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Leader-following consensus of multi-agent systems with jointly connected topology using distributed adaptive protocols
Available online at www.sciencedirect.com
Journal of the Franklin Institute 351 (2014) 5399–5410 www.elsevier.com/locate/jfranklin
Leader-following consensus of multi-agent systems with jointly connected topology using distributed adaptive protocols Xiaowu Mua,n, Xia Xiaoa, Kai Liua,b, Jian Zhanga a
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China b College of Science, Henan Institute of Engineering, Zhengzhou, Henan 451191, China
Received 30 June 2014; received in revised form 11 September 2014; accepted 30 September 2014 Available online 8 October 2014
Abstract The leader-following consensus problems for multi-agent systems with a linear and Lipschitz nonlinear dynamics are considered. Distributed adaptive protocols and Lipschitz distributed adaptive protocols are respectively designed for the linear and Lipschitz nonlinear cases, under which leader-following consensus is reached for jointly connected topology. Finally, a simulation example is provided to illustrate the theoretical results. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, coordination problem of multi-agent systems has received significant attention due to its broad applications in many areas including cooperative control of Sensor Networks [1], formation control [2], and traffic management [3]. A significant problem that appears frequently in the context of coordination of multi-agent systems is the consensus problem. Distributed strategies that achieve agreements have dramatic advances. Asynchronous asymptotic agreement problem for distributed decision-making systems considered in [4] is one of the pioneering works using distributed computation over networks. n
Corresponding author. E-mail address: [email protected] (X. Mu).
http://dx.doi.org/10.1016/j.jfranklin.2014.09.018 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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The average-consensus control problem for networks of agents with first-order dynamics was considered under fixed and switching topologies in [5]. Distributed observers were designed for tracking control of multi-agent networks with an active leader in [6]. The consensus of linear multiagent systems with switching directed topology by a new intermittent control was considered in [7]. The time-delay networked systems were discussed in [8,9]. The consensus problem of multi-agent systems was addressed in [10]. The distributed observer type protocols were given in [10], where there was a requirement that the smallest nonzero eigenvalue of the Laplacian matrix associated with the communication graph had to be known by each agent to determine the bound for the coupling weight. However, the smallest nonzero eigenvalue of the Laplacian matrix is global information. Therefore, the consensus protocols designed in [10] cannot be computed and implemented by the agents in a fully distributed fashion, i.e., using only the local information of its own and neighbors. To handle this problem, distributed adaptive static consensus protocols are proposed to adjust the coupling weights between neighboring agents in [11]. The effects of communication topology among the agents in the study of consensus problems are important, because it determines how the local behavior of each agent spreads throughout the group and has a strong impact on the stability of the multi-agent systems. When the interaction topology is time-varying, how to design control of multi-agent system is challenging. Under switching topologies, the consensus problem of multi-agent systems was discussed in [5,12–14]. The case of consensus with a leader which is also called distributed tracking or leader-following consensus is a particularly interesting topic, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones. A multi-agent leader-following problem with variable topology was considered in [15], where the state of the leader not only kept changing but also might not be measured. Wen [16] studied the distributed leader following consensus problem of linear multi-agent systems with switching directed topologies. Consensus tracking of nonlinear multi-agent systems with switching directed topology using M-matrix based approach was discussed in [17]. The leader-following consensus of non-linear multi-agent systems with jointly connected topology was considered in [18]. The leader-following consensus of multi-agent systems with adaptive dynamic protocol over switching communication topologies was mentioned in [19]. There are few results reported in the literature how to design distributed adaptive protocols to achieve the leader-following consensus with jointly connected topology. In this paper, we consider the leader-following consensus problem of the multi-agent systems with linear and nonlinear dynamics using linear and nonlinear distributed adaptive protocols under jointly connected topology. There are some main differences among this work and [11,19]. The consensus of multi-agent systems with fixed topology was studied in [11]. Li et al. [19] considered switching topologies, where the switching graphs were always connected in each switching interval. Here, only joint connectedness is assumed. Hence our restriction is more relaxed than the cases in [11,19]. In [11,19] leaderfollowing consensus of linear systems was considered. In our case, linear and nonlinear systems are also considered. The paper is organized as follows. In Section 2, some useful preliminary results are introduced and the problem formulation is presented. Section 3 contains the main results. Simulation results are presented in Section 4. The conclusion is given in Section 5. Notations: Throughout this paper, the following notations are used. Rn denotes the n-dimensional Euclidean space; AT stands for the transpose of the real matrix A; In represents an n n identity matrix; 1n ¼ ð1; …; 1ÞT A Rn ; diagfd1 ; d2 ; …; dn g denotes a diagonal matrix with diagonal elements being d 1 ; d 2 ; …; dn ; for real symmetric matrix P, P40ðP Z 0Þ means that matrix P is positive (semi-) definite; A B denotes the Kronecker product of matrices A and B.
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2. Preliminaries and problem formulation 2.1. Preliminaries Let G ¼ ðV; EÞ be an undirected graph of order N with the set of nodes V ¼ f1; …; Ng and the set of edges EDV V. ði; jÞA E means that there is an edge from node i to node j, i.e., agent i and j can receive information from each other. Accordingly, agent i and j are mutual neighbors. The set of neighbors of node i is denoted by N i ¼ fjA V : ðj; iÞA E; ja ig. A path is a sequence of connected edges in a graph. If there is a path between any two nodes of a graph G, then G is said to be connected, otherwise disconnected. Denote the adjacency matrix of graph G as A ¼ ½aij A RNN with aij ¼ 1 if ði; jÞA E and aij ¼ 0 otherwise. Note that aii ¼ 0. The degree matrix is denoted as Δ ¼ diagðΔ1 ; …; ΔN Þ, where Δi ¼ ∑j A N i aij . The Laplacian of graph G is defined as L ¼ Δ A, which is symmetric. We define graph G on nodes 0; 1; 2; …; N, which consists of graph G, node 0 and edges between the leader and its neighbors. If the leader is a neighbor of node i, then d i ¼ 1; otherwise, d i ¼ 0. We denote H ¼ L þ D, where D ¼ diagðd1 ; …; dN Þ. Lemma 1 (Godsil and Royle [20]). The Laplacian matrix L of graph G has a simple zero eigenvalue and all other eigenvalues are positive and real if and only if graph G is connected. Lemma 2 (Ni and Cheng [21]). ①The eigenvalues of the matrix H are non-negative. ②The matrix H is positive definite if and only if the graph G is connected. 2.2. Problem formulation Consider a multi-agent system consisting of N agents and a leader. The dynamics of the i-th agent is x_ i ðtÞ ¼ f i ðxi ðtÞ; ui ðtÞÞ;
i ¼ 1; 2; …; N;
ð1Þ
where xi ðtÞA R is the agent i's state, and ui ðtÞA R is the agent i's input, f i : R R -R is the function of xi ðtÞ; ui ðtÞ. The leader, indexed as i¼ 0, has dynamics as n
m
x_ 0 ðtÞ ¼ f 0 ðx0 ðtÞÞ;
n
m
n
ð2Þ
where x0 ðtÞA R is the leader's state, f0 is the function of x0 ðtÞ. Obviously, the leader's dynamics is independent of other agents. n
Definition 1. The leader-following consensus of system (1)–(2) is said to be achieved, if for each agent iA f1; 2; …; Ng, there is a protocol ui(t) such that the states of agents satisfy lim Jxi ðtÞ x0 ðtÞJ ¼ 0;
t-1
i ¼ 1; 2; …; N;
ð3Þ
for any initial condition xi ð0Þ; i ¼ 1; 2; …; N. Since the graph describing the interaction topology can vary with time, we need to consider all possible graphs fG p jp A Pg on the node set f0; 1; 2; …; Ng, where P is an index set. Accordingly, the subgraphs defined on the node set f1; 2; …; Ng can denote as fGp jp A Pg. We define a switching signal σðtÞ : ½0; 1Þ-P, which is a piecewise function. Therefore, the underlying graphs at time t on N þ 1 and N nodes are denoted as G σðtÞ and GσðtÞ . Obviously, the neighbor sets Ni of all agents, graph Laplacian L, all ðijÞ th entries aij of adjacency matrix A and the index
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number di describing the neighbors of the leader are all time varying. We use Ni(t), LσðtÞ , aij(t) and di(t) to denote their time varying versions, respectively. Then, H σðtÞ ¼ LσðtÞ þ DσðtÞ . Consider an infinite sequence of nonempty, bounded, non-overlapping and contiguous timeintervals ½t k ; t kþ1 Þ; k ¼ 0; 1; …, with t 0 ¼ 0, and t kþ1 t k r T for some constant T40. Suppose that in each interval ½t k ; t kþ1 Þ there is a sequence of non-overlapping subintervals w k 1 wk ½t 0k ; t 1k Þ; …; ½t jk ; t jþ1 ; t k Þ; k Þ; …; ½t k
t k ¼ t 0k ;
t kþ1 ¼ t wk k
j satisfying t jþ1 k t k Z τ; 0 r jr wk 1 for some integer wk Z 0 and a given constant τ, such that the time varying graph topology switches at time instants t 0k ; t 1k ; …; t wk k 1 . That is, during each j
time subinterval ½t jk ; t jþ1 k Þ the interconnection topology G σðtÞ is fixed and we denote it by G k . Note that in each of such subintervals, the subgraph G σðtÞ is permitted to be disconnected. We only require that the union of the switching graphs G σðtÞ on the time interval ½t k ; t kþ1 Þ is jointly connected in this paper. Definition 2. The union of a collection of graphs is a graph whose node and edge sets are the unions of the node and edge sets of all of the graphs in the collection. Moreover, such a collection is jointly connected if the union graph is a connected graph. The graphs are said to be jointly connected across the time interval ½t; t þ ΔtÞ if the union of graphs fG σðsÞ : s A½t; t þ ΔtÞ; Δt40g is jointly connected. Assumption 1. The collection of subgraphs G σðtÞ is jointly connected in each interval ½t k ; t kþ1 Þ, k ¼ 0; 1; … . Lemma 3 (Xu et al. [18]). Let matrices H j1 ; …; H jw be associated with the graphs G j1 ; …; G jw j respectively. If these graphs are jointly connected, then ∑iw¼ j1 H i is positive define. 3. Main results 3.1. Leader-following consensus of linear multi-agent systems Consider a multi-agent system consisting of N agents and a leader. The dynamics of each agent is described by x_ i ðtÞ ¼ Axi ðtÞ þ ui ðtÞ;
i ¼ 1; 2; …; N;
ð4Þ
where xi ðtÞA Rn is the agent i's state at time t, A A Rnn , and ui ðtÞ A Rn is the agent i's input. The dynamics of the leader, labeled as i¼ 0, is described by x_ 0 ðtÞ ¼ Ax0 ðtÞ;
ð5Þ
where x0 ðtÞA R is the leader's state. Here, we use the following adaptive protocol for agent i: " n
N
#
ui ðtÞ ¼ F ∑ kij ðtÞaij ðtÞðxj ðtÞ xi ðtÞÞ þ k i ðtÞd i ðtÞðx0 ðtÞ xi ðtÞÞ j¼1
k_ ij ðtÞ ¼ bij aij ðtÞðxi ðtÞ xj ðtÞÞT Γðxi ðtÞ xj ðtÞÞ k_ i ðtÞ ¼ bi d i ðtÞðxi ðtÞ x0 ðtÞÞT Γðxi ðtÞ x0 ðtÞÞ;
i ¼ 1; 2; …; N;
ð6Þ
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where bij ¼ bji and bi are positive constants, kij(t) denotes the time-varying coupling weight between agent i and agent j with k ij ð0Þ ¼ k ji ð0Þ, ki(t) denotes the time-varying coupling weight between the leader and the agent i, and F A Rnn and Γ A Rnn are the feedback gain matrices. The following theorem presents a sufficient condition for designing Eq. (6) to solve the leader following consensus problem. Theorem 1. Consider the multi-agent system (4), (5). Suppose Assumption 1 is satisfied and matrix A is Lyapunov stable. Then, all the agents follow the leader under the control law (6) with F ¼ P and Γ ¼ PT P, where P40 is a solution of the following linear matrix inequality: PA þ AT P r 0
ð7Þ
Moreover, each coupling weight kij(t) and ki(t) converges to finite steady-state value. Proof. Let ei ðtÞ ¼ xi ðtÞ x0 ðtÞ; i ¼ 1; 2; …; N, then the dynamics of the state error between the agent i and the leader is " # N
e_i ðtÞ ¼ Aei ðtÞ F ∑ kij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞd i ðtÞei ðtÞ j¼1
k_ ij ðtÞ ¼ bij aij ðtÞðei ðtÞ ej ðtÞÞT Γðei ðtÞ ej ðtÞÞ k_ i ðtÞ ¼ bi di ðtÞeTi ðtÞΓei ðtÞ; i ¼ 1; 2; …; N;
ð8Þ
Consider the Lyapunov function candidate N ðk ðtÞ αÞ2 ðkij ðtÞ αÞ2 i þ ∑ ; 2bij bi i ¼ 1 j ¼ 1;j a i i¼1
N
N
V ðt Þ ¼ ∑ eTi ðt ÞPei ðt Þ þ ∑ i¼1
N
∑
ð9Þ
where α is a positive constant. Obviously, V(t) is continuously differentiable at any time except for switching instants. Firstly, we want to show that V_ ðtÞ r 0 at any non-switching instants. We assume that subsystem p is active at time t. The time derivative of this Lyapunov candidate along the trajectory of system (8) is " # N N N T T T V_ ðtÞ ¼ ∑ ei ðtÞðPA þ A PÞei ðtÞ ∑ ei ðtÞPF ∑ K ij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞdi ðtÞei ðtÞ i¼1
N
i¼1
j¼1
N
þ ∑ ∑ ðk ij ðtÞ αÞaij ðtÞðei ðtÞej ðtÞÞT Γðei ðtÞ ej ðtÞÞ i¼1j¼1 N
ð10Þ
þ2 ∑ ðk i ðtÞαÞdi ðtÞeTi ðtÞΓei ðtÞ: i¼1
Because kij ð0Þ ¼ kji ð0Þ and bij ¼ bji , it follows from Eq. (8) that k ji ðtÞ ¼ kij ðtÞ, 8 t Z 0. Then we have N
∑
N
N
∑
i ¼ 1 j ¼ 1;j a i
Let
N
ðkij ðtÞ αÞaij ðtÞðei ðtÞ ej ðtÞÞT Γðei ðtÞ ej ðtÞÞ ¼ 2 ∑ ∑ ðkij ðtÞ αÞaij ðtÞeTi ðtÞΓðei ðtÞ ej ðtÞÞ:
eðtÞ ¼ ½eT1 ðtÞ; …; eTN ðtÞT . N
i¼1j¼1
Substituting F ¼ P and Γ ¼ PT P into Eq. (10), we can obtain " # N
N
i¼1
j¼1
V_ ðtÞ ¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPT P ∑ k ij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞd i ðtÞei ðtÞ i¼1
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N
N
þ2 ∑ ∑ ðk ij ðtÞ αÞaij ðtÞeTi ðtÞPT Pðei ðtÞ ej ðtÞÞ þ 2 ∑ ðk i ðtÞ αÞd i ðtÞeTi ðtÞPT Pei ðtÞ i¼1j¼1
i¼1
N
N
N
¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ 2α ∑ ∑ aij ðtÞeTi ðtÞPT Pðei ðtÞ i¼1
i¼1j¼1
N
ej ðtÞÞ 2α ∑ d i ðtÞeTi ðtÞPT Pei ðtÞ i¼1
¼ eT ðtÞ½I N ðPA þ AT PÞ 2αðLðtÞ þ DðtÞÞ ðPT PÞeðtÞ ¼ eT ðtÞ½I N ðPA þ AT PÞ 2αH p ðPT PÞeðtÞ r 2αeT ðtÞðH p PT PÞeðtÞ r0 Therefore, limt-1 VðtÞ exists. So, V(t) is bounded. In the following, we prove limt-1 eðtÞ ¼ 0. Considering the infinite sequence fVðt k Þ; k ¼ 0; 1; …g and using Cauchy's convergence criteria, we conclude that, 8 ϵ40, there exists a positive number K ϵ , such that for 8 k4K ϵ ; jVðt kþ1 Þ Vðt k Þjoϵ or equivalently, Rt j tkkþ1 V_ ðtÞ dtjoϵ. This integral can be rewritten as Z twþ1 wk 1 k ϵo ∑ V_ ðtÞ dt: w¼0
t wk
Because there are finite switches in the interval ½t k ; t kþ1 Þ, the number wk is finite for each k ¼ 0; 1; … . Thus, for 8k4K ϵ , we have Z twþ1 k V_ ðtÞ dt ϵr t wk
Z
r 2α o 2α
t wþ1 k
tw Z ktw þτ k t wk
eT ðtÞðH σðtwk Þ PT PÞeðtÞ dt eT ðtÞðH σðtwk Þ PT PÞeðtÞ dt;
w ¼ 0; 1; …; wk 1
or Z
t wk þτ
2α t wk
eT ðtÞðH σðtwk Þ PT PÞeðtÞ dtoϵ;
w ¼ 0; 1; …; wk 1:
which implies Z tþτ lim eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ dt ¼ 0: t-1
k
t
Since V_ ðtÞr 0, eðtÞ; k ij ðtÞ; ki ðtÞ are bounded and so is e_ ðtÞ from Eq. (8). Therefore, eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ is uniformly continuous. From the extension of Barbalat's Lemma, k we have lim eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ ¼ 0;
t-1
k
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From Lemma 3, ∑wwk¼01 H σðtwk Þ is positive definite due to the joint connectivity of the graphs through the time interval ½t k ; t kþ1 Þ. Thus, limt-1 ei ðtÞ ¼ 0; i ¼ 1; …; N. That is, the leader following consensus problem is solved. □ By noting Γ40, it can be seen from Eq. (8) that kij ðtÞ; ki ðtÞ are monotonically increasing. Since kij ðtÞ; k i ðtÞ are bounded. Then, the coupling weights k ij ðtÞ; ki ðtÞ converge to the finite steady-state values. 3.2. Leader-following consensus of nonlinear multi-agent systems We consider a multi-agent system consisting of N identical nonlinear agents and a nonlinear leader. The dynamics of each nonlinear agent is described by x_ i ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ ui ðtÞ;
i ¼ 1; 2; …; N;
ð11Þ
where xi ðtÞA R is the agent i's state at time t, A; B A R ui ðtÞA Rn is the agent i's input. The dynamics of the leader is described by n
nn
, f ðÞ is a nonlinear function, and
x_ 0 ðtÞ ¼ Ax0 ðtÞ þ Bf ðx0 ðtÞÞ:
ð12Þ
Here, we consider the following adaptive protocol for agent i: N
^ i ðtÞdi ðtÞ½gðx0 ðtÞÞ gðxi ðtÞÞ ui ðtÞ ¼ F^ ∑ kij ðtÞaij ðtÞ½gðxj ðtÞÞ gðxi ðtÞÞ þ Fk j¼1
^ k_ ij ðtÞ ¼ cij aij ðtÞðxi ðtÞ xj ðtÞÞT PF½gðx i ðtÞÞ gðxj ðtÞÞ T ^ ðtÞÞ gðx0 ðtÞÞ; k_ i ðtÞ ¼ ci di ðtÞðxi ðtÞ x0 ðtÞÞ PF½gðx i
i ¼ 1; 2; …; N;
ð13Þ
where cij ¼ cji and ci are positive constants, gðÞ is a nonlinear function, kij(t) and ki(t) are defined ^ Γ^ A Rnn are the feedback gain matrices. as in Eq. (6), F; Assumption 2. The nonlinear function f is assumed to satisfy the Lipschitz condition, i.e. J f ðxÞ f ðyÞJ r κ Jx y J;
8x; yA Rn ;
where κ Z 0 is a Lipschitz constant, and let Γ^ ¼ κI n . Assumption 3. For any positive semi-definite matrix Q A Rnn , and x; y A Rn , there exist positive constants ρ and γ such that non-linear function g satisfies the following conditions: ( ðxT yT ÞQðgðxÞ gðyÞÞ Z γðxT yT ÞQðx yÞ JgðxÞ gðyÞJ r ρ Jx y J: Theorem 2. Consider the multi-agent system (11), (12). Assume that Assumptions1, 2 and 3 are satisfied and matrix A is Lyapunov stable. Then, the leader following consensus is solved by the T control law (13) with the gain matrix F^ satisfying F^ P40, where P40 is a solution of the following linear matrix inequality: ! T AT P þ PA þ Γ^ Γ^ PB r 0: BT P I Moreover, each coupling weight kij(t) and ki(t) converges to finite steady-state value.
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Proof. Let ei ðtÞ ¼ xi ðtÞ x0 ðtÞ; i ¼ 1; 2; …; N, then the error dynamics can be described as e_ i ðtÞ ¼ Aei ðtÞ þ Bðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ " # N ^ þF ∑ k ij ðtÞaij ðtÞðgðxj ðtÞÞ gðxi ðtÞÞÞ þ ki ðtÞd i ðtÞðgðx0 ðtÞÞ gðxi ðtÞÞÞ j¼1
^ k_ ij ðtÞ ¼ cij aij ðtÞðei ðtÞ ej ðtÞÞT PFðgðx i ðtÞÞ gðxj ðtÞÞÞ _k i ðtÞ ¼ ci d i ðtÞeT ðtÞPFðgðx ^ i ¼ 1; 2; …; N: i ðtÞÞ gðx0 ðtÞÞÞ;
ð14Þ
i
Consider the Lyapunov function candidate N ðk ðtÞ βÞ2 ðk ij ðtÞ βÞ2 i þ ∑ ; 2c ci ij i ¼ 1 j ¼ 1;j a i i¼1
N
N
N
V ðt Þ ¼ ∑ eTi ðt ÞPei ðt Þ þ ∑ i¼1
∑
ð15Þ
where β is a positive constant. Since k ji ðtÞ ¼ k ij ðtÞ; 8t Z 0, then N
∑
N
^ ðkij ðtÞ βÞaij ðtÞðei ðtÞ ej ðtÞÞT PFðgðx i ðtÞÞ gðxj ðtÞÞÞ
∑
i ¼ 1 j ¼ 1;j a i N
N
^ ¼ 2 ∑ ∑ ðk ij ðtÞ βÞaij ðtÞeTi ðtÞPFðgðx i ðtÞÞ gðxj ðtÞÞÞ: i¼1j¼1
Let eðtÞ ¼ ½eT1 ðtÞ; …; eTN ðtÞT , we can obtain N
N
V_ ðtÞ ¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPBðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ i¼1 N
i¼1
N
þ2 ∑ ∑ kij ðtÞaij ðtÞeTi ðtÞF^ Pðgðxj ðtÞÞ gðxi ðtÞÞÞ T
i¼1j¼1 N
þ2 ∑ ki ðtÞdi ðtÞeTi ðtÞF^ Pðgðx0 ðtÞÞ gðxi ðtÞÞÞ T
i¼1 N N
þ2 ∑ ∑ ðk ij ðtÞ βÞaij ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðxj ðtÞÞÞ T
i¼1j¼1 N
þ2 ∑ ðki ðtÞ βÞd i ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðx0 ðtÞÞÞ T
i¼1
N
N
¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPBðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ i¼1
i¼1
N
N
2β ∑ ∑ aij ðtÞeTi ðtÞF^ i¼1j¼1
T
N
Pðgðxi ðtÞÞ gðxj ðtÞÞÞ 2β ∑ di ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðx0 ðtÞÞÞ T
i¼1
T ^ ¼ eT ðtÞðI N ðPA þ AT PÞÞeðtÞ þ 2eT ðtÞðI N PBÞf^ ðxðtÞÞ 2βeT ðtÞðH σðtÞ ðtÞ F^ PÞgðxðtÞÞ;
^ ¼ colðgðx1 ðtÞÞ gðx0 ðtÞÞ; …; where f^ ðxðtÞÞ ¼ colðf ðx1 ðtÞÞ f ðx0 ðtÞÞ; …; f ðxN ðtÞÞ f ðx 0 ðtÞÞÞ; gðxðtÞÞ gðxN ðtÞÞ gðx0 ðtÞÞÞ. At a non-switching time t, assume that the subgraph G p ; p A P is active. Then from Assumption 2, we obtain T T ^ ^ V_ ðtÞr eT ðtÞðI N ðPA þ AT P þ PBBT P þ Γ^ ΓÞeðtÞ 2βeT ðtÞðH p ðtÞ F^ PÞgðxðtÞÞ T T ^ r 2βγe ðtÞðH p ðtÞ F PÞeðtÞ
r0
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The rest steps of the proof are similar as those in Theorem 1. Thus, limt-1 ei ðtÞ ¼ 0; i ¼ 1; …; N. ^ i ðtÞ ej ðtÞÞ, k_ i ðtÞZ γci di ðtÞeTi ðtÞ From Eq. (14), we have k_ ij ðtÞZ γcij aij ðtÞðei ðtÞ ej ðtÞÞT PFðe ^ i ðtÞ, thus kij ðtÞ; ki ðtÞ are monotonically increasing. Since kij ðtÞ; ki ðtÞ are bounded. Then, each PFe coupling weight kij ðtÞ; k i ðtÞ converge to finite steady-state value. □ Remark 1. In the study of time-varying topologies, the most existing studies focused on consensus of linear multi-agent systems and generally required the time-varying topologies being connected or having a spanning tree. Jointly connected topology is a milder condition because it does not require the connection of the time-varying interconnection topology at any moment. In this paper, we consider the leader-following consensus of the linear and non-linear multi-agent systems and only require switching topologies to be jointly connected. Besides, distributed adaptive protocols and Lipschitz distributed adaptive protocols are introduced to solve such problems from new standpoints. However, the leader-following consensus for general linear and nonlinear multi-agent systems under jointly connected topology are still not comprehensively investigated. Remark 2. The consensus of multi-agent systems with nonlinear nodes and the synchronization problem of complex networks are essentially the same. In [22], a adaptive synchronization in tree-like dynamical networks was discussed, but the adaptive update law of coupling strength (coefficient) depending on global information was not distributed. In [23], under fixed undirected graph, a model of adaptive dynamical network for which the coupling and pinning strength as well as other parameters of each vertex were adapted depend only on the state information of its neighborhood and itself was discussed. The approaches used in [22,23] cannot be extended straightforward to network with switching topologies. Remark 3. Jointly connected topology is a special switching topology. The union topology in each time interval is connected, but the topology in each of the subintervals may be different and disconnected. Common Lyapunov function is proposed to address the topology in each time subinterval and the connected union topology in each time interval. Furthermore, we design distributed tracking protocols by constructing update law (adaptive control) of time-varying coupling weights kij ðtÞ; k i ðtÞ, common Lyapunov function is an appropriate tool to design adaptive control law. Remark 4. In this note, the matrix H σðtÞ associated with the graph G σðtÞ is positive semidefinite in each of the subintervals, which play an important role in proving leader following consensus. When the topology graph is switching and jointly connected directed, it cannot ensure that H σðtÞ is a nonsingular M-matrix. Therefore the approach in this paper or M-matrix based approach in [17] may be invalid. Furthermore, as pointed out in [17], how to construct distributed tracking protocols under fixed or switching directed topologies is still an open problem. 4. Simulation results In this section, we give one example to validate our theoretical results. We consider a nonlinear multi-agent system consisting of a leader and four agents with 0 1 0 1 0 1 0:1 0 0 11 10 0 10f 1 ðxi1 Þ B C B C B C 1 1 A; B ¼ @ 0 0:1 0 A; f ðxi ðtÞÞ ¼ @ 0 A¼@ 1 A 0
14 0
0
0
0:1
0
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in which f 1 ðxi1 Þ ¼ 0:0068xi1 0:00295ðJ xi1 þ 1 J Jxi1 þ 1J Þ. Let κ ¼ 0:127 and F^ ¼ 10I 3 . Using MatLab, we get 0 1 0:3193 0:6398 0:0816 B C 0:0780 A P ¼ @ 0:6398 9:4256 0:0816
0:0780
0:7187
Define the function g as ( 2x when J xJ 42 gð x Þ ¼ 1 2 x when J xJ r 2
ð16Þ
We suppose that possible interaction graphs are fG 1 ; G 2 ; G 3 ; G 4 ; G 5 ; G 6 g (see Fig. 1), and the interaction graphs are switched as G 1 -G 2 -G 3 -G 4 -G 5 -G 6 -G 1 -⋯, and each graph is active for 1 s. The switching signal σðtÞ is shown in Fig. 2. Note that both G 1 [ G 2 [ G 3 and G 4 [ G 5 [ G 6 are connected; therefore both fG 1 ; G 2 ; G 3 g and fG 4 ; G 5 ; G 6 g are jointly connected through some time intervals. Fig. 3 shows that the leader-following consensus is reached. The coupling weights kij ðtÞ; ki ðtÞ are shown in Fig. 4, which converge to finite steady-state values.
Fig. 1. Possible interaction topologies between leader and agents.
Fig. 2. Switching signal.
X. Mu et al. / Journal of the Franklin Institute 351 (2014) 5399–5410 2.5
3
2
2.5 2
1.5
1.5 1
0.5
xi2−x02
xi1−x01
1
0
0.5 0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
0
5
10
15
20
25
30
35
0
5
10
15
Time(s) 1
10
0.9
8
0.8
6
2
0.5
0 −4
0.3
−6
0.2
−8
0.1
10
15
35
20
25
30
0
35
0
x3(t)−x0(t)
0.4
−2
5
30
x (t)−x (t)
0.6
2
0
25
(t)−x (t) (t) xx11(t)−x 00
0.7
4 xi3−x03
20
Time(s)
12
−10
5409
x (t)−x (t) 4
0.2
0
0.4
0.6
0
0.8
1
Time(s)
Fig. 3. The error trajectories between the leader and all followers. 11
14
10
12
9 8
10
6
ki
k ij
7 5
8 6
4 4
3 2
2
1 0
0
5
10
15
20
25
30
35
40
0
0
5
t
10
15
20
25
30
35
40
t Fig. 4. Coupling weights kij(t) and ki(t).
5. Conclusion In this paper, we have addressed the leader-following consensus problems of multi-agent systems with a linear and Lipschitz nonlinear dynamics. Distributed adaptive protocols and Lipschitz distributed adaptive protocols are separately designed for the linear and Lipschitz
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nonlinear cases, under which leader-following consensus is reached for jointly connected topology. An interesting topic for future study is to consider the leader-following consensus for general linear and nonlinear multi-agent systems under jointly connected topology.
References [1] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, A survey on sensor networks, IEEE Commun. Mag. 40 (8) (2002) 102–114. [2] D.M. Stipanović, G. Inalhan, R. Teo, C.J. Tomlin, Decentralized overlapping control of a formation of unmanned aerial vehicles, Automatica 40 (8) (2004) 1285–1296. [3] J.L. Adler, G. Satapathy, V. Manikonda, B. Bowles, V.J. Blue, A multi-agent approach to cooperative traffic management and route guidance, Transp. Res. Part B : Methodol. 39 (4) (2005) 297–318. [4] J.N. Tsitsiklis, D.P. Bertsekas, M. Athans, Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Autom. Control 31 (9) (1986) 803–812. [5] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520–1533. [6] Y. Hong, G. Chen, L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica 44 (3) (2008) 846–850. [7] Wen G., Duan Z., Ren W., Chen G., Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications, Int. J. Robust Nonlinear Control 24 (16) (2014) 2438–2457. [8] L. Li, D. Ho, J. Lu, A unified approach to practical consensus with quantized data and time delay, IEEE Trans. Circuits Syst. I. Regular Pap. 60 (10) (2013) 2668–2678. [9] J. Lu, D. Ho, J. Cao, Synchronization in an array of nonlinearly coupled chaotic neural networks with delay coupling, Int. J. Bifurc. Chaos 18 (10) (2008) 3101–3111. [10] Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint, IEEE Trans. Circuits Syst. I: Regular Pap. 57 (1) (2010) 213–224. [11] Z. Li, W. Ren, X. Liu, M. Fu, Consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics using distributed adaptive protocols, IEEE Trans. Autom. Control 58 (7) (2013) 1786–1791. [12] K. You, Z. Li, L. Xie, Consensus condition for linear multi-agent systems over randomly switching topologies, Automatica 49 (10) (2013) 3125–3132. [13] A. Proskurnikov, Consensus in switching networks with sectorial nonlinear couplings: absolute stability approach, Automatica 49 (2) (2013) 488–495. [14] P. Lin, K. Qin, H. Zhao, M. Sun, A new approach to average consensus problems with multiple time-delays and jointly-connected topologies, J. Frankl. Inst. 349 (1) (2012) 293–304. [15] Y. Hong, J. Hu, L. Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica 42 (7) (2006) 1177–1182. [16] G. Wen, G. Hu, W. Yu, J. Cao, G. Chen, Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs, Syst. Control Lett. 62 (12) (2013) 1151–1158. [17] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies, IEEE Trans. Circuits Syst. I. Regular Pap. 61 (2) (2014) 499–511. [18] W. Xu, J. Cao, W. Yu, J. Lu, Leader-following consensus of non-linear multi-agent systems with jointly connected topology, IET Control Theory Appl. 8 (6) (2014) 432–440. [19] Z. Li, W. Ren, X. Liu, L. Xie, Distributed consensus of linear multi-agent systems with adaptive dynamic protocols, Automatica 49 (7) (2013) 1986–1995. [20] C.D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001. [21] W. Ni, D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Syst. Control Lett. 59 (3) (2010) 209–217. [22] J. Lu, J. Cao, Adaptive synchronization in tree-like dynamical networks, Nonlinear Anal.: Real World Appl. 8 (2007) 1252–1260. [23] W. Lu, Adaptive dynamical networks via neighborhood information: synchronization and pinning control, Chaos 17 (2007) 023122.
Journal of the Franklin Institute 351 (2014) 5399–5410 www.elsevier.com/locate/jfranklin
Leader-following consensus of multi-agent systems with jointly connected topology using distributed adaptive protocols Xiaowu Mua,n, Xia Xiaoa, Kai Liua,b, Jian Zhanga a
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China b College of Science, Henan Institute of Engineering, Zhengzhou, Henan 451191, China
Received 30 June 2014; received in revised form 11 September 2014; accepted 30 September 2014 Available online 8 October 2014
Abstract The leader-following consensus problems for multi-agent systems with a linear and Lipschitz nonlinear dynamics are considered. Distributed adaptive protocols and Lipschitz distributed adaptive protocols are respectively designed for the linear and Lipschitz nonlinear cases, under which leader-following consensus is reached for jointly connected topology. Finally, a simulation example is provided to illustrate the theoretical results. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In recent years, coordination problem of multi-agent systems has received significant attention due to its broad applications in many areas including cooperative control of Sensor Networks [1], formation control [2], and traffic management [3]. A significant problem that appears frequently in the context of coordination of multi-agent systems is the consensus problem. Distributed strategies that achieve agreements have dramatic advances. Asynchronous asymptotic agreement problem for distributed decision-making systems considered in [4] is one of the pioneering works using distributed computation over networks. n
Corresponding author. E-mail address: [email protected] (X. Mu).
http://dx.doi.org/10.1016/j.jfranklin.2014.09.018 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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The average-consensus control problem for networks of agents with first-order dynamics was considered under fixed and switching topologies in [5]. Distributed observers were designed for tracking control of multi-agent networks with an active leader in [6]. The consensus of linear multiagent systems with switching directed topology by a new intermittent control was considered in [7]. The time-delay networked systems were discussed in [8,9]. The consensus problem of multi-agent systems was addressed in [10]. The distributed observer type protocols were given in [10], where there was a requirement that the smallest nonzero eigenvalue of the Laplacian matrix associated with the communication graph had to be known by each agent to determine the bound for the coupling weight. However, the smallest nonzero eigenvalue of the Laplacian matrix is global information. Therefore, the consensus protocols designed in [10] cannot be computed and implemented by the agents in a fully distributed fashion, i.e., using only the local information of its own and neighbors. To handle this problem, distributed adaptive static consensus protocols are proposed to adjust the coupling weights between neighboring agents in [11]. The effects of communication topology among the agents in the study of consensus problems are important, because it determines how the local behavior of each agent spreads throughout the group and has a strong impact on the stability of the multi-agent systems. When the interaction topology is time-varying, how to design control of multi-agent system is challenging. Under switching topologies, the consensus problem of multi-agent systems was discussed in [5,12–14]. The case of consensus with a leader which is also called distributed tracking or leader-following consensus is a particularly interesting topic, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones. A multi-agent leader-following problem with variable topology was considered in [15], where the state of the leader not only kept changing but also might not be measured. Wen [16] studied the distributed leader following consensus problem of linear multi-agent systems with switching directed topologies. Consensus tracking of nonlinear multi-agent systems with switching directed topology using M-matrix based approach was discussed in [17]. The leader-following consensus of non-linear multi-agent systems with jointly connected topology was considered in [18]. The leader-following consensus of multi-agent systems with adaptive dynamic protocol over switching communication topologies was mentioned in [19]. There are few results reported in the literature how to design distributed adaptive protocols to achieve the leader-following consensus with jointly connected topology. In this paper, we consider the leader-following consensus problem of the multi-agent systems with linear and nonlinear dynamics using linear and nonlinear distributed adaptive protocols under jointly connected topology. There are some main differences among this work and [11,19]. The consensus of multi-agent systems with fixed topology was studied in [11]. Li et al. [19] considered switching topologies, where the switching graphs were always connected in each switching interval. Here, only joint connectedness is assumed. Hence our restriction is more relaxed than the cases in [11,19]. In [11,19] leaderfollowing consensus of linear systems was considered. In our case, linear and nonlinear systems are also considered. The paper is organized as follows. In Section 2, some useful preliminary results are introduced and the problem formulation is presented. Section 3 contains the main results. Simulation results are presented in Section 4. The conclusion is given in Section 5. Notations: Throughout this paper, the following notations are used. Rn denotes the n-dimensional Euclidean space; AT stands for the transpose of the real matrix A; In represents an n n identity matrix; 1n ¼ ð1; …; 1ÞT A Rn ; diagfd1 ; d2 ; …; dn g denotes a diagonal matrix with diagonal elements being d 1 ; d 2 ; …; dn ; for real symmetric matrix P, P40ðP Z 0Þ means that matrix P is positive (semi-) definite; A B denotes the Kronecker product of matrices A and B.
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2. Preliminaries and problem formulation 2.1. Preliminaries Let G ¼ ðV; EÞ be an undirected graph of order N with the set of nodes V ¼ f1; …; Ng and the set of edges EDV V. ði; jÞA E means that there is an edge from node i to node j, i.e., agent i and j can receive information from each other. Accordingly, agent i and j are mutual neighbors. The set of neighbors of node i is denoted by N i ¼ fjA V : ðj; iÞA E; ja ig. A path is a sequence of connected edges in a graph. If there is a path between any two nodes of a graph G, then G is said to be connected, otherwise disconnected. Denote the adjacency matrix of graph G as A ¼ ½aij A RNN with aij ¼ 1 if ði; jÞA E and aij ¼ 0 otherwise. Note that aii ¼ 0. The degree matrix is denoted as Δ ¼ diagðΔ1 ; …; ΔN Þ, where Δi ¼ ∑j A N i aij . The Laplacian of graph G is defined as L ¼ Δ A, which is symmetric. We define graph G on nodes 0; 1; 2; …; N, which consists of graph G, node 0 and edges between the leader and its neighbors. If the leader is a neighbor of node i, then d i ¼ 1; otherwise, d i ¼ 0. We denote H ¼ L þ D, where D ¼ diagðd1 ; …; dN Þ. Lemma 1 (Godsil and Royle [20]). The Laplacian matrix L of graph G has a simple zero eigenvalue and all other eigenvalues are positive and real if and only if graph G is connected. Lemma 2 (Ni and Cheng [21]). ①The eigenvalues of the matrix H are non-negative. ②The matrix H is positive definite if and only if the graph G is connected. 2.2. Problem formulation Consider a multi-agent system consisting of N agents and a leader. The dynamics of the i-th agent is x_ i ðtÞ ¼ f i ðxi ðtÞ; ui ðtÞÞ;
i ¼ 1; 2; …; N;
ð1Þ
where xi ðtÞA R is the agent i's state, and ui ðtÞA R is the agent i's input, f i : R R -R is the function of xi ðtÞ; ui ðtÞ. The leader, indexed as i¼ 0, has dynamics as n
m
x_ 0 ðtÞ ¼ f 0 ðx0 ðtÞÞ;
n
m
n
ð2Þ
where x0 ðtÞA R is the leader's state, f0 is the function of x0 ðtÞ. Obviously, the leader's dynamics is independent of other agents. n
Definition 1. The leader-following consensus of system (1)–(2) is said to be achieved, if for each agent iA f1; 2; …; Ng, there is a protocol ui(t) such that the states of agents satisfy lim Jxi ðtÞ x0 ðtÞJ ¼ 0;
t-1
i ¼ 1; 2; …; N;
ð3Þ
for any initial condition xi ð0Þ; i ¼ 1; 2; …; N. Since the graph describing the interaction topology can vary with time, we need to consider all possible graphs fG p jp A Pg on the node set f0; 1; 2; …; Ng, where P is an index set. Accordingly, the subgraphs defined on the node set f1; 2; …; Ng can denote as fGp jp A Pg. We define a switching signal σðtÞ : ½0; 1Þ-P, which is a piecewise function. Therefore, the underlying graphs at time t on N þ 1 and N nodes are denoted as G σðtÞ and GσðtÞ . Obviously, the neighbor sets Ni of all agents, graph Laplacian L, all ðijÞ th entries aij of adjacency matrix A and the index
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number di describing the neighbors of the leader are all time varying. We use Ni(t), LσðtÞ , aij(t) and di(t) to denote their time varying versions, respectively. Then, H σðtÞ ¼ LσðtÞ þ DσðtÞ . Consider an infinite sequence of nonempty, bounded, non-overlapping and contiguous timeintervals ½t k ; t kþ1 Þ; k ¼ 0; 1; …, with t 0 ¼ 0, and t kþ1 t k r T for some constant T40. Suppose that in each interval ½t k ; t kþ1 Þ there is a sequence of non-overlapping subintervals w k 1 wk ½t 0k ; t 1k Þ; …; ½t jk ; t jþ1 ; t k Þ; k Þ; …; ½t k
t k ¼ t 0k ;
t kþ1 ¼ t wk k
j satisfying t jþ1 k t k Z τ; 0 r jr wk 1 for some integer wk Z 0 and a given constant τ, such that the time varying graph topology switches at time instants t 0k ; t 1k ; …; t wk k 1 . That is, during each j
time subinterval ½t jk ; t jþ1 k Þ the interconnection topology G σðtÞ is fixed and we denote it by G k . Note that in each of such subintervals, the subgraph G σðtÞ is permitted to be disconnected. We only require that the union of the switching graphs G σðtÞ on the time interval ½t k ; t kþ1 Þ is jointly connected in this paper. Definition 2. The union of a collection of graphs is a graph whose node and edge sets are the unions of the node and edge sets of all of the graphs in the collection. Moreover, such a collection is jointly connected if the union graph is a connected graph. The graphs are said to be jointly connected across the time interval ½t; t þ ΔtÞ if the union of graphs fG σðsÞ : s A½t; t þ ΔtÞ; Δt40g is jointly connected. Assumption 1. The collection of subgraphs G σðtÞ is jointly connected in each interval ½t k ; t kþ1 Þ, k ¼ 0; 1; … . Lemma 3 (Xu et al. [18]). Let matrices H j1 ; …; H jw be associated with the graphs G j1 ; …; G jw j respectively. If these graphs are jointly connected, then ∑iw¼ j1 H i is positive define. 3. Main results 3.1. Leader-following consensus of linear multi-agent systems Consider a multi-agent system consisting of N agents and a leader. The dynamics of each agent is described by x_ i ðtÞ ¼ Axi ðtÞ þ ui ðtÞ;
i ¼ 1; 2; …; N;
ð4Þ
where xi ðtÞA Rn is the agent i's state at time t, A A Rnn , and ui ðtÞ A Rn is the agent i's input. The dynamics of the leader, labeled as i¼ 0, is described by x_ 0 ðtÞ ¼ Ax0 ðtÞ;
ð5Þ
where x0 ðtÞA R is the leader's state. Here, we use the following adaptive protocol for agent i: " n
N
#
ui ðtÞ ¼ F ∑ kij ðtÞaij ðtÞðxj ðtÞ xi ðtÞÞ þ k i ðtÞd i ðtÞðx0 ðtÞ xi ðtÞÞ j¼1
k_ ij ðtÞ ¼ bij aij ðtÞðxi ðtÞ xj ðtÞÞT Γðxi ðtÞ xj ðtÞÞ k_ i ðtÞ ¼ bi d i ðtÞðxi ðtÞ x0 ðtÞÞT Γðxi ðtÞ x0 ðtÞÞ;
i ¼ 1; 2; …; N;
ð6Þ
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where bij ¼ bji and bi are positive constants, kij(t) denotes the time-varying coupling weight between agent i and agent j with k ij ð0Þ ¼ k ji ð0Þ, ki(t) denotes the time-varying coupling weight between the leader and the agent i, and F A Rnn and Γ A Rnn are the feedback gain matrices. The following theorem presents a sufficient condition for designing Eq. (6) to solve the leader following consensus problem. Theorem 1. Consider the multi-agent system (4), (5). Suppose Assumption 1 is satisfied and matrix A is Lyapunov stable. Then, all the agents follow the leader under the control law (6) with F ¼ P and Γ ¼ PT P, where P40 is a solution of the following linear matrix inequality: PA þ AT P r 0
ð7Þ
Moreover, each coupling weight kij(t) and ki(t) converges to finite steady-state value. Proof. Let ei ðtÞ ¼ xi ðtÞ x0 ðtÞ; i ¼ 1; 2; …; N, then the dynamics of the state error between the agent i and the leader is " # N
e_i ðtÞ ¼ Aei ðtÞ F ∑ kij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞd i ðtÞei ðtÞ j¼1
k_ ij ðtÞ ¼ bij aij ðtÞðei ðtÞ ej ðtÞÞT Γðei ðtÞ ej ðtÞÞ k_ i ðtÞ ¼ bi di ðtÞeTi ðtÞΓei ðtÞ; i ¼ 1; 2; …; N;
ð8Þ
Consider the Lyapunov function candidate N ðk ðtÞ αÞ2 ðkij ðtÞ αÞ2 i þ ∑ ; 2bij bi i ¼ 1 j ¼ 1;j a i i¼1
N
N
V ðt Þ ¼ ∑ eTi ðt ÞPei ðt Þ þ ∑ i¼1
N
∑
ð9Þ
where α is a positive constant. Obviously, V(t) is continuously differentiable at any time except for switching instants. Firstly, we want to show that V_ ðtÞ r 0 at any non-switching instants. We assume that subsystem p is active at time t. The time derivative of this Lyapunov candidate along the trajectory of system (8) is " # N N N T T T V_ ðtÞ ¼ ∑ ei ðtÞðPA þ A PÞei ðtÞ ∑ ei ðtÞPF ∑ K ij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞdi ðtÞei ðtÞ i¼1
N
i¼1
j¼1
N
þ ∑ ∑ ðk ij ðtÞ αÞaij ðtÞðei ðtÞej ðtÞÞT Γðei ðtÞ ej ðtÞÞ i¼1j¼1 N
ð10Þ
þ2 ∑ ðk i ðtÞαÞdi ðtÞeTi ðtÞΓei ðtÞ: i¼1
Because kij ð0Þ ¼ kji ð0Þ and bij ¼ bji , it follows from Eq. (8) that k ji ðtÞ ¼ kij ðtÞ, 8 t Z 0. Then we have N
∑
N
N
∑
i ¼ 1 j ¼ 1;j a i
Let
N
ðkij ðtÞ αÞaij ðtÞðei ðtÞ ej ðtÞÞT Γðei ðtÞ ej ðtÞÞ ¼ 2 ∑ ∑ ðkij ðtÞ αÞaij ðtÞeTi ðtÞΓðei ðtÞ ej ðtÞÞ:
eðtÞ ¼ ½eT1 ðtÞ; …; eTN ðtÞT . N
i¼1j¼1
Substituting F ¼ P and Γ ¼ PT P into Eq. (10), we can obtain " # N
N
i¼1
j¼1
V_ ðtÞ ¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPT P ∑ k ij ðtÞaij ðtÞðei ðtÞ ej ðtÞÞ þ k i ðtÞd i ðtÞei ðtÞ i¼1
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N
N
þ2 ∑ ∑ ðk ij ðtÞ αÞaij ðtÞeTi ðtÞPT Pðei ðtÞ ej ðtÞÞ þ 2 ∑ ðk i ðtÞ αÞd i ðtÞeTi ðtÞPT Pei ðtÞ i¼1j¼1
i¼1
N
N
N
¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ 2α ∑ ∑ aij ðtÞeTi ðtÞPT Pðei ðtÞ i¼1
i¼1j¼1
N
ej ðtÞÞ 2α ∑ d i ðtÞeTi ðtÞPT Pei ðtÞ i¼1
¼ eT ðtÞ½I N ðPA þ AT PÞ 2αðLðtÞ þ DðtÞÞ ðPT PÞeðtÞ ¼ eT ðtÞ½I N ðPA þ AT PÞ 2αH p ðPT PÞeðtÞ r 2αeT ðtÞðH p PT PÞeðtÞ r0 Therefore, limt-1 VðtÞ exists. So, V(t) is bounded. In the following, we prove limt-1 eðtÞ ¼ 0. Considering the infinite sequence fVðt k Þ; k ¼ 0; 1; …g and using Cauchy's convergence criteria, we conclude that, 8 ϵ40, there exists a positive number K ϵ , such that for 8 k4K ϵ ; jVðt kþ1 Þ Vðt k Þjoϵ or equivalently, Rt j tkkþ1 V_ ðtÞ dtjoϵ. This integral can be rewritten as Z twþ1 wk 1 k ϵo ∑ V_ ðtÞ dt: w¼0
t wk
Because there are finite switches in the interval ½t k ; t kþ1 Þ, the number wk is finite for each k ¼ 0; 1; … . Thus, for 8k4K ϵ , we have Z twþ1 k V_ ðtÞ dt ϵr t wk
Z
r 2α o 2α
t wþ1 k
tw Z ktw þτ k t wk
eT ðtÞðH σðtwk Þ PT PÞeðtÞ dt eT ðtÞðH σðtwk Þ PT PÞeðtÞ dt;
w ¼ 0; 1; …; wk 1
or Z
t wk þτ
2α t wk
eT ðtÞðH σðtwk Þ PT PÞeðtÞ dtoϵ;
w ¼ 0; 1; …; wk 1:
which implies Z tþτ lim eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ dt ¼ 0: t-1
k
t
Since V_ ðtÞr 0, eðtÞ; k ij ðtÞ; ki ðtÞ are bounded and so is e_ ðtÞ from Eq. (8). Therefore, eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ is uniformly continuous. From the extension of Barbalat's Lemma, k we have lim eT ðtÞ½ðH σðt0k Þ þ ⋯ þ H σðtwk 1 Þ Þ PT PeðtÞ ¼ 0;
t-1
k
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From Lemma 3, ∑wwk¼01 H σðtwk Þ is positive definite due to the joint connectivity of the graphs through the time interval ½t k ; t kþ1 Þ. Thus, limt-1 ei ðtÞ ¼ 0; i ¼ 1; …; N. That is, the leader following consensus problem is solved. □ By noting Γ40, it can be seen from Eq. (8) that kij ðtÞ; ki ðtÞ are monotonically increasing. Since kij ðtÞ; k i ðtÞ are bounded. Then, the coupling weights k ij ðtÞ; ki ðtÞ converge to the finite steady-state values. 3.2. Leader-following consensus of nonlinear multi-agent systems We consider a multi-agent system consisting of N identical nonlinear agents and a nonlinear leader. The dynamics of each nonlinear agent is described by x_ i ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ ui ðtÞ;
i ¼ 1; 2; …; N;
ð11Þ
where xi ðtÞA R is the agent i's state at time t, A; B A R ui ðtÞA Rn is the agent i's input. The dynamics of the leader is described by n
nn
, f ðÞ is a nonlinear function, and
x_ 0 ðtÞ ¼ Ax0 ðtÞ þ Bf ðx0 ðtÞÞ:
ð12Þ
Here, we consider the following adaptive protocol for agent i: N
^ i ðtÞdi ðtÞ½gðx0 ðtÞÞ gðxi ðtÞÞ ui ðtÞ ¼ F^ ∑ kij ðtÞaij ðtÞ½gðxj ðtÞÞ gðxi ðtÞÞ þ Fk j¼1
^ k_ ij ðtÞ ¼ cij aij ðtÞðxi ðtÞ xj ðtÞÞT PF½gðx i ðtÞÞ gðxj ðtÞÞ T ^ ðtÞÞ gðx0 ðtÞÞ; k_ i ðtÞ ¼ ci di ðtÞðxi ðtÞ x0 ðtÞÞ PF½gðx i
i ¼ 1; 2; …; N;
ð13Þ
where cij ¼ cji and ci are positive constants, gðÞ is a nonlinear function, kij(t) and ki(t) are defined ^ Γ^ A Rnn are the feedback gain matrices. as in Eq. (6), F; Assumption 2. The nonlinear function f is assumed to satisfy the Lipschitz condition, i.e. J f ðxÞ f ðyÞJ r κ Jx y J;
8x; yA Rn ;
where κ Z 0 is a Lipschitz constant, and let Γ^ ¼ κI n . Assumption 3. For any positive semi-definite matrix Q A Rnn , and x; y A Rn , there exist positive constants ρ and γ such that non-linear function g satisfies the following conditions: ( ðxT yT ÞQðgðxÞ gðyÞÞ Z γðxT yT ÞQðx yÞ JgðxÞ gðyÞJ r ρ Jx y J: Theorem 2. Consider the multi-agent system (11), (12). Assume that Assumptions1, 2 and 3 are satisfied and matrix A is Lyapunov stable. Then, the leader following consensus is solved by the T control law (13) with the gain matrix F^ satisfying F^ P40, where P40 is a solution of the following linear matrix inequality: ! T AT P þ PA þ Γ^ Γ^ PB r 0: BT P I Moreover, each coupling weight kij(t) and ki(t) converges to finite steady-state value.
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Proof. Let ei ðtÞ ¼ xi ðtÞ x0 ðtÞ; i ¼ 1; 2; …; N, then the error dynamics can be described as e_ i ðtÞ ¼ Aei ðtÞ þ Bðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ " # N ^ þF ∑ k ij ðtÞaij ðtÞðgðxj ðtÞÞ gðxi ðtÞÞÞ þ ki ðtÞd i ðtÞðgðx0 ðtÞÞ gðxi ðtÞÞÞ j¼1
^ k_ ij ðtÞ ¼ cij aij ðtÞðei ðtÞ ej ðtÞÞT PFðgðx i ðtÞÞ gðxj ðtÞÞÞ _k i ðtÞ ¼ ci d i ðtÞeT ðtÞPFðgðx ^ i ¼ 1; 2; …; N: i ðtÞÞ gðx0 ðtÞÞÞ;
ð14Þ
i
Consider the Lyapunov function candidate N ðk ðtÞ βÞ2 ðk ij ðtÞ βÞ2 i þ ∑ ; 2c ci ij i ¼ 1 j ¼ 1;j a i i¼1
N
N
N
V ðt Þ ¼ ∑ eTi ðt ÞPei ðt Þ þ ∑ i¼1
∑
ð15Þ
where β is a positive constant. Since k ji ðtÞ ¼ k ij ðtÞ; 8t Z 0, then N
∑
N
^ ðkij ðtÞ βÞaij ðtÞðei ðtÞ ej ðtÞÞT PFðgðx i ðtÞÞ gðxj ðtÞÞÞ
∑
i ¼ 1 j ¼ 1;j a i N
N
^ ¼ 2 ∑ ∑ ðk ij ðtÞ βÞaij ðtÞeTi ðtÞPFðgðx i ðtÞÞ gðxj ðtÞÞÞ: i¼1j¼1
Let eðtÞ ¼ ½eT1 ðtÞ; …; eTN ðtÞT , we can obtain N
N
V_ ðtÞ ¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPBðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ i¼1 N
i¼1
N
þ2 ∑ ∑ kij ðtÞaij ðtÞeTi ðtÞF^ Pðgðxj ðtÞÞ gðxi ðtÞÞÞ T
i¼1j¼1 N
þ2 ∑ ki ðtÞdi ðtÞeTi ðtÞF^ Pðgðx0 ðtÞÞ gðxi ðtÞÞÞ T
i¼1 N N
þ2 ∑ ∑ ðk ij ðtÞ βÞaij ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðxj ðtÞÞÞ T
i¼1j¼1 N
þ2 ∑ ðki ðtÞ βÞd i ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðx0 ðtÞÞÞ T
i¼1
N
N
¼ ∑ eTi ðtÞðPA þ AT PÞei ðtÞ ∑ eTi ðtÞPBðf ðxi ðtÞÞ f ðx0 ðtÞÞÞ i¼1
i¼1
N
N
2β ∑ ∑ aij ðtÞeTi ðtÞF^ i¼1j¼1
T
N
Pðgðxi ðtÞÞ gðxj ðtÞÞÞ 2β ∑ di ðtÞeTi ðtÞF^ Pðgðxi ðtÞÞ gðx0 ðtÞÞÞ T
i¼1
T ^ ¼ eT ðtÞðI N ðPA þ AT PÞÞeðtÞ þ 2eT ðtÞðI N PBÞf^ ðxðtÞÞ 2βeT ðtÞðH σðtÞ ðtÞ F^ PÞgðxðtÞÞ;
^ ¼ colðgðx1 ðtÞÞ gðx0 ðtÞÞ; …; where f^ ðxðtÞÞ ¼ colðf ðx1 ðtÞÞ f ðx0 ðtÞÞ; …; f ðxN ðtÞÞ f ðx 0 ðtÞÞÞ; gðxðtÞÞ gðxN ðtÞÞ gðx0 ðtÞÞÞ. At a non-switching time t, assume that the subgraph G p ; p A P is active. Then from Assumption 2, we obtain T T ^ ^ V_ ðtÞr eT ðtÞðI N ðPA þ AT P þ PBBT P þ Γ^ ΓÞeðtÞ 2βeT ðtÞðH p ðtÞ F^ PÞgðxðtÞÞ T T ^ r 2βγe ðtÞðH p ðtÞ F PÞeðtÞ
r0
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The rest steps of the proof are similar as those in Theorem 1. Thus, limt-1 ei ðtÞ ¼ 0; i ¼ 1; …; N. ^ i ðtÞ ej ðtÞÞ, k_ i ðtÞZ γci di ðtÞeTi ðtÞ From Eq. (14), we have k_ ij ðtÞZ γcij aij ðtÞðei ðtÞ ej ðtÞÞT PFðe ^ i ðtÞ, thus kij ðtÞ; ki ðtÞ are monotonically increasing. Since kij ðtÞ; ki ðtÞ are bounded. Then, each PFe coupling weight kij ðtÞ; k i ðtÞ converge to finite steady-state value. □ Remark 1. In the study of time-varying topologies, the most existing studies focused on consensus of linear multi-agent systems and generally required the time-varying topologies being connected or having a spanning tree. Jointly connected topology is a milder condition because it does not require the connection of the time-varying interconnection topology at any moment. In this paper, we consider the leader-following consensus of the linear and non-linear multi-agent systems and only require switching topologies to be jointly connected. Besides, distributed adaptive protocols and Lipschitz distributed adaptive protocols are introduced to solve such problems from new standpoints. However, the leader-following consensus for general linear and nonlinear multi-agent systems under jointly connected topology are still not comprehensively investigated. Remark 2. The consensus of multi-agent systems with nonlinear nodes and the synchronization problem of complex networks are essentially the same. In [22], a adaptive synchronization in tree-like dynamical networks was discussed, but the adaptive update law of coupling strength (coefficient) depending on global information was not distributed. In [23], under fixed undirected graph, a model of adaptive dynamical network for which the coupling and pinning strength as well as other parameters of each vertex were adapted depend only on the state information of its neighborhood and itself was discussed. The approaches used in [22,23] cannot be extended straightforward to network with switching topologies. Remark 3. Jointly connected topology is a special switching topology. The union topology in each time interval is connected, but the topology in each of the subintervals may be different and disconnected. Common Lyapunov function is proposed to address the topology in each time subinterval and the connected union topology in each time interval. Furthermore, we design distributed tracking protocols by constructing update law (adaptive control) of time-varying coupling weights kij ðtÞ; k i ðtÞ, common Lyapunov function is an appropriate tool to design adaptive control law. Remark 4. In this note, the matrix H σðtÞ associated with the graph G σðtÞ is positive semidefinite in each of the subintervals, which play an important role in proving leader following consensus. When the topology graph is switching and jointly connected directed, it cannot ensure that H σðtÞ is a nonsingular M-matrix. Therefore the approach in this paper or M-matrix based approach in [17] may be invalid. Furthermore, as pointed out in [17], how to construct distributed tracking protocols under fixed or switching directed topologies is still an open problem. 4. Simulation results In this section, we give one example to validate our theoretical results. We consider a nonlinear multi-agent system consisting of a leader and four agents with 0 1 0 1 0 1 0:1 0 0 11 10 0 10f 1 ðxi1 Þ B C B C B C 1 1 A; B ¼ @ 0 0:1 0 A; f ðxi ðtÞÞ ¼ @ 0 A¼@ 1 A 0
14 0
0
0
0:1
0
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in which f 1 ðxi1 Þ ¼ 0:0068xi1 0:00295ðJ xi1 þ 1 J Jxi1 þ 1J Þ. Let κ ¼ 0:127 and F^ ¼ 10I 3 . Using MatLab, we get 0 1 0:3193 0:6398 0:0816 B C 0:0780 A P ¼ @ 0:6398 9:4256 0:0816
0:0780
0:7187
Define the function g as ( 2x when J xJ 42 gð x Þ ¼ 1 2 x when J xJ r 2
ð16Þ
We suppose that possible interaction graphs are fG 1 ; G 2 ; G 3 ; G 4 ; G 5 ; G 6 g (see Fig. 1), and the interaction graphs are switched as G 1 -G 2 -G 3 -G 4 -G 5 -G 6 -G 1 -⋯, and each graph is active for 1 s. The switching signal σðtÞ is shown in Fig. 2. Note that both G 1 [ G 2 [ G 3 and G 4 [ G 5 [ G 6 are connected; therefore both fG 1 ; G 2 ; G 3 g and fG 4 ; G 5 ; G 6 g are jointly connected through some time intervals. Fig. 3 shows that the leader-following consensus is reached. The coupling weights kij ðtÞ; ki ðtÞ are shown in Fig. 4, which converge to finite steady-state values.
Fig. 1. Possible interaction topologies between leader and agents.
Fig. 2. Switching signal.
X. Mu et al. / Journal of the Franklin Institute 351 (2014) 5399–5410 2.5
3
2
2.5 2
1.5
1.5 1
0.5
xi2−x02
xi1−x01
1
0
0.5 0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
0
5
10
15
20
25
30
35
0
5
10
15
Time(s) 1
10
0.9
8
0.8
6
2
0.5
0 −4
0.3
−6
0.2
−8
0.1
10
15
35
20
25
30
0
35
0
x3(t)−x0(t)
0.4
−2
5
30
x (t)−x (t)
0.6
2
0
25
(t)−x (t) (t) xx11(t)−x 00
0.7
4 xi3−x03
20
Time(s)
12
−10
5409
x (t)−x (t) 4
0.2
0
0.4
0.6
0
0.8
1
Time(s)
Fig. 3. The error trajectories between the leader and all followers. 11
14
10
12
9 8
10
6
ki
k ij
7 5
8 6
4 4
3 2
2
1 0
0
5
10
15
20
25
30
35
40
0
0
5
t
10
15
20
25
30
35
40
t Fig. 4. Coupling weights kij(t) and ki(t).
5. Conclusion In this paper, we have addressed the leader-following consensus problems of multi-agent systems with a linear and Lipschitz nonlinear dynamics. Distributed adaptive protocols and Lipschitz distributed adaptive protocols are separately designed for the linear and Lipschitz
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nonlinear cases, under which leader-following consensus is reached for jointly connected topology. An interesting topic for future study is to consider the leader-following consensus for general linear and nonlinear multi-agent systems under jointly connected topology.
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