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Collision-free consensus in multi-agent networks: A monotone systems perspective
Automatica 64 (2016) 217–225
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Brief paper
Collision-free consensus in multi-agent networks: A monotone systems perspective✩ Zhiqiang Miao a,b , Yaonan Wang a , Rafael Fierro b a
College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
b
Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131-0001, USA
article
info
Article history: Received 14 December 2014 Received in revised form 22 June 2015 Accepted 29 October 2015 Available online 7 December 2015 Keywords: Consensus Collision-free Order preservation Monotone systems Multi-agent systems Cooperative control
abstract This paper addresses the collision-free consensus problem in a network of agents with single-integrator dynamics. Distributed algorithms with local interactions are proposed to achieve consensus while guaranteeing collision-free among agents during the evolution of the multi-agent networks. The novelty of the proposed algorithms lies in the definition of neighbors for each agent, which is different from the usual sense that neighbors are selected by the distance between agents in the state space. In the proposed strategies, the neighbor set for each agent is determined by the distance or difference between agents in the index space after ordering and labeling all agents according to certain ordering rules including weighted order and lexicographic order. The consensus analysis of the proposed algorithms is presented with some existing results on algebraic graph theory and matrix analysis. Meanwhile, by realizing the relations between order preservation and collision-free, a systematic analysis framework on order preservation and hence collision-free for agents in arbitrary dimension is provided based on tools from monotone systems theory. Illustrated numerical examples are presented to validate the effectiveness of the proposed strategies. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction There has been a growing interest in distributed coordination and cooperative control of multi-agent systems in the past decade (Bai, Arcak, & Wen, 2011; Bullo, Cortes, & Martinez, 2009; Lewis, Zhang, Hengster-Movric, & Das, 2014; Mesbahi & Egerstedt, 2010; Qu, 2009; Ren & Beard, 2008; Ren & Cao, 2011; SemsarKazerooni & Khorasani, 2012). This board interest in multi-agent networks is motivated by the emerging applications including vehicle formations, cooperative robotics, sensor networks, and social networks. Consensus as one of the fundamental problems in the coordination of multi-agent systems, refers to the agreement of
✩ This work was supported in part by the National Natural Science Foundation of China (61573134, 61433016), National High Technology Research and Development Program of China (863 Program: 2012AA111004, 2012AA112312), Hunan Provincial Innovation Foundation for Postgraduate (521298960), and the fellowship from China Scholarship Council (CSC). The material in this paper was partially presented at the 54th IEEE Conference on Decision and Control, December 15–18, 2015, Osaka, Japan. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Z. Miao), [email protected] (Y. Wang), [email protected] (R. Fierro).
http://dx.doi.org/10.1016/j.automatica.2015.11.025 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
all agents upon certain quantities of interest. Consensus has been studied extensively in the literature (Cao, Yu, Ren, & Chen, 2013; Fax & Murray, 2004; Moreau, 2005; Olfati-Saber, Fax, & Murray, 2007), and the applications of consensus algorithms can be found in formation control (Lin, Francis, & Maggiore, 2005), rendezvous (Cortes, Martinez, & Bullo, 2006), flocking (Tanner, Jadbabaie, & Pappas, 2007), attitude alignment (Jadbabaie, Lin, & Morse, 2003), and sensor networks (Kar & Moura, 2010). Here we do not intend to provide a completed review on consensus problem, the interested reader is referred to the survey paper (Cao et al., 2013) and the mentioned monographs for more detailed discussions. The most well-known consensus algorithm is the local voting protocol which can be found in many of the aforementioned studies. Alternatively, cyclic pursuit strategies have recently been investigated for the distributed control multi-agent systems (Juang, 2013; Lin, Mireille, & Bruce, 2004; Marshall, Broucke, & Francis, 2006; Sinha & Ghose, 2006; Smith, Broucke, & Francis, 2005). In cyclic pursuit, each agent pursues its leading neighbor to form a ring network structure. The cyclic pursuit strategy is attractive because it inherently is decentralized and requires a small number of communication links. Cyclic pursuit strategies were utilized to address the agreement problem in Lin et al. (2004); Smith et al. (2005), formation control in Juang (2013) and Marshall
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(iii) A systematic analysis framework based on monotone systems theory is provided to investigate order preservation and hence collision-free for agents.
et al. (2006), and rendezvous problem in Sinha and Ghose (2006). Among these studies, a result in Lin et al. (2004) is closely relevant to the topic in this paper. It was showed in Lin et al. (2004) that if the agents in plane initially are ordered counterclockwise (or clockwise), then the order of agents will be preserved with cyclic pursuit, and therefore no collision would occur. Through the strategy proposed by Lin et al. (2004) achieved consensus while ensure collision-free among agents, the results are limited to agents in 2D, which render the applications involving unmanned aerial vehicles (UAV) in 3D difficult. Similar ideas in Lin et al. (2004) were reported in Wang, Xie, and Cao (2013) to solve the circle-forming problem for agents that move on a circle. In Wang et al. (2013), agents also are initially labeled counterclockwise on the circle, and each agent only receives information from its two immediate neighbors, resulting in an undirected ring network topology. The property of order preservation was studied as well to avoid inter-agent collisions. However, the result in Wang et al. (2013) is quite limited, as the motion of agents is restricted to onedimensional space of a circle. Most of the existing work on the distributed control of multiagent networks pays more attention on the final or ultimate state rather than the entire trajectory of the agents. Take the consensus task for example, all agents are expected to reach a common value eventually, disregarding the transient behavior of agents. Nevertheless, the transient behavior of agents needs to be restrained to meet certain constraints in some cases. Inter-agent collision-free is an example of such constraints. Intuitively, if there exist certain quantities or qualities of the network that remain unchanged during the evolution, then the information from these invariants can help us better understand the transient behavior of agents throughout evolution. Order preservation is a kind of invariants, which makes the transient behavior of agents more predictable. As observed by Lin et al. (2004) and Wang et al. (2013), the order preservation property is particularly useful to prevent collisions between agents. Order preservation also has been explored in some social networks like Krause’s opinion dynamics (Blondel, Hendrickx, & Tsitsiklis, 2009; Hendrickx, 2008; Yang, Dimarogonas, & Hu, 2014). A concise proof on order preservation was provided by Blondel et al. (2009) with simple arguments. However, the arguments only hold for agents in one dimension, which render the results rather conservative. A systematic analysis framework for arbitrary dimension remains to be provided. In this paper, order preservation will be studied in the context of monotone systems (Altafini, 2013; Angeli & Sontag, 2003; Hirsch & Smith, 2005; Smith, 2008). Monotone systems are orderpreserving systems, the trajectories of which preserve a given partial order through time. By realizing the relations between order preservation and collision-free, the problem of consensus with collision-free in a network of agents is addressed here. To be more specific, the contribution of this paper lies in designing distributed control laws such that all agents in the network reach the same value while guaranteeing inter-agent collision-free navigation. Preliminary results in this paper were presented in Miao, Wang, and Fierro (in press), where each agent is assumed to have the same number of neighbors. In this paper, we relax this assumption by generalizing the result to agents that have different number of neighbors. Moreover, we provide additional simulation case studies and a comparison of the proposed strategy with the cyclic pursuit strategy in Lin et al. (2004). The main contributions can be summarized as:
Lemma 1. The Laplacian matrix L has a simple zero eigenvalue with an associated eigenvector 1n , and all the other eigenvalues have positive real parts if and only if the digraph associated with has a directed spanning tree.
(i) Weighted order and lexicographic order are explored to define order for agents in arbitrary dimensional space; (ii) Neighbors for each agent are defined in the index space rather than in state space. Notice that agents in the network do not necessarily have the same number of neighbors.
Definition 1. Given a matrix S = [sij ]n×n , the associated digraph G(S ) of matrix S is a directed graph with n vertices indexed by 1, 2, . . . , n such that there is an edge in G(S ) from j to i if and only if sij ̸= 0.
The remainder of this paper is organized as follows. In Section 2, some mathematical preliminaries first are presented. Problem is formulated in Section 3. In Section 4, the proposed strategies and main results are stated. Simulation results for illustrating the effectiveness of the proposed strategies are given in Section 5. Section 6 concludes the paper. 2. Preliminaries 2.1. Notation The standard notations are used throughout this paper. N, R and C denote the sets of natural numbers, real numbers and complex numbers respectively. R+ denotes nonnegative real numbers, and Rn+ is the set of n-tuples for which all components belong to R+ . Let In ∈ Rn×n be the n-dimensional identity matrix; 1n ∈ Rn be the vector of all ones; 0n ∈ Rn be the vector of all zeros. For vectors x, y ∈ Rn , we denote x ≤ y if and only if xi ≤ yi for all i; x < y if and only if x ≤ y and x ̸= y; and x ≪ y if and only if xi < yi for all i. 2.2. Algebraic graph theory and matrix analysis In this subsection, some elements and results on graph theory and matrix analysis that will used in this paper are reviewed. These results can be found in Ren and Beard (2008) and Ren and Cao (2011), and references therein. Let G = {V , E , A} be a digraph with a node set V = {1, 2, . . . , n}, an edge set E ⊆ V × V , and an adjacency matrix A = [aij ]n×n with nonnegative elements. A directed edge denoted by (j, i) means that node i has access to node j, i.e., node i can receive information from node j. The elements of adjacency matrix A are defined as follows: If there is a directed link from node j to i (j ̸= i), then aij > 0; otherwise, aij = 0. We assume that aii = 0 for all i. The Laplacian matrix L = [lij ]n×n associated with the adjacency matrix A is defined as
−anij lij = aij
i ̸= j, i = j.
(1)
j=1,j̸=i
A directed path is a sequence of edges in the directed graph G with distinct nodes. A directed graph is strongly connected if, for any two distinct nodes j and i, there exists a directed path from node j to node i. A digraph with n nodes is called a directed tree if it has n − 1 edges and there exists a root node with directed paths to every other node. A directed spanning tree is a directed tree that includes all the nodes of the digraph. A digraph is said to have a directed spanning tree if there is at least one node having a directed path to every other node.
Z. Miao et al. / Automatica 64 (2016) 217–225
Lemma 2. A matrix S is irreducible if and only if its associated directed graph G(S ) is strongly connected.
1 0.9 0.8 0.7
States
Definition 2. Matrix S ∈ Rn×n is reducible if and only if for some permutation matrix P ∈ Rn×n , such that P T SP is in block triangular form; otherwise S is irreducible.
0.6 0.5 0.4 0.3 0.2 Collision
0.1 0
2.3. Monotone systems theory
0
x˙ = f (t , x).
(2)
Let Φt (x) : R+ × R → R denote the resulting semi-flow that describes the evolution of states in positive time. n
whenever x ≤ y and t ≥ 0,
(3)
whenever x < y and t > 0.
(4)
Definition 4. The system (2) is said to be cooperative if the off∂f diagonal elements of Jacobian matrix Df (t , x) = [ ∂ xi (t , x)]n×n are j
non-negative, i.e., for all (t , x) ∈ R+ × Rn ,
∂ fi (t , x) ≥ 0, ∂ xj
i ̸= j,
(5)
Lemma 3. Let system (2) be cooperative, and 0, then Φt (x) ≪ Φt (y). Hence Φt (x) is strongly monotone. 3. Problem formulation Consider n agents with single-integrator dynamics given by i = 1, 2, . . . , n m
(6) m
where xi ∈ R is the state of agent, and ui ∈ R is the control input of ith agent. The most common consensus algorithm called local voting protocol is given as ui =
aij (xj − xi ),
(7)
j∈Ni
where aij being the graph edge weights, Ni is the neighbor set of agent i. Then the closed-loop dynamics of agents is x˙ i =
3
4
5
1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
t [s]
To illustrate inter-agent collision, let us consider three agents in R. Denote x = [x1 , x2 , x3 ]T as the state vector, and the initial state of three agents are x(0) = [0, 0.3, 1]T . The consensus plots in Figs. 1 and 2 show the states xi in the consensus dynamics (8) with the graph topologies A1 = [0 1 1; 1 0 0; 1 0 0] and A2 = [0 1 0; 1 0 1; 0 1 0] respectively. It can be observed that agent x1 collides with agent x2 with graph topology A1 , while for agents with graph topology A2 , collision-free is ensured. In this paper, we will design the local interaction rules for agents represented by Eq. (6), so that the inter-agent collision-free consensus is achieved or reached in the sense that (i) consensus:
and system (2) is said to be irreducible, if the Jacobian matrix Df (t , x) is irreducible for all (t , x) ∈ R+ × Rn .
x˙ i = ui ,
t [s]
Fig. 2. States of three agents with graph topology A2 .
and strongly monotone if
Φt (x) ≪ Φt (y),
2
0.9
Definition 3. The map Φ is said to be monotone if
Φt (x) ≤ Φt (y),
1
Fig. 1. States of three agents with graph topology A1 .
States
The main analysis tools from monotone systems theory are now presented. These definitions and conditions on monotone systems are mainly based on the results in Smith (2008), and also can be found in Hirsch and Smith (2005). Given a function f (t , x) : R × Rn → Rn , piecewise continuous in t and of class C 1 with respect to x, the associated system of differential equations
n
219
aij (xj − xi ).
(8)
lim ∥xi (t ) − xj (t )∥ = 0,
t →∞
∀i, j.
(9)
(ii) collision-free: for ∀δ > 0, if ∥xi (0) − xj (0)∥ ≥ δ , then there exists a time-varying function d(t ) > 0 such that
∥xi (t ) − xj (t )∥ ≥ d(t ),
∀i ̸= j, t ≥ 0.
(10)
Remark 1. Collision avoidance is an important issue that has extensively been studied in the control design of multi-agent/robot systems. The collision avoidance problem has been studied in formation control (Dimarogonas, Loizou, Kyriakopoulos, & Zavlanos, 2006; Mastellone, Stipanovic, Graunke, Intlekofer, & Spong, 2008), coverage control (Hussein & Stipanovic, 2007), flocking (Tanner et al., 2007), and swarming (Gazi, 2005) of multi-agent systems. Comparing with the collision-free condition presented by Eq. (10), the collision avoidance condition in these studies is defined by ∥xi (t ) − xj (t )∥ ≥ d, ∀i ̸= j, t ≥ 0, where d > 0 is a positive constant. The most popular strategy utilized for collision avoidance is the potential field-based method, which consists of attraction to achieve the desired group behavior, and repulsion to avoid collision. However, it is evidently that the potential field-based method is inapplicable for avoiding collision in the consensus problem. 4. Methodology
j∈Ni
It has been shown in Ren and Beard (2008); Ren and Cao (2011) that consensus can be achieved with (8) provided that the associated digraph has a directed spanning tree. However, interagent collision may occur.
4.1. One dimensional case We first consider the special case when m = 1. For n agents xi ∈ R, i = 1, 2, . . . , n, we can always order them according to
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their different initial states. Without loss of generality, assume x1 (0) < x2 (0) < · · · < xi (0) < · · · < xn (0).
(11)
We assume that each agent only receives information from its neighbors. In this paper, neighbors are determined by the distance or difference between agents in the index space. Specifically, for agent xi , its neighbor Ni is defined as f
Ni = {xj | − Nib ≤ j − i ≤ Ni }, f Ni
f Nj ,
where ∈ N, ∈ N, and satisfy ≥ , ≤ if i < j. Furthermore, xi+k is called the kth nearest forward neighbor of xi ; while xi−k is called the kth nearest backward neighbor of xi . Now, consider the consensus algorithm (7) with communication protocol as follows: Nib
Njb
(i) Each agent i receives the information from Nib nearest f
backward neighbor and Ni nearest forward neighbor; (ii) Each agent sends its state information with fixed weight.
(13)
j=[i−Nib ]
p < 1, 1 ≤ p ≤ n, p>n
1 n
aij =
αj
e˙ i = −
j=[i−Nib ]
k=j
αj ei +
j=[i+1−Nib+1 ]
(14)
where αj > 0. The agents dynamics (13) can be written in a more compact form: x˙ = −Lx,
(15)
f
j=[i+1+Ni+1 ] k=j−1
+
f j=[i+Ni
αj ek .
(17)
]+1 k=i+1
The second term on the right side of Eq. (17) vanishes if [i − Nib ] =
[i + 1 − Nib+1 ]; and the third term vanishes if [i + Nif ] = [i + 1 + Nif+1 ]. Thus it can be obtained that e˙ = He,
(18)
(19)
e(0) ≪ 0n−1 ⇒ e(t ) ≪ 0n−1 ,
∀t ≥ 0 ,
where x = [x1 , x2 , . . . , xn ] is the state vector, L is the Laplacian matrix associated with matrix A = [aij ]n×n . We then can state the following result for the one dimensional case: Nnb
Theorem 1. Consider n agents in R. If + ≥ 1, then consensus with inter-agent collision free can be guaranteed with consensus dynamics (13).
e˙ i < −
αj ei .
(22)
j=[i+1−Nib+1 ]
Thus, for t ≥ 0, f
j=[i+Ni ]
|ei (t )| > exp
αj t |ei (0)|.
(23)
j=[i+1−Nib+1 ]
Then for ∀i < j and t ≥ 0, we have
|xi (t ) − xj (t )| =
· · · ≤ Nnf , we have if N1f ≥ 1, Nnb = 0 (Nnb ≥ 1, N1f = 0), then Nif ≥ 1 (Nib ≥ 1) for all i, which implies the digraph associated with
Laplacian matrix L or adjacency matrix A has a directed tree, and f f xn (x1 ) is the root node; if N1 ≥ 1, Nnb ≥ 1, then Ni ≥ 1, Nib ≥ 1 for all i, which implies the associated digraph is strongly connected. Thus consensus of all states xi (1 ≤ i ≤ n) can be achieved by applying Lemma 1. (ii) Collision-free: Define error vector e = [e1 , e2 , . . . , en−1 ], with ei = xi − xi+1 , i = 1, 2, . . . , n − 1, we have
e˙ i =
j=[i−Nib ]
αj (xj − xi ) −
j=[i+1−Nib+1 ]
αj (xj − xi+1 ).
>
|ek (t )|
k=i
f
l=[k+Nk ]
exp −
k=i
αl t |ek (0)|
l=[k+1−Nkb+1 ]
≥ exp(−λt )|xi (0) − xj (0)|, where λ = maxi≤k≤j−1 {
f l=[k+Nk ]
l=[k+1−Nkb+1 ]
(24)
αl } > 0, 1 ≤ i < j ≤ n.
It follows Eq. (24) that collision-free condition (10) holds for some exponential decay function. This completes the proof. Remark 2. The error vector e can be obtained by a coordinate transformation e = Px, where the matrix P ∈ R(n−1)×n is 1
(16)
k=j−1
|xk (t ) − xk+1 (t )| =
k=j−1
f
j=[i+1+Ni+1 ]
k=i
Proof. (i) Consensus: Consider agent dynamics given by Eq. (15). f f f Because N1 + Nnb ≥ 1, and N1b ≥ N2b ≥ · · · ≥ Nnb , N1 ≤ N2 ≤
f
(21)
f
k=j−1 f N1
(20)
Thus, the order of all agents is preserved. Furthermore, as ej (t ) < 0, ∀t ≥ 0, j = 1, 2, . . . , n − 1, Eq. (17) implies that
T
j=[i+Ni ]
α j ek
j=[i+Ni ]
j = i, −Nib ≤ j − i ≤ Nif
0
j=[i+1−Nib+1 ]−1 k=i−1
x1 (t ) < x2 (t ) < · · · < xi (t ) < · · · < xn (t ).
for ∀p ∈ N. This operator ensures the index would within the set {1, 2, . . . , n}. The weight aij satisfies
f
j=[i+Ni ]
following the order preservation property of system (18). Inequality (20) implies for all t ≥ 0,
where the operator [·] is defined as
[p ] = p
] and [i + ] ≤ [i + 1 + Nif+1 ]. If [i − Nib ] < [i + 1 − Nib+1 ], ] < [i + 1 + Nif+1 ], we have
and applying Lemma 3, one obtains
aij (xj − xi ),
[i + 1 − and [i +
ei (0) < 0 ⇒ e(0) ≪ 0n−1 ,
f
j=[i+Ni ]
f
f Ni
where H = [hij ](n−1)×(n−1) with hij ≥ 0, i ̸= j. Thus system (18) is cooperative by Definition 4. Using the condition
According to this protocol, the dynamics of each agent is
x˙ i =
Nib+1 f Ni
(12)
f Ni
Nib
f
Because Nib ≥ Nib+1 and Ni ≤ Ni+1 , it can be checked that [i − Nib ] ≤
P =
−1 1
−1 .. .
..
.
.
1
−1
Z. Miao et al. / Automatica 64 (2016) 217–225
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Table 1 Collision-free consensus algorithm. Input: xi (0) ∈ Rm , ui (t ) ∈ Rm , i = 1, 2, . . . , n, t ≥ 0. (i) Order agents according to weighted order (or lexicographic order) on their initial states; (ii) Determine the neighbors for each agent by set (12); (iii) Update agent’s state according to the voting protocol (13). Output: xi (t ) ∈ Rm , i = 1, 2, . . . , n, t ≥ 0.
Fig. 3. Order of n agents in R3 space.
It can be concluded that the monotonicity of error dynamics e is guaranteed by the monotonicity of state dynamics x, and the specific matrix P defined above. In the proposed algorithm, weights aij are assumed to satisfy aij = αj , ∀i, j, for the sake of simplicity. Actually, aij only needs to be non-increasing with the index f
f
distance or difference |i − j|. If Nib = 1, Ni = 0, or Ni = 1, Nib = 0 for all i, then the digraph forms a chain topology structure, which requires minimum number of communication links. Remark 3. In the proposed algorithm, agent xi cannot immediately determine whether another agent xj is its communication neighbor or not just through the information received from the agent xj . The agent xi may need some global information to acquire the order of agent xj in the group, and then judge if it is within its neighbor set defined by Eq. (12). However, the global information is required only at the initial time to determine the order for the agent group. Because the order of agents is preserved during the involution, the neighbor set for each agent remains unchanged. 4.2. Arbitrary dimensional case For n agents xi ∈ Rm , i = 1, 2, . . . , n, where m can be an arbitrary non-zero natural number, we cannot order them directly according to their initial states xi (0). Because two vectors xi (0) and xj (0) may unrelated, that is, neither xi (0) < xj (0) nor xj (0) < xi (0) holds. Nevertheless, we can map the vectors xi in Rm to χi in R by a weight vector:
χi = ωT xi ,
i = 1, 2, . . . , n,
(25)
where ω ∈ R is the non-zero weight vector, χi ∈ R can be viewed as the new weighted state of agent i. Define the n × m matrix of states m
T x1
xT 2 X = . , ..
(26)
and denote χ = [χ1 , χ2 , . . . , χn ]T , then Eq. (25) can be rewritten as (27)
Similar with the one dimensional case, we can now order the agents according to their weighted states without loss of generality, assume that
χ1 (0) ≤ χ2 (0) ≤ · · · ≤ χi (0) ≤ · · · ≤ χn (0).
X˙ = −LX ,
(29)
where the Laplacian matrix L is same as that in Eq. (15). Now, the main result can be stated in the following theorem: Theorem 2. Consider n agents in Rm . If either of the following two conditions holds, f
(C1) χ1 (0) < χn (0), and N1 ≥ 1, Nnb ≥ 1;
f
(C2) χ1 (0) < χ2 (0) < · · · < χn (0), and N1 + Nnb ≥ 1; then consensus with inter-agent collision free can be guaranteed by consensus algorithm shown in Table 1. Proof. Case C1 is considered first. The consensus and collision-free of consensus dynamics (29) will be investigated respectively. f f (i) Consensus: If N1 ≥ 1, Nnb ≥ 1, then Ni ≥ 1, Nib ≥ 1 for all i. It can be easily checked that the communication graph associated with the Laplacian matrix L in Eq. (29) is strongly connected. Thus consensus of all states xi (1 ≤ i ≤ n) is achieved with the aid of Lemma 1. Following Eq. (25), the consensus of states xi (1 ≤ i ≤ n) also implies the consensus of states χi (1 ≤ i ≤ n). (ii) Collision-free: Using Eqs. (27) and (29), the dynamics of χ can be obtained as
χ˙ = −Lχ .
(30)
Define e = [e1 , e2 , . . . , en−1 ], with ei = χi − χi+1 , i = 1, 2, . . . , n − 1, then along the same lines in the proof of Theorem 1, we have e˙ = He,
(31)
where the matrix H = [hij ](n−1)×(n−1) with hij ≥ 0 (i ̸= j) is the same as that in Eq. (18). Thus system (31) is cooperative. Because f Ni ≥ 1, Nib ≥ 1 for all i, it can be easily checked that the associated graph of matrix H is strongly connected. Thus, system (31) is also irreducible using Lemma 2. Since system (31) is cooperative and irreducible, and
χ1 (0) < χn (0) ⇒ e(0) ̸= 0n−1 ,
xTn
χ = X ω.
and χj (0) may be equal even through xi (0) is not equal to xj (0). An illustrating example in R3 is shown in Fig. 3, where χ3 (0) = χ4 (0). Analogously, we consider the same protocol as in one dimensional case. The proposed collision-free consensus algorithm for agents in arbitrary dimension is summarized in Table 1. Therefore, the consensus dynamics of agents in Rm is
(28)
In this paper, the order shown in (28) that obtained by comparing weighted states is referred as weighted order. It should be noted that ‘=’ is allowed in inequality (28), as two weighted states χi (0)
(32)
then applying Lemma 4, we have e(0) ≤ 0n−1 , e(0) ̸= 0n−1 ⇒ e(t ) ≪ 0n−1 ,
∀t > 0.
(33)
Inequality (33) implies for all t > 0,
χ1 (t ) < χ2 (t ) < · · · < χi (t ) < · · · < χn (t ).
(34)
Thus, the order of the agents is preserved. Furthermore, as ej (t ) < 0, ∀t > 0, j = 1, 2, . . . , n − 1, we have for all t > 0, f
j=[i+Ni ]
e˙ i < −
j=[i+1−Nib+1 ]
αj ei .
(35)
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Thus, for t ≥ t0 > 0,
|ei (t )| > exp −
f
j=[i+Ni ]
αj (t − t0 ) |ei (t0 )|.
(36)
j=[i+1−Nib+1 ]
Then it follows (36) that for ∀i < j and t ≥ t0 > 0,
|χi (t ) − χj (t )| > exp(−λ(t − t0 ))|χi (t0 ) − χj (t0 )|,
(37)
where λ is the same as that in Eq. (24). Considering the fact that
|χi (t ) − χj (t )| ≤ ∥ω∥∥xi (t ) − xj (t )∥,
(38)
Proof. We prove this claim by contradiction. Assume that such ω does not exists, then for ∀ ω ∈ Rm , it belongs to one or more of sets Cij with i, j = 1, 2, . . . , n: Cij = {ω ∈ Rm | ωT (xi (0) − xj (0)) = 0},
(40)
and Cij satisfies
which yields
∥xi (t ) − xj (t )∥ > ∥ω∥−1 exp(−λ(t − t0 ))|χi (t0 ) − χj (t0 )|.
Lemma 5. Given n agents xi ∈ Rm with different initial states xi (0), i = 1, 2, . . . , n. There always exists a weight vector ω ∈ Rm such that ωT xi (0) ̸= ωT xj (0) whenever xi (0) ̸= xj (0).
Cij = Rm .
(41)
1≤i,j≤n
(39)
Inequality (39) implies that collision-free is guaranteed. Now we consider Case C2. Along the same lines in the proof of Theorem 1, and using the fact (38), the proof for Case C2 can be easily derived. This completes the proof of Theorem 2. Remark 4. Because the order relationship is transitive (which means if a < b (a ≤ b), and b < c (b ≤ c ), then a < c (a ≤ c ), for a, b, c ∈ R), in order to give the order for agent group, we only need to define appropriate ordering rules to determine the order for each two agents. For agents in one dimension space xi ∈ R, i = 1, 2, . . . , n, the ordering rule for each agent pair xi , xj can be simply as: Denote xi < xj , whenever the value of state xi is less than the value of state xj . Whereas for agents in a high-dimensional space xi ∈ Rm , i = 1, 2, . . . , n, the ordering rule for each agent pair xi , xj is: Denote xi < xj , (xi ≤ xj ), whenever the value of weighted state ωT xi is less than (no larger than) the value of weighted state ωT xj , where ω ∈ Rm is the nonzero weight vector. Remark 5. The consensus dynamics (29) represents a linear system, thus the consensus or convergence of xi certainly is robust to disturbances. However, if there exist disturbances in the error dynamics (31), then the error ei may no longer always greater than zero. This implies the order preservation of xi probably is nonrobust to the disturbances. The reason is that order preservation or monotonicity of systems is an elegant mathematical property that established for systems with accurate models. A formal robustness analysis is an area of future research. Remark 6. The order of agents is determined by the vector ω, besides the initial states. Different weight vector may results in different order of agents, and thus different interaction topology. Because ‘=’ is allowed when ordering agents according to their initial weighted states, as shown in inequality (28). In such case, the initial order of agents may be not unique, even for a given weight vector. However, the non-uniqueness of order makes no difference on the correctness of Theorem 2, though it may be better to select an appropriate weight vector such that the order of agents is uniquely determined. 4.3. Lexicographic order: a special weighted order This subsection addresses the issue on the uniqueness of order for agents in arbitrary dimension. First, we show that there always exists an appropriate weight vector which ensures uniqueness of the order. Then the lexicographic order which always define unique order for agents is introduced, and interpreted as a special weighted order in the previous subsection. As mentioned before, the order of agents may be not unique even for a given weight vector. Thus, is it possible to find a specific weight vector such that the agents’ initial order is uniquely determined by their weighted states? A positive answer is given below:
Eq. (41) implies that there exists at least one set Ci0 j0 which contains infinite number of elements, that is
ωkT (xi0 (0) − xj0 (0)) = 0,
k = 1, 2, . . . .
(42)
The above infinite numbers of equations hold only if xi0 (0) = xj0 (0), however this is not true. This completes the proof. Lemma 5 ensures the existence of the weight vector that uniquely determines the order for a given group of agents. In the following, we will introduce a new order: lexicographic order, which can be viewed as a special weighted order that ensures the uniqueness of the order. Definition 5. Given two vectors a, b ∈ Rm , the lexicographic order is defined as a 0, ∀i < s, ai = bi and as < bs ,
(43)
where ‘iff’ represents ‘‘if and only if’’; and ai , bi respectively denote the ith element of vectors a, b. Before giving the interpretation of lexicographic order, some definitions first are presented. For n states xi ∈ Rm , let xki denote the kth element of vector xi . Given initial states xi (0) (1 ≤ i ≤ n), for ∀xri (0) ̸= xrj (0)(1 ≤ r ≤ m), we define γk (1 ≤ k ≤ m − 1) as
γk ≥
max
1≤i,j≤n;1≤r ≤m
|xir +k (0) − xrj +k (0)| |xri (0) − xrj (0)|
.
(44)
Denote
ω = [1, γ −1 , γ −2 , . . . , γ −(m−1) ]T ,
(45)
where γ satisfies 1
γ ≥ γ0 max (γk ) k ,
(46)
1≤k≤m−1
with γ0 ≥ 2. Based on these definitions, we have the following result: Lemma 6. Given n agents xi ∈ Rm with different initial states xi (0), i = 1, 2, . . . , n. For any two initial states xi (0) and xj (0) (1 ≤ i, j ≤ n), if xi (0)
∀ k < s,
xki (0) = xkj (0) and
xsi (0) < xsj (0).
(47)
Define eij (0) = ωT (xi (0) − xj (0)), then we have eij (0) =
k=m k=1
γ −(k−1) (xki (0) − xkj (0)).
(48)
Z. Miao et al. / Automatica 64 (2016) 217–225
(b) Agent’s states in (x1 , x2 ) plane.
(a) Initial states and interaction topology.
223
(c) Weighted states χ = x1 .
Fig. 4. Simulation results for Case A1 using weighted order with weight vector ω = [1, 0]T .
Denote dsij (0) = xsi (0) − xsj (0) for easy notation. If s = m, then k=m−1
eij (0) =
5. Simulation results
γ −(k−1) (xki (0) − xkj (0)) + γ −(m−1) dsij (0)
k=1
= γ −(m−1) dsij (0) < 0.
(49)
If s < m, then k=s−1
eij (0) =
γ −(k−1) (xki (0) − xkj (0)) + γ −(s−1) dsij (0)
k=1
+
k =m
γ −(k−1) (xki (0) − xkj (0))
k=s+1
= γ −(s−1) dsij (0) +
k=m
γ −(k−1) (xki (0) − xkj (0)).
(50)
k=s+1
Since dsij (0) < 0, and using (44), we have for k > s,
|xki (0) − xkj (0)| ≤ (γ /γ0 )k−s |dsij (0)|. k=m k=s+1
≤
γ −(k−1)
(51)
γ γ0
k−s
1 k 1− γ −(s−1) dsij (0). γ0 k=1
|dsij (0)|
k=m−s
(52)
When γ0 ≥ 2, it can be easily checked that
1 k > 0. γ0 k=1
k=m−s
1−
5.1. Case study The following two cases including two different initial conditions in 2D and 3D are considered:
Substituting (51) into (50), yields eij (0) ≤ γ −(s−1) dsij (0) −
To verify the effectiveness of the proposed control strategies, some numerical simulations are carried out in this section. From the results in the previous section, we know that the interaction topology for agents in the proposed strategies is determined by agents’ initial states and their ordering rules. Thus, agents with different initial states and different ordering rules both are considered. For the sake of simplicity, in all simulations of the proposed f algorithms, we assume that Nib = Ni = 1 for all i, and aij = 5 if agent i can receive information from agent j; otherwise aij = 0. In the following, we first investigate the proposed algorithm for agents in 2D and 3D respectively. Then a comparison study between the proposed algorithm and that in Lin et al. (2004) is carried out for further illustration.
(53)
Because dsij (0) < 0, we have eij (0) < 0 following (52). This completes the proof. Lexicographic order always defines an unique order for a group of agents, and Lemma 6 shows that it can be interpreted as a weighted order. Hence, we are ready to state the following result on lexicographic order: Theorem 3. Consider n agents in Rm . Assume that the agents’ initial order is given according to the lexicographic order defined by Eq. (43). f If N1 + Nnb ≥ 1, then consensus with inter-agent collision free can be guaranteed by consensus algorithm shown in Table 1. Proof. The proof of Theorem 3 can be provided following similar arguments in the proof of Theorems 1 and 2. The details are omitted to avoid redundance.
(A1) 9 agents uniformly distributed in the rectangle [0, 2]×[0, 1]; (A2) 1 agent in the center of square [0, 2]2 with x3 = 1; 4 agents in the vertexes of square [0, 2]2 with x3 = 0.5; and 5 agents randomly distributed in square [0, 2]2 with x3 = 0. For Case A1, two different orders namely weighted order with weight vector ω = [1, 0]T and lexicographic order are used to define orders for the given 9 agents. The simulation results are respectively shown in Figs. 4 and 5. As shown in Figs. 4(a) and 5(a), different orders result in different interaction topologies. With these two interaction topologies, the evolution of agents’ states over time are shown in Figs. 4(b) and 5(b). It can be observed that all agents will eventually converge to the same point, hence consensus of agents is achieved. Moreover, from the weighted states shown in Figs. 4(c) and 5(c), we can see that χ1 is always the smallest, χ9 is always the largest, and the order of χi , i = 1, 2, . . . , 9 remains unchanged over time. Order preservation and therefore collision-free are guaranteed in these two simulations. In order to show the possible applications of the proposed control strategies to agents in 3D like unmanned aerial vehicles (UAVs), the Case A2 in 3D also is simulated. The order for this case is determined by the weighted order with weight vector ω = [−0.05, −0.2, −1]T . The simulation results for Case A2 are illustrated in Fig. 6. As can be seen in these figures, the proposed strategies guarantee consensus as well as collision-free for the case in 3D. 5.2. Comparison study Finally, for better evaluation, we compare the proposed strategy with cyclic pursuit strategy in Lin et al. (2004) that also claimed to solve the collision-free consensus problem. Since the cyclic
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(a) Initial states and interaction topology.
(b) Agent’s states in (x1 , x2 ) plane.
(c) Weighted states χ = x1 +
1 2 x . 2
Fig. 5. Simulation results for Case A1 using lexicographic order or weighted order with weight vector ω = [1, 12 ]T .
(a) Initial states and order.
(b) Agent’s states in (x1 , x2 , x3 ) plane.
(c) Weighted states χ = −0.05x1 − 0.2x2 − x3 .
Fig. 6. Simulation results for Case A2 using weighted order with weight vector ω = [−0.05, −0.2, −1]T .
(a) Graph for agents that are ordered counterclockwise.
(b) Graph for agents that are ordered by weighted order.
(c) Agent’s trajectory using the cyclic pursuit strategy.
(d) Agent’s trajectory using the proposed strategy.
Fig. 7. Comparison results for agents with cyclic pursuit strategy and the proposed strategy.
pursuit strategy in Lin et al. (2004) is proposed for agents in plane, a case in 2D is considered for comparison. As shown in Fig. 7(a) and (b), agents are labeled counterclockwise with the algorithm in Lin et al. (2004); while in our algorithm, agents are
ordered by weighted order with weight vector ω = [−1, 0]T . With the different orders, the interaction topologies in two algorithms respectively are directed ring and bidirectional chain. Under different communication topology shown in Fig. 7(a) and (b), and
Z. Miao et al. / Automatica 64 (2016) 217–225
the same communication intensity, i.e., aij = 5 when agent i can receive information from agent j, the simulation results of two algorithms are illustrated in Fig. 7(c) and (d) respectively. Comparing the results, it can be observed that although both algorithms achieve collision-free consensus, the paths of agents with our proposed strategy are shorter than that with cyclic pursuit strategy in Lin et al. (2004). This phenomenon is cause by the ring structure of the interaction topology in Lin et al. (2004) which results in the rotation-like motion. Actually, through some calculations, it can be obtained that the Laplacian matrix of the graph in Lin et al. (2004) is a circulant matrix which has complex eigenvalues with nonzero imaginary parts. The complex eigenvalues would bring about oscillations in the motion of agents. 6. Conclusions In this paper, distributed control strategies were proposed for a multi-agent network to achieve consensus while avoiding interagent collisions. Two orders were introduced to define order for agents, and tools from algebraic graph theory, matrix theory, and monotone systems theory were provided to investigate consensus as well as collision-free for agent networks. Finally, some cases in 2D and 3D were studied to demonstrate the validity of the proposed algorithms. It is worth noticing that this work is only the first step on collision-free consensus problem. Some other issues are still untouched and need to be investigated. Further works may include collision-free consensus problem for agents with doubleintegrator dynamics, for agents with communication delays, and for agent team with leaders. References Altafini, C. (2013). Consensus problems on networks with antagonistic interactions. IEEE Transactions on Automatic Control, 58, 935–946. Angeli, D., & Sontag, E. (2003). Monotone control systems. IEEE Transactions on Automatic Control, 48, 1684–1698. Bai, H., Arcak, M., & Wen, J. (2011). Cooperative control design: a systematic, passivitybased approach. New York: Springer-Verlag. Blondel, V., Hendrickx, J., & Tsitsiklis, J. (2009). On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Transactions on Automatic Control, 54, 2586–2597. Bullo, F., Cortes, J., & Martinez, S. (2009). Distributed control of robotic networks. Princeton: Princeton University. Cao, Y., Yu, W., Ren, W., & Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9, 427–438. Cortes, J., Martinez, S., & Bullo, F. (2006). Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions. IEEE Transactions on Automatic Control, 51, 1289–1298. Dimarogonas, D., Loizou, S., Kyriakopoulos, K., & Zavlanos, M. (2006). A feedback stabilization and collision avoidance scheme for multiple independent nonpoint agents. Automatica, 42, 229–243. Fax, J., & Murray, R. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49, 1465–1476. Gazi, V. (2005). Swarm aggregations using artificial potentials and sliding-mode control. IEEE Transactions on Robotics, 21, 1208–1214. Hendrickx, J. (2008). Order preservation in a generalized version of Krause’s opinion dynamics model. Physica A. Statistical Mechanics and its Applications, 387, 5255–5262. Hirsch, M., & Smith, H. (2005). Monotone dynamical systems. In Handbook of differential equations, ordinary differential equations. Vol. 2. Amsterdam: Elsevier. Hussein, I. I., & Stipanovic, D. M. (2007). Effective coverage control for mobile sensor networks with guaranteed collision avoidance. IEEE Transactions on Control Systems Technology, 15, 642–657. Jadbabaie, A., Lin, J., & Morse, A. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48, 988–1001. Juang, J. C. (2013). On the formation patterns under generalized cyclic pursuit. IEEE Transactions on Automatic Control, 58, 2401–2405. Kar, S., & Moura, J. (2010). Distributed consensus algorithms in sensor networks: Quantized data and random link failures. IEEE Transactions on Signal Processing, 58, 1383–1400. Lewis, F., Zhang, H., Hengster-Movric, K., & Das, A. (2014). Cooperative control of multi-agent systems: optimal and adaptive design approaches. London: Springer. Lin, Z., Francis, B., & Maggiore, M. (2005). Necessary and sufficient graphical conditions for formation control of unicycles. IEEE Transactions on Automatic Control, 50, 121–127. Lin, Z., Mireille, B., & Bruce, F. (2004). Local control strategies for groups of mobile autonomous agents. IEEE Transactions on Automatic Control, 49, 622–629.
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Marshall, J., Broucke, M., & Francis, B. (2006). Pursuit formations of unicycles. Automatica, 42, 3–12. Mastellone, S., Stipanovic, D. M., Graunke, C. R., Intlekofer, K. A., & Spong, M. W. (2008). Formation control and collision avoidance for multi-agent nonholonomic systems: theory and experiments. International Journal of Robotics Research, 27, 107–126. Mesbahi, M., & Egerstedt, M. (2010). Graph theoretic methods for multi-agent networks. Princeton: Princeton University. Miao, Z., Wang, Y., & Fierro, R. (2015). Collision-free consensus via order preservation in multi-agent networks. In Proc of the 54th IEEE conference on decision and control (in press). Moreau, L. (2005). Stability of multiagent systems with time-dependent communication links. IEEE Transactions on Automatic Control, 50, 169–182. Olfati-Saber, R., Fax, J., & Murray, R. (2007). Consensus and cooperation in networked multi-agent systems. In Proceedings of the IEEE: 95 (pp. 215–233). Qu, Z. (2009). Cooperative control of dynamical systems: applications to autonomous vehicles. London: Springer-Verlag. Ren, W., & Beard, R. (2008). Distributed consensus in multi-vehicle cooperative control: theory and applications. London: Springer-Verlag. Ren, W., & Cao, Y. (2011). Distributed coordination of multi-agent networks. London: Springer-Verlag. Semsar-Kazerooni, E., & Khorasani, K. (2012). Team cooperation in a network of multi-vehicle unmanned systems: synthesis of consensus algorithms. New York: Springer. Sinha, A., & Ghose, D. (2006). Generalization of linear cyclic pursuit with application to rendezvous of multiple autonomous agents. IEEE Transactions on Automatic Control, 51, 1819–1824. Smith, H. (2008). Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems (2nd ed.). New York: American Mathematical Society. Smith, S., Broucke, M., & Francis, B. (2005). A hierarchical cyclic pursuit scheme for vehicle networks. Automatica, 41, 1045–1053. Tanner, H., Jadbabaie, A., & Pappas, G. (2007). Flocking in fixed and switching networks. IEEE Transactions on Automatic Control, 52, 863–868. Wang, C., Xie, G., & Cao, M. (2013). Forming circle formations of anonymous mobile agents with order preservation. IEEE Transactions on Automatic Control, 58, 3248–3254. Yang, Y., Dimarogonas, D., & Hu, X. (2014). Opinion consensus of modified Hegselmann-Krause models. Automatica, 50, 622–627. Zhiqiang Miao received the B.S. degree in Electrical and Information Engineering from Hunan University, Changsha, China, in 2010, where he is currently working toward the Ph.D. degree with the College of Electrical and Information Engineering. From September 2014 to September 2015, he was a Visiting Ph.D. Student with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, USA, supported by the China Scholarship Council. His research interests include robotics, distributed control of multi-agent systems, nonlinear systems and control, and neural networks. Yaonan Wang received the B.S. degree in Computer Engineering from East China Science and Technology University (ECSTU), Fuzhou, China, in 1981 and the M.S. and Ph.D. degrees in Electrical Engineering from Hunan University, Changsha, China, in 1990 and 1994, respectively. From 1994 to 1995, he was a postdoctoral Research Fellow with the National University of Defense Technology. From 1981 to 1994, he worked with ECSTU. From 1998 to 2000, he was a senior Humboldt Fellow in Germany, and from 2001 to 2004, he was a visiting professor with the University of Bremen, Bremen, Germany. He has been a professor at Hunan University since 1995. His research interests include robot control, intelligent control and information processing, industrial process control, and image processing. Rafael Fierro received the M.Sc. degree in control engineering from the University of Bradford, Bradford, UK, in 1990, and the Ph.D. degree in electrical engineering from the University of Texas at Arlington, in 1997. He is s currently a Professor of the Department of Electrical & Computer Engineering, University of New Mexico (UNM) where he has been since 2007. Prior to joining UNM, he held a postdoctoral appointment with the GRASP Laboratory at the University of Pennsylvania and a faculty position with the Department of Electrical and Computer Engineering at Oklahoma State University. His research interests include hybrid and embedded systems, heterogeneous multivehicle coordination, cooperative and distributed control of multi-agent systems, mobile sensor networks, and robotics. He directs the Multi-Agent, Robotics, Hybrid and Embedded Systems Laboratory at UNM. Dr. Fierro was the recipient of a Fulbright Scholarship, a 2004 National Science Foundation CAREER Award, and the 2007 International Society of Automation Transactions Best Paper Award. He is serving as associate editor for the Journal of Intelligent and Robotics Systems, IEEE Control Systems Magazine, and IEEE Transactions on Automation Science and Engineering.
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Brief paper
Collision-free consensus in multi-agent networks: A monotone systems perspective✩ Zhiqiang Miao a,b , Yaonan Wang a , Rafael Fierro b a
College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
b
Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131-0001, USA
article
info
Article history: Received 14 December 2014 Received in revised form 22 June 2015 Accepted 29 October 2015 Available online 7 December 2015 Keywords: Consensus Collision-free Order preservation Monotone systems Multi-agent systems Cooperative control
abstract This paper addresses the collision-free consensus problem in a network of agents with single-integrator dynamics. Distributed algorithms with local interactions are proposed to achieve consensus while guaranteeing collision-free among agents during the evolution of the multi-agent networks. The novelty of the proposed algorithms lies in the definition of neighbors for each agent, which is different from the usual sense that neighbors are selected by the distance between agents in the state space. In the proposed strategies, the neighbor set for each agent is determined by the distance or difference between agents in the index space after ordering and labeling all agents according to certain ordering rules including weighted order and lexicographic order. The consensus analysis of the proposed algorithms is presented with some existing results on algebraic graph theory and matrix analysis. Meanwhile, by realizing the relations between order preservation and collision-free, a systematic analysis framework on order preservation and hence collision-free for agents in arbitrary dimension is provided based on tools from monotone systems theory. Illustrated numerical examples are presented to validate the effectiveness of the proposed strategies. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction There has been a growing interest in distributed coordination and cooperative control of multi-agent systems in the past decade (Bai, Arcak, & Wen, 2011; Bullo, Cortes, & Martinez, 2009; Lewis, Zhang, Hengster-Movric, & Das, 2014; Mesbahi & Egerstedt, 2010; Qu, 2009; Ren & Beard, 2008; Ren & Cao, 2011; SemsarKazerooni & Khorasani, 2012). This board interest in multi-agent networks is motivated by the emerging applications including vehicle formations, cooperative robotics, sensor networks, and social networks. Consensus as one of the fundamental problems in the coordination of multi-agent systems, refers to the agreement of
✩ This work was supported in part by the National Natural Science Foundation of China (61573134, 61433016), National High Technology Research and Development Program of China (863 Program: 2012AA111004, 2012AA112312), Hunan Provincial Innovation Foundation for Postgraduate (521298960), and the fellowship from China Scholarship Council (CSC). The material in this paper was partially presented at the 54th IEEE Conference on Decision and Control, December 15–18, 2015, Osaka, Japan. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Z. Miao), [email protected] (Y. Wang), [email protected] (R. Fierro).
http://dx.doi.org/10.1016/j.automatica.2015.11.025 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
all agents upon certain quantities of interest. Consensus has been studied extensively in the literature (Cao, Yu, Ren, & Chen, 2013; Fax & Murray, 2004; Moreau, 2005; Olfati-Saber, Fax, & Murray, 2007), and the applications of consensus algorithms can be found in formation control (Lin, Francis, & Maggiore, 2005), rendezvous (Cortes, Martinez, & Bullo, 2006), flocking (Tanner, Jadbabaie, & Pappas, 2007), attitude alignment (Jadbabaie, Lin, & Morse, 2003), and sensor networks (Kar & Moura, 2010). Here we do not intend to provide a completed review on consensus problem, the interested reader is referred to the survey paper (Cao et al., 2013) and the mentioned monographs for more detailed discussions. The most well-known consensus algorithm is the local voting protocol which can be found in many of the aforementioned studies. Alternatively, cyclic pursuit strategies have recently been investigated for the distributed control multi-agent systems (Juang, 2013; Lin, Mireille, & Bruce, 2004; Marshall, Broucke, & Francis, 2006; Sinha & Ghose, 2006; Smith, Broucke, & Francis, 2005). In cyclic pursuit, each agent pursues its leading neighbor to form a ring network structure. The cyclic pursuit strategy is attractive because it inherently is decentralized and requires a small number of communication links. Cyclic pursuit strategies were utilized to address the agreement problem in Lin et al. (2004); Smith et al. (2005), formation control in Juang (2013) and Marshall
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(iii) A systematic analysis framework based on monotone systems theory is provided to investigate order preservation and hence collision-free for agents.
et al. (2006), and rendezvous problem in Sinha and Ghose (2006). Among these studies, a result in Lin et al. (2004) is closely relevant to the topic in this paper. It was showed in Lin et al. (2004) that if the agents in plane initially are ordered counterclockwise (or clockwise), then the order of agents will be preserved with cyclic pursuit, and therefore no collision would occur. Through the strategy proposed by Lin et al. (2004) achieved consensus while ensure collision-free among agents, the results are limited to agents in 2D, which render the applications involving unmanned aerial vehicles (UAV) in 3D difficult. Similar ideas in Lin et al. (2004) were reported in Wang, Xie, and Cao (2013) to solve the circle-forming problem for agents that move on a circle. In Wang et al. (2013), agents also are initially labeled counterclockwise on the circle, and each agent only receives information from its two immediate neighbors, resulting in an undirected ring network topology. The property of order preservation was studied as well to avoid inter-agent collisions. However, the result in Wang et al. (2013) is quite limited, as the motion of agents is restricted to onedimensional space of a circle. Most of the existing work on the distributed control of multiagent networks pays more attention on the final or ultimate state rather than the entire trajectory of the agents. Take the consensus task for example, all agents are expected to reach a common value eventually, disregarding the transient behavior of agents. Nevertheless, the transient behavior of agents needs to be restrained to meet certain constraints in some cases. Inter-agent collision-free is an example of such constraints. Intuitively, if there exist certain quantities or qualities of the network that remain unchanged during the evolution, then the information from these invariants can help us better understand the transient behavior of agents throughout evolution. Order preservation is a kind of invariants, which makes the transient behavior of agents more predictable. As observed by Lin et al. (2004) and Wang et al. (2013), the order preservation property is particularly useful to prevent collisions between agents. Order preservation also has been explored in some social networks like Krause’s opinion dynamics (Blondel, Hendrickx, & Tsitsiklis, 2009; Hendrickx, 2008; Yang, Dimarogonas, & Hu, 2014). A concise proof on order preservation was provided by Blondel et al. (2009) with simple arguments. However, the arguments only hold for agents in one dimension, which render the results rather conservative. A systematic analysis framework for arbitrary dimension remains to be provided. In this paper, order preservation will be studied in the context of monotone systems (Altafini, 2013; Angeli & Sontag, 2003; Hirsch & Smith, 2005; Smith, 2008). Monotone systems are orderpreserving systems, the trajectories of which preserve a given partial order through time. By realizing the relations between order preservation and collision-free, the problem of consensus with collision-free in a network of agents is addressed here. To be more specific, the contribution of this paper lies in designing distributed control laws such that all agents in the network reach the same value while guaranteeing inter-agent collision-free navigation. Preliminary results in this paper were presented in Miao, Wang, and Fierro (in press), where each agent is assumed to have the same number of neighbors. In this paper, we relax this assumption by generalizing the result to agents that have different number of neighbors. Moreover, we provide additional simulation case studies and a comparison of the proposed strategy with the cyclic pursuit strategy in Lin et al. (2004). The main contributions can be summarized as:
Lemma 1. The Laplacian matrix L has a simple zero eigenvalue with an associated eigenvector 1n , and all the other eigenvalues have positive real parts if and only if the digraph associated with has a directed spanning tree.
(i) Weighted order and lexicographic order are explored to define order for agents in arbitrary dimensional space; (ii) Neighbors for each agent are defined in the index space rather than in state space. Notice that agents in the network do not necessarily have the same number of neighbors.
Definition 1. Given a matrix S = [sij ]n×n , the associated digraph G(S ) of matrix S is a directed graph with n vertices indexed by 1, 2, . . . , n such that there is an edge in G(S ) from j to i if and only if sij ̸= 0.
The remainder of this paper is organized as follows. In Section 2, some mathematical preliminaries first are presented. Problem is formulated in Section 3. In Section 4, the proposed strategies and main results are stated. Simulation results for illustrating the effectiveness of the proposed strategies are given in Section 5. Section 6 concludes the paper. 2. Preliminaries 2.1. Notation The standard notations are used throughout this paper. N, R and C denote the sets of natural numbers, real numbers and complex numbers respectively. R+ denotes nonnegative real numbers, and Rn+ is the set of n-tuples for which all components belong to R+ . Let In ∈ Rn×n be the n-dimensional identity matrix; 1n ∈ Rn be the vector of all ones; 0n ∈ Rn be the vector of all zeros. For vectors x, y ∈ Rn , we denote x ≤ y if and only if xi ≤ yi for all i; x < y if and only if x ≤ y and x ̸= y; and x ≪ y if and only if xi < yi for all i. 2.2. Algebraic graph theory and matrix analysis In this subsection, some elements and results on graph theory and matrix analysis that will used in this paper are reviewed. These results can be found in Ren and Beard (2008) and Ren and Cao (2011), and references therein. Let G = {V , E , A} be a digraph with a node set V = {1, 2, . . . , n}, an edge set E ⊆ V × V , and an adjacency matrix A = [aij ]n×n with nonnegative elements. A directed edge denoted by (j, i) means that node i has access to node j, i.e., node i can receive information from node j. The elements of adjacency matrix A are defined as follows: If there is a directed link from node j to i (j ̸= i), then aij > 0; otherwise, aij = 0. We assume that aii = 0 for all i. The Laplacian matrix L = [lij ]n×n associated with the adjacency matrix A is defined as
−anij lij = aij
i ̸= j, i = j.
(1)
j=1,j̸=i
A directed path is a sequence of edges in the directed graph G with distinct nodes. A directed graph is strongly connected if, for any two distinct nodes j and i, there exists a directed path from node j to node i. A digraph with n nodes is called a directed tree if it has n − 1 edges and there exists a root node with directed paths to every other node. A directed spanning tree is a directed tree that includes all the nodes of the digraph. A digraph is said to have a directed spanning tree if there is at least one node having a directed path to every other node.
Z. Miao et al. / Automatica 64 (2016) 217–225
Lemma 2. A matrix S is irreducible if and only if its associated directed graph G(S ) is strongly connected.
1 0.9 0.8 0.7
States
Definition 2. Matrix S ∈ Rn×n is reducible if and only if for some permutation matrix P ∈ Rn×n , such that P T SP is in block triangular form; otherwise S is irreducible.
0.6 0.5 0.4 0.3 0.2 Collision
0.1 0
2.3. Monotone systems theory
0
x˙ = f (t , x).
(2)
Let Φt (x) : R+ × R → R denote the resulting semi-flow that describes the evolution of states in positive time. n
whenever x ≤ y and t ≥ 0,
(3)
whenever x < y and t > 0.
(4)
Definition 4. The system (2) is said to be cooperative if the off∂f diagonal elements of Jacobian matrix Df (t , x) = [ ∂ xi (t , x)]n×n are j
non-negative, i.e., for all (t , x) ∈ R+ × Rn ,
∂ fi (t , x) ≥ 0, ∂ xj
i ̸= j,
(5)
Lemma 3. Let system (2) be cooperative, and
(6) m
where xi ∈ R is the state of agent, and ui ∈ R is the control input of ith agent. The most common consensus algorithm called local voting protocol is given as ui =
aij (xj − xi ),
(7)
j∈Ni
where aij being the graph edge weights, Ni is the neighbor set of agent i. Then the closed-loop dynamics of agents is x˙ i =
3
4
5
1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
t [s]
To illustrate inter-agent collision, let us consider three agents in R. Denote x = [x1 , x2 , x3 ]T as the state vector, and the initial state of three agents are x(0) = [0, 0.3, 1]T . The consensus plots in Figs. 1 and 2 show the states xi in the consensus dynamics (8) with the graph topologies A1 = [0 1 1; 1 0 0; 1 0 0] and A2 = [0 1 0; 1 0 1; 0 1 0] respectively. It can be observed that agent x1 collides with agent x2 with graph topology A1 , while for agents with graph topology A2 , collision-free is ensured. In this paper, we will design the local interaction rules for agents represented by Eq. (6), so that the inter-agent collision-free consensus is achieved or reached in the sense that (i) consensus:
and system (2) is said to be irreducible, if the Jacobian matrix Df (t , x) is irreducible for all (t , x) ∈ R+ × Rn .
x˙ i = ui ,
t [s]
Fig. 2. States of three agents with graph topology A2 .
and strongly monotone if
Φt (x) ≪ Φt (y),
2
0.9
Definition 3. The map Φ is said to be monotone if
Φt (x) ≤ Φt (y),
1
Fig. 1. States of three agents with graph topology A1 .
States
The main analysis tools from monotone systems theory are now presented. These definitions and conditions on monotone systems are mainly based on the results in Smith (2008), and also can be found in Hirsch and Smith (2005). Given a function f (t , x) : R × Rn → Rn , piecewise continuous in t and of class C 1 with respect to x, the associated system of differential equations
n
219
aij (xj − xi ).
(8)
lim ∥xi (t ) − xj (t )∥ = 0,
t →∞
∀i, j.
(9)
(ii) collision-free: for ∀δ > 0, if ∥xi (0) − xj (0)∥ ≥ δ , then there exists a time-varying function d(t ) > 0 such that
∥xi (t ) − xj (t )∥ ≥ d(t ),
∀i ̸= j, t ≥ 0.
(10)
Remark 1. Collision avoidance is an important issue that has extensively been studied in the control design of multi-agent/robot systems. The collision avoidance problem has been studied in formation control (Dimarogonas, Loizou, Kyriakopoulos, & Zavlanos, 2006; Mastellone, Stipanovic, Graunke, Intlekofer, & Spong, 2008), coverage control (Hussein & Stipanovic, 2007), flocking (Tanner et al., 2007), and swarming (Gazi, 2005) of multi-agent systems. Comparing with the collision-free condition presented by Eq. (10), the collision avoidance condition in these studies is defined by ∥xi (t ) − xj (t )∥ ≥ d, ∀i ̸= j, t ≥ 0, where d > 0 is a positive constant. The most popular strategy utilized for collision avoidance is the potential field-based method, which consists of attraction to achieve the desired group behavior, and repulsion to avoid collision. However, it is evidently that the potential field-based method is inapplicable for avoiding collision in the consensus problem. 4. Methodology
j∈Ni
It has been shown in Ren and Beard (2008); Ren and Cao (2011) that consensus can be achieved with (8) provided that the associated digraph has a directed spanning tree. However, interagent collision may occur.
4.1. One dimensional case We first consider the special case when m = 1. For n agents xi ∈ R, i = 1, 2, . . . , n, we can always order them according to
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Z. Miao et al. / Automatica 64 (2016) 217–225
their different initial states. Without loss of generality, assume x1 (0) < x2 (0) < · · · < xi (0) < · · · < xn (0).
(11)
We assume that each agent only receives information from its neighbors. In this paper, neighbors are determined by the distance or difference between agents in the index space. Specifically, for agent xi , its neighbor Ni is defined as f
Ni = {xj | − Nib ≤ j − i ≤ Ni }, f Ni
f Nj ,
where ∈ N, ∈ N, and satisfy ≥ , ≤ if i < j. Furthermore, xi+k is called the kth nearest forward neighbor of xi ; while xi−k is called the kth nearest backward neighbor of xi . Now, consider the consensus algorithm (7) with communication protocol as follows: Nib
Njb
(i) Each agent i receives the information from Nib nearest f
backward neighbor and Ni nearest forward neighbor; (ii) Each agent sends its state information with fixed weight.
(13)
j=[i−Nib ]
p < 1, 1 ≤ p ≤ n, p>n
1 n
aij =
αj
e˙ i = −
j=[i−Nib ]
k=j
αj ei +
j=[i+1−Nib+1 ]
(14)
where αj > 0. The agents dynamics (13) can be written in a more compact form: x˙ = −Lx,
(15)
f
j=[i+1+Ni+1 ] k=j−1
+
f j=[i+Ni
αj ek .
(17)
]+1 k=i+1
The second term on the right side of Eq. (17) vanishes if [i − Nib ] =
[i + 1 − Nib+1 ]; and the third term vanishes if [i + Nif ] = [i + 1 + Nif+1 ]. Thus it can be obtained that e˙ = He,
(18)
(19)
e(0) ≪ 0n−1 ⇒ e(t ) ≪ 0n−1 ,
∀t ≥ 0 ,
where x = [x1 , x2 , . . . , xn ] is the state vector, L is the Laplacian matrix associated with matrix A = [aij ]n×n . We then can state the following result for the one dimensional case: Nnb
Theorem 1. Consider n agents in R. If + ≥ 1, then consensus with inter-agent collision free can be guaranteed with consensus dynamics (13).
e˙ i < −
αj ei .
(22)
j=[i+1−Nib+1 ]
Thus, for t ≥ 0, f
j=[i+Ni ]
|ei (t )| > exp
αj t |ei (0)|.
(23)
j=[i+1−Nib+1 ]
Then for ∀i < j and t ≥ 0, we have
|xi (t ) − xj (t )| =
· · · ≤ Nnf , we have if N1f ≥ 1, Nnb = 0 (Nnb ≥ 1, N1f = 0), then Nif ≥ 1 (Nib ≥ 1) for all i, which implies the digraph associated with
Laplacian matrix L or adjacency matrix A has a directed tree, and f f xn (x1 ) is the root node; if N1 ≥ 1, Nnb ≥ 1, then Ni ≥ 1, Nib ≥ 1 for all i, which implies the associated digraph is strongly connected. Thus consensus of all states xi (1 ≤ i ≤ n) can be achieved by applying Lemma 1. (ii) Collision-free: Define error vector e = [e1 , e2 , . . . , en−1 ], with ei = xi − xi+1 , i = 1, 2, . . . , n − 1, we have
e˙ i =
j=[i−Nib ]
αj (xj − xi ) −
j=[i+1−Nib+1 ]
αj (xj − xi+1 ).
>
|ek (t )|
k=i
f
l=[k+Nk ]
exp −
k=i
αl t |ek (0)|
l=[k+1−Nkb+1 ]
≥ exp(−λt )|xi (0) − xj (0)|, where λ = maxi≤k≤j−1 {
f l=[k+Nk ]
l=[k+1−Nkb+1 ]
(24)
αl } > 0, 1 ≤ i < j ≤ n.
It follows Eq. (24) that collision-free condition (10) holds for some exponential decay function. This completes the proof. Remark 2. The error vector e can be obtained by a coordinate transformation e = Px, where the matrix P ∈ R(n−1)×n is 1
(16)
k=j−1
|xk (t ) − xk+1 (t )| =
k=j−1
f
j=[i+1+Ni+1 ]
k=i
Proof. (i) Consensus: Consider agent dynamics given by Eq. (15). f f f Because N1 + Nnb ≥ 1, and N1b ≥ N2b ≥ · · · ≥ Nnb , N1 ≤ N2 ≤
f
(21)
f
k=j−1 f N1
(20)
Thus, the order of all agents is preserved. Furthermore, as ej (t ) < 0, ∀t ≥ 0, j = 1, 2, . . . , n − 1, Eq. (17) implies that
T
j=[i+Ni ]
α j ek
j=[i+Ni ]
j = i, −Nib ≤ j − i ≤ Nif
0
j=[i+1−Nib+1 ]−1 k=i−1
x1 (t ) < x2 (t ) < · · · < xi (t ) < · · · < xn (t ).
for ∀p ∈ N. This operator ensures the index would within the set {1, 2, . . . , n}. The weight aij satisfies
f
j=[i+Ni ]
following the order preservation property of system (18). Inequality (20) implies for all t ≥ 0,
where the operator [·] is defined as
[p ] = p
] and [i + ] ≤ [i + 1 + Nif+1 ]. If [i − Nib ] < [i + 1 − Nib+1 ], ] < [i + 1 + Nif+1 ], we have
and applying Lemma 3, one obtains
aij (xj − xi ),
[i + 1 − and [i +
ei (0) < 0 ⇒ e(0) ≪ 0n−1 ,
f
j=[i+Ni ]
f
f Ni
where H = [hij ](n−1)×(n−1) with hij ≥ 0, i ̸= j. Thus system (18) is cooperative by Definition 4. Using the condition
According to this protocol, the dynamics of each agent is
x˙ i =
Nib+1 f Ni
(12)
f Ni
Nib
f
Because Nib ≥ Nib+1 and Ni ≤ Ni+1 , it can be checked that [i − Nib ] ≤
P =
−1 1
−1 .. .
..
.
.
1
−1
Z. Miao et al. / Automatica 64 (2016) 217–225
221
Table 1 Collision-free consensus algorithm. Input: xi (0) ∈ Rm , ui (t ) ∈ Rm , i = 1, 2, . . . , n, t ≥ 0. (i) Order agents according to weighted order (or lexicographic order) on their initial states; (ii) Determine the neighbors for each agent by set (12); (iii) Update agent’s state according to the voting protocol (13). Output: xi (t ) ∈ Rm , i = 1, 2, . . . , n, t ≥ 0.
Fig. 3. Order of n agents in R3 space.
It can be concluded that the monotonicity of error dynamics e is guaranteed by the monotonicity of state dynamics x, and the specific matrix P defined above. In the proposed algorithm, weights aij are assumed to satisfy aij = αj , ∀i, j, for the sake of simplicity. Actually, aij only needs to be non-increasing with the index f
f
distance or difference |i − j|. If Nib = 1, Ni = 0, or Ni = 1, Nib = 0 for all i, then the digraph forms a chain topology structure, which requires minimum number of communication links. Remark 3. In the proposed algorithm, agent xi cannot immediately determine whether another agent xj is its communication neighbor or not just through the information received from the agent xj . The agent xi may need some global information to acquire the order of agent xj in the group, and then judge if it is within its neighbor set defined by Eq. (12). However, the global information is required only at the initial time to determine the order for the agent group. Because the order of agents is preserved during the involution, the neighbor set for each agent remains unchanged. 4.2. Arbitrary dimensional case For n agents xi ∈ Rm , i = 1, 2, . . . , n, where m can be an arbitrary non-zero natural number, we cannot order them directly according to their initial states xi (0). Because two vectors xi (0) and xj (0) may unrelated, that is, neither xi (0) < xj (0) nor xj (0) < xi (0) holds. Nevertheless, we can map the vectors xi in Rm to χi in R by a weight vector:
χi = ωT xi ,
i = 1, 2, . . . , n,
(25)
where ω ∈ R is the non-zero weight vector, χi ∈ R can be viewed as the new weighted state of agent i. Define the n × m matrix of states m
T x1
xT 2 X = . , ..
(26)
and denote χ = [χ1 , χ2 , . . . , χn ]T , then Eq. (25) can be rewritten as (27)
Similar with the one dimensional case, we can now order the agents according to their weighted states without loss of generality, assume that
χ1 (0) ≤ χ2 (0) ≤ · · · ≤ χi (0) ≤ · · · ≤ χn (0).
X˙ = −LX ,
(29)
where the Laplacian matrix L is same as that in Eq. (15). Now, the main result can be stated in the following theorem: Theorem 2. Consider n agents in Rm . If either of the following two conditions holds, f
(C1) χ1 (0) < χn (0), and N1 ≥ 1, Nnb ≥ 1;
f
(C2) χ1 (0) < χ2 (0) < · · · < χn (0), and N1 + Nnb ≥ 1; then consensus with inter-agent collision free can be guaranteed by consensus algorithm shown in Table 1. Proof. Case C1 is considered first. The consensus and collision-free of consensus dynamics (29) will be investigated respectively. f f (i) Consensus: If N1 ≥ 1, Nnb ≥ 1, then Ni ≥ 1, Nib ≥ 1 for all i. It can be easily checked that the communication graph associated with the Laplacian matrix L in Eq. (29) is strongly connected. Thus consensus of all states xi (1 ≤ i ≤ n) is achieved with the aid of Lemma 1. Following Eq. (25), the consensus of states xi (1 ≤ i ≤ n) also implies the consensus of states χi (1 ≤ i ≤ n). (ii) Collision-free: Using Eqs. (27) and (29), the dynamics of χ can be obtained as
χ˙ = −Lχ .
(30)
Define e = [e1 , e2 , . . . , en−1 ], with ei = χi − χi+1 , i = 1, 2, . . . , n − 1, then along the same lines in the proof of Theorem 1, we have e˙ = He,
(31)
where the matrix H = [hij ](n−1)×(n−1) with hij ≥ 0 (i ̸= j) is the same as that in Eq. (18). Thus system (31) is cooperative. Because f Ni ≥ 1, Nib ≥ 1 for all i, it can be easily checked that the associated graph of matrix H is strongly connected. Thus, system (31) is also irreducible using Lemma 2. Since system (31) is cooperative and irreducible, and
χ1 (0) < χn (0) ⇒ e(0) ̸= 0n−1 ,
xTn
χ = X ω.
and χj (0) may be equal even through xi (0) is not equal to xj (0). An illustrating example in R3 is shown in Fig. 3, where χ3 (0) = χ4 (0). Analogously, we consider the same protocol as in one dimensional case. The proposed collision-free consensus algorithm for agents in arbitrary dimension is summarized in Table 1. Therefore, the consensus dynamics of agents in Rm is
(28)
In this paper, the order shown in (28) that obtained by comparing weighted states is referred as weighted order. It should be noted that ‘=’ is allowed in inequality (28), as two weighted states χi (0)
(32)
then applying Lemma 4, we have e(0) ≤ 0n−1 , e(0) ̸= 0n−1 ⇒ e(t ) ≪ 0n−1 ,
∀t > 0.
(33)
Inequality (33) implies for all t > 0,
χ1 (t ) < χ2 (t ) < · · · < χi (t ) < · · · < χn (t ).
(34)
Thus, the order of the agents is preserved. Furthermore, as ej (t ) < 0, ∀t > 0, j = 1, 2, . . . , n − 1, we have for all t > 0, f
j=[i+Ni ]
e˙ i < −
j=[i+1−Nib+1 ]
αj ei .
(35)
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Z. Miao et al. / Automatica 64 (2016) 217–225
Thus, for t ≥ t0 > 0,
|ei (t )| > exp −
f
j=[i+Ni ]
αj (t − t0 ) |ei (t0 )|.
(36)
j=[i+1−Nib+1 ]
Then it follows (36) that for ∀i < j and t ≥ t0 > 0,
|χi (t ) − χj (t )| > exp(−λ(t − t0 ))|χi (t0 ) − χj (t0 )|,
(37)
where λ is the same as that in Eq. (24). Considering the fact that
|χi (t ) − χj (t )| ≤ ∥ω∥∥xi (t ) − xj (t )∥,
(38)
Proof. We prove this claim by contradiction. Assume that such ω does not exists, then for ∀ ω ∈ Rm , it belongs to one or more of sets Cij with i, j = 1, 2, . . . , n: Cij = {ω ∈ Rm | ωT (xi (0) − xj (0)) = 0},
(40)
and Cij satisfies
which yields
∥xi (t ) − xj (t )∥ > ∥ω∥−1 exp(−λ(t − t0 ))|χi (t0 ) − χj (t0 )|.
Lemma 5. Given n agents xi ∈ Rm with different initial states xi (0), i = 1, 2, . . . , n. There always exists a weight vector ω ∈ Rm such that ωT xi (0) ̸= ωT xj (0) whenever xi (0) ̸= xj (0).
Cij = Rm .
(41)
1≤i,j≤n
(39)
Inequality (39) implies that collision-free is guaranteed. Now we consider Case C2. Along the same lines in the proof of Theorem 1, and using the fact (38), the proof for Case C2 can be easily derived. This completes the proof of Theorem 2. Remark 4. Because the order relationship is transitive (which means if a < b (a ≤ b), and b < c (b ≤ c ), then a < c (a ≤ c ), for a, b, c ∈ R), in order to give the order for agent group, we only need to define appropriate ordering rules to determine the order for each two agents. For agents in one dimension space xi ∈ R, i = 1, 2, . . . , n, the ordering rule for each agent pair xi , xj can be simply as: Denote xi < xj , whenever the value of state xi is less than the value of state xj . Whereas for agents in a high-dimensional space xi ∈ Rm , i = 1, 2, . . . , n, the ordering rule for each agent pair xi , xj is: Denote xi < xj , (xi ≤ xj ), whenever the value of weighted state ωT xi is less than (no larger than) the value of weighted state ωT xj , where ω ∈ Rm is the nonzero weight vector. Remark 5. The consensus dynamics (29) represents a linear system, thus the consensus or convergence of xi certainly is robust to disturbances. However, if there exist disturbances in the error dynamics (31), then the error ei may no longer always greater than zero. This implies the order preservation of xi probably is nonrobust to the disturbances. The reason is that order preservation or monotonicity of systems is an elegant mathematical property that established for systems with accurate models. A formal robustness analysis is an area of future research. Remark 6. The order of agents is determined by the vector ω, besides the initial states. Different weight vector may results in different order of agents, and thus different interaction topology. Because ‘=’ is allowed when ordering agents according to their initial weighted states, as shown in inequality (28). In such case, the initial order of agents may be not unique, even for a given weight vector. However, the non-uniqueness of order makes no difference on the correctness of Theorem 2, though it may be better to select an appropriate weight vector such that the order of agents is uniquely determined. 4.3. Lexicographic order: a special weighted order This subsection addresses the issue on the uniqueness of order for agents in arbitrary dimension. First, we show that there always exists an appropriate weight vector which ensures uniqueness of the order. Then the lexicographic order which always define unique order for agents is introduced, and interpreted as a special weighted order in the previous subsection. As mentioned before, the order of agents may be not unique even for a given weight vector. Thus, is it possible to find a specific weight vector such that the agents’ initial order is uniquely determined by their weighted states? A positive answer is given below:
Eq. (41) implies that there exists at least one set Ci0 j0 which contains infinite number of elements, that is
ωkT (xi0 (0) − xj0 (0)) = 0,
k = 1, 2, . . . .
(42)
The above infinite numbers of equations hold only if xi0 (0) = xj0 (0), however this is not true. This completes the proof. Lemma 5 ensures the existence of the weight vector that uniquely determines the order for a given group of agents. In the following, we will introduce a new order: lexicographic order, which can be viewed as a special weighted order that ensures the uniqueness of the order. Definition 5. Given two vectors a, b ∈ Rm , the lexicographic order is defined as a
(43)
where ‘iff’ represents ‘‘if and only if’’; and ai , bi respectively denote the ith element of vectors a, b. Before giving the interpretation of lexicographic order, some definitions first are presented. For n states xi ∈ Rm , let xki denote the kth element of vector xi . Given initial states xi (0) (1 ≤ i ≤ n), for ∀xri (0) ̸= xrj (0)(1 ≤ r ≤ m), we define γk (1 ≤ k ≤ m − 1) as
γk ≥
max
1≤i,j≤n;1≤r ≤m
|xir +k (0) − xrj +k (0)| |xri (0) − xrj (0)|
.
(44)
Denote
ω = [1, γ −1 , γ −2 , . . . , γ −(m−1) ]T ,
(45)
where γ satisfies 1
γ ≥ γ0 max (γk ) k ,
(46)
1≤k≤m−1
with γ0 ≥ 2. Based on these definitions, we have the following result: Lemma 6. Given n agents xi ∈ Rm with different initial states xi (0), i = 1, 2, . . . , n. For any two initial states xi (0) and xj (0) (1 ≤ i, j ≤ n), if xi (0)
∀ k < s,
xki (0) = xkj (0) and
xsi (0) < xsj (0).
(47)
Define eij (0) = ωT (xi (0) − xj (0)), then we have eij (0) =
k=m k=1
γ −(k−1) (xki (0) − xkj (0)).
(48)
Z. Miao et al. / Automatica 64 (2016) 217–225
(b) Agent’s states in (x1 , x2 ) plane.
(a) Initial states and interaction topology.
223
(c) Weighted states χ = x1 .
Fig. 4. Simulation results for Case A1 using weighted order with weight vector ω = [1, 0]T .
Denote dsij (0) = xsi (0) − xsj (0) for easy notation. If s = m, then k=m−1
eij (0) =
5. Simulation results
γ −(k−1) (xki (0) − xkj (0)) + γ −(m−1) dsij (0)
k=1
= γ −(m−1) dsij (0) < 0.
(49)
If s < m, then k=s−1
eij (0) =
γ −(k−1) (xki (0) − xkj (0)) + γ −(s−1) dsij (0)
k=1
+
k =m
γ −(k−1) (xki (0) − xkj (0))
k=s+1
= γ −(s−1) dsij (0) +
k=m
γ −(k−1) (xki (0) − xkj (0)).
(50)
k=s+1
Since dsij (0) < 0, and using (44), we have for k > s,
|xki (0) − xkj (0)| ≤ (γ /γ0 )k−s |dsij (0)|. k=m k=s+1
≤
γ −(k−1)
(51)
γ γ0
k−s
1 k 1− γ −(s−1) dsij (0). γ0 k=1
|dsij (0)|
k=m−s
(52)
When γ0 ≥ 2, it can be easily checked that
1 k > 0. γ0 k=1
k=m−s
1−
5.1. Case study The following two cases including two different initial conditions in 2D and 3D are considered:
Substituting (51) into (50), yields eij (0) ≤ γ −(s−1) dsij (0) −
To verify the effectiveness of the proposed control strategies, some numerical simulations are carried out in this section. From the results in the previous section, we know that the interaction topology for agents in the proposed strategies is determined by agents’ initial states and their ordering rules. Thus, agents with different initial states and different ordering rules both are considered. For the sake of simplicity, in all simulations of the proposed f algorithms, we assume that Nib = Ni = 1 for all i, and aij = 5 if agent i can receive information from agent j; otherwise aij = 0. In the following, we first investigate the proposed algorithm for agents in 2D and 3D respectively. Then a comparison study between the proposed algorithm and that in Lin et al. (2004) is carried out for further illustration.
(53)
Because dsij (0) < 0, we have eij (0) < 0 following (52). This completes the proof. Lexicographic order always defines an unique order for a group of agents, and Lemma 6 shows that it can be interpreted as a weighted order. Hence, we are ready to state the following result on lexicographic order: Theorem 3. Consider n agents in Rm . Assume that the agents’ initial order is given according to the lexicographic order defined by Eq. (43). f If N1 + Nnb ≥ 1, then consensus with inter-agent collision free can be guaranteed by consensus algorithm shown in Table 1. Proof. The proof of Theorem 3 can be provided following similar arguments in the proof of Theorems 1 and 2. The details are omitted to avoid redundance.
(A1) 9 agents uniformly distributed in the rectangle [0, 2]×[0, 1]; (A2) 1 agent in the center of square [0, 2]2 with x3 = 1; 4 agents in the vertexes of square [0, 2]2 with x3 = 0.5; and 5 agents randomly distributed in square [0, 2]2 with x3 = 0. For Case A1, two different orders namely weighted order with weight vector ω = [1, 0]T and lexicographic order are used to define orders for the given 9 agents. The simulation results are respectively shown in Figs. 4 and 5. As shown in Figs. 4(a) and 5(a), different orders result in different interaction topologies. With these two interaction topologies, the evolution of agents’ states over time are shown in Figs. 4(b) and 5(b). It can be observed that all agents will eventually converge to the same point, hence consensus of agents is achieved. Moreover, from the weighted states shown in Figs. 4(c) and 5(c), we can see that χ1 is always the smallest, χ9 is always the largest, and the order of χi , i = 1, 2, . . . , 9 remains unchanged over time. Order preservation and therefore collision-free are guaranteed in these two simulations. In order to show the possible applications of the proposed control strategies to agents in 3D like unmanned aerial vehicles (UAVs), the Case A2 in 3D also is simulated. The order for this case is determined by the weighted order with weight vector ω = [−0.05, −0.2, −1]T . The simulation results for Case A2 are illustrated in Fig. 6. As can be seen in these figures, the proposed strategies guarantee consensus as well as collision-free for the case in 3D. 5.2. Comparison study Finally, for better evaluation, we compare the proposed strategy with cyclic pursuit strategy in Lin et al. (2004) that also claimed to solve the collision-free consensus problem. Since the cyclic
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(a) Initial states and interaction topology.
(b) Agent’s states in (x1 , x2 ) plane.
(c) Weighted states χ = x1 +
1 2 x . 2
Fig. 5. Simulation results for Case A1 using lexicographic order or weighted order with weight vector ω = [1, 12 ]T .
(a) Initial states and order.
(b) Agent’s states in (x1 , x2 , x3 ) plane.
(c) Weighted states χ = −0.05x1 − 0.2x2 − x3 .
Fig. 6. Simulation results for Case A2 using weighted order with weight vector ω = [−0.05, −0.2, −1]T .
(a) Graph for agents that are ordered counterclockwise.
(b) Graph for agents that are ordered by weighted order.
(c) Agent’s trajectory using the cyclic pursuit strategy.
(d) Agent’s trajectory using the proposed strategy.
Fig. 7. Comparison results for agents with cyclic pursuit strategy and the proposed strategy.
pursuit strategy in Lin et al. (2004) is proposed for agents in plane, a case in 2D is considered for comparison. As shown in Fig. 7(a) and (b), agents are labeled counterclockwise with the algorithm in Lin et al. (2004); while in our algorithm, agents are
ordered by weighted order with weight vector ω = [−1, 0]T . With the different orders, the interaction topologies in two algorithms respectively are directed ring and bidirectional chain. Under different communication topology shown in Fig. 7(a) and (b), and
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the same communication intensity, i.e., aij = 5 when agent i can receive information from agent j, the simulation results of two algorithms are illustrated in Fig. 7(c) and (d) respectively. Comparing the results, it can be observed that although both algorithms achieve collision-free consensus, the paths of agents with our proposed strategy are shorter than that with cyclic pursuit strategy in Lin et al. (2004). This phenomenon is cause by the ring structure of the interaction topology in Lin et al. (2004) which results in the rotation-like motion. Actually, through some calculations, it can be obtained that the Laplacian matrix of the graph in Lin et al. (2004) is a circulant matrix which has complex eigenvalues with nonzero imaginary parts. The complex eigenvalues would bring about oscillations in the motion of agents. 6. Conclusions In this paper, distributed control strategies were proposed for a multi-agent network to achieve consensus while avoiding interagent collisions. Two orders were introduced to define order for agents, and tools from algebraic graph theory, matrix theory, and monotone systems theory were provided to investigate consensus as well as collision-free for agent networks. Finally, some cases in 2D and 3D were studied to demonstrate the validity of the proposed algorithms. It is worth noticing that this work is only the first step on collision-free consensus problem. Some other issues are still untouched and need to be investigated. Further works may include collision-free consensus problem for agents with doubleintegrator dynamics, for agents with communication delays, and for agent team with leaders. References Altafini, C. (2013). Consensus problems on networks with antagonistic interactions. IEEE Transactions on Automatic Control, 58, 935–946. Angeli, D., & Sontag, E. (2003). Monotone control systems. IEEE Transactions on Automatic Control, 48, 1684–1698. Bai, H., Arcak, M., & Wen, J. (2011). Cooperative control design: a systematic, passivitybased approach. New York: Springer-Verlag. Blondel, V., Hendrickx, J., & Tsitsiklis, J. (2009). On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Transactions on Automatic Control, 54, 2586–2597. Bullo, F., Cortes, J., & Martinez, S. (2009). Distributed control of robotic networks. Princeton: Princeton University. Cao, Y., Yu, W., Ren, W., & Chen, G. (2013). An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics, 9, 427–438. Cortes, J., Martinez, S., & Bullo, F. (2006). Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions. IEEE Transactions on Automatic Control, 51, 1289–1298. Dimarogonas, D., Loizou, S., Kyriakopoulos, K., & Zavlanos, M. (2006). A feedback stabilization and collision avoidance scheme for multiple independent nonpoint agents. Automatica, 42, 229–243. Fax, J., & Murray, R. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49, 1465–1476. Gazi, V. (2005). Swarm aggregations using artificial potentials and sliding-mode control. IEEE Transactions on Robotics, 21, 1208–1214. Hendrickx, J. (2008). Order preservation in a generalized version of Krause’s opinion dynamics model. Physica A. Statistical Mechanics and its Applications, 387, 5255–5262. Hirsch, M., & Smith, H. (2005). Monotone dynamical systems. In Handbook of differential equations, ordinary differential equations. Vol. 2. Amsterdam: Elsevier. Hussein, I. I., & Stipanovic, D. M. (2007). Effective coverage control for mobile sensor networks with guaranteed collision avoidance. IEEE Transactions on Control Systems Technology, 15, 642–657. Jadbabaie, A., Lin, J., & Morse, A. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48, 988–1001. Juang, J. C. (2013). On the formation patterns under generalized cyclic pursuit. IEEE Transactions on Automatic Control, 58, 2401–2405. Kar, S., & Moura, J. (2010). Distributed consensus algorithms in sensor networks: Quantized data and random link failures. IEEE Transactions on Signal Processing, 58, 1383–1400. Lewis, F., Zhang, H., Hengster-Movric, K., & Das, A. (2014). Cooperative control of multi-agent systems: optimal and adaptive design approaches. London: Springer. Lin, Z., Francis, B., & Maggiore, M. (2005). Necessary and sufficient graphical conditions for formation control of unicycles. IEEE Transactions on Automatic Control, 50, 121–127. Lin, Z., Mireille, B., & Bruce, F. (2004). Local control strategies for groups of mobile autonomous agents. IEEE Transactions on Automatic Control, 49, 622–629.
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Tanner, H., Jadbabaie, A., & Pappas, G. (2007). Flocking in fixed and switching networks. IEEE Transactions on Automatic Control, 52, 863–868. Wang, C., Xie, G., & Cao, M. (2013). Forming circle formations of anonymous mobile agents with order preservation. IEEE Transactions on Automatic Control, 58, 3248–3254. Yang, Y., Dimarogonas, D., & Hu, X. (2014). Opinion consensus of modified Hegselmann-Krause models. Automatica, 50, 622–627. Zhiqiang Miao received the B.S. degree in Electrical and Information Engineering from Hunan University, Changsha, China, in 2010, where he is currently working toward the Ph.D. degree with the College of Electrical and Information Engineering. From September 2014 to September 2015, he was a Visiting Ph.D. Student with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, USA, supported by the China Scholarship Council. His research interests include robotics, distributed control of multi-agent systems, nonlinear systems and control, and neural networks. Yaonan Wang received the B.S. degree in Computer Engineering from East China Science and Technology University (ECSTU), Fuzhou, China, in 1981 and the M.S. and Ph.D. degrees in Electrical Engineering from Hunan University, Changsha, China, in 1990 and 1994, respectively. From 1994 to 1995, he was a postdoctoral Research Fellow with the National University of Defense Technology. From 1981 to 1994, he worked with ECSTU. From 1998 to 2000, he was a senior Humboldt Fellow in Germany, and from 2001 to 2004, he was a visiting professor with the University of Bremen, Bremen, Germany. He has been a professor at Hunan University since 1995. His research interests include robot control, intelligent control and information processing, industrial process control, and image processing. Rafael Fierro received the M.Sc. degree in control engineering from the University of Bradford, Bradford, UK, in 1990, and the Ph.D. degree in electrical engineering from the University of Texas at Arlington, in 1997. He is s currently a Professor of the Department of Electrical & Computer Engineering, University of New Mexico (UNM) where he has been since 2007. Prior to joining UNM, he held a postdoctoral appointment with the GRASP Laboratory at the University of Pennsylvania and a faculty position with the Department of Electrical and Computer Engineering at Oklahoma State University. His research interests include hybrid and embedded systems, heterogeneous multivehicle coordination, cooperative and distributed control of multi-agent systems, mobile sensor networks, and robotics. He directs the Multi-Agent, Robotics, Hybrid and Embedded Systems Laboratory at UNM. Dr. Fierro was the recipient of a Fulbright Scholarship, a 2004 National Science Foundation CAREER Award, and the 2007 International Society of Automation Transactions Best Paper Award. He is serving as associate editor for the Journal of Intelligent and Robotics Systems, IEEE Control Systems Magazine, and IEEE Transactions on Automation Science and Engineering.