretical 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 fractionalorder nonlinear systems can be studied by using the proposed method of this paper.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.