Consensus tracking problem for linear fractional multi-agent systems with initial state error

Abstract. In this paper, we discuss the consensus tracking problem by introducing two iterative learning control (ILC) protocols (namely, D-type and PD-type) with initial state error for fractional-order homogenous and heterogenous multi-agent systems (MASs), respectively. The initial state of each agent is fixed at the same position away from the desired one for iterations. For both homogenous and heterogenous MASs, the D-type ILC rule is first designed and analyzed, and the asymptotical convergence property is carefully derived. Then, an additional P -type component is added to formulate a PD-type ILC rule, which also guarantees the asymptotical consensus performance. Moreover, it turns out that the PD-type ILC rule can further adjust the final performance. Two numerical examples are provided to verify the theoretical results.


Introduction and problem formulation
With the developments in various fields such as unmanned aerial vehicle (UVA) formation, wireless sensor network, and micro-robot, multi-agent systems (MASs) have been became a hotly investigated topic in recent years.
In accordance to the agent dynamics, MASs are classified into homogenous and heterogenous types [6,31]. A majority of rich references concerns homogeneous MASs [2,4,19,24,28], where all agents are assumed to have the same time-domain dynamics. However, MASs are heterogeneous among numerous practical applications. That is, agents have different system dynamics. The conventional Kronecker product cannot be applied for the closed system dynamics because of different state evolution mechanisms. Thus, the design and analysis for heterogeneous MASs are more complex. Iterative learning control (ILC) has been proved an effective control strategy for various repetitive systems by trial and error to improve the system performance. As a consequence, ILC is also applied to achieve iteration-domain-based learning consensus for MASs whenever the MASs can repeat their coordination process. Indeed, the conventional MASs have been widely investigated in the past few years [11,17,18,30].
To achieve the perfect tracking of system, the initial states for all agents are same as the desired initial state or are set via proper updating rule [1,9]. Nevertheless, MASs can be regarded as "systems of systems", where the controllers are distributed in nature, and all agents are independent entities. Therefore, it is difficult to require all agents to realize the desired initial state because the communication among agents may not be integrated. Moreover, it is not feasible that all agents' initial states are adjusted arbitrarily even if all agents are conscious of the desired initial state. Thus, it is extremely important to relax the initial condition.
To describe memory and hereditary properties of the systems, in which some effects are ignored while using integer-order models, fractional-order differential equation filed a powerful tool [22]. Some systems such as electromagnetism, damped vibrations, and viscoelastic single mass systems should be described by fractional differential equations [5,20,[34][35][36]. For example, aircrafts in a particular formulation fly under the influence of granular materials such as rain, snow, fog, and dust [3]. As a consequence, grouping the above objectives clearly forms fractional-order multi-agent systems, which have become important in practice. Recently, fractional-order MASs are widely used in crossdisciplinary nature and attract much attention on the coordination and control problems [7,16,21,27,31,37]. Emerging results have been reported on the leader-follower solutions of fractional-order MASs [25,26]. Available literatures, authors study the consensus problem of fractional-order systems with input delays by Laplace transform in [21]. The convergence of the iterative process for fractional-order system in time domain is discussed in [10]. Luo et al. [12] designed both P -type and P I-type ILC update laws for linear fractional-order MAS, where a direct channel from the input to the output was assumed for facilitation of the control design. In [33], a fractional-order iterative learning control framework with initial state learning for the tracking problem of integrate linear time-varying systems is present, where both open-loop and closed-loop D α -type iterative learning updating laws are considered. Yang et al. in Chapter 4 of [30] address a leaderfollower tracking problem by ILC approach in the integral multi-agent systems with initial state error, which shows the PD-type updating rule is able to tune the final control performance than D-type one. In addition to ILC approach, there are many interesting results for fractional-order MAS by using frequency domain theory, linear matrix inequality, observer-based protocols, and sampled-data control [15,29,32].
How to establish a framework to describe the communication topology is the key problem in ILC problem for fractional-order MASs. In fact, their communication networks are much more complicated than the first-order MAS due the memory and hereditary properties. First, the information is transmitted crossly for each MAS for digital control, and different MAS makes some necessary communication. Second, how to design ILC laws to display the reliable information channels for each MAS is important. It is a difficult to design the suitable ILC laws to deal with the singularity for the output digital message deriving from the fractional-order MASs. Thus, many techniques of fractional differential equations have been involved in establishing the convergence results. According to this investigation, general results of ILC for fractional-order MASs with initial state error have not been reported. This observation strongly motivates the investigation of this paper. In other words, we investigate the consensus tracking problem by designing D α -type and PD α -type ILC laws with initial state error for fractional-order homogenous MASs and fractional-order heterogenous MASs, respectively. This will provide another possible way to realizing the consensus control for fractional-order MASs.
In MASs, the graph theory is an assistant tool to depict the communication topology among agents. In this paper, we use the symbol G (G = (V, E, A)) as a directed graph (or weighted graph), where V = {v 1 , v 2 , . . . , v N } ({v i , i = 1, 2, . . . , N } is node of the ith agent, N is the number of agent)is the set of vertex, E ⊆ V × V is the set of edges, and A is the adjacency matrix. A directed edge from v i to v j is indicated by an ordered pair v i , v j ∈ E, which denotes the ith agent transmits information to the jth agent. All neighboring nodes of the jth agent are denoted by is the adjacency matrix of G, where a ij is the weight of the edge v j , v i . If v j , v i ∈ E, we say a neighbour of the ith agent is the jth agent and a ij > 0. Else, a ij = 0.
There exists a kind of matrices which is called Laplacian matrix in the interaction of agents. The Laplacian matrix is defined that L = D − A, where the diagonal matrix D = diag(d in 1 , . . . , d in N ) is the degree matrix of the graph G, and d in i = N j=1 a ij is the in-degree of vertex v i . Obviously, L1 N = 0 (1 N is N dimension column vector whose all elements are "1").
We introduce a symbol "⊗". The symbol ⊗ represents Kronecker product whose definition and property can be seen in [12]. There are N heterogenous time-invariant dynamic agents, which interaction topology can be described by G = (V, E, A), and the ith agent (v i ∈ V) at kth iteration is governed by the following linear fractional timeinvariant model: where x k,i (t) ∈ R ni , u k,i (t) ∈ R m , and y k,i (t) ∈ R p are the state vector, input vector, and output vector, respectively. A i ∈ R ni×ni , B i ∈ R ni×m , and C i ∈ R p×ni are constant matrices. Then, the leader's state vector x d (t) and the leader's trajectory (the desired consensus trajectory) y d (t) defined on a finite-time interval [0, T ] are resolved from following equation: where u d (t) ∈ R m is the continuous and unique desired control input, the Here, we regard the desired trajectory y d (t) as a virtual leader and index it by vertex v 0 in the graph representation. Thus, the whole graph depicting the information flow among leader and agents can be expressedG = (V ∪ {v 0 },Ẽ,Ã), whereẼ andÃ are the set of edge and the weighted adjacency matrix ofG, respectively.
In this work, our contributions are listed as follows: (i) We consider the consensus tracking problem by proposing fractional-type ILC rules with initial state error for fractional-order homogenous and heterogenous MASs. This is a basic work for this topic and provide a general approach to deal with the continued topic in this fields. (ii) We establish two fractional versions of convergence results under mild assumptions by using fractional calculus techniques, which display the difference between fractional-order MASs and first-order MASs. One has to deal with the singular kernel involving the formula of the state function. (iii) We show that the PD α -type updating rule can adjust the final consensus performance, which display fractional ILC law is good strategy for fractional-order MASs. (iv) We reveal that the relationship between virtual input (output) and desired input (output) can be displayed via an appropriate equation for homogenous MASs but not for heterogeneous MASs.

Preliminary
In this paper, ρ(M ) = max 1 j n {|Λ j |}, and Λ j (j = 1, 2, . . . , n) is the eigenvalue of the n-dimensional square matrix M . The λ-norm of vector function h: To simplify the controller design and convergence analysis, we need the following assumptions, lemmas, and definitions. Assumption 1. The fixed and directed communication graphG involves a spanning tree, which takes the virtual leader as the root node. Assumption 2. The initial state of every agent is set to the same position, which is no equal to the desired state at every iteration, that is, Assumption 3. CB is of full column rank.
Lemma 2. Consider a positive sequence {ϕ i } satisfying where c 1 , c 2 , and ρ 1 are positive constants, 0 < ρ 0 < 1 and the q satisfied that Proof. The proof is similar to that of Lemma 3.2 in [30].

Main results
In this section, the learning law is first designed for MASs with identical agents in Section 3.1. Then, the results will be extended to heterogeneous MASs in Section 3.2.

Controller design for homogeneous MASs
Assume that in (1), each agent has identical dynamics. That is, In this section, we design suitable protocols and analyze convergence for the fractionalorder linear MASs (3). In Section 3.1.1, the distributed D α -type protocol is considered, and its convergence properties are fully analyzed. To improve the final performance, we add the P -type structure to the D α -type protocol to analyze the convergence properties in Section 3.1.2. http://www.journals.vu.lt/nonlinear-analysis

Distributed D α -type updating rule for homogenous agents
Considering the distributed construction of fractional-order MAS, we let η k,i (t) be the distribute error by the ith agent at the (k + 1)th iteration, which is defined as where s i = 1 if v 0 , v i ∈Ē (it means the ith agent directly links to the virtual leader), and s i = 0 otherwise. The actual tracking error is defined as e k,i (t) = y d (t) − y k,i (t), which cannot be utilized in the controller design due to two reasons. The first one is that such a tracking error is not available for all agents because only a small number of followers can access the leader's trajectory. In other words, the distributed error η k,i (t) rather than e k,i (t) is used in the following algorithms. The other reason is that we require a suitable causality between the input and tracking error according to (1) so that the input signals can be updated effectively [13,14]. In other words, it is the suitable derivation of the distributed error rather than itself will be employed.
Hence, we first adopt the D α -type ILC protocol: where satisfy the following virtual dynamics: and Theorem 1. The fractional-order homogeneous MAS (3), under Assumptions 1-3, the communication graphG, and distributed D α -type updating rule (5) are considered. If the learning rule W D α is chosen such that then the control input u k,i (t) and the output y k, respectively, as the iteration number tends to infinity.
Proof. According to the definition of the distributed measurement η k,i (t) in (4), condition (7), and the property of linearity in the fractional calculus, we can obtain that The virtual tracking error is defined asẽ k,i (t) =ỹ v,i (t) − y k,i (t), then (9) can be simplified as Define the following notations: From (5) and (10) we have Thus, Eq. (11) can be rewritten in the following columnar form: The result, which we take the Caputo fractional calculus from both sides of (6) combining with Lemma 1, does substraction with Eq. (1), Thus, (13) is rewritten by the column stack vectors united from (3), and we have Note thatẽ k (t) =ỹ v (t) − y k (t) = (I N ⊗ C)δx k (t). For (12), we have where In terms of Assumption 1, L + S must be a matrix of full rank.

Theorem 2.
If the D α -type rule converges and W D α CB is nonsingular, there exist uniquẽ u v,i (t) andỹ v,i (t) satisfying the virtual dynamics (6) and (7). Specifically, Proof. From (7) and the dynamics (6) and (2) we have On The solution of (24) is Therefore, from (23) we havẽ

Remark 2.
If W D α is of full column rank, theñ

Distributed PD α -type updating rule for homogenous agents
In this section, the PD α -type updating rule is considered to improve the final performance. The distributed measurement is consistent with the D α -type case. The proposed PD α -type updating rule for ith agent at kth iteration is where W PD α is a learning gain matrix with suitable dimension. The desired trajectory y d (t) and virtual output trajectory y v,i (t) satisfy the following formula: Remark 3. Condition (26) is inspired from [30,Chap. 4] for case of α = 1, i.e., Theorem 3. Consider the MAS (3), if the learning gain matrix W D α is chose such that then the ILC rule is stable, and the output trajectory of any follower converges to the virtual output trajectory as the iteration number tends to infinity. In particular, Proof. The proof is similar to that of Theorems 1 and 2.ẽ k,i (t), δu k,i (t), and δx k,i (t) are defined as in Theorem 1, as well as δũ k,i (t) and δx k,i (t) are defined as in Theorem 2.
Further, according to (26), we have This yields that Solving (29), we have

Controller design for heterogenous agents
This section is extension of Section 3.1. The model of MASs (1) consisting of N heterogeneous agents is rewritten as whereȂ = diag{A 1 , A 2 , . . . , A N },B = diag{B 1 , B 2 , . . . , B N }, andC = diag{C 1 , C 2 , . . . , C N } are constant matrices with suitable dimensions. The definition of three column stack vectors x k (t), y k (t), and u k (t) is consistent with the previous section.

Distributed D α -type updating rule for heterogeneous agents
The distribute measurement η k,i (t) is the same with (4). We still apply D α -type updating rule for every agent but the gain matrix of each agent is different: The P -type updating law is then added to the D α -type updating law to improve the final performance: Similarly, let thex v,i (t),ũ v,i (t), andỹ v,i (t) = C ixv,i (t) satisfy the following virtual dynamics: Theorem 4. Assume that Assumption 1-3 hold for the time-invariant linear MASs (30) with the D α -type updating law. If the learning gain matrixW D α (W D α = diag{W D α ,1 , W D α ,2 , . . . , W D α ,N }) satisfies the following condition: for someρ 0 ∈ (0, 1), then there exists a positive constant λ such that δu k λ → 0, which indicates that lim k→∞ u k,i =ũ v,i (t), and then lim k→∞ y k,i (t) =ỹ v,i (t) for all t ∈ [0, T ], i = 1, 2, . . . , N .
Proof. The procedure that prove this theorem is similar to that of Theorem 1.

Distributed PD α -type updating rule for heterogeneous agents
For the PD α -type updating rule, we have the following results.
Theorem 5. Assume that Assumptions 1-3 hold for the time-invariant linear fractionalorder MASs (30) with N heterogeneous agents and PD α -type updating law (32). If the learning gain W D α still choose that then the ILC rule is stable and the output trajectory of any follower convergence.
Proof. The steps for this theorem is omitted here because it can be derived similar to that of Theorems 3 and 4.
Remark 4. In Theorem 5, we need the following equation, which is also motivated by Remark 3: The relation between y d (t) andỹ v,i (t) (or between u d (t) andũ v,i (t)) cannot be showed via an equality due to different agent submits to different fractional-order dynamical system in Theorems 4 and 5.
Remark 5. The explicit solution of model (1) (see (17)) can also be obtained in the form of Mittag-Leffler functions.
Remark 6. Compared with the Chapter 4 in [30], this paper extends the integral-order model to the fractional-order model and corrects the inappropriateness, such as expressions for u i,j and y i,j in Theorem 4.3 of the original paper.

Simulation examples
In this section, two example are provided under uniform order and communication topology graph.
We select an arbitrary order in (0, 1), say α = 0.7, and initial input u 0 (t) = 0. Firstly, we consider the D α -type updating rule. Next, we add P -type updating rule to adjust the finial performance base on the example of the D α -type updating rule.
For Fig. 1, the Laplacian for followers is Example 1. Consider a group of five followers at kth iteration with their dynamics governed by the following model: Obviously, the initial condition for followers satisfies Assumption 2. rank(CB) = 1 satisfies Assumption 3.
We apply the updating rules (5) and (25) with learning gains matrix W D α = [0.0215, 0.0376], which is provide by Remark 1, and W PD α = diag (3,3) is chose correctly by many tests. Then the convergence condition is calculated: which satisfies the convergence requirement in Theorems 1 and 2. Figure 2 shows the output profiles of all agents at the 50th iteration under D α -type updating rule. The output of followers is capable to track the general trend of the leader. According to the property of linearity in fractional calculus, (7) is rewritten that W D α C 0 D α t (y d (t)− y v,i (t)) = 0. Equation (7) is easily demonstrated by it. But the deviations between the followers and the leader is large. Simple calculation shows that the characteristic value of F D α is σ(F D α ) = {0, −1.1908}.    Figure 3 shows the output profiles of all agents at the 50th iteration under a PD α -type updating rule. Different with the Fig. 2, we add the P -type updating rule to adjust the output profiles. It can be seen that the tracking error becomes smaller and smaller. The characteristic value of F PD α is σ(F PD α ) = {−1.1908, −4}. Due to the memorability of fractional calculus, the tracking trajectory of followers can roughly approach the leader's tracking trajectory with the growth of the iteration, but the deviation still exists. Figure 4 shows the error value between y d (t) and y 50 (t) under PD α -type updating rule. Obviously, the error value becomes smaller and smaller with the change of time. These pictures illustrate the PD α -type updating rule can improve the finial tracking trajectory from Figs. 3 and 4.
Example 2. In this example, we consider a network consisting of one leader and five heterogeneous follower agents, which communication topology is showed by Fig. 1. The model of the first agent at kth iteration is governed by the model of the second agent at kth iteration is governed by C 0 D α t x k,2 (t) = 0.5 x k,2 (t) + 0.8 u k,2 (t), y k,2 (t) = 0.7 x k,2 (t); the model of the third agent at kth iteration is governed by the model of the fourth agent at kth iteration is governed by The model of desired reference trajectory is governed that   Figure 5(a) shows the output trajectory of heterogeneous agents at the 100th iteration under the D α -type updating rule (31). Even though every agent is governed by different fractional-order model with larger the error, the output of followers is able to track the general trend of the leader. It easily shows that the formula (33) is equalled. Figure 5(b) shows the output profiles of all agent at the 100th iteration under PD αtype updating rule (32). Obviously, the tracking error which is adjusted by adding P -type updating rule becomes small and small. Similarly, in consider of the memorability of the fractional-order model, the error between the leader's tracking trajectory y d (t), and the followers' tracking trajectory y k (t) still is occurred. Figure 6 shows the error of tracking trajectory between the leader's trajectory y d (t), and the followers' trajectory y 100 (t) becomes smaller and smaller.
Evidently, the PD α -type updating rule which is introduced can adjust the final tracking performance than the D α updating rule from Figs. 5(a), 5(b).

Conclusion
This paper studies the learning consensus problem for both fractional-order homogeneous and heterogeneous multi-agent systems, where the initialization position of each agent is fixed away from the desired one for all iterations. This condition implies an imperfect state resetting. It is proved that the D α -type updating rule is convergent but the final trajectories are not the desired reference. To further improve tracking performance, the PD α -type updating rule is then proposed, which provides the designer more freedom to tune the final performance. The theoretical results are verified by two numerical simulations. For further research, it is of interest to consider the fractional-order nonlinear MASs. In addition to fractional-order differential equations to describe engineering systems under complex environments, fuzzy differential equations are also suitable to simulate the process of change under uncertain conditions which can more accurately reflect the reality. Multiagent systems using the fuzzy differential equation will be explored in the future.