On the Group Controllability of Leader-Based Continuous-Time Multiagent Systems

School of Information Engineering, Minzu University of China, Beijing 100081, China School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, 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 decades, distributed coordination control of networked MASs has become a hot and challenging issue in lots of areas, such as applied mathematics, control theory, mechanics, engineering, and neurobiology [1][2][3][4][5][6][7][8][9][10]. Studies in this area include several basic problems, such as stability, consensus and synchronization [11], containment [12], controllability [13], and formation control and tracking control [14]. e controllability problem is a key essential problem in modern control theory and attracts increasing attention due to its wide applications in engineering. An MAS is controllable if each dynamic follower can attain its desirable configuration from any initial state during finite time by regulating some leaders. In [15], Tanner first put forward the controllability problem of networked systems in a leader-following framework, where a certain agent was acted as the leader (the external control input), and an algebraic feature based on eigenvalues and eigenvectors of such system's Laplacian matrices was derived by nearest neighbor rules. Based on this, Liu et al. [16] discussed the controllability of discrete-time MASs with a single leader based on nearest neighbor rules and derived a simple controllable condition for such a system on switching topology. Afterwards, further studies on the controllability of MASs have mainly been concentrated from graph-theoretic and algebraic-theoretic points of view, respectively. At present, many works on the controllability of MASs from the perspective of graph theory have concentrated on the basis of partitions of graph topology, such as equitable partition/relaxed equitable partition/ external equitable partition in [17], connected component partition in [18], and selection of leaders [19]. Further research studies on the controllability were presented for some different special topology graphs, such as path graphs [20], cycle graphs [21], multichain topologies [22], stars and trees [23], two-time-scale topologies [24], and regular graphs [25]. Lots of algebraic controllable conditions of MASs were characterized in [26,27]. e aforementioned results on the controllability of MASs just contained a single group. However, in engineering practice, a single group can be compartmentalized into some subgroups with the improvement of MASs' complexity [28]. It is a very challenging work to study the controllability problem of MASs with multiple subgroups and multiple leaders considering the control law, the information topology structure between different subgroups, and the effect of dynamical leaders acting on the follower agents, which will be highlighted in this paper. More recently, the group controllability of continuous-time/discrete-time MASs leaderless with different topologies and communication restrictions in [29,30] was studied, respectively.
Motivated by the results of previous studies, this paper aims at the group controllability of continuous-time MASs consisting of some different subgroups by adjusting the leaders. e main contributions of this paper are summed up as follows: (1) Different from the group controllability problem of continuous-time MASs under the leaderless framework studied based on the fixed topology in [30], the current work has considered the group controllability of continuous-time MASs under the leaderfollower framework with fixed topology and switching topology, respectively, which can be expressed by the system matrices. It is obvious that different models can lead to completely different features for MASs with leaders. (2) e concepts of the group controllability of continuous-time MASs with multiple leaders are proposed based on switching and fixed topologies, respectively. (3) Sufficient and/or necessary algebraic-and graphtheoretic group controllable characterizations of continuous-time MASs with multiple leaders under the group consensus protocol are established from the system's Laplacian matrices. (4) e effects of subgroups and leaders on the group controllability are discussed. e rest of this work is arranged as follows. e problem formulation is stated in Section 2. Section 3 builds the group controllability of MASs with multiple leaders. Numerical example and simulations are given in Section 4. Finally, Section 5 summarises the conclusion.

Problem Formulation
Consider a continuous-time MAS consisting of N agents governed by where x i ∈ R is the state and u i ∈ R is the control input, respectively. In engineering practice, the whole group can be compartmentalized into a few subgroups. Without loss of generality, in this paper, such MAS consisting of m + n + l + k (m, n, l, k > 1) agents is compartmentalized into subgroup (G 1 , x 1 ) and subgroup (G 2 , x 2 ), as shown in Figure 1.
and N 2p , respectively, represent the leaders' neighbor sets of subgroups 1 and 2.

Remark 1.
It is noted that the (i, j)th entry of the system's adjacency matrix, denoted as a ij , in this paper, can be allowed to be negative, which makes it more difficult and complex to discuss the group controllability problem since there are negative factors in the coupling links between different subgroups. Supposed that . , x m ) T and x 2 ≜ (x m+1 , . . . , x m+n ) T are the state vectors of follower agents in G 1 and G 2 and y 1 ≜ (y 1 , . . . , y l ) T and y 2 ≜ (y l+1 , . . . , y l+p ) T are the state vectors of the leader agents in G 1 and G 2 , respectively. en, the dynamics of the followers in system (1) becomes

Complexity
where L 1 � [l ij ] ∈ R m×m and L 2 � [l ij ] ∈ R n×n are the Laplacian matrices of graphs G 1 and G 2 , respectively, Remark 2. Furthermore, because a ij can be allowed to be negative, nonzero controller gains can be appropriately selected as long as L 1 and L 2 are Laplacian matrices. In essence, the group controllability of continuous-time leaderbased MASs cannot be affected by the controller gains.
In order to discuss the group controllability problem of system (3), its equivalent augmented system can be described by where (x 2 , y 1 ) and (x 1 , y 2 ) are the inputs of subgroup (G 1 , x 1 ) and subgroup (G 2 , x 2 ), respectively.

Group Controllability Analysis
is section discusses the group controllability of continuous-time leader-based MASs and establishes the group controllability criteria by adjusting appropriate leaders with switching and fixed topologies, respectively.

Group Controllability on Switching
Topology. Similar to literature [29], corresponding system (5) with switching topology can be described as where the switching path σ(t): R + ⟶ 1, . . . , K { } can be described by a piecewise constant scalar function, which presents the coupling links of such time-variant system, and K is the number of probable switching topologies. Moreover, Some relevant important concepts will be introduced in the following, and more details can be seen in [29].

Lemma 1.
(see [29]). For matrices Definition 4 (see [29], cyclic invariant subspace). For A ∈ R N×N and a linear subspace For system (6), the subspace sequence is defined as Proof. Similar proof can be referred from that of Lemma 2 in [29]; here, it is omitted.
we have On the contrary, it is easy to know W 1m ⊆ R m . erefore, we can have W 1m � R m . For the subspace W 2n , we can also have the similar result W 2n � R n . From Lemmas 1 and 2, the assertion holds.
eorem 1 provides an important and simple method to check the controllability of continuous-time MASs with leaders by designing a switching path. At the same time, it is noted that the group controllability of continuous-time MASs with leaders can depend on leaderto-follower information communications (i.e., matrices B i ) and the subgroup-to-subgroup information communications (i.e., matrices C i ) regardless of the internal information communications between subgroups (i.e., matrices A i ) whether the topology of the internal network is fixed or switching, that is, the controllability of the subgroups is not required, which provides an important convenience for designing a switching path to ensure the group controllability for continuous-time MASs with switching topology.
In the following, some special important cases are discussed.
e result is obvious from Definition 5.
□ Remark 4. From Lemma 3, it is too complex to compute the controllability matrices of system (5). On this basis, the group controllability of such MAS with leaders is shown by the technique of PBH rank test.
Moreover, we can have Since α ≠ 0, then there must be rank(Q 1 ) < m, which implies that system (5) is uncontrollable, contradicting to the assertion that system (5) attains the group controllability.
e necessity of (1) is proved.

Complexity
Sufficiency: by contradiction, assumed that system (5) is uncontrollable, then ∃λ ∈ C of A 1 , which corresponds to the eigenvector β( ≠ 0) satisfying and then β ′ [λI − A 1 , C 1 , B 1 ] � 0 so that rank(λI− A 1 , C 1 , B 1 ) < m. is contradicts to rank(sI − A 1 , C 1 , B 1 ) � m for ∀s ∈ C. e sufficiency of (1) is proved. (5) where the columns of U 1 and the diagonal matrix Λ 1 are made up of orthogonal eigenvectors and eigenvalues of A 1 , respectively. Moreover, U 1 U 1 T � I; then, the controllability matrix of such a system can be expressed as where Since U 1 consists of the orthogonal eigenvectors of A 1 , then U 1 is nonsingular, which implies that rank(Q 1 ) � rank(Q 1 ). Let λ i and η i be the eigenvalues and their corresponding eigenvectors of A 1 , respectively, for i � 1, 2, . . . , m.
Note that condition L T i � L i implies that the information weight from agent i to agent j is the same as that from agent j to agent i in the same subgroup, that is, the topological structure is symmetric for the subgroups.  (5) is uncontrollable if subgroups (G, x 1 ) and (G, x 2 ) are both complete graphs (see Figure 2) and a ij � b iq (∀i, j ∈ ℓ 1 , ℓ 2 , ∀q ∈ ℓ l ), regardless of how to connect (G, x 1 ) and (G, x 2 ).
Here presents an algorithm for designing the leaders.

Complexity
Algorithm 1 (algorithm for designing leaders). For the given initial and desired states x(0) and x(t 1 ), MASs can reach the desired state during [0, t 1 ], where t 1 > 0 is the finial time. Suppose that MAS (5) is controllable; then, its Gram matrix is where t ∈ [0, t 1 ]. Since W c [0, t 1 ] is invertible, we can design a set of inputs (leaders) as en, a solution of system (5) is which can make the system state from x(0) to x(t 1 ) during [0, t 1 ]. Notice that t 1 is the longest time to get a set of inputs.

Example and Simulations
A nine-agent system with followers 4 and a leader as subgroup 1 and followers 3 and a leader as subgroup 2 is described by Figure 4 with a 12 � a 21 � 1,a 23 � a 32 � 2,a 34 � a 43 � 1,a 45 � a 54 � 1,a 56 � a 65 � 1,a 67 � a 76 � 1; otherwise, a ij � 0. From Figure 4, the system matrices are as follows: which mean all the eigenvectors of A 2 are unorthogonal to any one column of B 2 or C 2 . ose imply that system (5) described by Figure 4 can attain the group controllability. Figures 5 and 6 depict the initial states, final states, and moving trajectories of the followers of subgroup 1 and subgroup 2 described by the black star dots and the black circular dots, respectively. Beginning from random initial states, the followers of subgroup 1 and subgroup 2 can be finally governed to a straight-line alignment and a trapezoid alignment, respectively.

Conclusion
is paper has discussed the group controllability of continuous-time MASs with multiple leaders on switching and fixed topologies, respectively. Some useful and effective results of group controllability are obtained by the rank test and the PBH test. Specially, the group controllability of continuous-time MASs for some special topology graphs has also been studied.

Data Availability
In this paper, no data are needed; only mathematical derivation is needed.

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