Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol †

: This paper investigates the containment control problem of discrete-time ﬁrst-order multiagent system composed of multiple leaders and followers, and we propose a proportional-integral (PI) coordination control protocol. Assume that each follower has a directed path to one leader, and we consider several cases according to different topologies composed of the followers. Under the general directed topology that has a spanning tree, the frequency-domain analysis method is used to obtain the sufﬁcient convergence condition for the followers achieving the containment-rendezvous that all the followers reach an agreement value in the convex hull formed by the leaders. Specially, a less conservative sufﬁcient condition is obtained for the followers under symmetric and connected topology. Furthermore, it is proved that our proposed protocol drives the followers with unconnected topology to converge to the convex hull of the leaders. Numerical examples show the correctness of the theoretical results.


Introduction
Distributed coordination control of multi-agent systems has been an active research area for its widespread potential engineering applications including cooperative surveillance, sensor networks, spacecraft formation flying, etc. The most fundamental problem studied in coordination control is consensus problem [1][2][3][4][5][6][7][8][9][10][11] which means that each agent reaches an agreement based on the information from its relative agents. According to the different number of leaders in the multi-agent system, the consensus problem is generally divided into leaderless case [12,13], single-leader-follower case [14][15][16], and multipleleader-follower case.
The containment control problem [17][18][19][20][21][22][23][24] is considered as a special multiple-leaderfollower consensus problem for multi-agent system with multiple leaders and followers, and it requires all followers to converge into the convex hull spanned by leaders. Many valuable results on the containment control problem of first-order multi-agent systems, which are investigated in this paper, have been obtained in recent decades. Basic results for realizing containment of continuous-time first-order multi-agent systems with stationary leaders have been given by Liu in [25], and the convergence conditions under fixed and switching topologies are dependent on the topology structure. Wang and their colleagues [26] investigated the containment problem of first-order multi-agent system with communication noises, and designed a time-varying gain to reduce noises. A PD-type control protocol was introduced and a parameter condition was given to guarantee the containment achievement under input delays in Rong's work [27]. Mu and partners [28] provided necessary and sufficient criteria for containment convergence if the communication

Graph Theory
Consider a multi-agent system with n agents is denoted by a graph G(V, E), where V = {1, 2, . . . , n} and E ⊆ V × V stand for the vertex set and edge set, respectively. An edge (i, j) ∈ E represents that agent j is able to access information of agent i and means that vertex i is a neighbor of vertex j. If agent j has no neighbor, it is called a leader, otherwise, it is a follower. The index set is denoted as N j = {i ∈ V : (i, j) ∈ E, i = j}. A directed path from i to j is a sequence of edges in a graph of the form (i, h 0 ), (h 1 , h 2 ), . . . , (h k , j), where h k ∈ V. The adjacency matrix is a nonnegative matrix A = [a ij ] ∈ R n×n defined as a ji > 0 if (i, j) ∈ E, and a ji = 0 otherwise. Furthermore, self edges are not allowed in this paper, i.e., a ii = 0. The Laplacian matrix is defined as L = [l ij ] ∈ R n×n , where l ii = ∑ n j=1,j =i a ij and l ij = −a ij , i = j.
A directed graph is called a directed tree if each node in graph has exactly one parent except for one node which is called the root, and the root has directed paths to each other node. A directed spanning tree of a directed graph is a direct tree that contains all nodes of the directed graph. A directed graph has a spanning tree if there exists a directed spanning tree as a subset of the directed graph.

Agents' Dynamics and Coordination Protocol
Investigate a discrete-time multi-agent system consisting of m leaders and n − m followers labelled by 1, . . . , m and m + 1, . . . , n, respectively. The dynamic model of agent i is given by where x i (k) ∈ R p and u i (k) ∈ R p are the state and control input of agent i, respectively. According to the PI control strategy, we use the following PI coordination control algorithm for the first-order agents, where a ij is the (i, j) entry of the adjacency matrix A, r i (k) represents the integral term, γ 1 and γ 2 are positive gain parameters to be decided for the proportional term and integral term, respectively.
The states of leaders (1) remain static since the inputs of leaders are always zero, so we only investigate the followers' dynamics here. With the protocol (2), the dynamics of follower i are rewritten as In order to analyze the convergence performance of system (3), we introduce two topologies, one of which is named as leader-follower topology composed of the leaders and followers, and the other one is named as follower topology composed of the followers. The Laplacian matrix of leader-follower topology is L given by where L 1 represents the topology between leaders and followers, and L 2 represents the topology among followers. Meanwhile, the Laplacian matrix corresponding to the follower topology is L F formulated as where the diagonal matrix D is defined as a m+1,j , . . . , m ∑ j=1 a n,j }.
Generally, we need the basic assumption on the leader-follower topology of system (3) as follows.

Assumption 1.
For each of the followers in the leader-follower topology, there is at least one leader that has a directed path to the follower.
On the basis of Assumption 1, we make further assumptions on the follower topology as follows.

Assumption 2.
The follower topology has a directed spanning tree. Assumption 3. The edges of the follower topology are bidirectional, i.e., Laplacian matrix L F is symmetric, and it has a directed spanning tree.
Then we have the following lemma.

System (3) is reformulated in a vector form as
where , and we get Let Y(k) = [X T F (k),R T (k)] T , and the system (5) is expressed in a compact form as To continue the convergence analysis of system (6), some useful lemmas are listed firstly.

Lemma 2 ([39]
). Let P(z) be a polynomial of order two with complex coefficients in the form of P(z) = z 2 + (p 1 + jq 1 )z + p 2 + jq 2 , where j is the imaginary unit. The polynomial P(z) has all its zeros in the open left half of the z-complex plane if and only if p 1 > 0 and p 2 where λ(·) denotes matrix eigenvalue, ρ(·) denotes matrix spectral radius and Co(·) denotes the convex hull.
Next, we will obtain the convergence conditions of the system (6) according to different follower topologies.

General Directed Follower Topology
Theorem 1. Consider the multi-agent system (3) with leader-follower and follower topologies satisfying Assumptions 1 and 2, respectively. With condition ∑ n i=m+1 q i r i (0) = 0, all the followers reach containment-rendezvous asymptotically that the followers converge to an agreement value in the convex hull spanned by the leaders, if γ 1 and γ 2 satisfy . . , n represent the eigenvalues of the matrix L F + D.
Proof. According to the properties of the Kronecker product, we set the agents' state dimension as p = 1 in the following proof, and system (6) is written aŝ Meanwhile, we divide the proof into two steps including convergence analysis and the analysis of final rendezvous state.
Step 1: To analyze the convergence performance of system (8), we investigate the characteristic equation of (8) as follows, and it can be reformulated from Lemma 3 as Evidently, it is obtained from Lemma 1 that the Equation (10) has a root at z = 1. Before analyzing Equation (10), we first pay attention to the following equation Obviously, (11) is equivalent to Instead of studying Equation (12) directly, we apply the bilinear transformation s = z+1 z−1 to it and get Thus, Equation (12) where Based on Lemma 2, the polynomial has all zeros in the open left half complex plane if and only if the gains γ 1 and γ 2 satisfy which is equivalent to Thus, the roots of Equation (11) all lie inside the unit circle if and only if there exist gain parameters γ 1 and γ 2 satisfying condition (16).
Back to Equation (10), we reformulate it as where . Under condition (16), it is obvious that if condition |λ(γ 2 DΘ −1 (e jω ))| < 1 holds with ω ∈ (0, π], the roots of Equation (17) lie inside the unit circle except for one root at z = 1. Thus, the characteristic Equation (9) has all roots within or on the unit circle and we have finally proved the asymptotic convergence of the system under Assumption 1, i.e.
where X d ∈ R n−m andR d ∈ R n−m are constant vectors.
Step 2: We will prove that all followers reach an agreement value in convex hull spanned by leaders.
Under Assumption 1,we get from Lemma 1 and Equation (19) lim where x d ∈ R is a constant. From Equation (20), it is clear that followers converge to the same value. Taking the z transformation of Equation (4) with p = 1, we get Re-express Equation (21) as Since the convergence of the system has been proved, using the final value theorem yields Letting Multiplying Q = [q m+1 , q m+2 , . . . , q n ] on both sides of (24), we get where Q is the left eigenvector of L F corresponding to eigenvalue 0. Then, we take the limit of Equation (25) as z → 1 and have Equation (26) is rewritten as and we finally get with ∑ n i=m+1 q i r i (0) = 0. Apparently, x d is in the convex hull formed by leaders. When p = 1, the proof is the same while the final state x d is a constant vector instead of a constant. Hence, we can draw the conclusion that all followers will eventually reach an agreement value in the convex hull spanned by leaders under our proposed protocol. Theorem 1 is proved.
. . , n is the (i, i) entry of the diagonal matrix D. Let F(z) = det(I + G(z)L F ). According to the generalized Nyquist stability criterion [41], the zeros of F(z) are in the unit circle when G(z) does not have poles out of the unit circle, if λ(G(e jω )L F ) does not enclose the point (−1, j0) for ω ∈ [−π, π]. Because of the symmetry of the Laplacian matrix L F , it follows from Lemma 4 that where For calculating conveniently, we assume that γ 2 ≤ γ 1 . In order to ensure that all the poles of G(e jω ) are within or on the unit circle, we have Equation (33) is rewritten as where a = 2 cos 2 ω + (γ 1 θ i − 2) cos ω − γ 1 θ i and b = sin ω(2 cos ω + γ 1 θ i − 2). To analyze the intersections of g i (e jω ) on the real axis, we get For ω ∈ (0, π], Equation (36) has only one solution ω = π and It is evident that ρ(L F )Co(0 ∪ g i (e jω )) does not enclose the point (−1, j0), if all the points (ρ(L F )g i (e jπ ), j0) are on the right side of the point (−1, j0) for i = m + 1, . . . , n, i.e., Hence, λ(G(e jω )L F ) does not enclose the point (−1, j0) and G(z) has no poles out of the unit circle, if we choose the gain parameters γ 1 and γ 2 satisfying γ 2 ≤ γ 1 ≤ 2 . . , n. Thus, F(z) has all zeros within the unit circle, which means that the roots of characteristic Equation (9) are within or on the unit circle. Hence, the convergence of the system is proved.
The analysis of final rendezvous state is omitted here for it is almost same as the proof of Theorem 1. The only difference is that the left eigenvector of the symmetric Laplacian Matrix L F corresponding to eigenvalue 0 becomes [1, . . . , 1], and the final value is Theorem 2 is proved.

Unconnected Follower Topology
Considering the follower topology is unconnected, i.e., it has no spanning tree, the containment-rendezvous cannot be achieved by our proposed PI coordination control protocol. In this case, we can divide the follower topology into several connected parts, and obtain the convergence conditions based on the results in Theorem 1.
In order to analyze the convergence behavior, we divide the unconnected follower topology into N connected parts, each of which has a spanning tree or has only one agent, and we labelled the parts as 1, . . . , N. For each part, the state and integral vectors of followers are defined as X F l (k) and R l (k), l = 1, 2, · · · , N. Hence, the dynamic models of followers are formulated as where L 1 l , L 2 l and L F l are same as the definitions of above L 1 , L 2 and L F , respectively. For each part l, the characteristic equation is given by where D l is same as the definition of above D. Similar to Theorem 1, we take into account the following equation Then, in the light of Theorem 1, we come to the following results.
Proof. Divide the unconnected follower topology into N connected parts (40), and the state and the integral term of followers are expressed as X F (k) = [X F 1 (k), . . . , X F N (k)] T and R(k) = [R 1 (k), . . . , R N (k)] T . It is evident the the dynamics (40) of each part have completely same form as (4).
According to the proof of Theorem 1, the followers in one part reach the containmentrendezvous under condition in Theorem 3. In each part. the followers reach an agreement value in convex hull spanned by the leaders. Evidently, the convex hull spanned by the leaders in each part must be contained in the convex hull composed of all the leaders in the system. Hence, all followers converge to the convex hull composed of all the leaders.
General Topology. The general follower topology satisfying Assumptions 1 and 2 is shown in Figure 1.     Since the condition is easier to calculate, we are able to give a more specific range of the parameters. Under the given topology, we have γ 1 ≤ 0.67 and we choose γ 1 as 0.2 here. With the condition γ 1 = 0.2, we then have γ 2 > 0.1. Finally, we choose γ 1 = 0.2 and γ 2 = 0.11 to guarantee that the conditions in Theorem 2 hold. Then, all followers reach the containment-rendezvous asymptotically (see Figures 5 and 6).   Unconnected Topology. The unconnected follower topology shown in Figure 7 satisfies Assumption 1. It is evident that the topology can be divided into two connected follower topologies as shown in Figure 8.
The gain parameters are set as γ 1 = 0.2 and γ 2 = 0.1 satisfying the requirement in Theorem 3. It is seen from Figures 9 and 10, all followers are divided into two group and followers in each group reach own agreement value in the convex hull spanned by leaders.

Conclusions
In this paper, containment-rendezvous problem of discrete-time first-order multiagent systems is analyzed. The proposed control protocol includes a proportional term and an integral term. The proportional term ensures the realization of the containment, and the integral term guarantees the rendezvous. According to the frequency-domain analysis and numerical example, the effectiveness of our proposed protocol under the general connected follower topology is proved. For the symmetric and connected follower topology, a simpler convergence condition is presented. The containment control problem under unconnected follower topology is further discussed. Notably, the unconnected follower topology can be divided into several connected ones, so the followers still converge to the containment formed by all the leaders. Since the work in our paper is only a theoretic research and do not consider the trajectory of the agents, compared with some practical works [42][43][44], we will continue to study this question in a more practical way in our future work.