On the Controllability of Discrete-Time Leader-Follower Multiagent Systems with Two-Time-Scale and Heterogeneous Features

.is paper investigates the controllability of discrete-time leader-follower multiagent systems (MASs) with two-time-scale and heterogeneous features, motivated by the fact that many real systems are operating in discrete-time. In this study, singularly perturbed difference systems are used to model the two-time-scale heterogeneous discrete-time MASs. To avoid the ill-posedness problem caused by the singular perturbation parameter when using the classical control theory to study the model, the singular perturbation method was first applied to decompose the system into two subsystems with slow-time-scale and fast-time-scale feature. .en, from the perspective of algebra and graph theory, several easier-to-use controllability criteria for the related MASs are proposed. Finally, the effectiveness of the main results is verified by simulation.


Introduction
In recent years, with the wide application of MASs in aircraft formation, multirobot cooperative control, traffic vehicle control, network resource allocation, and other fields, scholars have become particularly interested in the distributed cooperative control [1][2][3][4] of these systems. MASs are systems composed of some dynamic agents with the certain autonomous ability through information communication and interaction. e ultimate goal of studying them is reflected in the control that people can have, which makes the research on the controllability of MASs extremely important. Controllability features of MASs are related to the agents that can reach all desired final states from any initial states within a limited time through controlling a specific portion of the agents.
In the 1960s, Kalman [5] introduced the concept of controllability of linear time-invariant (LTI) dynamic systems and pioneered the effective Kalman rank criterion for discriminating controllability of LTI dynamic systems. en, Tanner [6] extended the concept of controllability to MASs, aiming to understand more about the issue of single-integrator continuous-time in such systems. An algebraic controllability criterion was obtained under the assumption of one leader and nearest-neighbor communication protocol and made a great contribution to the field. Rahmani and Mesbahi [7] proposed the relationship between graph symmetry and system controllability based on Tanner's results and achieved conditions that could prove that the corresponding system is uncontrollable when the topological structure graph linked to the leader agent is symmetrical. Furthermore, in [8,9], the nontrivial equitable partition method is used to deeply explore the relationship between the controllability and network structure of MASs with multiple leaders. is is a research branch based on graph theory that should definitely be considered when regarding the issue of the controllability of MASs. Since then, a large number of studies [10,11] on this realm have been conducted, achieving different outcomes.
Most of the existing research studies on the controllability of MASs from the perspective of algebra consider that the agents constituting the systems are of a single type. Ni et al. [12] investigated the controllability of the first-order MASs and obtained some controllability criteria when the controllability was decoupled into two independent parts, one is about the controllability of each individual node, and the other is completely determined by the network topology. Taking the switching topology into consideration, Tian et al. [13] minutely studied the controllability of first-order MASs composed of continuous-time subsystems and discrete-time subsystems. Using the concepts of invariant subspace and controllable state set, a sufficient and necessary condition for the controllability of switched MASs was obtained. Based on the Jordan standard form of the Laplacian matrix, the effects of topology, communication strength, and the number of external inputs on the controllability of the first-order leader-based MASs are discussed in [14]. Meanwhile, a topological structure that is completely controllable regardless of the positions and number of leaders is also proposed. is structure is extremely relevant for system design in engineering practice.
As the discrete-time system is ubiquitous in life, it should not be ignored, specially considering a scenario of rapid development of information and communication technologies. Liu et al. studied the controllability of discrete-time MASs with one leader under fixed and switched topologies in [15] and concluded that when one of the agents is selected as the leader appropriately, the interconnected system is completely controllable even though each subsystem cannot be controlled. Furthermore, in [16], the concept of group controllability of multiagent systems is proposed first, and the controllability criteria of a class of first-order multiagent systems are explored only when the agents constituting the MASs are divided into different subgroups.
ese subgroups are divided according to different control objectives. However, the reality is that usually there are different types of individuals with different abilities in the same system. For example, heterogeneous MASs composed of unmanned air vehicles with different capabilities can often exhibit more superior performance [17]. Based on this, some recent studies have been considering the controllability of MASs and its heterogeneous characteristics. Guan et al. [18] did that and concluded that the controllability of these systems was completely dependent on the controllability of its underlying topology under the choice of specific leaders. Tian et al. [19] further studied the same features of heterogeneous MASs with switching topologies. Based on the concept of invariant subspace, the authors pointed out that if the union of all possible topologies is controllable, then so are these systems.
All the research results mentioned above consider that the agents that make up the system work on the same timescale. However, it is common that different timescales coexist in the same system. Due to the mutual influence between the different timescale components, they cannot be analyzed separately. For example, in the field of robots, the dynamics of flexible manipulator includes two major timescales: macro rigid motion and micro flexible vibration [20]. Almost all large-scale systems have a dynamic coexistence phenomenon with large timescale differences. Prandtl [21] first proposed a singularly perturbed model to describe the two-time-scale system when studying the fluid dynamic systems. In 1968, Kokotovic and Sannuti [22] used this new model to describe system dynamics with different timescales and proposed a fastslow combination control strategy, which established the basic control strategy of two-time-scale dynamic system. e singular perturbation method [23,24] is a major means of study used to understand more about singularly perturbed systems. e core idea is to decompose this specific system into fasttime-scale and slow-time-scale subsystems. Specifically, it is assumed that the slow variable remains unchanged during the response period of the fast variable. When analyzing the response of the slow variable, it is considered that the fast variable has reached stability state value. Many researchers had studied issues related to the singularly perturbed system and method (e.g., feedback control of the two-time-scale system [25]). Su et al. [26] first studied the controllability of discretetime first-order MASs with the two-time-scale feature and obtained necessary and/or sufficient controllability criteria based on the matrix theory. Furthermore, the controllability of continuous-time and discrete-time second-order MASs with the two-time-scale feature is discussed in [27,28], respectively.
However, few studies have considered the controllability of MASs with both heterogeneous and two-time-scale features and only one study, conducted by Long et al. [29], considers such features with continuous-time. With the increasing development of cyber-physical systems, discussions in discrete-time have been ascending more and more. Inspired by this, this paper focuses on the controllability of discrete-time leader-follower MASs with heterogeneous and two-time-scale features and tries to solve this research gap that requires more attention. e essential difference between discrete-time systems and continuous-time systems will lead to different modelling methods as well as different transformation derivation methods instead of a simple generalization. e research content of this article has the potential to supplement existing results in this research field. Specifically, the significance and innovation of this research are summarized as follows: (1) e definition of controllability of discrete-time leaderfollower MASs with heterogeneous and two-time-scale features is proposed for the first time.
(2) A singularly perturbed difference system is used to model the discrete-time leader-follower MASs with heterogeneous and two-time-scale features, and the singular perturbation method is applied to decouple the model. is is done to avoid the ill-posedness problem when the classical control method is directly used to study the controllability of these systems.

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(3) Several easier-to-use necessary and/or sufficient conditions for the controllability of discrete-time leader-follower MASs were obtained with heterogeneous and two-time-scale features. e rest of this paper is arranged as follows. Section 2 gives the preliminary knowledge, describes the problem to be studied, and models the discrete-time leader-follower MASs with heterogeneous and two-time-scale features. In Section 3, we first decompose the systems by the singular perturbation method to eliminate the singular perturbation parameter. en, we define the controllability of the discrete-time leader-follower MASs with two-time-scale and heterogeneous features, and several necessary and/or sufficient conditions for controllability are stated. e effectiveness of the proposed criterion is verified by the simulation in Section 4. Conclusions are drawn in Section 5.

Preliminaries.
A graph G(V, E) composed of the vertex set V and edge set E can be used to represent a concrete MAS abstractly. Each agent is represented by a vertex, and the information interaction between agents is represented as an edge. e number of vertices is recorded as N � |V|, and the number of edges is recorded as M � |E|. Each edge in the set E has a pair of vertices in the set V corresponding to it. If any vertex pair (i, j) and (j, i) correspond to the same edge, the graph is labelled as an undirected graph. If not, it is called a directed graph. e adjacency matrix A � [a ij ] is usually used to represent the topological structure of the graph. If there is an edge from vertex j to vertex i, then a ij ≠ 0 and a neighbor of i is j. Otherwise, a ij � 0. All neighbor vertices of the vertex i are recorded as N i . If the graph G is undirected, then A is a symmetric matrix. e Laplacian matrix of the graph G is defined as L � D − A, where D is the degree matrix of G. Because the sum of each row of the Laplacian matrix L is 0, regardless of the graph being an undirected or a directed one, 0 is the eigenvalue of L, and the corresponding eigenvector is 1 (a column vector where every element is 1).
e Laplacian matrix corresponding to the undirected graph is a positive semidefinite matrix, and the algebraic multiple of its eigenvalue 0 is the number of connected components of the graph.

Lemma 1 (see [30]). If all eigenvalues λ(e
where A is a constant matrix with appropriate dimensions. Lemma 2 (see [30]). If all eigenvalues λ(e A ) of e A satisfying |λ(e A )| < 1, then when ε ⟶ 0, there is where A is a constant matrix with appropriate dimensions.
Next, R, C, and I are used to represent the real number set, the complex number set, and the identity matrix with a suitable dimension. e symbol ⊗ denotes the Kronecker product, and ε(0 < ε ≤ 1) is a singular perturbation parameter used to distinguish two timescales.

Problem Formulation.
e discrete-time MAS with heterogeneous and two-time-scale features under a leader-follower framework in consideration is described below. First of all, the heterogeneity reflects that m + m l agents in the MAS are first-order integrators, and the remaining n + n l agents are second-order integrators. m l represents the number of leaders among the first-order integrators agents, and m is the number of followers in the first-order integrators agent cluster. Similarly, n l and n are the numbers of leaders and followers among the second-order integrator agents, respectively. In this way, the whole system is divided into four parts: first-order follower agent cluster, first-order leader agent cluster, second-order follower agent cluster, and second-order leader agent cluster, which are represented by G(f 1 ), G(l 1 ), G(f 2 ), and G(l 2 ) in Figure 1. e interaction between all agents can be expressed by the following matrix L corresponding to the entire MAS: where Secondly, each first-order agent i operates on two different timescales simultaneously. Specifically, vectors x i ∈ R n x ×1 and z i ∈ R n z ×1 are used to represent the position state vectors of the first-order agent i on the slow-time-scale and the fast-time-scale. Each second-order agent o has vectors x o ∈ R n x ×1 and w o ∈ R n x ×1 to represent the position and velocity states of slow-time-scale while vectors z o ∈ R n z ×1 and d o ∈ R n z ×1 are used to represent the position Mathematical Problems in Engineering 3 and velocity states of fast-time-scale. e dynamic model of all agents in this MAS is modelled by Here, the input matrices of two different timescales are B 1 ∈ R n x ×q and B 2 ∈ R n z ×q .

Decomposition of Singularly Perturbed Systems.
In view of the existence of the singular perturbation parameter ε representing different timescales, systems (4a) and (4b) are labelled as a singularly perturbed system. If the classical controllability theory is used to process this system, it will cause the ill-posedness problem. Inspired by [25], systems (4a) and (4b) should first be decomposed to eliminate the singular perturbation parameter ε so that it can be discussed with the traditional controllability research method. Since ε exists in the slow-time-scale equation, n is the timescale of the fast-time-scale subsystem. If l is used to represent the timescale of the slow-time-scale subsystem, then Accordingly, the first attempt was to decouple the equation of the slow-time-scale subsystem. In this process, a reasonable assumption is that the states of the fast-time-scale subsystem have reached steady-states. Based on the matrix φ 22 is invertible, formula (8b) can be written as Mathematical Problems in Engineering r 2 (n) � φ 21 r 1 (n) + I + φ 22 r 2 (n) + ϕ 2 u(n), (11) which further leads to Substituting (12) into (8a) yields Simultaneously, the states of the agents and the input signals of the slow-time-scale subsystem are represented as r s (n) and u s (n). So, r s (n + 1) � I + εφ s r s (n) + εϕ s u s (n).
Let us describe the decomposition process in detail. During this process, the state vector r 1 of (8a) and (8b) can be assumed to remain unchanged. By adding φ −1 22 φ 21 r 1 (n) to both sides of equation (8b), it gives So far, formula (17b) can be obtained.

Controllability Analysis
Definition 1. For the discrete-time leader-follower MAS with two-time-scale and heterogeneous features as (4a) and (4b) discussed in this paper, if any nonzero states r s and r f of systems (17a) and (17b) meet the following conditions simultaneously, then systems (4a) and (4b) can be said to be controllable: (1) For the initial nonzero state r s (0) � r s , there exists a piecewise input u s so that it can reach the zero state within a finite time T 1 (i.e., r s (T 1 ) � 0). (2) For the initial nonzero state r f (0) � r f , there exists a piecewise input u f so that it can reach the zero state within a finite time T 2 (i.e., r f (T 2 ) � 0).

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Proof of Proposition (1) that is, Furthermore, en, Since the vector ρ is nonzero, we obtain According to Lemma 3, it can be concluded that MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. e necessity is proved.
Sufficiency: with λ s and μ f representing the eigenvalues of matrices φ s and φ f , our purpose is to prove that if rank[λ s I − φ s , ϕ s ] � (m + 2n) × n x and rank[μ f I − φ f , ϕ f ] � (m + 2n) × n z , then MAS (4a) and (4b) are controllable. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable.
Motivated by (24), there holds ρ T [λ s I − φ s , ϕ s ] � 0. is means rank[λ s I − φ s , ϕ s ] < (m + 2n) × n x . A similar conclusion can be drawn that rank[μ f I − φ f , ϕ f ] < (m + 2n) × n z . Because the contrapositive proposition of the original proposition is true, the original proposition is also true. e sufficiency is proved. □ Proof of Proposition (2). Necessity: our purpose is to prove that if MAS (4a) and (4b) are controllable, the vectors ς and ζ that satisfy ς T φ s � σς T , ς T ϕ s � 0, ζ T φ f � cζ T , and ζ T ϕ f � 0 must be zero vectors, where ς ∈ C (m+2n)n x ×1 and ζ ∈ C (m+2n)n z ×1 . Let us look at its contrapositive proposition. Suppose that there is a nonzero vector ς ∈ C (m+2n)n x ×1 and σ ∈ C satisfy ς T φ s � σς T and ς T ϕ s � 0; then, which further results in Since the vector ς is nonzero, we obtain , and then MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. e necessity is proved.
Sufficiency: we want to prove that if the vectors ς and ζ that satisfy ς T φ s � σς T , ς T ϕ s � 0, ζ T φ f � cζ T , and ζ T ϕ f � 0 are all zero vectors, then MAS (4a) and (4b) are controllable, where ς ∈ C (m+2n)n x ×1 , ζ ∈ C (m+2n)n z ×1 , σ ∈ C, and c ∈ C. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, rank[ϕ s , φ s ϕ s , . . . , φ (m+2n)×n x −1 s ϕ s ]<(m + 2n) × n x . Similar to the sufficiency proof of Proposition (1), we can find that there must be a nonzero vector ς ∈ C (m+2n)n x ×1 and σ ∈ C such that ς T φ s � σς T and ς T ϕ s � 0. Similar conclusions can be drawn about the matrices φ f and ϕ f . Because the contrapositive proposition of the original proposition is true, the original proposition is also true. e sufficiency is proved.

Theorem 2. If the two matrices φ s and φ f do not have the same eigenvalues with Q, the discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable, where
Proof. Let us look at its contrapositive proposition. Suppose that the slow-time-scale subsystem of MAS (4a) and (4b) is uncontrollable. So, under Proposition (2) of eorem 1, it gives that there exists a nonzero vector ς ∈ C (m+2n)n x ×1 such that ς T φ s � ας T and ς T ϕ s � 0, where α is an eigenvalue of the matrix φ s . Defining a new vector [ς T , 0, 0, 0], then Obviously, φ s and Q have the same eigenvalue α. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. A similar conclusion about the matrices φ f and Q can be obtained with a similar proof process, so it is omitted here. eorem 2 is proved. □

Theorem 3. e discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is
controllable if the eigenvalues of φ s and φ f are all different, and all rows of matrices P − 1 and R − 1 are not orthogonal to at least one column of matrices ϕ s and ϕ f , respectively, where P and R are, respectively, composed of the eigenvectors of matrices φ s and φ f .
Proof. According to the knowledge of matrix theory, since the eigenvalues of φ s are different, φ s can be similarly diagonalized. Let λ s (∀s ∈ 1, 2, . . . , (m + 2n)n x ) represent the eigenvalues of φ s and P as an invertible matrix composed of corresponding eigenvectors, it follows that φ s � PΛP − 1 with en, Based on the conclusion of matrix theory that the elementary transformation of matrix does not change the rank, rankΓ � rank , diag η 1, m l +2n l ( )nx+ m l +2n l ( )nz , η 2, m l +2n l ( )nx+ m l +2n l ( )nz , . . . , η (m+2n)n x , m l +2n l ( )nx+ m l +2n l ( )nz extended to the case that when there is such a subgraph with 0 indegree and the subgraph has no directed path from any leader agent, then the MAS corresponding to graph G is uncontrollable.
Remark 3. eorem 4 and Corollary 1 are more graph-focused criteria than the previous theorems.

Remark 4.
eorem 4 and Corollary 1 can be simply and visually understood as if there is one agent or multiple agents in the system without any directed path from the input signal. In this case, the system will not be completely controllable. is is consistent with the conclusion in the linear system; that is, if the system is controllable, every vertex must be reachable from one input, regardless of the dynamic form. Otherwise, the system is reducible and part of it cannot be controlled. It is noteworthy that similar properties have been demonstrated in structural controllability.

Simulation
e following examples are all results from MATLAB R2018a software used in a Windows 7 operating system environment.
Here, a discrete-time leader-follower MAS is considered, with two-time-scale and heterogeneous features consisting of seven agents. e interaction between agents is shown in Figure 2 and, as it can be seen, the system consists of four first-order agents and three second-order ones. In these two types of agent clusters, all agents act as followers except for the one that acts as the leader, that is, m � 3, m l � 1, n � 2, and n l � 1. e interaction information between these agents can be expressed as the following matrix: e matrices B 1 , B 2 , F 1 , F 2 , E 1 , and E 2 in system (7) are chosen as B 1 � 1 2 , B 2 � 3 1 , Evidently, the eigenvalues of φ s and φ f are different from those of Q. According to eorem 2, the conclusion is that this MAS is controllable.
By setting ε � 0.1, results on the evolution process of the state errors of the follower agents on the slow-time-scale and fast-time-scale can be obtained as shown in Figures 3-6. e differences between the current states and the desired states of each follower agent will eventually converge to 0 under the control of the leader agents, which also indicates that this MAS is controllable. It is also simple to identify that the states of the follower agents on the fast-time-scale tend to reach the expected values more quickly than the states on the slow-time-scale.
To illustrate the controllability of this MAS more intuitively, the trajectories of five follower agents are shown in Figure 7. e first-and second-order agents are represented by circles and asterisks, respectively, and the x-axis and y-axis coordinates of each agent correspond to its position state on slow-time-scale and fast-time-scale. e initial states of all agents were randomly selected from (0, 1) × (0, 1). e dotted line represents their trajectories, and their ultimate control goal is to form a triangle on the two-dimensional plane.

Conclusion
is paper studies the controllability of discrete-time MASs with two-time-scale and heterogeneous features based on the leader-follower structure. Considering the essential difference between a discrete-time system and a continuous-time system, the modelling and analysis methods applied here are also different. e content of this paper can supplement existing results regarding this issue. In this paper, the singularly perturbed difference system is first applied to model the system, so it later can be decomposed into a slow-time-scale subsystem and a fast-time-scale subsystem. is process is done using the singular perturbation method before the controllability analysis, which avoids the ill-posedness problem when the classical controllability method is used to analyze the system directly. Due to the computational burden caused by using the rank of controllability matrix to judge controllability, some more practical and operational controllability criteria were obtained based on the matrix and the graph theory. ese methods only depend on the characteristics of submatrices of the system matrix and the input matrix. Finally, the validity could be verified by simulation. For future studies related to this one, a new focus will be shaped towards the controllability of multitime-scale MASs and the robustness of controllability.

Data Availability
No data were used to support the findings of this study.

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