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.
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. 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. 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 graph composed of the vertex set and edge set 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 , and the number of edges is recorded as . Each edge in the set has a pair of vertices in the set corresponding to it. If any vertex pair and correspond to the same edge, the graph is labelled as an undirected graph. If not, it is called a directed graph. The adjacency matrix is usually used to represent the topological structure of the graph. If there is an edge from vertex to vertex , then and a neighbor of is . Otherwise, . All neighbor vertices of the vertex are recorded as . If the graph is undirected, then is a symmetric matrix. The Laplacian matrix of the graph is defined as , where is the degree matrix of . Because the sum of each row of the Laplacian matrix is 0, regardless of the graph being an undirected or a directed one, 0 is the eigenvalue of , and the corresponding eigenvector is (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 eigenvalues of satisfying , then when , there iswhere is a constant matrix with appropriate dimensions.
Lemma 2 (see [30]). If all eigenvalues of satisfying , then when , there iswhere is a constant matrix with appropriate dimensions.
Next, , , and 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 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 that agents in the MAS are first-order integrators, and the remaining agents are second-order integrators. represents the number of leaders among the first-order integrators agents, and is the number of followers in the first-order integrators agent cluster. Similarly, and 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 , , , and in Figure 1. The interaction between all agents can be expressed by the following matrix corresponding to the entire MAS:where represents the elements of the matrix and is the block matrix of , which embodies the connection between and if or within (if ).
Secondly, each first-order agent operates on two different timescales simultaneously. Specifically, vectors and are used to represent the position state vectors of the first-order agent on the slow-time-scale and the fast-time-scale. Each second-order agent has vectors and to represent the position and velocity states of slow-time-scale while vectors and are used to represent the position and velocity states of fast-time-scale. The dynamic model of all agents in this MAS is modelled bywhere and . and are control inputs. Here, the input matrices of two different timescales are and .
Inspired by the consensus protocol, the following communication protocol for MAS (4a) and (4b) was designed:with , , , and representing the position state coupling matrices and the velocity state coupling matrices, accordingly.
Let , , , , , , , , , , , . By sorting out formulas (4a), (4b)–(6), one haswhere , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Then, using , , and yieldswhere
3. Main Results
3.1. Decomposition of Singularly Perturbed Systems
In view of the existence of the singular perturbation parameter 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 so that it can be discussed with the traditional controllability research method. Since exists in the slow-time-scale equation, is the timescale of the fast-time-scale subsystem. If 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 is invertible, formula (8b) can be written aswhich further leads to
Substituting (12) into (8a) yields
Let and . Simultaneously, the states of the agents and the input signals of the slow-time-scale subsystem are represented as and . So,
From the perspective of the timescale , the control input signal of the slow-time-scale subsystem remains unchanged during the period , that is . Based on this, combined with formula (14), it gives
Using Lemmas 1 and 2 when the value is large enough, and letting and stand for and yields
The decoupled slow-time-scale subsystem becomesby letting and ignoring the infinitesimal term.
Secondly, the decoupled fast-time-scale subsystem is obtained as follows:where , , , and .
Let us describe the decomposition process in detail. During this process, the state vector of (8a) and (8b) can be assumed to remain unchanged. By adding to both sides of equation (8b), it gives
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 and 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 , there exists a piecewise input so that it can reach the zero state within a finite time (i.e., ).(2)For the initial nonzero state , there exists a piecewise input so that it can reach the zero state within a finite time (i.e., ).In the equation, and .
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., and .
Proof. From Definition 1 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)Let and represent the eigenvalues of matrices and , respectively, then and , where and .(2)The vectors and satisfying , , , and must be zero vectors, where and .
Proof of Proposition (1). Necessity: one has to prove that if MAS (4a) and (4b) are controllable, then and , where and represent the eigenvalues of matrices and , respectively. Let us look at its contrapositive proposition. By contradiction, suppose there is an eigenvalue of such that , which means that the rows of the matrix are linearly dependent. Consequently, there must exist a nonzero vector such thatthat is,Furthermore,Then,Since the vector is nonzero, we obtain . A similar conclusion can be drawn for the matrix . 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: with and representing the eigenvalues of matrices and , our purpose is to prove that if and , 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, . Then, there exists a nonzero left eigenvector corresponding to the eigenvalue of the matrix , so thatThen,Motivated by (24), there holds . This means . A similar conclusion can be drawn that . 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 and that satisfy , , , and must be zero vectors, where and . Let us look at its contrapositive proposition. Suppose that there is a nonzero vector and satisfy and ; then,which further results inSince the vector is nonzero, we obtain . A similar conclusion can be drawn for the matrix , 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 vectors and that satisfy , , , and are all zero vectors, then MAS (4a) and (4b) are controllable, where , , , and . Let us look at its contrapositive proposition. By contradiction, suppose that MAS (4a) and (4b) are uncontrollable. So, . Similar to the sufficiency proof of Proposition (1), we can find that there must be a nonzero vector and such that and . Similar conclusions can be drawn about the matrices and . 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 matrices and do not have the same eigenvalues with , 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 Theorem 1, it gives that there exists a nonzero vector such that and , where is an eigenvalue of the matrix .
Defining a new vector , thenObviously, and have the same eigenvalue . Because the contrapositive proposition of the original proposition is true, the original proposition is also true. A similar conclusion about the matrices and 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 and are all different, and all rows of matrices and are not orthogonal to at least one column of matrices and , respectively, where and are, respectively, composed of the eigenvectors of matrices and .
Proof. According to the knowledge of matrix theory, since the eigenvalues of are different, can be similarly diagonalized. Let represent the eigenvalues of and as an invertible matrix composed of corresponding eigenvectors, it follows that with .
Then,Let , then motivated by that is an invertible matrix, there holds . By expanding as with for , one hasBased on the conclusion of matrix theory that the elementary transformation of matrix does not change the rank,whereSince the eigenvalues of the matrix are different, then is a full row rank matrix. If we let and , then for and , , that is, is the inner product of vectors and . Meanwhile, since all rows of matrices are not orthogonal to at least one column of , there must exist a diagonal matrix whose elements are all not zero. Using this information, we can conclude the full row rank of the matrix . According to the previous derivation, this is equivalent to that the row rank of the controllability matrix is full. Regarding and , 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 Theorems 2 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 Theorems 2 and 3 give the conditions that can satisfy the relevant matrices , , , and for the systems to be controllable. These matrices are related to , , , , , , and 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 is , we discuss two cases according to the type of agent :(1)Suppose the agent is a first-order integrator. Then, corresponds to the rows to of . Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain to rows of are all zero. Let denote an eigenvector of corresponding to eigenvalue and let , then it gives and as such, and since the eigenvector is a nonzero vector, . 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 agent is a second-order integrator. Then, corresponds to the rows to of . Let us look at its contrapositive proposition. By contradiction, suppose that this follower agent does not have any input from leader agents, we obtain to rows of are all zero. Let denote two different eigenvectors of the matrix corresponding to eigenvalue and , 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, Theorem 4 is proved.
Corollary 1. If several follower agents and their connections are regarded as a subgraph of , 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 is uncontrollable.
Remark 3. Theorem 4 and Corollary 1 are more graph-focused criteria than the previous theorems.
Remark 4. Theorem 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. 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 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, , , , and .
The interaction information between these agents can be expressed as the following matrix:
The matrices , , , , , and in system (7) are chosen as
Based on the previous derivation, it was possible to calculate the eigenvalues of matrices , , and as {11.32, −8.1, 2.62, −0.28, −1.13, −0.970.31 }, {−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, −0.930.27, −16.383.49}, respectively. Evidently, the eigenvalues of and are different from those of . According to Theorem 2, the conclusion is that this MAS is controllable.
By setting , 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.
To illustrate the controllability of this MAS more intuitively, the trajectories of five follower agents are shown in Figure 7. 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 . The dotted line represents their trajectories, and their ultimate control goal is to form a triangle on the two-dimensional plane.
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.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors thank Xiaoling Wang and Mingkang Long for discussions. This work was supported by the National Natural Science Foundation of China (grant nos. 61873326 and 61672298).