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

Authors: Mengqi Gu; Guo-Ping Jiang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This 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. Then, 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.


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## Body

## 1. 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 [1–4] of these systems. MASs are systems composed of some dynamic agents with the certain autonomous ability through information communication and interaction. The 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. Then, 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. This 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. This 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. These 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 fast-slow combination control strategy, which established the basic control strategy of two-time-scale dynamic system. The singular perturbation method [23, 24] is a major means of study used to understand more about singularly perturbed systems. The core idea is to decompose this specific system into fast-time-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 discrete-time 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. The 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. The 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)
The definition of controllability of discrete-time leader-follower 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. This is done to avoid the ill-posedness problem when the classical control method is directly used to study the controllability of these systems.(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.The rest of this paper is arranged as follows. Section2 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. Then, 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. The effectiveness of the proposed criterion is verified by the simulation in Section 4. Conclusions are drawn in Section 5.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

A graphGV,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. The 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. The adjacency matrix A=aij is usually used to represent the topological structure of the graph. If there is an edge from vertex j to vertex i, then aij≠0 and a neighbor of i is j. Otherwise, aij=0. All neighbor vertices of the vertex i are recorded as Ni. If the graph G is undirected, then A is a symmetric matrix. The 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). The 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 eigenvaluesλeA of eA satisfying λeA<1, then when ε⟶0, there is(1)ε∑p=01/ε−1I+εA1/ε−1−p=∫01eAtdt+Oε,where A is a constant matrix with appropriate dimensions.Lemma 2 (see [30]).
If all eigenvaluesλeA of eA satisfying λeA<1, then when ε⟶0, there is(2)I+εA1/ε∼eA+Oε,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. The symbol ⊗ denotes the Kronecker product, and ε0<ε≤1 is a singular perturbation parameter used to distinguish two timescales.


### 2.2. Problem Formulation

The 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 thatm+ml agents in the MAS are first-order integrators, and the remaining n+nl agents are second-order integrators. ml 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, nl 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 Gf1, Gl1, Gf2, and Gl2 in Figure 1. The interaction between all agents can be expressed by the following matrix L corresponding to the entire MAS:(3)L=lij=Lf1f1Lf1l1Lf1f2Lf1l2Ll1f1Ll1l1Ll1f2Ll1l2Lf2f1Lf2l1Lf2f2Lf2l2Ll2f1Ll2l1Ll2f2Ll2l2=L11L12L13L14L21L22L23L24L31L32L33L34L41L42L43L44,where lij=−aij,i≠j,∑j∈Niaij,i=j represents the elements of the matrix L and Lp,qp,q∈f1,f2,l1,l2 is the block matrix of L, which embodies the connection between Gp and Gq if p≠q or within Gp (if p=q).Figure 1
Legend for discrete-time leader-follower MAS with two-time-scale and heterogeneous features.Secondly, each first-order agenti operates on two different timescales simultaneously. Specifically, vectors xi∈ℝnx×1 and zi∈ℝnz×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 xo∈ℝnx×1 and wo∈ℝnx×1 to represent the position and velocity states of slow-time-scale while vectors zo∈ℝnz×1 and do∈ℝnz×1 are used to represent the position and velocity states of fast-time-scale. The dynamic model of all agents in this MAS is modelled by(4a)xin+1=xin+εB1uin,zin+1=zin+B2uin,i∈g1,(4b)xon+1=xon+εwon,won+1=won+εB1uon,zon+1=zon+don,don+1=don+B2uon,o∈g2,where g1≜1,2,…,m+ml and g2≜m+ml+1,m+ml+2,…,m+ml+n+nl. uit∈ℝq×1 and uot∈ℝq×1 are control inputs. Here, the input matrices of two different timescales are B1∈ℝnx×q and B2∈ℝnz×q.Inspired by the consensus protocol, the following communication protocol for MAS (4a) and (4b) was designed:(5)ui=F1∑j∈Niaijxjn−xin+F2∑j∈Niaijzjn−zin,i∈g1,(6)uo=F1∑j∈Noaojxjn−xon+F2∑j∈Noaojzjn−zon+E1∑j∈Noaojwjn−won+E2∑j∈Noaojdjn−don,o∈g2,with F1∈ℝq×nx, F2∈ℝq×nz, E1∈ℝq×nx, and E2∈ℝq×nz representing the position state coupling matrices and the velocity state coupling matrices, accordingly.Letxf1=x1T,x2T,…,xmTT∈ℝmnx×1, xf2=xm+ml+1T,xm+ml+2T,…,xm+ml+nTT∈ℝnnx×1, xl1=xm+1T,xm+2T,…,xm+mlTT∈ℝmlnx×1, wl2=wm+ml+n+1T,wm+ml+n+2T,…,wm+ml+n+nlTT∈ℝnlnx×1, zf1=z1T,z2T,…,zmTT∈ℝmnz×1, xl2=xm+ml+n+1T,xm+ml+n+2T,…,xm+ml+n+nlTT∈ℝnlnx×1, zf2=zm+ml+1T,zm+ml+2T,…,zm+ml+nTT∈ℝnnz×1, zl2=zm+ml+n+1T,zm+ml+n+2T,…,zm+ml+n+nlTT∈ℝnlnz×1, wf2=wm+ml+1T,wm+ml+2T,…,wm+ml+nTT∈ℝnnx×1, df2=dm+ml+1T,dm+ml+2T,…,dm+ml+nTT∈ℝnnz×1, dl2=dm+ml+n+1T,dm+ml+n+2T,…,dm+ml+n+nlTT∈ℝnlnz×1, zl1=zm+1T,zm+2T,…,zm+mlTT∈ℝmlnz×1. By sorting out formulas (4a), (4b)–(6), one has(7)xf1n+1xf2n+1wf2n+1zf1n+1zf2n+1df2n+1=I+εθ11εθ120εθ14εθ1500IεI000εθ31εθ32I+εθ33εθ34εθ35εθ36θ41θ420I+θ44θ4500000IIθ61θ62θ63θ64θ65I+θ66×xf1nxf2nwf2nzf1nzf2ndf2n+εϑ11εϑ120εϑ14εϑ150000000εϑ31εϑ32εϑ33εϑ34εϑ35εϑ36ϑ41ϑ420ϑ44ϑ450000000ϑ61ϑ62ϑ63ϑ64ϑ65ϑ66×xl1nxl2nwl2nzl1nzl2ndl2n,where θ11=−L11⊗B1F1, θ12=−L13⊗B1F1, θ14=−L11⊗B1F2, θ15=−L13⊗B1F2, θ31=−L31⊗B1F1, θ32=−L33⊗B1F1, θ33=−L33⊗B1E1, θ34=−L31⊗B1F2, θ35=−L33⊗B1F2, θ36=−L33⊗B1E2, θ41=−L11⊗B2F1, θ42=−L13⊗B2F1, θ44=−L11⊗B2F2θ45=−L13⊗B2F2, θ61=−L31⊗B2F1, θ62=−L33⊗B2F1, θ63=−L33⊗B2E1, θ64=−L31⊗B2F2, θ65=−L33⊗B2F2, θ66=−L33⊗B2E2, ϑ11=−L12⊗B1F1, ϑ12=−L14⊗B1F1, ϑ14=−L12⊗B1F2, ϑ15=−L14⊗B1F2, ϑ31=−L32⊗B1F1, ϑ32=−L34⊗B1F1, ϑ33=−L34⊗B1E1, ϑ34=−L32⊗B1F2, ϑ35=−L34⊗B1F2, ϑ36=−L34⊗B1E2, ϑ41=−L12⊗B2F1, ϑ42=−L14⊗B2F1, ϑ44=−L12⊗B2F2, ϑ45=−L14⊗B2F2, ϑ61=−L32⊗B2F1, ϑ62=−L34⊗B2F1, ϑ63=−L34⊗B2E1, ϑ64=−L32⊗B2F2, ϑ65=−L34⊗B2F2, and ϑ66=−L34⊗B2E2.Then, usingr1=xf1T,xf2T,wf2TT∈ℝmnx+2nnx×1, r2=zf1T,zf2T,df2TT∈ℝmnz+2nnz×1, and u=xl1T,xl2T,wl2T,zl1T,zl2T,dl2TT∈ℝml+2nlnx+ml+2nlnz×1 yields(8a)r1n+1=I+εφ11r1n+εφ12r2n+εϕ1u,(8b)r2n+1=φ21r1n+I+φ22r2n+ϕ2u,where(9)φ11=θ11θ12000Iθ31θ32θ33,φ12=θ14θ150000θ34θ35θ36,φ21=θ41θ420000θ61θ62θ63,φ22=θ44θ45000Iθ64θ65θ66,ϕ1=ϑ11ϑ120ϑ14ϑ150000000ϑ31ϑ32ϑ33ϑ34ϑ35ϑ36,ϕ2=ϑ41ϑ420ϑ44ϑ450000000ϑ61ϑ62ϑ63ϑ64ϑ65ϑ66.


## 2.1. Preliminaries

A graphGV,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. The 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. The adjacency matrix A=aij is usually used to represent the topological structure of the graph. If there is an edge from vertex j to vertex i, then aij≠0 and a neighbor of i is j. Otherwise, aij=0. All neighbor vertices of the vertex i are recorded as Ni. If the graph G is undirected, then A is a symmetric matrix. The 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). The 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 eigenvaluesλeA of eA satisfying λeA<1, then when ε⟶0, there is(1)ε∑p=01/ε−1I+εA1/ε−1−p=∫01eAtdt+Oε,where A is a constant matrix with appropriate dimensions.Lemma 2 (see [30]).
If all eigenvaluesλeA of eA satisfying λeA<1, then when ε⟶0, there is(2)I+εA1/ε∼eA+Oε,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. The symbol ⊗ denotes the Kronecker product, and ε0<ε≤1 is a singular perturbation parameter used to distinguish two timescales.


## 2.2. Problem Formulation

The 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 thatm+ml agents in the MAS are first-order integrators, and the remaining n+nl agents are second-order integrators. ml 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, nl 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 Gf1, Gl1, Gf2, and Gl2 in Figure 1. The interaction between all agents can be expressed by the following matrix L corresponding to the entire MAS:(3)L=lij=Lf1f1Lf1l1Lf1f2Lf1l2Ll1f1Ll1l1Ll1f2Ll1l2Lf2f1Lf2l1Lf2f2Lf2l2Ll2f1Ll2l1Ll2f2Ll2l2=L11L12L13L14L21L22L23L24L31L32L33L34L41L42L43L44,where lij=−aij,i≠j,∑j∈Niaij,i=j represents the elements of the matrix L and Lp,qp,q∈f1,f2,l1,l2 is the block matrix of L, which embodies the connection between Gp and Gq if p≠q or within Gp (if p=q).Figure 1
Legend for discrete-time leader-follower MAS with two-time-scale and heterogeneous features.Secondly, each first-order agenti operates on two different timescales simultaneously. Specifically, vectors xi∈ℝnx×1 and zi∈ℝnz×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 xo∈ℝnx×1 and wo∈ℝnx×1 to represent the position and velocity states of slow-time-scale while vectors zo∈ℝnz×1 and do∈ℝnz×1 are used to represent the position and velocity states of fast-time-scale. The dynamic model of all agents in this MAS is modelled by(4a)xin+1=xin+εB1uin,zin+1=zin+B2uin,i∈g1,(4b)xon+1=xon+εwon,won+1=won+εB1uon,zon+1=zon+don,don+1=don+B2uon,o∈g2,where g1≜1,2,…,m+ml and g2≜m+ml+1,m+ml+2,…,m+ml+n+nl. uit∈ℝq×1 and uot∈ℝq×1 are control inputs. Here, the input matrices of two different timescales are B1∈ℝnx×q and B2∈ℝnz×q.Inspired by the consensus protocol, the following communication protocol for MAS (4a) and (4b) was designed:(5)ui=F1∑j∈Niaijxjn−xin+F2∑j∈Niaijzjn−zin,i∈g1,(6)uo=F1∑j∈Noaojxjn−xon+F2∑j∈Noaojzjn−zon+E1∑j∈Noaojwjn−won+E2∑j∈Noaojdjn−don,o∈g2,with F1∈ℝq×nx, F2∈ℝq×nz, E1∈ℝq×nx, and E2∈ℝq×nz representing the position state coupling matrices and the velocity state coupling matrices, accordingly.Letxf1=x1T,x2T,…,xmTT∈ℝmnx×1, xf2=xm+ml+1T,xm+ml+2T,…,xm+ml+nTT∈ℝnnx×1, xl1=xm+1T,xm+2T,…,xm+mlTT∈ℝmlnx×1, wl2=wm+ml+n+1T,wm+ml+n+2T,…,wm+ml+n+nlTT∈ℝnlnx×1, zf1=z1T,z2T,…,zmTT∈ℝmnz×1, xl2=xm+ml+n+1T,xm+ml+n+2T,…,xm+ml+n+nlTT∈ℝnlnx×1, zf2=zm+ml+1T,zm+ml+2T,…,zm+ml+nTT∈ℝnnz×1, zl2=zm+ml+n+1T,zm+ml+n+2T,…,zm+ml+n+nlTT∈ℝnlnz×1, wf2=wm+ml+1T,wm+ml+2T,…,wm+ml+nTT∈ℝnnx×1, df2=dm+ml+1T,dm+ml+2T,…,dm+ml+nTT∈ℝnnz×1, dl2=dm+ml+n+1T,dm+ml+n+2T,…,dm+ml+n+nlTT∈ℝnlnz×1, zl1=zm+1T,zm+2T,…,zm+mlTT∈ℝmlnz×1. By sorting out formulas (4a), (4b)–(6), one has(7)xf1n+1xf2n+1wf2n+1zf1n+1zf2n+1df2n+1=I+εθ11εθ120εθ14εθ1500IεI000εθ31εθ32I+εθ33εθ34εθ35εθ36θ41θ420I+θ44θ4500000IIθ61θ62θ63θ64θ65I+θ66×xf1nxf2nwf2nzf1nzf2ndf2n+εϑ11εϑ120εϑ14εϑ150000000εϑ31εϑ32εϑ33εϑ34εϑ35εϑ36ϑ41ϑ420ϑ44ϑ450000000ϑ61ϑ62ϑ63ϑ64ϑ65ϑ66×xl1nxl2nwl2nzl1nzl2ndl2n,where θ11=−L11⊗B1F1, θ12=−L13⊗B1F1, θ14=−L11⊗B1F2, θ15=−L13⊗B1F2, θ31=−L31⊗B1F1, θ32=−L33⊗B1F1, θ33=−L33⊗B1E1, θ34=−L31⊗B1F2, θ35=−L33⊗B1F2, θ36=−L33⊗B1E2, θ41=−L11⊗B2F1, θ42=−L13⊗B2F1, θ44=−L11⊗B2F2θ45=−L13⊗B2F2, θ61=−L31⊗B2F1, θ62=−L33⊗B2F1, θ63=−L33⊗B2E1, θ64=−L31⊗B2F2, θ65=−L33⊗B2F2, θ66=−L33⊗B2E2, ϑ11=−L12⊗B1F1, ϑ12=−L14⊗B1F1, ϑ14=−L12⊗B1F2, ϑ15=−L14⊗B1F2, ϑ31=−L32⊗B1F1, ϑ32=−L34⊗B1F1, ϑ33=−L34⊗B1E1, ϑ34=−L32⊗B1F2, ϑ35=−L34⊗B1F2, ϑ36=−L34⊗B1E2, ϑ41=−L12⊗B2F1, ϑ42=−L14⊗B2F1, ϑ44=−L12⊗B2F2, ϑ45=−L14⊗B2F2, ϑ61=−L32⊗B2F1, ϑ62=−L34⊗B2F1, ϑ63=−L34⊗B2E1, ϑ64=−L32⊗B2F2, ϑ65=−L34⊗B2F2, and ϑ66=−L34⊗B2E2.Then, usingr1=xf1T,xf2T,wf2TT∈ℝmnx+2nnx×1, r2=zf1T,zf2T,df2TT∈ℝmnz+2nnz×1, and u=xl1T,xl2T,wl2T,zl1T,zl2T,dl2TT∈ℝml+2nlnx+ml+2nlnz×1 yields(8a)r1n+1=I+εφ11r1n+εφ12r2n+εϕ1u,(8b)r2n+1=φ21r1n+I+φ22r2n+ϕ2u,where(9)φ11=θ11θ12000Iθ31θ32θ33,φ12=θ14θ150000θ34θ35θ36,φ21=θ41θ420000θ61θ62θ63,φ22=θ44θ45000Iθ64θ65θ66,ϕ1=ϑ11ϑ120ϑ14ϑ150000000ϑ31ϑ32ϑ33ϑ34ϑ35ϑ36,ϕ2=ϑ41ϑ420ϑ44ϑ450000000ϑ61ϑ62ϑ63ϑ64ϑ65ϑ66.


## 3. Main Results

### 3.1. Decomposition of Singularly Perturbed Systems

In view of the existence of the singular perturbation parameterε 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(10)n=lε,l=1,2,….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 matrixφ22 is invertible, formula (8b) can be written as(11)r2n=φ21r1n+I+φ22r2n+ϕ2un,which further leads to(12)r2n=−φ22−1φ21r1n−φ22−1ϕ2un.Substituting (12) into (8a) yields(13)r1n+1=I+εφ11−φ12φ22−1φ21r1n+εϕ1−φ12φ22−1ϕ2un.Letφs=φ11−φ12φ22−1φ21 and ϕs=ϕ1−φ12φ22−1ϕ2. Simultaneously, the states of the agents and the input signals of the slow-time-scale subsystem are represented as rsn and usn. So,(14)rsn+1=I+εφsrsn+εϕsusn.From the perspective of the timescalel, the control input signal of the slow-time-scale subsystem remains unchanged during the period l1/ε≤n<l+11/ε, that is usn=u^sl1/ε. Based on this, combined with formula (14), it gives(15)rsl1ε+1=I+εφsrsl1ε+εϕsu^sl1ε,rsl1ε+2=I+εφsrsl1ε+1+εϕsu^sl1ε+1=I+εφs2rsl1ε+εI+εφsϕsu^sl1ε+εϕsu^sl1ε,…rsl+11ε=I+εφs1/εrsl1ε+ε∑p=01/ε−1I+εφs1/ε−1−pϕsu^sl1ε.Using Lemmas1 and 2 when the value n=1/ε is large enough, and letting φ^s and ϕ^s stand for eφs and ∫01eφstdtϕs yields(16)rsl+11ε=φ^s+Oεrsl1ε+ϕ^s+Oεu^sl1ε.The decoupled slow-time-scale subsystem becomes(17a)r^sl+1=φ^sr^sl+ϕ^su^sl,by letting r^sl=rsl1/ε and ignoring the infinitesimal term.Secondly, the decoupled fast-time-scale subsystem is obtained as follows:(17b)rfn+1=φfrfn+ϕfufn,where rfn=r2n+φ22−1φ21r1n, φf=I+φ22, ϕf=ϕ2, and uf=u.Let us describe the decomposition process in detail. During this process, the state vectorr1 of (8a) and (8b) can be assumed to remain unchanged. By adding φ22−1φ21r1n to both sides of equation (8b), it gives(18)r2n+1+φ22−1φ21r1n=r2n+1+φ22−1φ21r1n+1=φ21r1n+I+φ22r2n+ϕ2u+φ22−1φ21r1n=φ22+Iφ22−1φ21r1n+I+φ22r2n+ϕ2u=I+φ22r2n+φ22−1φ21r1n+ϕ2u.So far, formula (17b) can be obtained.


### 3.2. 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 rf 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 stater^s0=r^s, there exists a piecewise input u^s so that it can reach the zero state within a finite time T1 (i.e., r^sT1=0).(2)
For the initial nonzero staterf0=rf, there exists a piecewise input uf so that it can reach the zero state within a finite time T2 (i.e., rfT2=0).
In the equation,T2≪T1 and T1T2∈1,2,….Lemma 3.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if and only if the controllability matrices corresponding to two subsystems (17a) and (17b) with different timescales are all full row rank, i.e., rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=m+2n×nx and rankϕf,φfϕf,…,φfm+2n×nz−1ϕf=m+2n×nz.Proof.
From Definition1 and the Kalman rank criterion [5], the assertion is obvious.Theorem 1.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if and only if one of the following statements is satisfied:(1)
Letλs and μf represent the eigenvalues of matrices φ^s and φf, respectively, then rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, where s∈1,2,…,m+2n×nx and f∈1,2,…,m+2n×nz.(2)
The vectorsς∈ℂm+2nnx×1 and ζ∈ℂm+2nnz×1 satisfying ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 must be zero vectors, where σ∈ℂ and γ∈ℂ.Proof of Proposition (1).
Necessity: one has to prove that if MAS (4a) and (4b) are controllable, then rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, where λs and μf represent the eigenvalues of matrices φ^s and φf, respectively. Let us look at its contrapositive proposition. By contradiction, suppose there is an eigenvalue λs of φ^s such that rankλsI−φ^s,ϕ^s<m+2n×nx, which means that the rows of the matrix λsI−φ^s,ϕ^s are linearly dependent. Consequently, there must exist a nonzero vector ρ such that(19)ρTλsI−φ^s,ϕ^s=0,that is,(20)ρTϕ^s=0,λsρT=ρTφ^s.
Furthermore,(21)ρTϕ^s=0,ρTφ^sϕ^s=0,…,ρTφ^sm+2n×nx−1ϕ^s=0.
Then,(22)ρTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Since the vectorρ is nonzero, we obtain rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. A similar conclusion can be drawn for the matrix ϕf,φfϕf,…,φfm+2n×nz−1ϕf. 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. The necessity is proved.
Sufficiency: withλs and μf representing the eigenvalues of matrices φ^s and φf, our purpose is to prove that if rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, then MAS (4a) and (4b) are controllable. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. Then, there exists a nonzero left eigenvector ρT corresponding to the eigenvalue λs of the matrix φ^s, so that(23)ρTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Then,(24)ρTϕ^s=0,ρTφ^sϕ^s=λsρTϕ^s=0,…ρTφ^sm+2n×nx−1ϕ^s=0.
Motivated by (24), there holds ρTλsI−φ^s,ϕ^s=0. This means rankλsI−φ^s,ϕ^s<m+2n×nx. A similar conclusion can be drawn that rankμfI−φf,ϕf<m+2n×nz. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The 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 ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 must be zero vectors, where ς∈ℂm+2nnx×1 and ζ∈ℂm+2nnz×1. Let us look at its contrapositive proposition. Suppose that there is a nonzero vector ς∈ℂm+2nnx×1 and σ∈ℂ satisfy ςTφ^s=σςT and ςTϕ^s=0; then,(25)ςTϕ^s=0,ςTφ^sϕ^s=σςTϕ^s=0,ςTφ^s2ϕ^s=σςTφ^sϕ^s=0,…ςTφ^sm+2n×nx−1ϕ^s=0,which further results in(26)ςTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Since the vectorς is nonzero, we obtain rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. A similar conclusion can be drawn for the matrix ϕf,φfϕf,…,φfm+2n×nz−1ϕf, and then MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The necessity is proved.
Sufficiency: we want to prove that if the vectorsς and ζ that satisfy ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 are all zero vectors, then MAS (4a) and (4b) are controllable, where ς∈ℂm+2nnx×1, ζ∈ℂm+2nnz×1, σ∈ℂ, and γ∈ℂ. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. Similar to the sufficiency proof of Proposition (1), we can find that there must be a nonzero vector ς∈ℂm+2nnx×1 and σ∈ℂ such that ςTφ^s=σςT and ςTϕ^s=0. Similar conclusions can be drawn about the matrices φf and ϕf. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The sufficiency is proved.Theorem 2.
If the two matricesφ^s and φf 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(27)Q=φ^sϕ^s00ϕ^sTI0000φfϕf00ϕfTI.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 Theorem 1, it gives that there exists a nonzero vector ς∈ℂm+2nnx×1 such that ςTφ^s=αςT and ςTϕ^s=0, where α is an eigenvalue of the matrix φ^s.
Defining a new vectorςT,0,0,0, then(28)ςT000Q=ςT000φ^sϕ^s00ϕ^sTI0000φfϕf00ϕfTI=ςTφ^sςTϕ^s00=αςT000=αςT000.
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 φf and Q can be obtained with a similar proof process, so it is omitted here. Theorem 2 is proved.Theorem 3.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if the eigenvalues of φ^s and φf 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 ϕf, respectively, where P and R are, respectively, composed of the eigenvectors of matrices φ^s and φf.Proof.
According to the knowledge of matrix theory, since the eigenvalues ofφ^s are different, φ^s can be similarly diagonalized. Let λs∀s∈1,2,…,m+2nnx represent the eigenvalues of φ^s and P as an invertible matrix composed of corresponding eigenvectors, it follows that φ^s=PΛP−1 with Λ=diagλ1,λ2,…,λm+2nnx.
Then,(29)ϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=PP−1ϕ^s,PΛP−1ϕ^s,…,PΛm+2n×nx−1P−1ϕ^s=PP−1ϕ^s,ΛP−1ϕ^s,…,Λm+2n×nx−1P−1ϕ^s.
LetΓ=P−1ϕ^s,ΛP−1ϕ^s,…,Λm+2n×nx−1P−1ϕ^s, then motivated by thatP is an invertible matrix, there holds rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=rankΓ. By expanding P−1ϕ^s as η1,η2,…,ηml+2nlnx+ml+2nlnz with ηi=η1i,η2i,…,ηm+2nnx,iT for ∀i∈1,2,…,ml+2nlnx+ml+2nlnz, one has(30)Γ=η1,η2,…,ηml+2nlnx+ml+2nlnz,Λη1,η2,…,ηml+2nlnx+ml+2nlnz,…,Λm+2nnx−1η1,η2,…,ηml+2nlnx+ml+2nlnz.
Based on the conclusion of matrix theory that the elementary transformation of matrix does not change the rank,(31)rankΓ=rankη1,η2,…,ηml+2nlnx+ml+2nlnz,Λη1,η2,…,ηml+2nlnx+ml+2nlnz,…,Λm+2nnx−1η1,η2,…,ηml+2nlnx+ml+2nlnz=rankη1,Λη1,…,Λm+2nnx−1η1,η2,Λη2,…,Λm+2nnx−1η2,…,ηml+2nlnx+ml+2nlnz,Ληml+2nlnx+ml+2nlnz,…,Λm+2nnx−1ηml+2nlnx+ml+2nlnz=rankdiagη11,η21,…,ηm+2nnx,1K,…,diagη1,ml+2nlnx+ml+2nlnz,η2,ml+2nlnx+ml+2nlnz,…,ηm+2nnx,ml+2nlnx+ml+2nlnzK,where(32)K=1λ1⋯λ1m+2nnx−11λ2⋯λ2m+2nnx−1⋮⋮⋱⋮1λm+2nnx⋯λm+2nnxm+2nnx−1.
Since the eigenvalues of the matrixφ^s are different, then K is a full row rank matrix. If we let P−1=p1p2⋮pm+2nnx∈ℝm+2nnx×m+2nnx and ϕ^s=ϕ1,ϕ2,…,ϕml+2nlnx+ml+2nlnz∈ℝm+2nnx×ml+2nlnx+ml+2nlnz, then for ∀i∈1,2,…,m+2nnx and ∀j∈1,2,…,ml+2nlnx+ml+2nlnz, ηi,j=pi,ϕj, that is, ηi,j is the inner product of vectors pi and ϕj. Meanwhile, since all rows of matrices P−1 are not orthogonal to at least one column of ϕ^s, there must exist a diagonal matrix diagη1i,η2i,…,ηm+2nnx,i whose elements are all not zero. Using this information, we can conclude the full row rank of the matrix Γ. According to the previous derivation, this is equivalent to that the row rank of the controllability matrix ϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s is full. Regarding φf and ϕf, similar results can be obtained.
So far, it has been proved that under the two conditions mentioned above, the discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable.Remark 1.
Compared with calculation of the rank of controllability matrices, the criteria proposed in Theorems2 and 3 are considered to be more practical, because they are only related to the eigenvalues of some submatrices of the system matrix and the input matrix in (7). These eigenvalues are relatively simpler to calculate.Remark 2.
The conclusions of Theorems2 and 3 give the conditions that can satisfy the relevant matrices φ^s, φf, ϕ^s, and ϕf for the systems to be controllable. These matrices are related to B1, B2, F1, F2, E1, E2, and aij of system models (4a) and (4b). Therefore, in essence, given Theorems 2 and 3 can be used to provide specific guidance for the design of the control protocol of the system.Theorem 4.
If the discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable and there is a follower agent of the fast-time-scale subsystem whose indegree is zero, it can be concluded that this follower agent must have control input from leader agents.Proof.
Assuming that the index of the agent with zero indegree in the follower agent cluster of the fast-time-scale subsystem isi, we discuss two cases according to the type of agent i:(1)
Suppose the agenti is a first-order integrator. Then, 0,…,0,Inz,0,…,0 corresponds to the rows i−1nz+1th to inzth of φf. Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain i−1nz+1th to inzth rows of ϕf are all zero. Let ξ0∈ℝnz×1 denote an eigenvector of Inz corresponding to eigenvalue λ=1 and let ξT=0,…,0,ξ0T,0,…,0∈ℝ1×mnz+2nnz, then it gives(33)ξTφf=0,…,0,ξ0T,0,…,0φf=0,…,0,λξ0T,0,…,0=λ0,…,0,ξ0T,0,…,0=λξT,(34)ξTϕf=0,…,0,ξ0T,0,…,0ϕf=0,and as such,(35)ξTφfϕf=λξTϕf=0,ξTφf2ϕf=λξTφfϕf=0,…ξTφfm+2nnz−1ϕf=0.(36)ξTϕf,φfϕf,φf2ϕf,…,φfm+2nnz−1ϕf=0,and since the eigenvectorξT is a nonzero vector, rankϕf,φfϕf,φf2ϕf,…,φfm+2nnz−1ϕf<m+2nnz. From Lemma 3, we can see that MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is true.(2)
Assume that agenti is a second-order integrator. Then, 0,…,0,Inz,0,…,0,Inz,0,…,0 corresponds to the rows i−1nz+1th to inzth of φf. Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain i−1nz+1th to inzth rows of ϕf are all zero. Let ξ0δ0∈ℝnz×1 denote two different eigenvectors of the matrix I corresponding to eigenvalue λ=1 and ξT=0,…,0,ξ0T,0,…,0,δ0T,0,…,0∈ℝ1×mnz+2nnz, and we can discover the same result as the case (1); that is, MAS (4a) and (4b) are also uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true.
In summary, Theorem4 is proved.Corollary 1.
If several follower agents and their connections are regarded as a subgraph ofG, then Theorem 4 can be 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.
Theorem4 and Corollary 1 are more graph-focused criteria than the previous theorems.Remark 4.
Theorem4 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. This 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.


## 3.1. Decomposition of Singularly Perturbed Systems

In view of the existence of the singular perturbation parameterε 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(10)n=lε,l=1,2,….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 matrixφ22 is invertible, formula (8b) can be written as(11)r2n=φ21r1n+I+φ22r2n+ϕ2un,which further leads to(12)r2n=−φ22−1φ21r1n−φ22−1ϕ2un.Substituting (12) into (8a) yields(13)r1n+1=I+εφ11−φ12φ22−1φ21r1n+εϕ1−φ12φ22−1ϕ2un.Letφs=φ11−φ12φ22−1φ21 and ϕs=ϕ1−φ12φ22−1ϕ2. Simultaneously, the states of the agents and the input signals of the slow-time-scale subsystem are represented as rsn and usn. So,(14)rsn+1=I+εφsrsn+εϕsusn.From the perspective of the timescalel, the control input signal of the slow-time-scale subsystem remains unchanged during the period l1/ε≤n<l+11/ε, that is usn=u^sl1/ε. Based on this, combined with formula (14), it gives(15)rsl1ε+1=I+εφsrsl1ε+εϕsu^sl1ε,rsl1ε+2=I+εφsrsl1ε+1+εϕsu^sl1ε+1=I+εφs2rsl1ε+εI+εφsϕsu^sl1ε+εϕsu^sl1ε,…rsl+11ε=I+εφs1/εrsl1ε+ε∑p=01/ε−1I+εφs1/ε−1−pϕsu^sl1ε.Using Lemmas1 and 2 when the value n=1/ε is large enough, and letting φ^s and ϕ^s stand for eφs and ∫01eφstdtϕs yields(16)rsl+11ε=φ^s+Oεrsl1ε+ϕ^s+Oεu^sl1ε.The decoupled slow-time-scale subsystem becomes(17a)r^sl+1=φ^sr^sl+ϕ^su^sl,by letting r^sl=rsl1/ε and ignoring the infinitesimal term.Secondly, the decoupled fast-time-scale subsystem is obtained as follows:(17b)rfn+1=φfrfn+ϕfufn,where rfn=r2n+φ22−1φ21r1n, φf=I+φ22, ϕf=ϕ2, and uf=u.Let us describe the decomposition process in detail. During this process, the state vectorr1 of (8a) and (8b) can be assumed to remain unchanged. By adding φ22−1φ21r1n to both sides of equation (8b), it gives(18)r2n+1+φ22−1φ21r1n=r2n+1+φ22−1φ21r1n+1=φ21r1n+I+φ22r2n+ϕ2u+φ22−1φ21r1n=φ22+Iφ22−1φ21r1n+I+φ22r2n+ϕ2u=I+φ22r2n+φ22−1φ21r1n+ϕ2u.So far, formula (17b) can be obtained.


## 3.2. 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 rf 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 stater^s0=r^s, there exists a piecewise input u^s so that it can reach the zero state within a finite time T1 (i.e., r^sT1=0).(2)
For the initial nonzero staterf0=rf, there exists a piecewise input uf so that it can reach the zero state within a finite time T2 (i.e., rfT2=0).
In the equation,T2≪T1 and T1T2∈1,2,….Lemma 3.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if and only if the controllability matrices corresponding to two subsystems (17a) and (17b) with different timescales are all full row rank, i.e., rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=m+2n×nx and rankϕf,φfϕf,…,φfm+2n×nz−1ϕf=m+2n×nz.Proof.
From Definition1 and the Kalman rank criterion [5], the assertion is obvious.Theorem 1.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if and only if one of the following statements is satisfied:(1)
Letλs and μf represent the eigenvalues of matrices φ^s and φf, respectively, then rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, where s∈1,2,…,m+2n×nx and f∈1,2,…,m+2n×nz.(2)
The vectorsς∈ℂm+2nnx×1 and ζ∈ℂm+2nnz×1 satisfying ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 must be zero vectors, where σ∈ℂ and γ∈ℂ.Proof of Proposition (1).
Necessity: one has to prove that if MAS (4a) and (4b) are controllable, then rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, where λs and μf represent the eigenvalues of matrices φ^s and φf, respectively. Let us look at its contrapositive proposition. By contradiction, suppose there is an eigenvalue λs of φ^s such that rankλsI−φ^s,ϕ^s<m+2n×nx, which means that the rows of the matrix λsI−φ^s,ϕ^s are linearly dependent. Consequently, there must exist a nonzero vector ρ such that(19)ρTλsI−φ^s,ϕ^s=0,that is,(20)ρTϕ^s=0,λsρT=ρTφ^s.
Furthermore,(21)ρTϕ^s=0,ρTφ^sϕ^s=0,…,ρTφ^sm+2n×nx−1ϕ^s=0.
Then,(22)ρTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Since the vectorρ is nonzero, we obtain rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. A similar conclusion can be drawn for the matrix ϕf,φfϕf,…,φfm+2n×nz−1ϕf. 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. The necessity is proved.
Sufficiency: withλs and μf representing the eigenvalues of matrices φ^s and φf, our purpose is to prove that if rankλsI−φ^s,ϕ^s=m+2n×nx and rankμfI−φf,ϕf=m+2n×nz, then MAS (4a) and (4b) are controllable. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. Then, there exists a nonzero left eigenvector ρT corresponding to the eigenvalue λs of the matrix φ^s, so that(23)ρTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Then,(24)ρTϕ^s=0,ρTφ^sϕ^s=λsρTϕ^s=0,…ρTφ^sm+2n×nx−1ϕ^s=0.
Motivated by (24), there holds ρTλsI−φ^s,ϕ^s=0. This means rankλsI−φ^s,ϕ^s<m+2n×nx. A similar conclusion can be drawn that rankμfI−φf,ϕf<m+2n×nz. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The 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 ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 must be zero vectors, where ς∈ℂm+2nnx×1 and ζ∈ℂm+2nnz×1. Let us look at its contrapositive proposition. Suppose that there is a nonzero vector ς∈ℂm+2nnx×1 and σ∈ℂ satisfy ςTφ^s=σςT and ςTϕ^s=0; then,(25)ςTϕ^s=0,ςTφ^sϕ^s=σςTϕ^s=0,ςTφ^s2ϕ^s=σςTφ^sϕ^s=0,…ςTφ^sm+2n×nx−1ϕ^s=0,which further results in(26)ςTϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=0.
Since the vectorς is nonzero, we obtain rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. A similar conclusion can be drawn for the matrix ϕf,φfϕf,…,φfm+2n×nz−1ϕf, and then MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The necessity is proved.
Sufficiency: we want to prove that if the vectorsς and ζ that satisfy ςTφ^s=σςT, ςTϕ^s=0, ζTφf=γζT, and ζTϕf=0 are all zero vectors, then MAS (4a) and (4b) are controllable, where ς∈ℂm+2nnx×1, ζ∈ℂm+2nnz×1, σ∈ℂ, and γ∈ℂ. Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s<m+2n×nx. Similar to the sufficiency proof of Proposition (1), we can find that there must be a nonzero vector ς∈ℂm+2nnx×1 and σ∈ℂ such that ςTφ^s=σςT and ςTϕ^s=0. Similar conclusions can be drawn about the matrices φf and ϕf. Because the contrapositive proposition of the original proposition is true, the original proposition is also true. The sufficiency is proved.Theorem 2.
If the two matricesφ^s and φf 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(27)Q=φ^sϕ^s00ϕ^sTI0000φfϕf00ϕfTI.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 Theorem 1, it gives that there exists a nonzero vector ς∈ℂm+2nnx×1 such that ςTφ^s=αςT and ςTϕ^s=0, where α is an eigenvalue of the matrix φ^s.
Defining a new vectorςT,0,0,0, then(28)ςT000Q=ςT000φ^sϕ^s00ϕ^sTI0000φfϕf00ϕfTI=ςTφ^sςTϕ^s00=αςT000=αςT000.
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 φf and Q can be obtained with a similar proof process, so it is omitted here. Theorem 2 is proved.Theorem 3.
The discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable if the eigenvalues of φ^s and φf 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 ϕf, respectively, where P and R are, respectively, composed of the eigenvectors of matrices φ^s and φf.Proof.
According to the knowledge of matrix theory, since the eigenvalues ofφ^s are different, φ^s can be similarly diagonalized. Let λs∀s∈1,2,…,m+2nnx represent the eigenvalues of φ^s and P as an invertible matrix composed of corresponding eigenvectors, it follows that φ^s=PΛP−1 with Λ=diagλ1,λ2,…,λm+2nnx.
Then,(29)ϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=PP−1ϕ^s,PΛP−1ϕ^s,…,PΛm+2n×nx−1P−1ϕ^s=PP−1ϕ^s,ΛP−1ϕ^s,…,Λm+2n×nx−1P−1ϕ^s.
LetΓ=P−1ϕ^s,ΛP−1ϕ^s,…,Λm+2n×nx−1P−1ϕ^s, then motivated by thatP is an invertible matrix, there holds rankϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s=rankΓ. By expanding P−1ϕ^s as η1,η2,…,ηml+2nlnx+ml+2nlnz with ηi=η1i,η2i,…,ηm+2nnx,iT for ∀i∈1,2,…,ml+2nlnx+ml+2nlnz, one has(30)Γ=η1,η2,…,ηml+2nlnx+ml+2nlnz,Λη1,η2,…,ηml+2nlnx+ml+2nlnz,…,Λm+2nnx−1η1,η2,…,ηml+2nlnx+ml+2nlnz.
Based on the conclusion of matrix theory that the elementary transformation of matrix does not change the rank,(31)rankΓ=rankη1,η2,…,ηml+2nlnx+ml+2nlnz,Λη1,η2,…,ηml+2nlnx+ml+2nlnz,…,Λm+2nnx−1η1,η2,…,ηml+2nlnx+ml+2nlnz=rankη1,Λη1,…,Λm+2nnx−1η1,η2,Λη2,…,Λm+2nnx−1η2,…,ηml+2nlnx+ml+2nlnz,Ληml+2nlnx+ml+2nlnz,…,Λm+2nnx−1ηml+2nlnx+ml+2nlnz=rankdiagη11,η21,…,ηm+2nnx,1K,…,diagη1,ml+2nlnx+ml+2nlnz,η2,ml+2nlnx+ml+2nlnz,…,ηm+2nnx,ml+2nlnx+ml+2nlnzK,where(32)K=1λ1⋯λ1m+2nnx−11λ2⋯λ2m+2nnx−1⋮⋮⋱⋮1λm+2nnx⋯λm+2nnxm+2nnx−1.
Since the eigenvalues of the matrixφ^s are different, then K is a full row rank matrix. If we let P−1=p1p2⋮pm+2nnx∈ℝm+2nnx×m+2nnx and ϕ^s=ϕ1,ϕ2,…,ϕml+2nlnx+ml+2nlnz∈ℝm+2nnx×ml+2nlnx+ml+2nlnz, then for ∀i∈1,2,…,m+2nnx and ∀j∈1,2,…,ml+2nlnx+ml+2nlnz, ηi,j=pi,ϕj, that is, ηi,j is the inner product of vectors pi and ϕj. Meanwhile, since all rows of matrices P−1 are not orthogonal to at least one column of ϕ^s, there must exist a diagonal matrix diagη1i,η2i,…,ηm+2nnx,i whose elements are all not zero. Using this information, we can conclude the full row rank of the matrix Γ. According to the previous derivation, this is equivalent to that the row rank of the controllability matrix ϕ^s,φ^sϕ^s,…,φ^sm+2n×nx−1ϕ^s is full. Regarding φf and ϕf, similar results can be obtained.
So far, it has been proved that under the two conditions mentioned above, the discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable.Remark 1.
Compared with calculation of the rank of controllability matrices, the criteria proposed in Theorems2 and 3 are considered to be more practical, because they are only related to the eigenvalues of some submatrices of the system matrix and the input matrix in (7). These eigenvalues are relatively simpler to calculate.Remark 2.
The conclusions of Theorems2 and 3 give the conditions that can satisfy the relevant matrices φ^s, φf, ϕ^s, and ϕf for the systems to be controllable. These matrices are related to B1, B2, F1, F2, E1, E2, and aij of system models (4a) and (4b). Therefore, in essence, given Theorems 2 and 3 can be used to provide specific guidance for the design of the control protocol of the system.Theorem 4.
If the discrete-time leader-follower MAS with two-time-scale and heterogeneous features (4a) and (4b) is controllable and there is a follower agent of the fast-time-scale subsystem whose indegree is zero, it can be concluded that this follower agent must have control input from leader agents.Proof.
Assuming that the index of the agent with zero indegree in the follower agent cluster of the fast-time-scale subsystem isi, we discuss two cases according to the type of agent i:(1)
Suppose the agenti is a first-order integrator. Then, 0,…,0,Inz,0,…,0 corresponds to the rows i−1nz+1th to inzth of φf. Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain i−1nz+1th to inzth rows of ϕf are all zero. Let ξ0∈ℝnz×1 denote an eigenvector of Inz corresponding to eigenvalue λ=1 and let ξT=0,…,0,ξ0T,0,…,0∈ℝ1×mnz+2nnz, then it gives(33)ξTφf=0,…,0,ξ0T,0,…,0φf=0,…,0,λξ0T,0,…,0=λ0,…,0,ξ0T,0,…,0=λξT,(34)ξTϕf=0,…,0,ξ0T,0,…,0ϕf=0,and as such,(35)ξTφfϕf=λξTϕf=0,ξTφf2ϕf=λξTφfϕf=0,…ξTφfm+2nnz−1ϕf=0.(36)ξTϕf,φfϕf,φf2ϕf,…,φfm+2nnz−1ϕf=0,and since the eigenvectorξT is a nonzero vector, rankϕf,φfϕf,φf2ϕf,…,φfm+2nnz−1ϕf<m+2nnz. From Lemma 3, we can see that MAS (4a) and (4b) are uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is true.(2)
Assume that agenti is a second-order integrator. Then, 0,…,0,Inz,0,…,0,Inz,0,…,0 corresponds to the rows i−1nz+1th to inzth of φf. Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain i−1nz+1th to inzth rows of ϕf are all zero. Let ξ0δ0∈ℝnz×1 denote two different eigenvectors of the matrix I corresponding to eigenvalue λ=1 and ξT=0,…,0,ξ0T,0,…,0,δ0T,0,…,0∈ℝ1×mnz+2nnz, and we can discover the same result as the case (1); that is, MAS (4a) and (4b) are also uncontrollable. Because the contrapositive proposition of the original proposition is true, the original proposition is also true.
In summary, Theorem4 is proved.Corollary 1.
If several follower agents and their connections are regarded as a subgraph ofG, then Theorem 4 can be 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.
Theorem4 and Corollary 1 are more graph-focused criteria than the previous theorems.Remark 4.
Theorem4 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. This 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.


## 4. Simulation

The 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. The interaction between agents is shown in Figure2 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, ml=1, n=2, and nl=1.Figure 2
Topology of the leader-based discrete-time MAS with heterogeneous and two-time-scale features consisting of seven agents as described in the example.The interaction information between these agents can be expressed as the following matrix:(37)L11=3−10020−103,L12=0−1−1,L13=−1−1−10−10,L14=000,L21=000,L22=0,L23=00,L24=0,L31=00−1000,L32=−10,L33=4−1−12,L34=−1−1,L41=000,L42=0,L43=00,L44=0.The matricesB1, B2, F1, F2, E1, and E2 in system (7) are chosen asB1=12,B2=31,(38)F1=11,F2=21,E1=33,E2=13.Based on the previous derivation, it was possible to calculate the eigenvalues of matricesφ^s, φf, and Q as {11.32, −8.1, 2.62, −0.28, −1.13, −0.97±0.31i }, {−23.71, −21.68, −18.63, −13.21, −7.43, −0.03, −0.31}, and {38.15, 18.85, 7.51, 3.08, 1.12, 1, −0.13, −0.27, −1.35, −8.3, −20.22, −26.08, −26.59, −30.66, −0.93±0.27i, −16.38±3.49i}, respectively. Evidently, the eigenvalues of φ^s and φf are different from those of Q. According to Theorem 2, the conclusion is that this MAS is controllable.By settingε=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. The 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.Figure 3
The position state errors of all follower agents on slow-time-scale. Variablesx1, x2, x3, x5, and x6, respectively, correspond to the position states of agents 1, 2, 3, 5, and 6 in the slow-time-scale subsystem. ep represents the deviation between the current position state and the target position state.Figure 4
The velocity state errors of second-order follower agents on slow-time-scale. Variablesw5 and w6, respectively, correspond to the velocity states of agents 5 and 6 in the slow-time-scale subsystem. ev represents the deviation between the current velocity state and the target velocity state.Figure 5
The position state errors of all follower agents on fast-time-scale. Variablesz1, z2, z3, z5, and z6, respectively, correspond to the position states of agents 1, 2, 3, 5, and 6 in the fast-time-scale subsystem. The meaning of ep is the same as in Figure 3.Figure 6
The velocity state errors of second-order follower agents on fast-time-scale. Variablesd5 and d6, respectively, correspond to the velocity states of agents 5 and 6 in the fast-time-scale subsystem. ev has the same meaning as in Figure 4.To illustrate the controllability of this MAS more intuitively, the trajectories of five follower agents are shown in Figure7. The 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. The initial states of all agents were randomly selected from 0,1×0,1. The dotted line represents their trajectories, and their ultimate control goal is to form a triangle on the two-dimensional plane.Figure 7
The trajectories of follower agents.


## 5. Conclusion

This 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. The 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. This 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. These 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 multi-time-scale MASs and the robustness of controllability.



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