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

Li, Xungen; Lv, Shuaishuai; Pan, Mian; Ma, Qi; Cai, Wenyu

doi:https://doi.org/10.1155/2020/9130875

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Finite-time Control of Complex Systems and Their Applications

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Research Article | Open Access

Volume 2020 | Article ID 9130875 | https://doi.org/10.1155/2020/9130875

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

Xungen Li,^1,2Shuaishuai Lv ,^1,2Mian Pan,¹Qi Ma,¹and Wenyu Cai¹

Academic Editor: Oh-Min Kwon

Received11 Jun 2020

Revised19 Sept 2020

Accepted01 Oct 2020

Published28 Oct 2020

Abstract

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.

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 [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 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 et al. proposed a FOILC framework with initial state learning and presented sufficient and necessary conditions for open-loop and closed-loop D^α-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. Section 2 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

Consider multiagents with the same dynamic. The direct graph is used to describe the information transfer between multiagents, where is the node set, is the edge set, and is the adjacency matrix of the direct graph. is a direct edge of the agents k and i. The set of neighbors of the ith agent is denoted by . The matrix element represents node k passing information to node i; otherwise, . Here, the communication topology graph has no self-loop phenomenon, namely, . is defined as the degree matrix, where , and 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 . is the identity matrix. is defined as a function set and the mth derivative of # is continuous over a finite time interval . and are the sets of real and natural numbers. . Denote the Kronecker product by , for some matrices ,, , and , the following properties will be satisfied such that

Definition 1. Assuming the continuous vector function , the Lebesgue-p norm of is defined as

Lemma 1 (see [19]). Assuming the functions and , then the convolution generalized Yong inequality of the functions and iswhere , , and is the convolution integral of and . In particular, if , the inequality is converted to .

2.3. Fractional Calculus

Definition 2 (see [21]). The [21, 22] Riemann–Liouville fractional integrals of with order are defined aswhere is gamma function. The left- and right-sided Caputo derivatives arewhere and means the integral part of .

Lemma 2 (see [20]). Suppose the functions , are continuous in , and , exist, then the fractional integration by parts is

Definition 3. (see [20, 23]). The Mittag–Leffler function can be described asParticularly, when , we can obtain

Lemma 3 (see [20]). Let ; then we have

Lemma 4 (see [23]). For the initial value problem

The Volterra-type nonlinear integral equation can be obtained as

Property 1. If , then , where .

3. Problem Description

Considering N 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:where , is the left-sided -order derivative of , . is the state vectors, and are the input and output vectors, respectively, and are constant matrices with , , and .

The expected trajectory on the finite-time interval is generated by the virtual leader and it is described aswhere 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 , where and are the new edge set and the new adjacency matrix of . 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.

is defined as the distributed information of the jth agent, which is measured or received from other agents at the ith iteration. Considerwhere is the entry of adjacency matrix , if the jth agent can obtain the desired trajectory, and otherwise.

The tracking error of the jth agent is defined as . Then, equation (17) can be reorganized as

Define column stack vectors in the ith iteration

According to (19), (18) can be reorganized in a compact formwhere L is the Laplacian matrix of graph , is unit matrix, and .

Similarly, equation (15) can be rearranged as

3.1. Open-Loop -type FOILC

For FOMASs described by (15), considering the nonzero initial state, the open-loop -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]). The graph 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 , if Assumption 1 and 2 are satisfied. Distributed -type updating rule (23) is applied to the FOMASs (15). If the matrices and the learning gains and satisfy the following condition:where , then . Namely, the output converges uniformly to the desired trajectory as .

Proof. The 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 obtainwhere is a vector in which all entries are 1.
From Lemma 2 and 3, we can see thatTaking (27) into (26), we further getAccording to Lemma 1, taking Lebesgue-p norm on both sides of (28), we achievewhereRecalling the condition of , it deduces thatSo, as the iterations number increases, i.e., , we obtainIt shows that the tracking errors of all the agents tend to reach zero in finite time when . The proof is completed. When , the open-loop PD^α-type fractional-order algorithm degenerates into the D^α-type fractional-order algorithm, which has the following form:Thus, the following corollary can be obtained.

Corollary 1. Consider the FOMASs (15) and under the communication graph , if Assumptions 1 and 2 are satisfied. Distributed Dα-type updating rule (33) is applied to the FOMASs (15). Assuming thatholds for all [0, T], where , then . Namely, the output converges uniformly to the desired trajectory as .

Proof. The process of proof is similar to Theorem 1.

3.2. Closed-Loop -Type FOILC

The closed-loop -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 aswhere and are the same as defined in (20).

Theorem 2. Consider the FOMASs (15) under a directed graph , if Assumptions 1 and 2 hold. The closed-loop -type FOILC described in (36) is applied for the system (15). If learning gains and satisfywherethen . Hence, the system outputs can fully track the desired trajectory in a finite time when for all ; that is, .

Proof. From (15) and (36), we can getwhere is a vector in which all entries are 1.
Similar to the derivation of (27), one can conclude thatSubstituting (40) into (39), it yieldsTherefore,According to Assumption 1, one can find a feedback gain matrix of differentiation such that is a nonsingular matrix. Therefore, premultiplying by on both sides of (42), taking Lebesgue-p norm, and adopting the generalized Young inequality of convolution integral, it can be concluded thatwhereFurtherRecalling the condition of , according to inequality (43), it is deduced thatFrom (45), when the number of iterations is large enough, i.e., , we obtainSo, it can be proved that the errors of all the fractional-order agents tend to zero as . For the FOMASs (15), if in (37), then the -type FOILC will become -type FOILC.Thus, according to Theorem 2, we can obtain a corollary as follows.

Corollary 2. For the FOMASs (15) under a directed graph , suppose Assumptions 1 and 2 hold. If the learning gain in (48) is chosen such thatwhereThen the tracking error satisfies . Namely, the outputs of the FOMASs (15) converge to the desired trajectory uniformly in a finite time when , i.e., .

Proof. The proof process of the corollary is similar to Theorem 2.

3.3. Open-Closed-Loop -Type FOILC

Considering the FOMASs (15), an open-closed-loop -type FOILC is designed as

Similar to (25), the updating law (51) can be rewritten by the Kronecker product aswhere and are the same as defined in (20) and (36).

Theorem 3. Consider the FOMASs (15) under a directed graph , if Assumptions 1 and 2 hold. Let the distributed closed-loop -type FOILC described in (52) be applied for the system with learning gains ,, and satisfyingwhereThen . Thus, the system outputs of the fractional-order agents converge to when for all ; that is, .

Remark 3. According to the conditions of Theorems 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.

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 Figure 1, 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.

From Figure 1, the Laplacian matrix and the information transfer matrix of the leader to the followers can be obtained as follows:

The dynamic model of the jth agent is described asHere, , .

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 , , , and . The control objective of the initial state is and the initial control is set as for all agents.

Case 1. Open-loop PD^α-type: the open-loop PD^α-type is applied to the multiagent system (1). Based on Theorem 1, the gains are selected as ,. Thus, we can calculate , which satisfies the convergence condition (26).
The simulation results are shown in Figures 2–4. The initial states of the followers at the first iteration are , , , and . And the desired initial states of the four followers are zero; that is 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 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 and for . 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.

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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 ,. Clearly, ; thus, the convergence condition can be satisfied.
Figures 5–7 show the trajectory tracking performances employing the closed-loop -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 -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 PD^α-type, the closed-loop FOILC performs better and has faster convergence speed than the open-loop one.

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Case 3. Open-closed-loop -type
In this simulation, the initial states and inputs are the same as Case 1 and Case 2. According to Theorem 3, the learning gain matrix can be obtained as follows:Clearly, ; thus, the convergence condition in Theorem 3 can be satisfied.
The simulation results with open-closed-loop PD^α-type FOILC are presented in Figures 8–10. The results are similar to those of the open-loop and closed-loop PD^α-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 PD^α-type FOILC and closed-loop PD^α-type, applying open-closed-loop PD^α-type FOILC has better performance in the initial state and for the outputs.

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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 PD^α type, closed-loop PD^α type, and open-closed-loop PD^α 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.

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study has been supported by the National Natural Science Foundation of China (no. 61871163), the Zhejiang Provincial Natural Science Foundation of China (no. LQ19E070003), and Zhejiang Provincial Key Lab of Equipment Electronics.

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Copyright © 2020 Xungen Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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