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Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays
Communicated by Xudong Zhao
Accepted Manuscript
Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays Huiyang Liu, Guangming Xie PII: DOI: Reference:
S0925-2312(18)30070-5 10.1016/j.neucom.2018.01.045 NEUCOM 19247
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
21 August 2017 3 December 2017 19 January 2018
Please cite this article as: Huiyang Liu, Guangming Xie, Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.01.045
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Huiyang Liua,∗, Guangming Xieb a School
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Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays✩
of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China. b Intelligent Control Laboratory, College of Engineering, Peking University, Beijing, 100871, China
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Abstract
This paper addresses containment control of multi-agent systems with multiple interactive leaders. The containment control objective is two-fold: the leaders converge to a desired formation, and the followers move into the convex hull spanned by the leaders’ final positions. Some sampled-data based protocols are
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proposed, which are effective when the communication environment suffers from intermit information, switching topology and time-delays. For the leaders, the
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convergence analysis can be easily obtained by using the properties of the SIA matrices. The geometric configuration of the desired formation is not affected by the switching topology or time-delays. For the followers, the convergence
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analysis is much complicated since the properties of the SIA matrices cannot be used. Based on the relationship between the topology and matrices, it is proved that the followers will move into the convex hull of the desired forma-
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tion, and then change their positions in the convex hull as the time evolves under the switching topology; however, the final states of the followers are not
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affected by the time-delays. Finally, the theoretical results are illustrated by some simulations. Keywords: Containment control; multi-agent systems; directed graphs; switching topology; time-delays. ∗ Corresponding
author Email address: [email protected] (Huiyang Liu )
Preprint submitted to Neurocomputing
January 27, 2018
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1. Introduction
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In the past decades, inspired by the work [1, 2] and [3], many researchers have devoted themselves to the study on cooperative control of multi-agent
systems. The reasons lie in its wide applications in many fields such as coordi5
nation of multiple robots, collaboration of human society, and formation control of multiple aircrafts. As a kind of cooperative control, the containment control
strategies, motivated by numerous natural and society phenomena, have been
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developed by several researchers by using neighbor information. For examples,
the natural phenomena include shepherd dogs, the guard geese and so on. In 10
human society, the containment control phenomena can be found in the avoiding danger, encirclement and suppression, and even the opinions of a group of individuals.
In the containment control problems, there are always two types of agents:
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leaders and followers. They play different roles in achieving the coordination control objective. Therefore, it is important to design different coordination
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control protocols for them. The containment control problem in this paper concerns with the formation control for the leaders and consensus for the followers. Many researchers have studied formation control problems in the past decades
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[4, 5, 6]. The study of consensus has a history dated back to several decades [7, 8], and has recently been studied extensively (e.g., switching topology [9, 10],
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communication noises [11], sampled-data or discrete-time systems [12, 13, 14], tracking [15, 16], fractional-order [17] and so on). Next, we make a brief summary on the containment control problems of
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multi-agent systems. In [18], by exploiting the theory of partial difference equa-
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tions, a hybrid control scheme based on stop-go rules was proposed for a collection of mobile robots. In [19], containment control in leader-follower networks with switching communication topologies was investigated. Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks was studied in [20]. In [21], containment control of multi2
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agent systems with stochastic topologies was studied. In the past several years, containment control problems of multi-agent systems have been developing very fast. Several research topics have been addressed, such as intermittent infor-
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mation [22, 23, 24], finite-time containment control [25, 26, 27], hybrid model predictive control scheme [28], noises environment [29]. The results are vari35
ous regarding the varieties of agents’ dynamics, tasks demanded, the control strategies and the topology structures.
In most of the above results, it is assumed that there is no communication among the leaders. However, it may be more practical to study containment
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control with multiple interactive leaders. The leaders also need to communicate with each other to reach their desired objectives. In [30], containment con-
trol for multiple unicycle agents under undirected connected graphs was introduced. Formation-containment control problems for general linear multi-agent systems without time-delays and with time-varying delays under fixed directed topologies were studied in [31, 32]. However, in a communication network, the communication channel may break off because of the cost saving or the commu-
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nication constraint, or get well again by remediation. This results in a switching
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topology. For consensus problems, the switching topology case has been widely studied, e.g., [9, 10, 33]. Different from the consensus problems, the properties of SIA matrices are invalid for the containment control with switching topology. In addition, time-delay is a ubiquitous phenomenon and widely studied in the
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field of systems and control [34, 35, 36]. The existence of the time-delays always
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leads to the instability of the whole systems. In multi-agent systems, many researchers have studied the coordination control problems with time-delays, e.g., [3, 10, 33, 37]. The methods pertain to frequency domain analysis, properties of SIA matrix, Lyapunov function method and so on. We will employ a new
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method to solve the containment control problem with time-delays. Therefore, it is significant to study containment control with interactive leaders, switching topology and time-delays. Motivated by the progress in this field, we will investigate sampled-data
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based containment control for continuous-time multi-agent systems with switch3
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ing topology and time-delays. The leaders are interactive, and they can communicate with each other. The leaders are expected to form a desired formation, and the followers are expected to move into the convex hull spanned by the final
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positions of the leaders. According to the different objectives that the leaders and the followers are to achieve, we will design a formation control protocol
that is independent of the followers’ states for the leaders, and a consensus pro-
tocol that depends on the neighbor information for the followers. The desired formation for the leaders can be achieved by using the properties of the SIA
matrices. The geometric configuration of the desired formation is not affected by the switching topology and time-delays. Since the properties of SIA ma-
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trices are invalid for the convergence analysis of the followers, we analyze the behaviors of the followers by using the relationship between the topology and matrices. We will show that, for the switching topology case, the followers first converge to the convex hull spanned by the final positions of the leaders, and 75
then change their positions as the time evolves; for the time-delays case, the
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final states of the followers are not affected by the time-delays, but determined by the topology structure and the final states of the leaders. We will also give
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some examples to illustrate the theoretic results. The outline of this paper is given as follows: we next establish some of the 80
basic notations. We then, in Section 2, give some preliminaries, followed by the
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problem formulation in Section 3. In Section 4, the main results are presented. Section 5 gives some simulation results. Then, the conclusions are given in
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Section 6.
Notations: Let 0m×n (or 0) denote an m × n (or compatible dimensional)
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all-zero matrix. In represents the n−dimensional identity matrix. 1n represents
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the n−dimensional all 1 vector. Matrix A is said to be nonnegative if all its
entries are nonnegative. A nonnegative matrix A ∈ Cn×n with the property that all its row sums are +1 is said to be a (row) stochastic matrix. A stochastic
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matrix A ∈ Cn×n is called indecomposable and aperiodic (SIA) (or ergodic) Qk if there exists a column vector ν such that limk→∞ Ak = 1n ν T . i=1 Ai =
Ak Ak−1 · · · A1 denotes the left product of matrices. 4
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2. Preliminaries The notation G = (V(G), E(G), A) represents a weighted directed graph, where V(G) = {v1 , v2 , · · · , vn } is the set of vertices, E(G) ⊆ {(vi , vj ) : vi , vj ∈ V(G)} is the set of edges, and A = [aij ] is a weighted adjacent matrix with
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nonnegative adjacent entry aij , i, j = 1, 2, · · · , n, [38]. In an edge eij = (vj , vi ) ∈ E(G), vj is called the parent vertex of vi and vi is the child vertex of vj . If
eij ∈ E(G), i 6= j, is an edge, then the corresponding adjacent entry aij > 0. Otherwise, aii = 0 for all vi ∈ V(G). The neighbor set of vertex vi is denoted 100
by Ni = {vj ∈ V(G) : (vj , vi ) ∈ E(G), j 6= i}. A directed path in G is a sequence
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vi1 , vi2 , . . . , vik of vertices such that (vis , vis+1 ) ∈ E(G) for s = 1, 2, · · · , k − 1.
A directed graph G contains a spanning tree if there exists at least one vertex that has a directed path to all the other vertices in G. 105
P The in-degree and out-degree of vi are defined as: degin (vi ) = vj ∈Ni aij P and degout (vi ) = vj ∈Ni aji , respectively. The degree matrix of G is a diagonal
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matrix defined as: D = [dij ], where dij = degin (vi ) for i = j, otherwise, dij = 0. The Laplacian of G is defined as: L = D − A.
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Lemma 1. ([9]) For a directed graph G, 0 is an eigenvalue of L, and 1n is the corresponding right eigenvector; 0 is a simple eigenvalue of L, and all the other eigenvalues have positive real parts if and only if G has a spanning tree.
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Definition 1. ([39]) A set K ⊂ Rm is said to be convex if (1 − γ)x + γy ∈
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K whenever x, y ∈ K and 0 ≤ γ ≤ 1. The convex hull of a finite set of Pn points x1 , x2 , . . . , xn ∈ Rm is denoted by co{x1 , x2 , . . . , xn } = { i=1 αi xi |αi ∈ Pn R≥0 , i=1 αi = 1}.
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3. Problem formulation In this paper, a group of n coordinating agents are considered. We use vertex
vi to represent the ith agent, and edge eij ∈ E(G) to represent an available
information channel from agent j to agent i. The dynamics of each agent is
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modeled by a continuous-time system: χ˙ i (t) = ςi (t), i = 1, 2, · · · , n,
where χi (t), ςi (t) ∈ R are the position and the control input of the ith agent,
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(1)
respectively.
Definition 2. In a multi-agent team, an agent i is called a follower, or i ∈ F, if (vj , vi ) ∈ E(G) for at least one vj ∈ V(G); an agent i is called a leader, or i ∈ L, if (vj , vi ) ∈ / E(G) for each vj ∈ V(F).
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We assume that the leader set is L = {1, 2, · · · , m} and the follower set is
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F = {m + 1, m + 2, · · · , n}. From the definitions of leader and follower, L can be partitioned as:
L=
LF F
LF L
0m×(n−m)
LLL
,
(2)
where LF F ∈ R(n−m)×(n−m) , LF L ∈ R(n−m)×m and LLL ∈ Rm×m . It is obvious 130
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that if the directed topology G has a spanning tree, then the topology for the leaders, denoted by GL , also has a spanning tree. From Lemma 1, LF F is
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invertible.
Definition 3. A formation of the leader set is a vector ηL = [η1 , η2 , · · · , ηm ]T ∈
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Rm . The states of the m leaders, denoted by χL = [χ1 , χ2 , · · · , χm ]T , converge to the formation ηL if χi (t) − ηi → 0, as t → +∞, for i ∈ L. 4. Main results
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4.1. Containment control under switching topology In a communication network of mobile agents, the communication channel
may break off for cost saving or being destroyed and get well again. It results in a variation of the communication topology. Suppose that the set of all possible
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communication graphs is finite, denoted by G¯ = {G1 , G2 , · · · , GM }, where M is a finite constant. Let h > 0 be the sampling period. Denote the interaction topology at the `th sampling time as G(`h), ` = 0, 1, 2, · · ·. The union of a 6
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¯ i = `1 , `1 + 1, · · · , `2 − 1, where `1 < `2 , is a group of directed graphs G(ih) ⊂ G, directed graph with vertices given by vi , i = 1, 2, · · · , n, and edges given by the 145
union of the edges of G(ih).
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We propose the following time-varying protocol for the system (1): P j∈L∪F aij (`h)(χj (`h) − χi (`h)), i ∈ F ςi (t) = if t ∈ [`h, `h+h), P j∈L aij (`h)((χj (`h) − ηj ) − (χi (`h) − ηi )), i ∈ L (3) where aij (`h) > 0 if agent i can receive information from agent j at the `th
sampling time, and ηi , i = 1, 2, · · · , m, are constants, which represent the for150
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mation pattern of the leaders. Denote χ(`h) = [χ1 (`h), χ2 (`h), · · · , χn (`h)]T . The agent dynamics can be summarized as:
χ(`h + h) = (In − hL(`h))χ(`h) +
0(n−m)×1
hLLL (`h)ηL
,
(4)
where L(`h) denotes the Laplacian matrix of directed graph G at the `th sam-
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pling time, ` = 0, 1, 2, · · ·, and L(0) = 0n×n .
According to the partition of the Laplacian matrix, the leaders’ dynamics
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can be written as:
χL (`h + h) = (Im − hLLL (`h))χL (`h) + hLLL (`h)ηL . 155
(5)
Note that the set of all the possible communication graphs of the leaders is also
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finite, denoted by G¯L = {G1 , G2 , · · · , GP }, where P ≤ M . The following proposition can be derived immediately from the existing re-
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sults by defining a new variable yi (`h) = χi (`h) − ηi , i = 1, 2, · · · , m, and using the SIA matrix theory. The verification can refer to [9, 33]. We will not verify it.
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Proposition 1. Let GL (`h), ` = 0, 1, 2, · · ·, be a sequence of switching commu-
nication graphs of the leaders. Assume that h <
1 maxi=1,···,m,j=1,2,···,P dii (j) ,
where
dii (j) is the in-degree of the ith leader in the jth possible communication graph in G¯L . Then all the leaders converge to a desired formation if the union of di-
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rected graphs GL (`h) ⊂ G¯L , ` = `s , `s + 1, · · · , `s+1 − 1, s = 0, 1, 2, · · ·, `0 = 1, 7
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L2
L1
F1
L2
F1 (b)
(a)
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L1
Figure 1: Two directed graphs, where the weights are assumed to be 1
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has a spanning tree. Furthermore, the final formation of the leaders is given as χ∗L = 1m z T (χL (0) − ηL ) + ηL for some vector z ∈ Rm .
Remark 1. The final states of the leaders may be affected by the switching topology. However, the geometric configuration of the desired formation are 170
not affected by the switching topology. For example, consider a multi-agent
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team with 3 agents, where L1 and L2 are the leaders, and F 1 is the follower. Obviously, the union of the two directed graphs has a spanning tree. Assume that the states of the agents are in R, thus the agents move in the χ−axis.
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The initial states of the leaders are chosen as χ1 (0) = 4 and χ2 (0) = 5. The desired formation is given by h = [3, 1]T . We consider two special cases. Firstly, the switching signal is chosen such that the communication topology is always
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modeled by the directed graph (a) at all the sampling times. In this case, z1 = [1, 0]T . According to Proposition 1, the final states of L1 and L2 are given by
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χ∗1 = 4 and χ∗2 = 2.
Secondly, the switching signal is chosen such that the communication topol-
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ogy is always modeled by the directed graph (b) at all the sampling times. In
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this case, z2 = [0, 1]T . According to Proposition 1, the final states of L1 and L2
are given by χ∗1 = 7 and χ∗2 = 5.
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It is obvious that the final states of the leaders may be different if the switch-
ing signals are different. However, the distance of the leaders is always 2. Now, we will study the dynamics of the followers. Noticing that for each 8
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L(`h), ` = 0, 1, 2, · · ·, it can be written as L(`h) =
LFF (`h)
LF L (`h)
0m×(n−m)
LLL (`h)
We have the following lemma.
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1 maxi=1,···,n,j=1,···,M dii (jh) ,
.
where dii (jh) is the in-
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Lemma 2. Suppose that h <
¯ If the degree of the ith agent in the jth possible communication graph in G. union of directed graphs G(`s h), G(`s h+h), · · · , G(`s+1 h−h), s = 0, 1, 2, · · ·, `0 =
1, has a spanning tree, then lim`→+∞ (In−m − hLF F (`h))(In−m − hLF F (`h − h)) · · · (In−m − hLF F (h)) = 0(n−m)×(n−m) .
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Proof. Denote Ψ(s) , (In − hL(` s+1 h − h)) · · · (In −hL(`s h + h))(In − ΨF F (s) ΨF L (s) , where ΨF F (s) = hL(`s h)). It is noted that Ψ(s) = 0m×(n−m) ΨLL (s) Q`s+1 −1 Q`s+1 −1 (In−m − hLF F (ih)), ΨLL (s) = i=` (Im − hLLL (ih)), and i=`s s ΨF L (s) `s+1 −1
Y
(In−m − hLF F (ih))(−hLF L (`s h))
Y
(In−m − hLF F (ih))(−hLF L (`s h + h))(Im − hLLL (`s h)) + · · · +
M
=
+
`s+1 −1
i=`s +2
+
`s+1 −3
(In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))
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+
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i=`s +1
(−hLF L (`s+1 h − h))
`s+1 −2
Y
i=`s
Y
i=`s
(Im − hLLL (ih))
(Im − hLLL (ih)).
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From the partition of Laplacian matrix, we have LF F (`h)1n−m + LF L (`h)1m = 0n×1 , L(`h)1n = LLL (`h)1m
and then
(In − hL(`h))1n =
(In−m − hLF F (`h))1n−m + (−hLF L (`h))1m (Im − hLLL (`h))1m
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= 1n .
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Furthermore, we have Ψ(s)1n = 1n , i.e., ΨF F (s)1n−m + ΨF L (s)1m = 1n−m and ΨLL (s)1m = 1m . Thus, we have =
`s+1 −1
Y
(In−m − hLF F (ih))(−hLF L (`s h))1m
i=`s +1
+
`s+1 −1
Y
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ΨF L (s)1m
(In−m − hLF F (ih))(−hLF L (`s h + h))1m + · · · +
i=`s +2
+(In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))1m +(−hLF L (`s+1 h − h))1m .
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Let us analyze the entries of ΨF L (s)1m . We first consider the last term (−hLF L (`s+1 h − h))1m . Since the sampling period h > 0 and −LF L (`s+1 h − h)) = AF L (`s+1 h − h)), the nonzero components of the vector (−hLF L (`s+1 h −
h))1m correspond to the followers who can receive information from the leaders directly at the (`s+1 − 1)th sampling time. The nonzero components of the vector (In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))1m correspond to the followers who can receive the leaders’ information directly (1-level follow-
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ers) at the (`s+1 − 2)th sampling time and the followers (named 2-level fol-
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lowers) who can not receive information from the leaders directly, but, at the (`s+1 − 1)th sampling time, they can receive information from one of the followers who can receive information from the leaders directly at some sampling times in `s , `s + 1, · · · , `s+1 . For any integer `s < `s+1 , the nonzero components of the
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vector (In−m − hLF F (`s+1 h − h)) · · · (In−m − hLF F (`s h + h))(−hLF L (`s h))1m
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correspond to the followers who can receive the leaders’ information directly at `s th sampling time, the followers who can not receive the leaders’ information directly, but can receive information at the `s + 1th sampling time, from one of the followers who can receive information from the leaders directly at
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some sampling times, · · ·, and the followers ((`s+1 − `s )-level followers) who can receive information at the `s+1 − 1 sampling time, through `s+1 − 1 − `s followers who can receive the leaders’ information at some sampling times directly or indirectly. Each row sum of ΨF L (s) is larger than 0 if the union of
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G(`s h), G(`s h + h), . . . , G(`s+1 h − h) ⊂ G¯ has a spanning tree. The condition 10
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h <
1 maxi=1,···,n,j=1,···,M dii (jh)
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Figure 2: Directed graphs for a group of agents with 2 leaders and 3 followers.
guarantees that each term of ΨF F (s) is a non-
negative matrix with positive diagonal entries, i.e., (In−m − hLF F (ih)), i = `s , `s + 1, · · · , `s+1 − 1, is nonnegative with positive diagonal entries. From the fact that ΨF F (s)1n−m + ΨF L (s)1m = 1n−m , we know that each row sum of ΨF F (s) must be less than 1. From Proposition 2 and Lemma 6 in [2], we can
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obtain lim`→+∞ ΨF F (`h) = lim`→+∞ (In−m − hLF F (`h))(In−m − hLF F (`h − ` −1 h)) · · · (In−m − hLF F (`s` h))Πss=0 ΨF F (s)) = 0(n−m)×(n−m) .
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We give an example to illustrate the entries of ΨF L (s)1m . Example 1. A directed graph is formed by 5 agents, shown by Figure 2. The vertex 4 and vertex 5 represent the leaders, and the vertex 1, vertex 2, and vertex
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3 represent the followers. We assume that the communication network switches
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with a sequence I − II − III − I − · · ·, and switches every 1s to the next topology. The union of the communication graphs has a spanning tree in each time interval
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of 3s. The Laplacian matrices of the three graphs can be easily written and par-
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and LIII
=
0
0 0 0 0 0 0 −1 , LII = 0 ··· ··· ··· ··· ··· .. 0 0 0.0 0 .. 0 0 0.0 0 .. −1. 0 0 .. 0.0 0 . 0 .. 0 0 . ··· ··· ··· .. 0 . 1 −1 .. 0.0 0 0
0 0 ··· 0 0
1
0
0
0
0
0
···
···
0
0
0 1 0 ···
. 0 .. 0 . 0 ..−1 . 0 .. 0 ··· ··· . 0 .. 0 . 0 .. 0
0 0 0 , ··· 0 0
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titioned: LI =
. 0 .. 0 . 0 .. 0 . 1 .. 0
0 0
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0 0 From the expression of ΨF L (s), we have
= (I3 − hLF F (`s + 2))(I3 − hLF F (`s + 1))(−hLF L (`s ))1m
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ΨF L (s)1m
+(I3 − hLF F (`s + 2))(−hLF L (`s + 1))1m
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+(−hLF L (`s + 2))1m .
Without loss of generality, when s = 0, =
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ΨF L (0)1m
(I3 − hLF F (3))(I3 − hLF F (2))(−hLF L (1))1m
+(I3 − hLF F (3))(−hLF L (2))1m +(−hLF L (3))1m .
By calculation, (−hLF L (3))1m =
0 0 0
iT
. It means that none of the
followers receive the leaders’ information directly at the 3th sampling time. h iT (I3 − hLF F (3))(−hLF L (2))1m = h 0 1 0 means that the follower 2 can
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h
receive the leaders’ information at the 2th sampling time. (I3 − hLF F (3))(I3 − h iT hLF F (2))(−hLF L (1))1m = h h 0 1 means that the follower 3 can receive the leaders’ information directly at the 1th sampling and the follower 1 can
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receive the information from follower 3 at the 3th sampling time. The union of 12
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the graphs at the 1th, 2th, and 3th sampling times has a spanning tree, then all h iT the entries of ΨF L (0)1m are positive. In fact, ΨF L (0)1m = h2 h h . ΨF F (0)1n−m are positive and smaller than 1.
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From the fact that ΨF F (0)1n−m + ΨF L (0)1m = 1n−m , then the entries of
The following theorem gives the main result of containment control under
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switching topology. Theorem 1. Suppose that h <
1 maxi=1,2,···,n,j=1,2,···,M dii (jh) ,
where dii (j) is the
¯ Then conin-degree of ith agent in jth possible communication graph in G. 255
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tainment control is achieved if the union of directed graphs G(`s h), G(`s h + h), · · · , G(`s+1 h − h), s = 0, 1, 2, · · ·, `0 = 1, has a spanning tree. Proof. From (4), we have lim χ(`h + h)
`→+∞
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0(n−m)×1 ) = lim ((In − hL(`h))(In −hL(`h − h)) · · · (In −hL(h))(χ(0)− `→+∞ ηL `−1 Y ` X LF F (jh − h) LF L (jh − h) 0(n−m)×1 ) (In − hL(jh)) −h 0m×(n−m) 0m×m ηL i=1 j=i+1 LF F (`h) LF L (`h) 0(n−m)×1 0(n−m)×1 ) + . −h 0m×(n−m) 0m×m ηL ηL
From the results in [9, 33], there exists a vector µ ∈ Rn×1 such that
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lim ((In − hL(`h))(In − hL(`h − h)) · · · (In − hL(h)) = 1n µT .
`→+∞
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Therefore,
lim χ(`h + h)
`→+∞
= 1n µT (χL (0) − ηL ) +
0(n−m)×1 ηL
`−1 Y ` X LF F (jh − h) LF L (jh − h) 0(n−m)×1 −h lim { (In −hL(jh)) `→+∞ 0 0 η m×m L i=1j=i+1 m×(n−m) 13
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LF L (`h)
0(n−m)×1
} 0m×m ηL 0 (n−m)×1 = 1n µT (χL (0) − ηL ) + ηL `−1 ` P Q (I − hL (jh))L (jh − h)η + L (`h)η n−m FF FL L FL L −h lim i=1 j=i+1 . `→+∞ 0m×1 0m×(n−m)
`−1 P
Let Ξ , −h lim (
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+
LF F (`h)
` Q
(In−m −hLF F (jh))LF L (jh−h)+LF L (`h)). We have 1n−m µT (χL (0) − ηL ) + ΞηL . From the definition of lim`→+∞ χ(`h + h) = 1m µT (χL (0) − ηL ) + ηL convex hull, we know that if Ξ satisfies:
260
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`→+∞ i=1 j=i+1
(1) Ξ1m = 1n−m ;
M
(2) all the entries of Ξ are nonnegative, then
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1n−m µT (χL (0) − ηL ) + ΞηL = Ξ(1m µT (χL (0) − ηL ) + ηL ) = Ξχ∗L . That is, all the followers will move into the convex hull spanned by the final states of the leaders.
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Now, let us verify these two conditions.
265
Firstly, we show that Ξ1m = 1n−m . From the property of Laplacian matrix,
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we have LF F (`h)1n−m + LF L (`h)1m = 0(n−m)×1 . It equals to LF L (`h)1m =
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−LF F (`h)1n−m . Note that −h[
=
−[
= −[
`−1 Y ` X
(In−m − hLF F (jh))LF L (jh − h) + LF L (`h)]1m
i=1 j=i+1
`−1 Y ` X
(In−m − hLF F (jh))h(−LF F (jh − h)) + h(−LF F (`h))]1n−m
i=1 j=i+1
`−1 Y ` X
(In−m − hLF F (jh))(In−m − hLF F (jh − h))
i=1 j=i+1
14
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− =
`−1 Y ` X
i=1 j=i+1
−(
` Y
(In−m − hLF F (jh)) + (In−m − hLF F (`h)) − In−m ]1n−m
(In−m − hLF F (ih)) − In−m )1n−m .
From Lemma 2, we have lim`→+∞ 270
Therefore, Ξ1m = lim −( `→+∞
` Y
Q`
i=1 (In−m
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i=1
− hLF F (ih)) = 0(n−m)×(n−m) .
(In−m − hLF F (ih)) − In−m )1n−m = 1n−m .
i=1
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Secondly, we show that all the entries of Ξ are nonnegative. From the choice of h, we know that In−m − hLF F (jh) > 0. From the fact that all the entries of LF L (jh) are non-positive, we can easily obtain that Ξ is nonnegative. Therefore, the containment control objective is achieved.
275
4.2. Containment control under fixed topology and time-delays
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Suppose that the communication time-delays exist in the information transmission. The sequence of communication delays τij (`h), i 6= j, ` = 0, 1, 2, · · ·,
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have an upper bound, i.e., τij (`h) < τmax (i 6= j). Here, τmax denotes the maximal allowable time-delay. We propose the following containment control protocol: P aij (χj (`h − τij (`h)) − χi (`h)), i ∈ F j∈L∪F if t ∈ [`h, `h + h) ςi (t) = P aij ((χj (`h − τij (`h)) − ηj ) − (χi (`h) − ηi )), i ∈ L
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280
j∈L
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(6)
where τij (`h) may be larger than one sampling period.
T Denote ξ(`h) = [χT (`h), χT (`h − h), · · · , χT (`h − d τmax h eh)] , where χ(`h −
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jh) = [χ1 (`h − jh), χ2 (`h − jh), · · · , χn (`h − jh)]T , j = 0, 1, · · · , d τmax h e, and
d τmax h e is the minimum integer no smaller than
285
τmax h .
We assume that χ(ih) =
χ(0) for i < 0. Then the system (1) under protocol (6) can be summarized as follows: ξ(`h + h) = Φξ(`h) + Υ
15
(7)
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hA1
···
hAd τmax e−1 h
0
···
0
In .. .
··· .. .
0 .. .
0
···
In
The Laplacian matrix L = D −
Pd τmax e h i=0
0 hAd τmax e h 0 hLLL ηL 0 0 . , Υ = .. .. . . 0 0
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where In − hD + hA0 In 0 Φ = .. . 0
Ai .
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From the definitions of leader and follower, we have the following partitions: LF F LF L DF AF F i AF Li , D = , Ai = . L= 0 LLL DL 0 ALLi
Theorem 2. Suppose that the sampling period satisfies h <
1 maxi=1,···,n dii .
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ing protocol (6) for (1), containment control is achieved if the directed graph G 290
has a spanning tree.
M
T Proof. Denote ξF (`h) = [χTF (`h), χTF (`h − h), · · · , χTF (`h − d τmax h eh)] and
T ξL (`h) = [χTL (`h), χTL (`h − h), · · · , χTL (`h − d τmax h eh)] . Then the dynamics of
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the leaders can be written as:
ξL (`h + h) = ΦLL ξL (`h) + ΥL ,
=
Im − hDL + hALL0
hALL1
···
hALLd τmax e−1 h
hALLd τmax e h
Im
0
···
0
0
0 .. .
Im .. .
··· .. .
0 .. .
0 .. .
0
···
Im
0
AC
CE
ΦLL
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where
ΥL
(8)
=
0
hLLL ηL 0 0 .. . 0
, 16
,
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295
and the dynamics of the followers can be written as: ξF (`h + h) = ΦF F ξF (`h) + ΦF L ξL (`h),
ΦF L
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ΦF F
τmax In−m − hDF +hAFF 0 hAFF 1 · · · hAFF dτmax hA FF d h e h e−1 I 0 · · · 0 0 n−m 0 In−m · · · 0 0 = , .. .. .. .. .. . . . . . 0 0 ··· In−m 0 τmax hAF L0 hAF L1 · · · hAF Ld τmax hA F Ld e−1 e h h 0 0 ··· 0 0 0 0 ··· 0 0 = . .. .. .. .. .. . . . . . 0 0 ··· 0 0
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where
(9)
M
Define a new variable βi (·) = χi (·) − ηi , i = 1, 2, · · · , m. By using the properties of SIA matrices, from the leaders’ dynamics (8), it can be easily
∗ ξL
,
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obtained:
lim ξL (`h + h)
`→∞
T = 1m×(d τmax e+1) γ (ξL (0) − 1d τmax e+1 ⊗ ηL ) + 1d τmax e+1 ⊗ ηL h h h
for some vector γ ∈ Rm×(d
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300
τmax h
e+1)
. Consider the first m rows and note that
χL (ih) = χL (0) for i < 0. The final states of the leaders are given by χ∗L =
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1m γ T [1d τmax e+1 ⊗ (χL (0) − ηL )] + ηL . h Noticing that h <
1 maxi=1,···,n dii ,
then ρ(ΦF F ) ≤ 1. Suppose that 1 is an
eigenvalue of ΦF F , then there exists an eigenvector α = [α0T , α1T , · · · , αdTτmax e ]T 6= h
0 such that ΦF F α = 1 · α, i.e.,
AC
305
(In−m −hDF + hAF F 0 )α0 + hAF F 1 α1 + · · · hAF F d τmax e = α0 , e αd τmax h h
17
α0
= .. .
α1
αd τmax e−1 h
=
αd τmax e. h
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We can obtain that (In−m −hDF +hAF F 0 +hAF F 1 +· · · hAF F d τmax e )α0 = α0 , h that is, (In−m − hLF F )α0 = α0 ⇔ hLF F α0 = 0. Since h > 0 and LF F is Therefore, all the eigenvalues of ΦF F are less than 1.
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invertible, we have α0 = 0. It follows that α = 0. This is a contradiction.
After the leaders achieving the desired formation, we can obtain
310
lim ξF (`h + h)
`→∞
=
` ∗ lim [Φ`+1 F F ξF (0) + (ΦF F ΦF L + · · · + ΦF F ΦF L + ΦF L )ξL ]
`→∞
` X
∗ ΦiF F ΦF L ξL ]
L−1 F F AF L0
L−1 F F AF L1
lim [Φ`+1 F F ξF (0) +
`→∞
i=0
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=
=
−1 ∗ ΦF L ξ L (I(n−m)×(d τmax e+1) − ΦF F ) h
=
∗ ΓξL
where Γ = 1d τmax e+1 ⊗ h
h
final positions of the followers are given by: =
τ L−1 e F F AF Ld max h
i
. The
−1 −1 ∗ τ [L−1 e ]ξL F F AF L0 LF F AF L1 · · · LF F AF Ld max h
M
χ∗F
···
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∗ = L−1 F F AF L χL .
−1 It is easy to verify that L−1 F F AF L 1m = 1n−m and all the entries of LF F AF L
are non-negative. From the definition of convex hull, we know that all the followers converge to the convex hull spanned by the leaders’ final states.
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315
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5. Simulations
In the following directed graphs, all the weights of the directed graphs are
AC
assumed to be 1. Example 2. In this example, we assume that the communication topologies are
320
modeled by Figure 3, Figure 4, Figure 5 and Figure 6, where Li, i = 1, 2, · · · , 5, represent the leaders and F i, i = 1, 2, · · · , 7, represent the followers, respectively. The communication network switches stochastically, and switches every 1s to the next graph. The union of the communication graphs Ga ∪ Gb ∪ Gc ∪ Gd has a 18
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spanning tree in each time interval of 4s. From Theorem 1, the sampling period
330
should satisfy h < 0.5. Suppose that all the agents move in the x − y plane. The initial conditions h iT are chosen as: x(0) = 25 18 26 0 0 20 10 25 8 7 23 33 , h iT y(0) = 2 28 30 0 27 9 22 20 30 22 12 6 and the desired h iT formation of the leaders in the x−y plane are given by η1 = 16 13 17 8 8 , h iT η2 = 12 25 20 12 21 . The sampling period is chosen as h = 0.35.
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325
For system (4), Figure 7 and Figure 8 show the trajectories of all the agent
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under two different stochastic switching signals and the positions of all the agent at 40s, respectively. It is obvious that after some time, the leaders form a formation and then keep stationary. The trajectories of all the agents are affected 335
by the switching signal.
Figure 9 and Figure 10 show the trajectories of all the agents under the same switching signal and the positions of all the agents at 60s and 80s, respec-
M
tively. Influenced by the switching topology, the positions of all the followers are dynamic.
From Figure 7, Figure 8, Figure 9 and Figure 10, we can also see that the
340
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final states of the leaders may be different under different switching signals. However, the geometric configuration of the desired formation is not affected by
PT
the switching signals.
Example 3. Suppose the communication topology is shown in Figure 11, where Li, i = 1, 2, 3, denote the leaders and F i, i = 1, 2, 3, 4, denote the followers.
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345
It is noted that the directed G has a spanning tree if the time-delays are not
AC
considered. Assume that all the agents move in the x − y plane. The desired h iT h iT formation are given as: η1 = −0.5 −3 −1 , η2 = 1 3 7 . The h iT initial conditions are chosen as x(0) = , y(0) = 1 6 6 1 3 6 2 h iT . 10 2 7 −1 0 3 6
350
We assume that the information channels F 2 → F 1, F 3 → F 2, L2 → F 3,
F 3 → F 4, L1 → L2, and L2 → L3 have no time-delays, L1 → F 1, L2 → F 2, 19
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L5
L5 L3
L1
L2
L4
F1
L3
L4
F1
F7
F7 F6
F6 F4
L1
F2
F4 F3
F3
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L2
F2
F5
F5
Figure 4: Gb
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Figure 3: Ga
L5 L2
L3
L1
L4
F1
L3
F2
F4
F3
L1
L4
F1
F7 F6
F4
L2
L5
F7 F6
F2
F3
F5
Figure 6: Gd
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Figure 5: Gc
M
F5
F 2 → F 3, and L3 → L1 have time-delay τ1 = 1, and F 1 → F 2, L3 → F 3, and 355
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L3 → F 4 have time-delay τ2 = 2. The simulation results using (6) for (1) are shown by Figure 12 and Figure 13, where h = 0.2 in Figure 12 and h = 0.05 in Figure 13. We can see that all followers ultimately converge to the convex hull
CE
formed by the final positions of the leaders. We assume that the information channels L1 → F 1, L2 → F 2, F 2 → F 3,
L3 → F 3, have no time-delays, F 2 → F 1, F 3 → F 2, L2 → F 3, and F 3 → F 4
have time-delay τ1 = 1, and F 1 → F 2, L3 → F 3, L2 → L3, and L3 → L1 have
AC 360
time-delay τ2 = 2. The simulation results using (6) for (1) are shown by Figure 14 and Figure 15, where h = 0.2 in Figure 14 and h = 0.05 in Figure 15. From the simulations, we obtain that the final positions of the followers are
not affected by the sampling period and the time-delays, but determined by the
20
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35
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
25
y
20
15
25
20
15
10
10
5
5
0
0
5
10
15
20
25
30
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
y
30
0
35
0
5
x
15
20
25
30
Figure 7: Position trajectories of all
Figure 8: Position trajectories of all
the agents under the first switching
the agents under the second switching
signal at 40s
signal at 40s
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35
y
20
15
10
5
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
25
20
y
25
0
35
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
15
10
5
0
5
10
15
20
25
30
0
35
M
x
0
5
10
15
20
25
30
x
Figure 10: Position trajectories of all
the agents at 60s
the agents at 80s
ED
Figure 9: Position trajectories of all
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PT
F1
AC
10
x
35
365
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35
F4
F3
F2
L2
L3
L1
Figure 11: A directed graph with 7 agents
topology structure and the final positions of the leaders.
21
35
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Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
8
6
6
4
4
y
2
2
0
0
-2
-2
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
1
1.5
2
x
Figure 12:
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y
8
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
2.5
3
3.5
4
4.5
5
5.5
6
x
Trajectories of all the
Figure 13:
agents with h = 0.2
Trajectories of all the
agents with h = 0.05
10
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
8
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8
6
6
y
y
4
2
0
-2
-4
0
1
2
3
4
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
5
x
Trajectories of all the
agents with h = 0.2
2
0
-2
6
M
Figure 14:
4
0
Figure 15:
1
2
3
4
5
x
Trajectories of all the
agents with h = 0.05
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6. Conclusion
In this paper,we have studied sampled-data based containment control prob-
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lems of continuous-time multi-agent systems with multiple interactive leaders. The communication networks are assumed to be switching or fixed with timedelays. We have proposed some protocols to overcome the switching topology
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370
and time-delays. For the leaders, we have shown that, the final states may be affected by the switching topology, but the geometric configuration of the de-
AC
sired formation is not affected by the switching topology. The followers move into the convex hull spanned by the final positions of the leaders first, and then
375
change their positions in the convex hull under the switching topology. For the time-delay case, the final positions of the leaders and the followers are not affected by the time-delays. Our future work will focus on containment control for multi-agent systems with more complex dynamics. 22
6
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Acknowledgements This work was supported by the National Natural Science Foundation of
380
Central Universities (Grant No. FRF-TP-17-022A2).
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Huiyang Liu received her B.S. degree in Information and Computing Sci-
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ence, her M.S. degree in Applied Mathematics from Zhengzhou University in 2004 and 2007, respectively. She received her Ph.D. degree in Control The-
500
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ory from Peking University in 2012. She is currently a lecture in School of Mathematics and Physics, University of Science and Technology Beijing. Her current research interests are in the fields of coordination of multi-agent sys-
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tems, consensus problems, containment control and swarm dynamics. Email:
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[email protected]
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Guangming Xie received his B.S. degrees in Applied Mathematics and
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Computer Science and Technology, his M. E. degree in Control Theory and Control Engineering, and his Ph.D. degree in Control Theory and Control Engineering from Tsinghua University, Beijing, China in 1996, 1998, and 2001,
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respectively. Now he is a Professor of Dynamics and Control with the College of Engineering, Peking University, Beijing. His research interests include hybrid
and switched systems, networked control systems, and multi-agent systems,
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multi-robot systems, and bio-mimetic robotics. Email: [email protected]
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Accepted Manuscript
Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays Huiyang Liu, Guangming Xie PII: DOI: Reference:
S0925-2312(18)30070-5 10.1016/j.neucom.2018.01.045 NEUCOM 19247
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
21 August 2017 3 December 2017 19 January 2018
Please cite this article as: Huiyang Liu, Guangming Xie, Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.01.045
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Huiyang Liua,∗, Guangming Xieb a School
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Sampled-data based containment control of continuous-time multi-agent systems with switching topology and time-delays✩
of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China. b Intelligent Control Laboratory, College of Engineering, Peking University, Beijing, 100871, China
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Abstract
This paper addresses containment control of multi-agent systems with multiple interactive leaders. The containment control objective is two-fold: the leaders converge to a desired formation, and the followers move into the convex hull spanned by the leaders’ final positions. Some sampled-data based protocols are
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proposed, which are effective when the communication environment suffers from intermit information, switching topology and time-delays. For the leaders, the
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convergence analysis can be easily obtained by using the properties of the SIA matrices. The geometric configuration of the desired formation is not affected by the switching topology or time-delays. For the followers, the convergence
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analysis is much complicated since the properties of the SIA matrices cannot be used. Based on the relationship between the topology and matrices, it is proved that the followers will move into the convex hull of the desired forma-
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tion, and then change their positions in the convex hull as the time evolves under the switching topology; however, the final states of the followers are not
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affected by the time-delays. Finally, the theoretical results are illustrated by some simulations. Keywords: Containment control; multi-agent systems; directed graphs; switching topology; time-delays. ∗ Corresponding
author Email address: [email protected] (Huiyang Liu )
Preprint submitted to Neurocomputing
January 27, 2018
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1. Introduction
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In the past decades, inspired by the work [1, 2] and [3], many researchers have devoted themselves to the study on cooperative control of multi-agent
systems. The reasons lie in its wide applications in many fields such as coordi5
nation of multiple robots, collaboration of human society, and formation control of multiple aircrafts. As a kind of cooperative control, the containment control
strategies, motivated by numerous natural and society phenomena, have been
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developed by several researchers by using neighbor information. For examples,
the natural phenomena include shepherd dogs, the guard geese and so on. In 10
human society, the containment control phenomena can be found in the avoiding danger, encirclement and suppression, and even the opinions of a group of individuals.
In the containment control problems, there are always two types of agents:
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leaders and followers. They play different roles in achieving the coordination control objective. Therefore, it is important to design different coordination
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control protocols for them. The containment control problem in this paper concerns with the formation control for the leaders and consensus for the followers. Many researchers have studied formation control problems in the past decades
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[4, 5, 6]. The study of consensus has a history dated back to several decades [7, 8], and has recently been studied extensively (e.g., switching topology [9, 10],
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communication noises [11], sampled-data or discrete-time systems [12, 13, 14], tracking [15, 16], fractional-order [17] and so on). Next, we make a brief summary on the containment control problems of
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multi-agent systems. In [18], by exploiting the theory of partial difference equa-
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tions, a hybrid control scheme based on stop-go rules was proposed for a collection of mobile robots. In [19], containment control in leader-follower networks with switching communication topologies was investigated. Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks was studied in [20]. In [21], containment control of multi2
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agent systems with stochastic topologies was studied. In the past several years, containment control problems of multi-agent systems have been developing very fast. Several research topics have been addressed, such as intermittent infor-
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mation [22, 23, 24], finite-time containment control [25, 26, 27], hybrid model predictive control scheme [28], noises environment [29]. The results are vari35
ous regarding the varieties of agents’ dynamics, tasks demanded, the control strategies and the topology structures.
In most of the above results, it is assumed that there is no communication among the leaders. However, it may be more practical to study containment
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control with multiple interactive leaders. The leaders also need to communicate with each other to reach their desired objectives. In [30], containment con-
trol for multiple unicycle agents under undirected connected graphs was introduced. Formation-containment control problems for general linear multi-agent systems without time-delays and with time-varying delays under fixed directed topologies were studied in [31, 32]. However, in a communication network, the communication channel may break off because of the cost saving or the commu-
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nication constraint, or get well again by remediation. This results in a switching
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topology. For consensus problems, the switching topology case has been widely studied, e.g., [9, 10, 33]. Different from the consensus problems, the properties of SIA matrices are invalid for the containment control with switching topology. In addition, time-delay is a ubiquitous phenomenon and widely studied in the
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field of systems and control [34, 35, 36]. The existence of the time-delays always
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leads to the instability of the whole systems. In multi-agent systems, many researchers have studied the coordination control problems with time-delays, e.g., [3, 10, 33, 37]. The methods pertain to frequency domain analysis, properties of SIA matrix, Lyapunov function method and so on. We will employ a new
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method to solve the containment control problem with time-delays. Therefore, it is significant to study containment control with interactive leaders, switching topology and time-delays. Motivated by the progress in this field, we will investigate sampled-data
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based containment control for continuous-time multi-agent systems with switch3
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ing topology and time-delays. The leaders are interactive, and they can communicate with each other. The leaders are expected to form a desired formation, and the followers are expected to move into the convex hull spanned by the final
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positions of the leaders. According to the different objectives that the leaders and the followers are to achieve, we will design a formation control protocol
that is independent of the followers’ states for the leaders, and a consensus pro-
tocol that depends on the neighbor information for the followers. The desired formation for the leaders can be achieved by using the properties of the SIA
matrices. The geometric configuration of the desired formation is not affected by the switching topology and time-delays. Since the properties of SIA ma-
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trices are invalid for the convergence analysis of the followers, we analyze the behaviors of the followers by using the relationship between the topology and matrices. We will show that, for the switching topology case, the followers first converge to the convex hull spanned by the final positions of the leaders, and 75
then change their positions as the time evolves; for the time-delays case, the
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final states of the followers are not affected by the time-delays, but determined by the topology structure and the final states of the leaders. We will also give
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some examples to illustrate the theoretic results. The outline of this paper is given as follows: we next establish some of the 80
basic notations. We then, in Section 2, give some preliminaries, followed by the
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problem formulation in Section 3. In Section 4, the main results are presented. Section 5 gives some simulation results. Then, the conclusions are given in
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Section 6.
Notations: Let 0m×n (or 0) denote an m × n (or compatible dimensional)
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all-zero matrix. In represents the n−dimensional identity matrix. 1n represents
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the n−dimensional all 1 vector. Matrix A is said to be nonnegative if all its
entries are nonnegative. A nonnegative matrix A ∈ Cn×n with the property that all its row sums are +1 is said to be a (row) stochastic matrix. A stochastic
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matrix A ∈ Cn×n is called indecomposable and aperiodic (SIA) (or ergodic) Qk if there exists a column vector ν such that limk→∞ Ak = 1n ν T . i=1 Ai =
Ak Ak−1 · · · A1 denotes the left product of matrices. 4
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2. Preliminaries The notation G = (V(G), E(G), A) represents a weighted directed graph, where V(G) = {v1 , v2 , · · · , vn } is the set of vertices, E(G) ⊆ {(vi , vj ) : vi , vj ∈ V(G)} is the set of edges, and A = [aij ] is a weighted adjacent matrix with
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nonnegative adjacent entry aij , i, j = 1, 2, · · · , n, [38]. In an edge eij = (vj , vi ) ∈ E(G), vj is called the parent vertex of vi and vi is the child vertex of vj . If
eij ∈ E(G), i 6= j, is an edge, then the corresponding adjacent entry aij > 0. Otherwise, aii = 0 for all vi ∈ V(G). The neighbor set of vertex vi is denoted 100
by Ni = {vj ∈ V(G) : (vj , vi ) ∈ E(G), j 6= i}. A directed path in G is a sequence
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vi1 , vi2 , . . . , vik of vertices such that (vis , vis+1 ) ∈ E(G) for s = 1, 2, · · · , k − 1.
A directed graph G contains a spanning tree if there exists at least one vertex that has a directed path to all the other vertices in G. 105
P The in-degree and out-degree of vi are defined as: degin (vi ) = vj ∈Ni aij P and degout (vi ) = vj ∈Ni aji , respectively. The degree matrix of G is a diagonal
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matrix defined as: D = [dij ], where dij = degin (vi ) for i = j, otherwise, dij = 0. The Laplacian of G is defined as: L = D − A.
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Lemma 1. ([9]) For a directed graph G, 0 is an eigenvalue of L, and 1n is the corresponding right eigenvector; 0 is a simple eigenvalue of L, and all the other eigenvalues have positive real parts if and only if G has a spanning tree.
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Definition 1. ([39]) A set K ⊂ Rm is said to be convex if (1 − γ)x + γy ∈
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K whenever x, y ∈ K and 0 ≤ γ ≤ 1. The convex hull of a finite set of Pn points x1 , x2 , . . . , xn ∈ Rm is denoted by co{x1 , x2 , . . . , xn } = { i=1 αi xi |αi ∈ Pn R≥0 , i=1 αi = 1}.
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3. Problem formulation In this paper, a group of n coordinating agents are considered. We use vertex
vi to represent the ith agent, and edge eij ∈ E(G) to represent an available
information channel from agent j to agent i. The dynamics of each agent is
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modeled by a continuous-time system: χ˙ i (t) = ςi (t), i = 1, 2, · · · , n,
where χi (t), ςi (t) ∈ R are the position and the control input of the ith agent,
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(1)
respectively.
Definition 2. In a multi-agent team, an agent i is called a follower, or i ∈ F, if (vj , vi ) ∈ E(G) for at least one vj ∈ V(G); an agent i is called a leader, or i ∈ L, if (vj , vi ) ∈ / E(G) for each vj ∈ V(F).
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We assume that the leader set is L = {1, 2, · · · , m} and the follower set is
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F = {m + 1, m + 2, · · · , n}. From the definitions of leader and follower, L can be partitioned as:
L=
LF F
LF L
0m×(n−m)
LLL
,
(2)
where LF F ∈ R(n−m)×(n−m) , LF L ∈ R(n−m)×m and LLL ∈ Rm×m . It is obvious 130
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that if the directed topology G has a spanning tree, then the topology for the leaders, denoted by GL , also has a spanning tree. From Lemma 1, LF F is
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invertible.
Definition 3. A formation of the leader set is a vector ηL = [η1 , η2 , · · · , ηm ]T ∈
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Rm . The states of the m leaders, denoted by χL = [χ1 , χ2 , · · · , χm ]T , converge to the formation ηL if χi (t) − ηi → 0, as t → +∞, for i ∈ L. 4. Main results
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4.1. Containment control under switching topology In a communication network of mobile agents, the communication channel
may break off for cost saving or being destroyed and get well again. It results in a variation of the communication topology. Suppose that the set of all possible
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communication graphs is finite, denoted by G¯ = {G1 , G2 , · · · , GM }, where M is a finite constant. Let h > 0 be the sampling period. Denote the interaction topology at the `th sampling time as G(`h), ` = 0, 1, 2, · · ·. The union of a 6
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¯ i = `1 , `1 + 1, · · · , `2 − 1, where `1 < `2 , is a group of directed graphs G(ih) ⊂ G, directed graph with vertices given by vi , i = 1, 2, · · · , n, and edges given by the 145
union of the edges of G(ih).
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We propose the following time-varying protocol for the system (1): P j∈L∪F aij (`h)(χj (`h) − χi (`h)), i ∈ F ςi (t) = if t ∈ [`h, `h+h), P j∈L aij (`h)((χj (`h) − ηj ) − (χi (`h) − ηi )), i ∈ L (3) where aij (`h) > 0 if agent i can receive information from agent j at the `th
sampling time, and ηi , i = 1, 2, · · · , m, are constants, which represent the for150
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mation pattern of the leaders. Denote χ(`h) = [χ1 (`h), χ2 (`h), · · · , χn (`h)]T . The agent dynamics can be summarized as:
χ(`h + h) = (In − hL(`h))χ(`h) +
0(n−m)×1
hLLL (`h)ηL
,
(4)
where L(`h) denotes the Laplacian matrix of directed graph G at the `th sam-
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pling time, ` = 0, 1, 2, · · ·, and L(0) = 0n×n .
According to the partition of the Laplacian matrix, the leaders’ dynamics
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can be written as:
χL (`h + h) = (Im − hLLL (`h))χL (`h) + hLLL (`h)ηL . 155
(5)
Note that the set of all the possible communication graphs of the leaders is also
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finite, denoted by G¯L = {G1 , G2 , · · · , GP }, where P ≤ M . The following proposition can be derived immediately from the existing re-
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sults by defining a new variable yi (`h) = χi (`h) − ηi , i = 1, 2, · · · , m, and using the SIA matrix theory. The verification can refer to [9, 33]. We will not verify it.
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Proposition 1. Let GL (`h), ` = 0, 1, 2, · · ·, be a sequence of switching commu-
nication graphs of the leaders. Assume that h <
1 maxi=1,···,m,j=1,2,···,P dii (j) ,
where
dii (j) is the in-degree of the ith leader in the jth possible communication graph in G¯L . Then all the leaders converge to a desired formation if the union of di-
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rected graphs GL (`h) ⊂ G¯L , ` = `s , `s + 1, · · · , `s+1 − 1, s = 0, 1, 2, · · ·, `0 = 1, 7
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L2
L1
F1
L2
F1 (b)
(a)
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L1
Figure 1: Two directed graphs, where the weights are assumed to be 1
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has a spanning tree. Furthermore, the final formation of the leaders is given as χ∗L = 1m z T (χL (0) − ηL ) + ηL for some vector z ∈ Rm .
Remark 1. The final states of the leaders may be affected by the switching topology. However, the geometric configuration of the desired formation are 170
not affected by the switching topology. For example, consider a multi-agent
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team with 3 agents, where L1 and L2 are the leaders, and F 1 is the follower. Obviously, the union of the two directed graphs has a spanning tree. Assume that the states of the agents are in R, thus the agents move in the χ−axis.
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The initial states of the leaders are chosen as χ1 (0) = 4 and χ2 (0) = 5. The desired formation is given by h = [3, 1]T . We consider two special cases. Firstly, the switching signal is chosen such that the communication topology is always
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modeled by the directed graph (a) at all the sampling times. In this case, z1 = [1, 0]T . According to Proposition 1, the final states of L1 and L2 are given by
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χ∗1 = 4 and χ∗2 = 2.
Secondly, the switching signal is chosen such that the communication topol-
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ogy is always modeled by the directed graph (b) at all the sampling times. In
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this case, z2 = [0, 1]T . According to Proposition 1, the final states of L1 and L2
are given by χ∗1 = 7 and χ∗2 = 5.
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It is obvious that the final states of the leaders may be different if the switch-
ing signals are different. However, the distance of the leaders is always 2. Now, we will study the dynamics of the followers. Noticing that for each 8
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L(`h), ` = 0, 1, 2, · · ·, it can be written as L(`h) =
LFF (`h)
LF L (`h)
0m×(n−m)
LLL (`h)
We have the following lemma.
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1 maxi=1,···,n,j=1,···,M dii (jh) ,
.
where dii (jh) is the in-
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Lemma 2. Suppose that h <
¯ If the degree of the ith agent in the jth possible communication graph in G. union of directed graphs G(`s h), G(`s h+h), · · · , G(`s+1 h−h), s = 0, 1, 2, · · ·, `0 =
1, has a spanning tree, then lim`→+∞ (In−m − hLF F (`h))(In−m − hLF F (`h − h)) · · · (In−m − hLF F (h)) = 0(n−m)×(n−m) .
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Proof. Denote Ψ(s) , (In − hL(` s+1 h − h)) · · · (In −hL(`s h + h))(In − ΨF F (s) ΨF L (s) , where ΨF F (s) = hL(`s h)). It is noted that Ψ(s) = 0m×(n−m) ΨLL (s) Q`s+1 −1 Q`s+1 −1 (In−m − hLF F (ih)), ΨLL (s) = i=` (Im − hLLL (ih)), and i=`s s ΨF L (s) `s+1 −1
Y
(In−m − hLF F (ih))(−hLF L (`s h))
Y
(In−m − hLF F (ih))(−hLF L (`s h + h))(Im − hLLL (`s h)) + · · · +
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=
+
`s+1 −1
i=`s +2
+
`s+1 −3
(In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))
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+
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i=`s +1
(−hLF L (`s+1 h − h))
`s+1 −2
Y
i=`s
Y
i=`s
(Im − hLLL (ih))
(Im − hLLL (ih)).
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From the partition of Laplacian matrix, we have LF F (`h)1n−m + LF L (`h)1m = 0n×1 , L(`h)1n = LLL (`h)1m
and then
(In − hL(`h))1n =
(In−m − hLF F (`h))1n−m + (−hLF L (`h))1m (Im − hLLL (`h))1m
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= 1n .
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Furthermore, we have Ψ(s)1n = 1n , i.e., ΨF F (s)1n−m + ΨF L (s)1m = 1n−m and ΨLL (s)1m = 1m . Thus, we have =
`s+1 −1
Y
(In−m − hLF F (ih))(−hLF L (`s h))1m
i=`s +1
+
`s+1 −1
Y
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ΨF L (s)1m
(In−m − hLF F (ih))(−hLF L (`s h + h))1m + · · · +
i=`s +2
+(In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))1m +(−hLF L (`s+1 h − h))1m .
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Let us analyze the entries of ΨF L (s)1m . We first consider the last term (−hLF L (`s+1 h − h))1m . Since the sampling period h > 0 and −LF L (`s+1 h − h)) = AF L (`s+1 h − h)), the nonzero components of the vector (−hLF L (`s+1 h −
h))1m correspond to the followers who can receive information from the leaders directly at the (`s+1 − 1)th sampling time. The nonzero components of the vector (In−m − hLF F (`s+1 h − h))(−hLF L (`s+1 h − 2h))1m correspond to the followers who can receive the leaders’ information directly (1-level follow-
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ers) at the (`s+1 − 2)th sampling time and the followers (named 2-level fol-
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lowers) who can not receive information from the leaders directly, but, at the (`s+1 − 1)th sampling time, they can receive information from one of the followers who can receive information from the leaders directly at some sampling times in `s , `s + 1, · · · , `s+1 . For any integer `s < `s+1 , the nonzero components of the
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vector (In−m − hLF F (`s+1 h − h)) · · · (In−m − hLF F (`s h + h))(−hLF L (`s h))1m
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correspond to the followers who can receive the leaders’ information directly at `s th sampling time, the followers who can not receive the leaders’ information directly, but can receive information at the `s + 1th sampling time, from one of the followers who can receive information from the leaders directly at
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some sampling times, · · ·, and the followers ((`s+1 − `s )-level followers) who can receive information at the `s+1 − 1 sampling time, through `s+1 − 1 − `s followers who can receive the leaders’ information at some sampling times directly or indirectly. Each row sum of ΨF L (s) is larger than 0 if the union of
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G(`s h), G(`s h + h), . . . , G(`s+1 h − h) ⊂ G¯ has a spanning tree. The condition 10
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h <
1 maxi=1,···,n,j=1,···,M dii (jh)
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Figure 2: Directed graphs for a group of agents with 2 leaders and 3 followers.
guarantees that each term of ΨF F (s) is a non-
negative matrix with positive diagonal entries, i.e., (In−m − hLF F (ih)), i = `s , `s + 1, · · · , `s+1 − 1, is nonnegative with positive diagonal entries. From the fact that ΨF F (s)1n−m + ΨF L (s)1m = 1n−m , we know that each row sum of ΨF F (s) must be less than 1. From Proposition 2 and Lemma 6 in [2], we can
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obtain lim`→+∞ ΨF F (`h) = lim`→+∞ (In−m − hLF F (`h))(In−m − hLF F (`h − ` −1 h)) · · · (In−m − hLF F (`s` h))Πss=0 ΨF F (s)) = 0(n−m)×(n−m) .
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We give an example to illustrate the entries of ΨF L (s)1m . Example 1. A directed graph is formed by 5 agents, shown by Figure 2. The vertex 4 and vertex 5 represent the leaders, and the vertex 1, vertex 2, and vertex
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3 represent the followers. We assume that the communication network switches
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with a sequence I − II − III − I − · · ·, and switches every 1s to the next topology. The union of the communication graphs has a spanning tree in each time interval
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of 3s. The Laplacian matrices of the three graphs can be easily written and par-
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and LIII
=
0
0 0 0 0 0 0 −1 , LII = 0 ··· ··· ··· ··· ··· .. 0 0 0.0 0 .. 0 0 0.0 0 .. −1. 0 0 .. 0.0 0 . 0 .. 0 0 . ··· ··· ··· .. 0 . 1 −1 .. 0.0 0 0
0 0 ··· 0 0
1
0
0
0
0
0
···
···
0
0
0 1 0 ···
. 0 .. 0 . 0 ..−1 . 0 .. 0 ··· ··· . 0 .. 0 . 0 .. 0
0 0 0 , ··· 0 0
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titioned: LI =
. 0 .. 0 . 0 .. 0 . 1 .. 0
0 0
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0 0 From the expression of ΨF L (s), we have
= (I3 − hLF F (`s + 2))(I3 − hLF F (`s + 1))(−hLF L (`s ))1m
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ΨF L (s)1m
+(I3 − hLF F (`s + 2))(−hLF L (`s + 1))1m
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+(−hLF L (`s + 2))1m .
Without loss of generality, when s = 0, =
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ΨF L (0)1m
(I3 − hLF F (3))(I3 − hLF F (2))(−hLF L (1))1m
+(I3 − hLF F (3))(−hLF L (2))1m +(−hLF L (3))1m .
By calculation, (−hLF L (3))1m =
0 0 0
iT
. It means that none of the
followers receive the leaders’ information directly at the 3th sampling time. h iT (I3 − hLF F (3))(−hLF L (2))1m = h 0 1 0 means that the follower 2 can
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h
receive the leaders’ information at the 2th sampling time. (I3 − hLF F (3))(I3 − h iT hLF F (2))(−hLF L (1))1m = h h 0 1 means that the follower 3 can receive the leaders’ information directly at the 1th sampling and the follower 1 can
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receive the information from follower 3 at the 3th sampling time. The union of 12
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the graphs at the 1th, 2th, and 3th sampling times has a spanning tree, then all h iT the entries of ΨF L (0)1m are positive. In fact, ΨF L (0)1m = h2 h h . ΨF F (0)1n−m are positive and smaller than 1.
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From the fact that ΨF F (0)1n−m + ΨF L (0)1m = 1n−m , then the entries of
The following theorem gives the main result of containment control under
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switching topology. Theorem 1. Suppose that h <
1 maxi=1,2,···,n,j=1,2,···,M dii (jh) ,
where dii (j) is the
¯ Then conin-degree of ith agent in jth possible communication graph in G. 255
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tainment control is achieved if the union of directed graphs G(`s h), G(`s h + h), · · · , G(`s+1 h − h), s = 0, 1, 2, · · ·, `0 = 1, has a spanning tree. Proof. From (4), we have lim χ(`h + h)
`→+∞
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0(n−m)×1 ) = lim ((In − hL(`h))(In −hL(`h − h)) · · · (In −hL(h))(χ(0)− `→+∞ ηL `−1 Y ` X LF F (jh − h) LF L (jh − h) 0(n−m)×1 ) (In − hL(jh)) −h 0m×(n−m) 0m×m ηL i=1 j=i+1 LF F (`h) LF L (`h) 0(n−m)×1 0(n−m)×1 ) + . −h 0m×(n−m) 0m×m ηL ηL
From the results in [9, 33], there exists a vector µ ∈ Rn×1 such that
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lim ((In − hL(`h))(In − hL(`h − h)) · · · (In − hL(h)) = 1n µT .
`→+∞
AC
Therefore,
lim χ(`h + h)
`→+∞
= 1n µT (χL (0) − ηL ) +
0(n−m)×1 ηL
`−1 Y ` X LF F (jh − h) LF L (jh − h) 0(n−m)×1 −h lim { (In −hL(jh)) `→+∞ 0 0 η m×m L i=1j=i+1 m×(n−m) 13
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LF L (`h)
0(n−m)×1
} 0m×m ηL 0 (n−m)×1 = 1n µT (χL (0) − ηL ) + ηL `−1 ` P Q (I − hL (jh))L (jh − h)η + L (`h)η n−m FF FL L FL L −h lim i=1 j=i+1 . `→+∞ 0m×1 0m×(n−m)
`−1 P
Let Ξ , −h lim (
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+
LF F (`h)
` Q
(In−m −hLF F (jh))LF L (jh−h)+LF L (`h)). We have 1n−m µT (χL (0) − ηL ) + ΞηL . From the definition of lim`→+∞ χ(`h + h) = 1m µT (χL (0) − ηL ) + ηL convex hull, we know that if Ξ satisfies:
260
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`→+∞ i=1 j=i+1
(1) Ξ1m = 1n−m ;
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(2) all the entries of Ξ are nonnegative, then
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1n−m µT (χL (0) − ηL ) + ΞηL = Ξ(1m µT (χL (0) − ηL ) + ηL ) = Ξχ∗L . That is, all the followers will move into the convex hull spanned by the final states of the leaders.
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Now, let us verify these two conditions.
265
Firstly, we show that Ξ1m = 1n−m . From the property of Laplacian matrix,
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we have LF F (`h)1n−m + LF L (`h)1m = 0(n−m)×1 . It equals to LF L (`h)1m =
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−LF F (`h)1n−m . Note that −h[
=
−[
= −[
`−1 Y ` X
(In−m − hLF F (jh))LF L (jh − h) + LF L (`h)]1m
i=1 j=i+1
`−1 Y ` X
(In−m − hLF F (jh))h(−LF F (jh − h)) + h(−LF F (`h))]1n−m
i=1 j=i+1
`−1 Y ` X
(In−m − hLF F (jh))(In−m − hLF F (jh − h))
i=1 j=i+1
14
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− =
`−1 Y ` X
i=1 j=i+1
−(
` Y
(In−m − hLF F (jh)) + (In−m − hLF F (`h)) − In−m ]1n−m
(In−m − hLF F (ih)) − In−m )1n−m .
From Lemma 2, we have lim`→+∞ 270
Therefore, Ξ1m = lim −( `→+∞
` Y
Q`
i=1 (In−m
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i=1
− hLF F (ih)) = 0(n−m)×(n−m) .
(In−m − hLF F (ih)) − In−m )1n−m = 1n−m .
i=1
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Secondly, we show that all the entries of Ξ are nonnegative. From the choice of h, we know that In−m − hLF F (jh) > 0. From the fact that all the entries of LF L (jh) are non-positive, we can easily obtain that Ξ is nonnegative. Therefore, the containment control objective is achieved.
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4.2. Containment control under fixed topology and time-delays
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Suppose that the communication time-delays exist in the information transmission. The sequence of communication delays τij (`h), i 6= j, ` = 0, 1, 2, · · ·,
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have an upper bound, i.e., τij (`h) < τmax (i 6= j). Here, τmax denotes the maximal allowable time-delay. We propose the following containment control protocol: P aij (χj (`h − τij (`h)) − χi (`h)), i ∈ F j∈L∪F if t ∈ [`h, `h + h) ςi (t) = P aij ((χj (`h − τij (`h)) − ηj ) − (χi (`h) − ηi )), i ∈ L
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280
j∈L
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(6)
where τij (`h) may be larger than one sampling period.
T Denote ξ(`h) = [χT (`h), χT (`h − h), · · · , χT (`h − d τmax h eh)] , where χ(`h −
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jh) = [χ1 (`h − jh), χ2 (`h − jh), · · · , χn (`h − jh)]T , j = 0, 1, · · · , d τmax h e, and
d τmax h e is the minimum integer no smaller than
285
τmax h .
We assume that χ(ih) =
χ(0) for i < 0. Then the system (1) under protocol (6) can be summarized as follows: ξ(`h + h) = Φξ(`h) + Υ
15
(7)
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hA1
···
hAd τmax e−1 h
0
···
0
In .. .
··· .. .
0 .. .
0
···
In
The Laplacian matrix L = D −
Pd τmax e h i=0
0 hAd τmax e h 0 hLLL ηL 0 0 . , Υ = .. .. . . 0 0
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where In − hD + hA0 In 0 Φ = .. . 0
Ai .
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From the definitions of leader and follower, we have the following partitions: LF F LF L DF AF F i AF Li , D = , Ai = . L= 0 LLL DL 0 ALLi
Theorem 2. Suppose that the sampling period satisfies h <
1 maxi=1,···,n dii .
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ing protocol (6) for (1), containment control is achieved if the directed graph G 290
has a spanning tree.
M
T Proof. Denote ξF (`h) = [χTF (`h), χTF (`h − h), · · · , χTF (`h − d τmax h eh)] and
T ξL (`h) = [χTL (`h), χTL (`h − h), · · · , χTL (`h − d τmax h eh)] . Then the dynamics of
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the leaders can be written as:
ξL (`h + h) = ΦLL ξL (`h) + ΥL ,
=
Im − hDL + hALL0
hALL1
···
hALLd τmax e−1 h
hALLd τmax e h
Im
0
···
0
0
0 .. .
Im .. .
··· .. .
0 .. .
0 .. .
0
···
Im
0
AC
CE
ΦLL
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where
ΥL
(8)
=
0
hLLL ηL 0 0 .. . 0
, 16
,
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295
and the dynamics of the followers can be written as: ξF (`h + h) = ΦF F ξF (`h) + ΦF L ξL (`h),
ΦF L
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ΦF F
τmax In−m − hDF +hAFF 0 hAFF 1 · · · hAFF dτmax hA FF d h e h e−1 I 0 · · · 0 0 n−m 0 In−m · · · 0 0 = , .. .. .. .. .. . . . . . 0 0 ··· In−m 0 τmax hAF L0 hAF L1 · · · hAF Ld τmax hA F Ld e−1 e h h 0 0 ··· 0 0 0 0 ··· 0 0 = . .. .. .. .. .. . . . . . 0 0 ··· 0 0
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where
(9)
M
Define a new variable βi (·) = χi (·) − ηi , i = 1, 2, · · · , m. By using the properties of SIA matrices, from the leaders’ dynamics (8), it can be easily
∗ ξL
,
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obtained:
lim ξL (`h + h)
`→∞
T = 1m×(d τmax e+1) γ (ξL (0) − 1d τmax e+1 ⊗ ηL ) + 1d τmax e+1 ⊗ ηL h h h
for some vector γ ∈ Rm×(d
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300
τmax h
e+1)
. Consider the first m rows and note that
χL (ih) = χL (0) for i < 0. The final states of the leaders are given by χ∗L =
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1m γ T [1d τmax e+1 ⊗ (χL (0) − ηL )] + ηL . h Noticing that h <
1 maxi=1,···,n dii ,
then ρ(ΦF F ) ≤ 1. Suppose that 1 is an
eigenvalue of ΦF F , then there exists an eigenvector α = [α0T , α1T , · · · , αdTτmax e ]T 6= h
0 such that ΦF F α = 1 · α, i.e.,
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305
(In−m −hDF + hAF F 0 )α0 + hAF F 1 α1 + · · · hAF F d τmax e = α0 , e αd τmax h h
17
α0
= .. .
α1
αd τmax e−1 h
=
αd τmax e. h
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We can obtain that (In−m −hDF +hAF F 0 +hAF F 1 +· · · hAF F d τmax e )α0 = α0 , h that is, (In−m − hLF F )α0 = α0 ⇔ hLF F α0 = 0. Since h > 0 and LF F is Therefore, all the eigenvalues of ΦF F are less than 1.
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invertible, we have α0 = 0. It follows that α = 0. This is a contradiction.
After the leaders achieving the desired formation, we can obtain
310
lim ξF (`h + h)
`→∞
=
` ∗ lim [Φ`+1 F F ξF (0) + (ΦF F ΦF L + · · · + ΦF F ΦF L + ΦF L )ξL ]
`→∞
` X
∗ ΦiF F ΦF L ξL ]
L−1 F F AF L0
L−1 F F AF L1
lim [Φ`+1 F F ξF (0) +
`→∞
i=0
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=
=
−1 ∗ ΦF L ξ L (I(n−m)×(d τmax e+1) − ΦF F ) h
=
∗ ΓξL
where Γ = 1d τmax e+1 ⊗ h
h
final positions of the followers are given by: =
τ L−1 e F F AF Ld max h
i
. The
−1 −1 ∗ τ [L−1 e ]ξL F F AF L0 LF F AF L1 · · · LF F AF Ld max h
M
χ∗F
···
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∗ = L−1 F F AF L χL .
−1 It is easy to verify that L−1 F F AF L 1m = 1n−m and all the entries of LF F AF L
are non-negative. From the definition of convex hull, we know that all the followers converge to the convex hull spanned by the leaders’ final states.
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315
CE
5. Simulations
In the following directed graphs, all the weights of the directed graphs are
AC
assumed to be 1. Example 2. In this example, we assume that the communication topologies are
320
modeled by Figure 3, Figure 4, Figure 5 and Figure 6, where Li, i = 1, 2, · · · , 5, represent the leaders and F i, i = 1, 2, · · · , 7, represent the followers, respectively. The communication network switches stochastically, and switches every 1s to the next graph. The union of the communication graphs Ga ∪ Gb ∪ Gc ∪ Gd has a 18
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spanning tree in each time interval of 4s. From Theorem 1, the sampling period
330
should satisfy h < 0.5. Suppose that all the agents move in the x − y plane. The initial conditions h iT are chosen as: x(0) = 25 18 26 0 0 20 10 25 8 7 23 33 , h iT y(0) = 2 28 30 0 27 9 22 20 30 22 12 6 and the desired h iT formation of the leaders in the x−y plane are given by η1 = 16 13 17 8 8 , h iT η2 = 12 25 20 12 21 . The sampling period is chosen as h = 0.35.
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325
For system (4), Figure 7 and Figure 8 show the trajectories of all the agent
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under two different stochastic switching signals and the positions of all the agent at 40s, respectively. It is obvious that after some time, the leaders form a formation and then keep stationary. The trajectories of all the agents are affected 335
by the switching signal.
Figure 9 and Figure 10 show the trajectories of all the agents under the same switching signal and the positions of all the agents at 60s and 80s, respec-
M
tively. Influenced by the switching topology, the positions of all the followers are dynamic.
From Figure 7, Figure 8, Figure 9 and Figure 10, we can also see that the
340
ED
final states of the leaders may be different under different switching signals. However, the geometric configuration of the desired formation is not affected by
PT
the switching signals.
Example 3. Suppose the communication topology is shown in Figure 11, where Li, i = 1, 2, 3, denote the leaders and F i, i = 1, 2, 3, 4, denote the followers.
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345
It is noted that the directed G has a spanning tree if the time-delays are not
AC
considered. Assume that all the agents move in the x − y plane. The desired h iT h iT formation are given as: η1 = −0.5 −3 −1 , η2 = 1 3 7 . The h iT initial conditions are chosen as x(0) = , y(0) = 1 6 6 1 3 6 2 h iT . 10 2 7 −1 0 3 6
350
We assume that the information channels F 2 → F 1, F 3 → F 2, L2 → F 3,
F 3 → F 4, L1 → L2, and L2 → L3 have no time-delays, L1 → F 1, L2 → F 2, 19
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L5
L5 L3
L1
L2
L4
F1
L3
L4
F1
F7
F7 F6
F6 F4
L1
F2
F4 F3
F3
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L2
F2
F5
F5
Figure 4: Gb
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Figure 3: Ga
L5 L2
L3
L1
L4
F1
L3
F2
F4
F3
L1
L4
F1
F7 F6
F4
L2
L5
F7 F6
F2
F3
F5
Figure 6: Gd
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Figure 5: Gc
M
F5
F 2 → F 3, and L3 → L1 have time-delay τ1 = 1, and F 1 → F 2, L3 → F 3, and 355
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L3 → F 4 have time-delay τ2 = 2. The simulation results using (6) for (1) are shown by Figure 12 and Figure 13, where h = 0.2 in Figure 12 and h = 0.05 in Figure 13. We can see that all followers ultimately converge to the convex hull
CE
formed by the final positions of the leaders. We assume that the information channels L1 → F 1, L2 → F 2, F 2 → F 3,
L3 → F 3, have no time-delays, F 2 → F 1, F 3 → F 2, L2 → F 3, and F 3 → F 4
have time-delay τ1 = 1, and F 1 → F 2, L3 → F 3, L2 → L3, and L3 → L1 have
AC 360
time-delay τ2 = 2. The simulation results using (6) for (1) are shown by Figure 14 and Figure 15, where h = 0.2 in Figure 14 and h = 0.05 in Figure 15. From the simulations, we obtain that the final positions of the followers are
not affected by the sampling period and the time-delays, but determined by the
20
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35
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
25
y
20
15
25
20
15
10
10
5
5
0
0
5
10
15
20
25
30
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
y
30
0
35
0
5
x
15
20
25
30
Figure 7: Position trajectories of all
Figure 8: Position trajectories of all
the agents under the first switching
the agents under the second switching
signal at 40s
signal at 40s
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35
y
20
15
10
5
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
25
20
y
25
0
35
Final positions of the leaders L1 L2 L3 L4 L5 Final positions of the followers F1 F2 F3 F4 F5 F6 F7
30
15
10
5
0
5
10
15
20
25
30
0
35
M
x
0
5
10
15
20
25
30
x
Figure 10: Position trajectories of all
the agents at 60s
the agents at 80s
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Figure 9: Position trajectories of all
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PT
F1
AC
10
x
35
365
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35
F4
F3
F2
L2
L3
L1
Figure 11: A directed graph with 7 agents
topology structure and the final positions of the leaders.
21
35
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Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
8
6
6
4
4
y
2
2
0
0
-2
-2
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
1
1.5
2
x
Figure 12:
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y
8
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
2.5
3
3.5
4
4.5
5
5.5
6
x
Trajectories of all the
Figure 13:
agents with h = 0.2
Trajectories of all the
agents with h = 0.05
10
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
10
8
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8
6
6
y
y
4
2
0
-2
-4
0
1
2
3
4
Final positions of the followers F1 F2 F3 F4 Final positions of the leaders L1 L2 L3
5
x
Trajectories of all the
agents with h = 0.2
2
0
-2
6
M
Figure 14:
4
0
Figure 15:
1
2
3
4
5
x
Trajectories of all the
agents with h = 0.05
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6. Conclusion
In this paper,we have studied sampled-data based containment control prob-
PT
lems of continuous-time multi-agent systems with multiple interactive leaders. The communication networks are assumed to be switching or fixed with timedelays. We have proposed some protocols to overcome the switching topology
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370
and time-delays. For the leaders, we have shown that, the final states may be affected by the switching topology, but the geometric configuration of the de-
AC
sired formation is not affected by the switching topology. The followers move into the convex hull spanned by the final positions of the leaders first, and then
375
change their positions in the convex hull under the switching topology. For the time-delay case, the final positions of the leaders and the followers are not affected by the time-delays. Our future work will focus on containment control for multi-agent systems with more complex dynamics. 22
6
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Acknowledgements This work was supported by the National Natural Science Foundation of
380
Central Universities (Grant No. FRF-TP-17-022A2).
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[37] Y. Sun and L. Wang, Consensus problems in networks of agents with double-integrator dynamics and time-delays, 82(10) (2009) 1937-1945.
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[38] N. Biggs, Algebraic Graph Theory, Cambridge, U.K.: Cambridge Univ. Press (1974).
[39] R.T. Rockafellar, Convex analysis, New Jersey: Princeton University Press,
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1972.
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Huiyang Liu received her B.S. degree in Information and Computing Sci-
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ence, her M.S. degree in Applied Mathematics from Zhengzhou University in 2004 and 2007, respectively. She received her Ph.D. degree in Control The-
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ory from Peking University in 2012. She is currently a lecture in School of Mathematics and Physics, University of Science and Technology Beijing. Her current research interests are in the fields of coordination of multi-agent sys-
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tems, consensus problems, containment control and swarm dynamics. Email:
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[email protected]
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Guangming Xie received his B.S. degrees in Applied Mathematics and
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Computer Science and Technology, his M. E. degree in Control Theory and Control Engineering, and his Ph.D. degree in Control Theory and Control Engineering from Tsinghua University, Beijing, China in 1996, 1998, and 2001,
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respectively. Now he is a Professor of Dynamics and Control with the College of Engineering, Peking University, Beijing. His research interests include hybrid
and switched systems, networked control systems, and multi-agent systems,
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multi-robot systems, and bio-mimetic robotics. Email: [email protected]
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