A New Perspective to Algebraic Characterization on Controllability of Multiagent Systems

School of Information Engineering, Minzu University of China, Beijing 100081, China Artificial Intelligence School, Wuchang University of Technology, Wuhan 430223, China Key Laboratory of Imaging Processing and Intelligence Control, School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China College of Science, North China University of Technology, Beijing 100144, China


Introduction
In recent years, cooperative and coordinated control of networked multiagent systems has become a surge of research activities. e controllability is a basic and important problem in modern control theory, which plays a key role in the analysis and synthesis of networked multiagent systems and has wide applications and advantages in formation control, pinning control, containment control and tracking control, etc [1][2][3][4][5][6][7][8][9][10]. A controllable multiagent system under a leader-follower framework means that all the followers can be governed to any desirable state (configuration) from any given initial state (configuration) in finite time by designing control protocol or algorithm exerted on leaders for every follower.. Controllability problem for a group of systems was first put forward by Tanner [11] in 2004 from the viewpoint of algebra, in which singleintegrator continuous-time model with a leader in terms of nearest neighbor rules was formulated, and an algebraic controllable criterion in view of eigenvalues and right eigenvectors of submatrices of Laplacian matrix was obtained under a fixed topology. Studies in this algebraic point of view have provided a theoretical basis for understanding internal relationships and interactions among graph structures, evolutionary protocol, and controllability. Afterwards, considerable efforts on the controllability of complex dynamical networks have been devoted from algebraic perspectives. Liu et al. [12] first proposed the concept of the controllability for a leader-follower dynamic network with discrete-time state and established some algebraic criteria on the controllability under different topologies, respectively. Immediately after that, the authors, respectively, investigated the switching controllability [13,14] and the group controllability [15,16] of discrete-time/continuous-time networked multiagent systems with/without multiple leaders and coupling time delays. Wang et al. [17] studied the controllability of multiagent systems with the consensus protocols under directed topologies for high-order-integrator dynamic agents and general linear dynamic intelligent agents, respectively, in which the authors illustrated that the controllability congruously depended on the interconnection topology among agents and dynamics. Guan and Wang [18] studied the controllability of a group of mobile autonomous agents based on absolute protocol under fixed and switching topologies, respectively. In addition, some other important necessary and/or sufficient conditions on the controllability in multiagent networks were studied for special topologies, such as path graphs and cycle graphs [19,20], multichain graphs [21], star graphs and tree graphs [22], grid graphs [23], symmetric structures [24], circulant networks [25], and two-time-scale networks [26][27][28]. Controllability can also be analyzed by the eigenvectors of Laplacian as in [29]. Moreover, the authors discussed the controllability of dynamic-edge multiagent networks and established PBH-like conditions in [30].
A parallel research method is from graphical perspectives, which depicts the graph-theoretic features of making multiagent systems controllable under different models formulated for structural controllability. e work in [31] first put forward structural controllability for multiagent networks from the viewpoint of graph, where a single-leader multiagent network was introduced and graph-theoretic criterion on controllability was derived under an undirected topology, and these similar problems were extended to high-order dynamic agents in [32]. Rahmani and Ji [33] made use of the relaxed equitable partition for information communication topology to establish some controllable conditions for multiagent systems. e structural controllability was discussed for real-world complex networks by graph-theoretic technique [34]. e controllable characterizations of multiagent networks were given from graphical opinion [35]. Guan et al. studied the structural controllability for heterogeneous multiagent networks based on directed and weighted topology from algebraic and graphical perspectives [36].
At present, the controllability of multiagent networks is still in the initial stage of development. e problem of controllability, whether in the establishment of mathematical model or in theoretical analysis, will face the influence of many factors such as topological structure, evolution protocols, communication distance, information quantization, external disturbance, communication restriction, environmental noise, parameter uncertainty, and so on, which make the problem of controllability of multiagent networks present new characteristics and theoretical difficulties lacking more effective tools for theoretical analysis. In modern control theory, the PBH rank test plays an important role in judging whether a system is controllable, while λ-matrix exists in the PBH rank test. In this paper, we introduce λ-matrix to determine the controllability of the generic linear multiagent systems, while λ-matrix is derived from the PBH. It is well known that the PBH rank test is very useful because it only relies on the eigenvalues of the system matrices. In fact, for multiple delayed and high-dimensional multiagent systems, it is too complex and difficult to compute the rank of controllable matrices of such system. Yet, by Matlab, it is very easy to calculate the eigenvalues of the system via the PBH rank test. However, the existing results on controllability are seldom studied via λ-matrix, especially from the viewpoint of determinant factors and invariant factors. To the best of our knowledge, in higher algebra, the theory of λ-matrix is quite mature, while determinant factors and invariant factors can be used to judge whether two matrices are similar or not. Furthermore, if two matrices are equivalent to each other, then they will have the same determinant factors and invariant factors. erefore, the controllability of multiagent networks via λ-matrix technique will be highlighted here. is paper concentrates on the controllability of generic linear multiagent networks with absolute protocol on fixed topology. Compared with the existing works, the main contributions of this paper are summed up as follows. Firstly, the model of generic linear multiagent networks based on absolute protocols under a leader-follower framework is proposed. Secondly, the definitions and properties of λ-matrix, determinant factors, and invariant factors are first introduced to investigate the controllability of multiagent networks by elementary transformation for λ-matrix. Finally, sufficient and/or necessary algebraic criteria on the controllability via determinant factors and invariant factors are established for generic linear multiagent networks. Furthermore, the controllable conditions for multiagent networks with special topological graphs by λ-matrix are also established. Compared with other existing methods, the λ-matrix method has the following advantages. (3) Various methods of judgement: for the same system, we can, respectively, use its Smith standard form, determinant factors, and invariant factors to determine whether the system can be controllable or not. In particular, for special graphs, their corresponding λ-matrix is simpler and special to judge the controllability. Moreover, eorems 1-4 in the following are sufficient and effective to judge the controllability of such system, which is theoretically shown via λ-matrix. To the best of our knowledge, the λ-matrix and determinant factors are not directly used to verify the controllable properties of a linear system, including a multiagent system, in the current literature. In this paper, we can get the λ-matrix from the PBH rank test and deal with λ-matrix to obtain eorems 1-4 in the following by using some special skills and methods, so that our conclusions are not ordinary, which cannot be directly deduced from the existing conclusions. Compared with other traditional approaches, λ-matrix and determinant factors have some advantages such as intuitive form, simple calculation, and various judging methods. erefore, some new algebra-theoretic necessary and/or sufficient conditions of the controllability for generic linear multiagent systems are established via λ-matrix. So, the introduction of λ-matrix is of practical significance. e remainder of the paper is organized as follows. e problem formulation and mathematical preliminaries are given in Section 2. e main results for controllability of generic linear multiagent systems are presented in Section 3. Section 4 shows numerical simulation examples, and a conclusion is given in Section 5.

Problem Formulation and Preliminaries
Consider a generic linear multiagent network consisting of n + n l dynamic agents described as where x i ∈ R m is the state of the i-th agent, A ∈ R m×m , B ∈ R m×p , n + n l ≜ 1, 2, . . . , n + n l , n is the followers' number, and n l denotes the number of leaders influenced by external control 2 Complexity inputs, for example, in Figure 1, u i ∈ R p is the control input. e control agreement protocol similar to reference [37] from the perspective of the controllability is governed by where the feedback gains K 1 , K 2 ∈ R p×m are to be designed. a ij ≥ 0 represents the edge weight from j to i, and N i indicates the neighbor set of agent i. e information topology links between agents are described by an undirected For such network in the leader-follower structure (see Figure 1), the Laplacian matrix can be divided into where L f and L l stand for the indices of followers and leaders, respectively, and L fl and L lf stand for the communications from the leaders to the followers and from the followers to the leaders, respectively. More details can be seen in literature [38]. where Remark 1. It is easy to see that the dimension of system (5) is higher than that of system (1). Obviously, it is more difficult and complex to analyze the controllability of multiagent networks so that it is always a challenging task since such system is high dimensional. Next, we will concentrate on the controllability of generic linear networked multiagent systems with high dimension, which is theoretically shown from a new perspective to algebraic features on the controllability.

Controllability Analysis Using λ-Matrix
Here, some useful concepts and symbols about λ-matrix in the study are reviewed briefly (see references [38,39] for details), and the mathematical definitions and the classical criterion of the controllability for multiagent systems are given.
Definition 1 (see [39]) (λ-matrix). If the element of a matrix is a polynomial of λ, that is, it is the element of P[λ], where P is a number field and P[λ] is a polynomial ring under the number field P, then such matrix is called λ-matrix. Since Definition 2 (see [39]) (the rank of λ-matrix). For λ-matrix which is not zero and all (r + 1)-order determinant A(λ) are zero (if there still exist), then the rank of λ-matrix is r.
Definition 3 (see [39]) (determinant factor). Suppose that the rank of polynomial matrix A(λ) is r(≥1); for positive integer k, 1 ≥ k ≥ r, and the maximum common factor of all k-order subdeterminants of A(λ) with first coefficient being 1 is defined as the k-order determinant factor of A(λ), denoted as D k (λ). As k > r, it is easy to know that D k (λ) � 0 from the definition of the rank of λ-matrix. In addition, for the convenience of discussion, let D 0 (λ) � 1.
Definition 5 (controllability). System (5) is said to be controllable if for any initial state x f (t 0 ) and any final state x f , there exists a finite time t > t 0 such that x f (t) � x f by adjusting the leaders' movement.
Definition 6 (controllability matrix). e controllability matrix of system (5) is defined as where Q ∈ R mn×m 2 nl . From Definitions 5 and 6, we have the following propositions immediately. (5) is controllable if one of the following statements is satisfied: where Φ stands for complex number field and λ i is the eigenvalue of A for i � 1, 2, . . ., mn.

Complexity
According to Definitions 2-3, we know that the rank of A(λ) as well as its determinant factors and invariant factors can keep unchanged under elementary transformation. So, we will characterize the controllability of system (5) by using determinant factors and invariant factors.   for r > 2; therefore, (D mn 1 (s), D mn 2 (s), . . . , D mn r (s)) � 1. From eorem 1, system (5) is controllable. is completes the proof.

Remark 3.
eorem 2 provides a simple way to judge the controllability of multiagent networks with lower dimension. Especially, if there exists an mn-order determinant factor of |[sI − A, B]|, which is a nonzero constant, then the system must be controllable.
From Definition 7, the following result can be obtained immediately. Remark 4. Notice that the unimodular matrix is easier to check and calculate so that it can play a key role in judging the controllability of multiagent networks.
Lemma 1 (see [39] From Definition 8 and Lemma 1, the following results are easily available.

Corollary 2. System (5) is controllable if there exists an mnorder reversible matrix for [sI − A, B].
It is generally known that, in modern control theory, the nonsingular linear transformation does not change the controllability of the system, and the nonsingular linear transformation is the result of some elementary transformation synthesis.

Theorem 4. System (5) is controllable if mn invariant factors of [sI − A, B] are all equal to 1.
Proof. Based on eorem 1 and Lemma 3, we know that system (5)  □ From eorem 4 and Lemmas 2-4, the following simple and easy result can be obtained.

Corollary 3. System (5) is controllable if
Note that eorem 4 is derived from invariant factors that make up the standard form of A(λ). erefore, we will give a brief algorithm to get invariant factors by determinant factors (see Algorithm 1).

Some Special Graphs.
is section will present the controllable conditions for multiagent networks with special topological graphs through λ-matrix and find that it is more effective, simpler, and easier to compute, test, and verify the results.
From eorem 2 of literature [17], with leaders selected in advance, the controllability for the generic linear multiagent network is decided by matrix pair (− L f , − L fl ), where L f and L fl show the information flows among follower agents and those from leaders to followers, respectively.

Lemma 5 (see [17]). System (5) is controllable if matrix pair
is provides more and better methods to make generic linear multiagent systems controllable.

Theorem 5. Path graph is controllable.
Proof. For Figure 2(a), we can have en, Removing the first column of G(s), Step 1: calculate the rank of A(λ). Suppose that rank(A(λ)) � r; then, there are r determinant factors in A(λ).
Step 4: by step 3, A(λ) turns into its Smith standard form Λ, and it is easy to get the k-order determinant factor of A(λ) as 1, 2, . . ., r).

Remark 6.
Notice that from path graph, it is more obvious to see that its λ-matrix is n × (n + 1), in which we only need to delete the first column of λ-matrix to get an n × n lower triangular matrix with constant diagonal elements. It is easy to see that the determinant of the n × n lower triangular matrix is nonzero constant. Immediately, we can know that path graph is controllable.
is is the simplest and most intuitive way of judging the controllability by the various methods available.

Theorem 6. Complete graph is uncontrollable.
Proof. For Figure 3, we can have en, Removing the i-th column of G(s), 6 Complexity   for i � 1, 2, . . ., n; by computing, we can get that det(G i (s)) � (− 1) n+i c(s − na) n− 1 is not a constant, and if we remove the (n + 1)-th column, then

Complexity
By computing, we can get that det(G n+1 (s)) � s(s − na) n− 1 is also not a constant. us, from eorem 3, complete graph is uncontrollable.

Theorem 7. Star graph is uncontrollable.
Proof. For Figure 4, we can have en, Removing the i-th column of G(s),  for i � 1, 2, . . ., n; by computing, we can get that det(G i (s)) � (− 1) n+i c(s − na) (s − a) n− 2 is not a constant, and if we remove the (n + 1)-th column, then By computing, we can get det(G n+1 (s)) � s(s − na) (s − a) n− 2 is also not a constant. us, from eorem 3, star graph is uncontrollable.

Remark 7.
In context, the system matrices with special topological graphs have simple and special forms, so the corresponding λ-matrix-an n × (n + 1) matrix is also simple and special. Subsequently, it only needs to delete one column of λ-matrix to get an n × n matrix, whose determinant is intuitive and easy to compute. us, by introducing λ-matrix, we can easily judge whether the system can be controllable or not. erefore, the λ-matrix method is more intuitive and easier to compute, test, verify, and judge the controllability for generic linear multiagent systems.

Example 1.
In this example, we consider a five-agent network with followers 1-3 and leaders 4-5 described by Figure 5, and the corresponding Laplacian matrices of Figure 5 can be given by Let and DesignK 1 and K 2 as en, By computing, we can get D 6 (s) � 1 of G(s) � [sI − A, B]. Based on eorem 1, such system is controllable. Figures 6 and 7 are the followers' trajectories, where the stars are random initial positions (configuration) and the circular dots are desired positions (configuration), such as a straight-line configuration and a triangle configuration, respectively.
From this example, we can find that we can design appropriate K 1 and K 2 based on matrix pair (− L f , − L fl ).

Example 2.
In this example, we consider a four-agent network with followers 1-3 and leader 4 described by Figure 8, and the corresponding Laplacian matrices of Figure 8 can be given by 10 Complexity It is easy to see D 3 (s) � 1 − (s + 2) 2 ≠ 1, and such network is uncontrollable from eorem 1. Figures 9 and 10 are the followers' trajectories, where the stars are random initial positions (configuration) and the target (desired) positions (configuration), such as a straightline configuration and a triangle configuration, respectively. e circular dots are final (actual) positions (configuration). Obviously, the target positions and final positions are different, and thus system is uncontrollable.

Remark 8.
It is easy to see from the numerical examples 1-2 that the method using λ− matrix is easier to judge. Especially for low-order matrices, almost no calculation is required.

Conclusion
is paper has studied the controllability problem of the generic linear multiagent networks from the perspective of λ-matrix.
e results have shown that the determinant factors play a key role in characterizing the controllability of multiagent networks. Moreover, invariant factors and unimodular matrix can also provide simpler and easier methods on the controllability of such system.
is new perspective provides a new way to further explore the controllability of multiagent systems.

Data Availability
No data were used.

Conflicts of Interest
e authors declare that they have no conflicts of interest.