Consensus Tracking Via Iterative Learning Control for Singular Fractional-Order Multi-Agent Systems Under Iteration-Varying Topologies and Initial State Errors

This paper investigates the leader-following consensus tracking problems via iterative learning control for singular fraction-order multi-agent systems in the presence of iteration-varying switching topologies and initial state errors. First, in order to eliminate the impulsive effect of singular systems and handle iteration-varying topologies, the closed-loop <inline-formula> <tex-math notation="LaTeX">${\mathcal{ D}}^\alpha $ </tex-math></inline-formula>-type iterative learning control protocol is proposed. To deal with initial state errors, the initial state learning laws are introduced in light of the initial output errors of each follower agent. The developed <inline-formula> <tex-math notation="LaTeX">${\mathcal{ D}}^\alpha $ </tex-math></inline-formula>-type learning protocols based on initial state learning laws can guarantee each follower track perfectly the leader agent in the fixed time interval. Next, the sufficient convergent conditions of consensus tracking errors are provided. Moreover, the <inline-formula> <tex-math notation="LaTeX">${\mathcal{ D}}^\alpha $ </tex-math></inline-formula>-type learning protocols are extended to nonlinear singular fraction-order multi-agent systems with iteration-varying topologies and initial state errors. Finally, two numerical examples are presented to verify the validity of the proposed <inline-formula> <tex-math notation="LaTeX">${\mathcal{ D}}^\alpha $ </tex-math></inline-formula>-type learning scheme in this paper.


I. INTRODUCTION
During the past decade, consensus analysis and cooperative control of multi-agent systems (MASs) have attracted extensive attention from scholars of different fields on account of their potential applications in several areas such as cooperative transportation by mobile robots [1], flocking [2], and formation control of vehicles [3], and so on. Consensus control has become a fundamental research topic for cooperative control, aiming to drive all follower agents to reach an agreement via the proper consensus protocol after a fixed time [4]. Actually, in some practical consensus scenes, such as satellite formation keeping [5], synchronisation of sensor networks [6], which needs to be achieved as perfectly as possible over a fixed time interval [7].
The associate editor coordinating the review of this manuscript and approving it for publication was Chao-Yang Chen .
Iterative learning control (ILC) has been widely utilized to cope with the repeated tracking control with high precision requirement in the fixed time interval due to its simplicity and effectiveness [8], [9]. Hence, ILC has been successfully implemented to many kinds of multi-agent systems in recent references, such as high-order nonlinear MASs [10], singular MASs [11], fractional-order MASs [12], and distributed parameter MASs [13]- [15], etc.. In [16], [17], the formation control problems of nonlinear MASs under switching interaction topologies were addressed by employing the ILC scheme. To handle the consensus tracking without a priori knowledge of the control direction, a new adaptive iterative learning control protocol was developed for uncertain nonlinear multi-agent systems under the fixed topology in [18]. Recently, the authors have investigated the problem of quantized iterative learning [19]- [21], and some practical factors including the finite-leveled quantizer with random packet losses was considered for the continuous-time MASs in [22]. Besides, in order to eliminate initial state errors and perform accurate consensus tracking, the initial state learning incorporated with ILC protocol was developed for MASs in [23]. However, it is worth noting that the whole aforesaid studies about ILC are focused on normal MASs. Singular systems, also referred to as generalized state-space systems, semi-state systems and differentialalgebraic systems, which can naturally represent a larger class of systems than the normal linear system model [24]. Applications of this class of systems can be extensively found in modeling and control of mechanical systems, interconnected systems, chemical processes, and other fields [25], [27]. Meanwhile, singular MASs, distinguishing from the norml MASs, possesses the characteristics of regularity and impulse behaviour [27]. And the consensus control of singular linear or nonlinear MASs has been reported in few literature, for instance, the non-fragile consensus control [4], the admissible consensus for homogenous descriptor MASs [29], the guaranteed-cost consensus for singular MASs under switching topologies [28]. It should be pointed out that all the aforementioned published works achieve consensus task after a finite time [28]. To accomplish the consensus task over a fixed time interval, the unified D-type iterative learning algorithm was firstly designed for a class of linear singular MASs in both continuous-time and discrete-time domain to ensure the outputs of followers converge to the leader's trajectory [11]. As is well known, fractional calculus has a long history which can be dated back to the 17th century, many researchers from physics, engineering and biology observe that a fruit number of systems can be modelled by fractional-order differential equations, such as, battery behavior, electromagnetic systems, etc. [30], [31]. Meanwhile, the consensus control of factional-order MASs have been widely concerned from different aspects [30]- [32]. Based on the memory property of fractional-order derivative, the D α -type and PI β -type iterative learning control protocols were applied to handle consensus tacking for nonlinear fractional-order MASs with fixed and iteration-varying communicating graphs, respectively [12], [32].
In the aforesaid references, the dynamic of MASs was governed by a normal system, a singular system or a fractional system. Recently, a novel system called singular fractional-order (SFO) system has been proposed [33], which can be considered as generalizations of singular MASs or fractional-order MASs and also has significant practical background, for instance, electrical networks with supercapacitors [34], swamp-floating plants [35], etc.. Till now, only a few meaningful results on above system have been achieved in [35]- [39]. As for the result of SFOMASs, the consensus problem of fractional-order singular MASs with uncertainties under fixed topology was firstly studied by virtue of robust admissible consensus protocols in [33]. However, there are still a great number of challenging and unsolved issues in the filed of SFOMASs, such as the switching topologies and different initial state errors, and so on. To the best of our knowledge, the consensus tracking of SFOMASs via iterative learning control in the presence of iteration-varying switching topologies and initial state errors has not been addressed in the literature yet.
In view of the above discussion, the main purpose of this paper is that the closed-loop D α -type iterative learning update controllers with initial state learning laws are constructed for linear and nonlinear SFOMASs to achieve perfectly consensus tracking performances of the follower agents under iteration-varying switching topologies and initial state errors over a finite time interval. The distinctive features of this paper can be summarized as follows: 1) The consensus tracking of singular fractional-order MASs with iteration-varying switching topologies and initial state errors have been investigated accurately for the first time in this paper. Then the consensus tracking objective and the attenuating ability of impulse effect have been gradually achieved for SFOMASs.
2) The closed-loop D α -type iterative learning control protocols based on initial state lerning laws via the outputs of each follower agent is proposed, which will be more practical than the distributed state protocol due to the fact that outputs are easier to measure. It is worth pointing out that we do not require that the singular fractional-order systems be impulse-free due to the effect of the developed closed-loop D α -type learning algorithm.
3) The sufficient convergence conditions of consensus tracking errors of each follower agent under the proposed D αtype ILC law are derived firstly for linear SFOMASs under fixed communication topology, and then extend the results to nonlinear SFOMASs under iteration-varying switching topologies case.
The layout of the paper is arranged as follows. The necessary preliminary about graph theory, fractional calculus, and useful lemmas are presented in Section 2. In Section 3, the developed ILC protocols are designed and main results on sufficient consensus tracking conditions are shown, respectively. Some numerical examples will be completed to verify that the achieved results are efficient in Section 4. Finally, some conclusions are drawn in Section 5.
Notations: R n denotes n-dimensional Euclidean space. The superscript T represents the matrix transposition. The Kronecker product is ⊗ and 1 describes the column vector with each entry being 1. I is an identity matrix with appropriate dimensions. For A is matrix equipped with the matrix norm edges E ⊆ V × V and the adjacency matrix A . Here V also be the index set representing the agents in the interaction topology. A direct edge from i to j can be depicted by an ordered pair (i, j) ∈ E , which means that the agent i can transmit information into the agent j. A = (a ij ) ∈ R N ×N denotes the weighted adjacency matrix of the graph G , which is defined as a ii = 0 and a ij > 0 ⇔ (j, i) ∈ E . Accordingly, denote L = D − A be the Laplacian matrix of the digraph A graph is said to contain a spanning tree, that there is a vertex called as the root such that exists a directed path from the root to all other vertex in the graph G .

2) FRACTIONAL-ORDER INTEGRALS AND DERIVATIVES
Introducing a positive real number α, the Riemann-Liouville fractional-order integral is defined as where (·) denotes the Gamma function. An alternative definition for the fractional-order derivative is introduced by Caputo as follow:

B. PROBLEM FORMULATION
Consider the singular fractional-order multi-agent systems (SFOMASs) consisting of N agents. At the kth iteration, the dynamics of the ith agent is described by: where k denotes the iteration index; i ∈ V represents the ith follower agent; t ∈ [0, T ] is the time variable; D α t (·) denotes the Caputo fractional derivative; z k,i (t) ∈ R n , u k,i (t) ∈ R m and y k,i (t) ∈ R m represent the state, control input and output, respectively; E ∈ R n×n is a singular matrix and 0 < rank(E) = r < n; A ∈ R n×n , B ∈ R n×m and C ∈ R m×n are the constant matrices. Throughout this study, the SFOMASs (1) is assumed to satisfy the regularity condition, i.e. det(s α E −A) is not identically zero. Regarding above N agents, the interaction topology among them is described by The desired trajectory for consensus tracking y d (t) is defined on a finite-time interval [0, T ], which is generated by the following dynamics: where z d (t) and u d (t) are the desired state and control input, respectively. In particular, we assume that only a subset of followers know the desired trajectory, which can be regarded as a virtual leader and indexed by vertex 0. Together with G , the extended topology including both leader and all followers can be depicted byG = (V ∪ {0} ,Ẽ ,Ã ), whereẼ is the corresponding edge set andÃ is the adjacency matrix ofG . Before addressing the consensus tracking problem of SFO-MASs, the basic Assumptions are given firstly.
Assumption 1: The graphG contains a spanning tree with the leader as its root. The control objective of this paper is to design an appropriate iterative algorithm to generate a control input sequence u k,i (t) such that each follower agent can track the leader's trajectory perfectly (3)

III. MAIN RESULTS
This section contains two subsections. The D α -type ILC updating law and its convergence properties are adequately revealed for linear SFOMASs with initial state errors and the fixed graph in Section III-A. Then, the iteration-varying topology condition is considered. In Section III-B, the theoretical results of consensus tracking are extended to the nonlinear SFOMASs.

A. CONVERGENCE ANALYSIS OF LINEAR SFOMAS
Based on the fixed topology, denote the available information γ k,i (t) at the (k + 1)th iteration for the agent i as where s i is the weight between agent i and the leader. If agent i can access the desired trajectory, then s i = 1; otherwise, s i = 0. Let e k,i (t) = y d (t) − y k,i (t) be the tracking error, then Eq.(4) can be rewritten as In order to handle the consensus tracking problem of SFO-MASs (1), the closed-loop D α -type iterative learning control algorithms for i ∈ V are constructed as follow: and the initial state learning mechanism is designed by where 1 ∈ R m×m and 2 ∈ R m×m denote two learning gain matrices to be designed. Remark 1: It has been known from Zhang [24,26,39] that impulse terms exist in the response of the singular fractional-order system. The impulse terms may result in control saturation or even deteriorate the system performance, and thereby it is expected to eliminate. Furthermore, with the developed closed-loop D α -type ILC law, the singular fractional-order system can be transformed into a normal system, where the impulsive effects can be removed. For more details, please refer to [26,39].
For simplicity, define the following column stack vectors: Then, Eqs.(5)-(7) via using Kronecker product, one obtains , and it can be obtained form Eq.(1) and Eq. (2) that Define the following vectors: Then, Eq.(11) can be rewritten in the compact form To investigate the variation of δu k (t) between two consecutive iterations from the Eq.(9), one gets At the same time, it can be obtained from Eq.(10) that Substituting Eq.(13) into Eq.(12) results in which implies There exists a learning gain matrix 1 such that (I N ⊗ B). Now, the following useful Lemmas are given, which will be utilized in the proof of main theorems.
Lemma 1: (see [31]) If the function f (x, t) is continuous, then the initial value problem is equivalent to the following nonlinear Volterra integral equation and its solutions are continuous. Lemma 2: (see [40]) Suppose that two non-negative real series {a k } ∞ Accordingly, one has Substituting Eq.(13) and Eq. (14) into Eq.(16) results in Introducing Furthermore, multiplying e −λt on both sides of Ineq. (18), one can obtain According to Hölder inequality, select an appropriate p ∈ (1, 1 where 1 p + 1 q = 1 and p, q > 0. Combining Ineq. (19) and Ineq. (20), one gives Next, taking supremum for Ineq.(21) w.r.t. t, one has δz k λ a 1 e k (0) e −λT + a 2 e k λ + a 3 (α) Obviously, for selecting λ sufficient large enough, i.e., Then, one has Based on the Assumptions and Lemmas given above, the main results of this paper are presented as follow.
Theorem 1: For the linear SFOMASs (1) under the iterative learning algorithm Eq. (7), if the learn gain 2 for initial state learning process satisfies Then, it can be rewritten as . (27) Substituting Eq. (17) Introducing the following notations and taking norm on Eq. (28) derives Then, multiplying e −λt on both sides of Ineq. (29) yields, which implies that By using Lemma 3, it indicates that Finally, selecting some sufficient large λ such that both Ineq. (23) and the following one meets, Therefore, Since the initial state learning mechanism Eq. (14), According to Lemma 2, if b 2 < 0.5, one has lim k→∞ e k λ =0.
The proof of Theorem 2 is completed. Actually, the fixed topology is difficult and restrictive to achieve for multi-agent systems. The fixed graph extended to the iteration-varying graph, which means that the controller is more robust to topology variations. The iteration-varying graphs are defined as follows: where L k is the Laplacian matrix of G in kth iteration, S k = diag s k,1 , s k,2 , · · · s k,N . If agent i can access the desired trajectory, then s k,i = 1; otherwise, s k,i = 0, i = 1, 2, . . . , N is associated withG . Now Eqs.(8)- (10) are written in an iteration-varying form as follows:

Corollary 1: For the system (1) with the iterative learning algorithm Eq.(9) and the initial state learning mechanism Eq.(10), Assumption 1 is satisfied. Then the consensus tracking objective (3) holds if there exists the gain matrix satisfying
The proof is similar to Theorem 2. Firstly, one can obtain the following agents' state trajectories via Eqs. (38) and (39): From Eq.(41), one obtains Taking λ norm on both sides on Eq.(43) yields, Finally, according to Lemma 2 and Eq.(40), we deduce lim k→∞ e k λ =0. The proof is completed.

B. EXTENSION TO NONLINEAR SFOMAS
As an extension of the previous subsection, the D αtype updating law for nonlinear SFOMASs with the fixed/iteration-varying graph and initial state error is formulated in this section.
The dynamics of the ith agent at kth iteration as follow where f (z k,i (t), t) ∈ R n represents a continuous nonlinear function about z k,i (t), the following assumption is satisfied: Assumption 2: f (z, t) satisfies Lipschitz conditions, which means that for any u, v ∈ R n , there exists a constant l f > 0 such that The desired consensus tracking trajectory is generated by the following dynamics: (46) , t) and defining the following vectors: It follows from Eq.(45) and Eq.(46) that There exists a learning gain matrix 1 such that Then, one has For simplicity, denoteā 3 = Ã + l f F . And taking norm and multiplying the factor e −λt on both sides of above expression, one has Obviously, choosing some λ large enough, i.e., Furthermore, one has For the nonlinear SFOMASs, the following convergence result is given: Theorem 3: For the nonlinear SFOMASs (45) with the D αtype iterative learning algorithm Eq. (9) and the initial state learning mechanism applied with learning gain 2 in Eq. (10) meets Theorem 1, and Assumptions 1-2 are satisfied. If there exists the gain matrix 1 satisfying Taking norm on Eq.(57) derives   which implies the following results According to Lemma 4, it indicates that Finally, for some sufficient large λ, satisfying Ineq.(55) and the following one, i.e., Furthermore, one yields e k+1 λ 2b 2 e k λ + (b 1 + The rest of proof is similar to Theorem 2. VOLUME 8, 2020  is given by and S = diag{1, 0, 0, 0}. It can be seen that the graphG has a spanning tree with the leader as its root, which satisfies Assumption 1. According to the D α -type iterative learning algorithm (9) and (10), the gain matrix is taken as  increases, all output of each follower agent approach consistently to the leader's trajectory. Moreover, Fig. 6 (a) depicts the initial errors along the iteration axis. It is observed that the initial states converge to the desired initial state. It can be seen from Fig. 6 (b) that the consensus tracking is achieved by the proposed learning learning algorithm (9) and (10).
Example 2: In Example 2, the nonlinear SFOMASs with iteration-varying graph and initial state error are considered.
Considering the ith agent model as follows: where the remain parameter settings are the same as Example 1. Nevertheless, the nonlinear term is expressed as f (z k,i (t), t) = 0.5 sin(z 1k,i (t)) − 0.8z 2k,i (t) 0.3z 1k,i (t) + 0.2 sin(z 2k,i (t)) .
The iteration-varying topologies are depicted in Fig. 2, in which there are four graphs. In each iteration, the graph VOLUME 8, 2020  is chosen by the selection function k = 4h + j, j = 1, 2, 3, 4, and h is a non-negative integer.   increases, all output profiles can track the desired trajectory. In view of Fig. 10 (a), all initial states can converge to the desired initial state during the initial state learning process. Fig. 10 (b) depicts the tracking errors in each iteration, which demonstrates the precise tracking performance over a finite time interval.

V. CONCLUSION
This paper has investigated the consensus tracking for both linear and nonlinear singular fractional-order MASs under iteration-varying topologies and initial state errors over a finite time interval. The closed-loop D α -type ILC protocols based on initial state learning laws have been proposed for the follower agents and the update laws depend on the information available from the neighbour agents. Then, with the developed D α -type ILC protocols, the sufficient convergence conditions have been presented and the perfect tracking can be achieved for both linear and nonlinear SFOMASs asymptotically. Two simulation examples demonstrate that the designed D α -type ILC protocols with initial state learning laws in this paper are effective. Future efforts will now turn to add from the respect of robustness against uncertainties which include not only external disturbances but also find the the feasibility condition of Theorem 3 cases.
JINGJING WANG received the B.S. degree in automation from Zhengzhou University, Zhengzhou, China, in 2013, and the M.S. degree in control theory and control engineering from the Guangxi University of Science and Technology, Liuzhou, China, in 2017. He has been with the Guangxi Science and Technology Normal University, Laibin, China, since 2018. His research interests include mobile robot path planning technology and multi-agent systems.
JIAQUAN ZHOU received the M.S. degree in control theory and control engineering from the Guangxi University of Science and Technology, Liuzhou, China, in 2017. He has been engaged in frontline teaching since graduation, carrying out teaching and scientific research in the direction of computer algorithms. His main research interests include intelligent control, learning control theory, and robotics.
YAJING MO is currently pursuing the degree with the Guangxi Science and Technology Normal University, Laibin, Guangxi. Her research interests include iterative learning control and mobile robot theory.
LINXIN LI was born in Huangshi, Hubei. He is currently pursuing the B.A. degree with the Guangxi Science and Technology Normal University. His research interests include computer network application technology and multiagent systems.