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Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays
Neurocomputing 118 (2013) 289–300
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Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays Bo Liu a, Xiaoling Wang a, Housheng Su b,n, Yanping Gao c, Li Wang d a
College of Science, North China University of Technology, Beijing 100144, PR China Department of Control Science and Engineering, Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Huazhong University of Science and Technology, Wuhan 430074, PR China c College of Computer and Information Engineering, Beijing Technology and Business University, Beijing 100048, PR China d Beijing Key Lab of Urban Intelligent Traffic Control Technology, North China University of Technology, Beijing 100144, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2012 Received in revised form 18 January 2013 Accepted 12 February 2013 Communicated by H. Dong Available online 3 May 2013
This paper investigates second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays by introducing novel decentralized adaptive strategies to both the coupling strengths and the feedback gains. Based on the Lyapunov stability theory, it is proved that the position and the velocity of each agent can converge to those of the virtual leader respectively, when the network is connected and at least one agent is informed, without any global information of the multiagent system. Finally, some numerical simulations are presented to verify the theoretical results. & 2013 Elsevier B.V. All rights reserved.
Keywords: Adaptive control Second-order consensus Pinning control Time-varying delays Virtual leader
1. Introduction The consensus of multi-agent systems is ubiquitous in nature, such as flocking of birds, schooling of fish, etc. Nowadays, the consensus problem of multi-agent systems is an important developing direction of control theory due to the broad application of the consensus in control engineering and sensor networks [1–4]. Consensus problems are investigated for single-integrator kinematics [5,6], double-integrator dynamics [7–12] and high-orderintegrator dynamics [13], respectively. So far, many techniques, such as the pinning control and the adaptive control have been brought to drive the system to achieve consensus. The pinning control, signifying that only a small fraction of agents can receive information from the desired state, is a valid method to study the consensus of multi-agent systems. Wang and Chen [14] compared the effectiveness of specifically and randomly pinning schemes for the stabilization problem of the scale-free dynamical networks. In [15], the authors investigated the flocking of multi-agent systems with a virtual leader by virtue of pinning control and found that an uninformed agent can track the virtual leader if it can receive information from the informed
n
Corresponding author. Tel.: +86 27 87540210. E-mail address: [email protected] (H. Su).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.02.038
agents directly or indirectly from time to time. The adaptive strategy is usually used for the coordinated control algorithms when the global information of complex dynamical networks is unavailable. In [16], a robust adaptive control approach was proposed to solve the consensus problem of multi-agent systems. Two local adaptive strategies, that is, vertex-based strategy and edge-based strategy, were discussed for the synchronization of complex networks in [17]. In [18], the authors discussed the adaptive synchronization of first-order complex dynamical networks, while second-order consensus was investigated in [19,20] by introducing adaptive strategies to both the coupling strengths and feedback gains. Time delays exist widely in practice due to the jams of channel, noise, etc. Therefore, we should pay attention to the influence of the time delays when the consensus of multi-agent systems is considered. Average consensus of dynamical agents with timevarying delays was studied in both fixed and switching networks by applying a linear matrix inequality method [6]. The stability of a class of neural networks with multiple time delays was solved in [21]. In [8], Su et al. presented two kinds of algorithms to investigate the second-order consensus of multi-agent systems with identical time delay in fixed directed networks. Refs. [9,10] studied the second-order consensus of multi-agent systems with nonuniform time-delays, and Ref. [11] provided the consensus criteria in the form of linear matrix inequality (LMI) for the
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multi-agent systems with time-varying delays under both static interconnection topology and switching topologies, respectively. Motivated by the works above, in this paper, we focus on addressing the second-order consensus of multi-agent systems with nonlinear dynamics and time-varying delays via pinning control. The main contributions of the paper are threefold: (1) The nonlinear intrinsic dynamics of all the agents and the virtual leader are heterogeneous, (2) the position and the velocity of the agents are both with time-varying delays, and (3) novel decentralized adaptive strategies are introduced to solve the second-order consensus of multi-agent systems. For the bounded time-varying delays, we investigate the second-order consensus of two cases that the weighted coupling configuration matrix is symmetric and the weighted coupling configuration matrix is asymmetric by adding novel adaptive decentralized strategies to both the coupling strengths and feedback gains, and obtain that the position and the velocity of each agent can converge to those of the virtual leader respectively when the network is connected even though only one agent is informed, without requiring global information of the multi-agent system which is needed in Refs. [6,8], and calculating any LMI as Ref. [22]. The rest of the paper is organized as follows. Section 2 states the second-order model investigated and introduces some preliminaries to solve the second-order consensus. Section 3 provides theoretical analysis to the second-order consensus of multi-agent systems with nonlinear dynamics and time-varying delays. Section 4 presents the numerical simulations to the main results. Section 5 concludes this paper. Notation: Throughout this paper, the notation AT means the transpose of the matrix A, ∥ ∥ is the Euclidean norm, H ¼ diagfh1 ; h2 ; …; hN g.
2. Preliminaries and problem statement Consider a multi-agent system consisting of N agents and each agent moves according to the following dynamics: 8_ q i ðtÞ ¼ pi ðtÞ > > >_ > > p ðtÞ ¼ f i ðpi ðtÞÞ þ ∑ αij ðtÞaij ðqj ðt−τðtÞÞ−qi ðt−τðtÞÞ > < i j∈N i ð1Þ þ ∑ βij ðtÞaij ðpj ðt−τðtÞÞ−pi ðt−τðtÞÞ > > > j∈Ni > > > : þui ðtÞ; i ¼ 1; …; N; where qi ðtÞ ¼ ðqi1 ðtÞ; qi2 ðtÞ; …; qin ðtÞÞT ∈Rn (i ¼ 1; 2; …; N) describes the position vector of the i-th agent, and pi ðtÞ ¼ ðpi1 ðtÞ; pi2 ðtÞ; …; pin ðtÞÞT ∈Rn (i ¼ 1; 2; …; N) is its velocity vector. f i : Rn -Rn is a continuous function and f i ðpi ðtÞÞ describes the intrinsic dynamics of the ith agent, τðtÞ represents time-varying delays in transmission between agent i and agent j. αij ðtÞ and βij ðtÞ denote the coupling strengths of the position and the velocity between agent i and agent j, respectively. The matrix A ¼ ðaij Þ∈RNN with aii ¼ −∑N j ¼ 1;j≠i aij represents the weighted coupling configuration of system (1) and aij is the weight of the edge which connects agent i and agent j. G(A) describes the topology structure of G. If there exists communication channel between agent i and agent j, then aij 4 0, otherwise, aij ¼ 0. Ni is the neighbor set of agent i. The virtual leader regulates itself according to the following dynamics: ( q_ 0 ðtÞ ¼ p0 ðtÞ ; ð2Þ p_ 0 ðtÞ ¼ f 0 ðp0 ðtÞÞ where q0 ðtÞ and p0 ðtÞ are the position and velocity of the virtual leader, f 0 : Rn -Rn is a continuous function and f 0 ðp0 ðtÞÞ is the intrinsic dynamics of the virtual leader. Note that this virtual
leader is an external command (an external signal or a desired destination) to the system. The control input is designed as ui ðtÞ ¼ −hi ci ðtÞðqi ðt−τðtÞÞ−q0 ðtÞÞ−hi di ðtÞðpi ðt−τðtÞÞ−p0 ðtÞÞ;
i ¼ 1; …; N; ð3Þ
where hi is a binary value, if the ith agent is controlled, then hi ¼ 1, otherwise hi ¼ 0. ci(t) and di(t) are the feedback gains of position and velocity. Based on (1) and (3), the adaptive strategies on the coupling strengths of position and velocity are designed respectively as α_ ij ðtÞ ¼ aij kij fðqi ðt−τðtÞÞ−qj ðt−τðtÞÞÞT ðqi ðt−τðtÞÞ−qj ðt−τðtÞÞÞ þðq_ i ðtÞ−q_ j ðtÞÞT ðq_ i ðtÞ−q_ j ðtÞÞg;
ð4Þ
and β_ ij ðtÞ ¼ aij kij fðpi ðt−τðtÞÞ−pj ðt−τðtÞÞÞT ðpi ðt−τij ðtÞÞ−pj ðt−τij ðtÞÞÞ þðp_ i ðtÞ−p_ j ðtÞÞT ðp_ i ðtÞ−p_ j ðtÞÞg;
ð5Þ
where kij 4 0 is the weight of αij ðtÞ and βij ðtÞ with αij ð0Þ≥0 and βij ð0Þ≥0. The adaptive strategies on the feedback gains of position and velocity are designed respectively as c_ i ðtÞ ¼ hi ki ½ðqi ðt−τðtÞÞ−q0 ðtÞÞT ðqi ðt−τðtÞÞ−q0 ðtÞÞ þðq_ i ðtÞ−q_ 0 ðtÞÞT ðq_ i ðtÞ−q_ 0 ðtÞÞ;
ð6Þ
and d_ i ðtÞ ¼ hi ki ½ðpi ðt−τðtÞÞ−p0 ðtÞÞT ðpi ðt−τðtÞÞ−p0 ðtÞÞ þðp_ i ðtÞ−p_ 0 ðtÞÞT ðp_ i ðtÞ−p_ 0 ðtÞÞ;
ð7Þ
where ki 4 0 is the weight The weighted coupling and the velocity of system 2 a11 α11 a12 α12 ⋯ 6 a21 α21 a22 α22 ⋯ 6 B¼6 ⋮ ⋱ 4 ⋮ aN1 αN1
aN2 αN2
of ci(t) and di(t), ci ð0Þ≥0 and di ð0Þ≥0. configuration matrices of the position (1) are given respectively as follows: 3 a1N α1N a2N α2N 7 7 NN 7∈R ⋮ 5 ⋯ aNN αNN
with aii αii ¼ −∑N j ¼ 1;j≠i aij αij ; 2 a11 β11 a12 β12 ⋯ a1N β1N 6 6 a21 β21 a22 β22 ⋯ a2N β2N C¼6 6 ⋮ ⋮ ⋱ ⋮ 4 aN1 βN1
aN2 βN2
⋯
aNN βNN
3 7 7 NN 7∈R 7 5
with aii βii ¼ −∑N j ¼ 1;j≠i aij βij . Remark 1. Eq. (1) describes a second-order consensus algorithm for the multi-agent system with time-varying delays, and each agent uses the delayed information to update its position and velocity. Remark 2. In Eq. (1), the time-varying delays are in the presence of both the position and the velocity because the sensors have the broadcast delays, and this is different from [12] in which only the position has time-varying delays. To end this section, some necessary assumptions, propositions, definitions and lemmas are stated. Assumption 1. 0 ≤ τðtÞ ≤τ, t≥0, where τ is a positive constant. Remark 3. Differing from the existed results, such as [22], which required the derivative of the time delay satisfying 0 ≤ τ_ ðtÞ ≤1, in this paper, any information about the derivative of the time delay τðtÞ need not be known.
B. Liu et al. / Neurocomputing 118 (2013) 289–300
Assumption 2 (Su et al. [19]). The vector field f i : Rn -Rn ði ¼ 1; 2; …; NÞ satisfies ðx−yÞT ½f i ðxÞ−f i ðyÞ ≤ðx−yÞT Δðx−yÞ;
∀x; y∈Rn ;
i ¼ 0; 1; 2; …; N;
for some positive definite diagonal constant matrix Δ. Assumption 3. The coupling strengths and feedback gains of the position and the velocity are bounded, that is, ∥αij ðtÞ∥ ≤ αij ;
∥βij ðtÞ∥ ≤βij ;
∥ci ðtÞ∥ ≤ci ;
∥di ðtÞ∥ ≤di ;
where αij ; βij ; ci ; di , are positive constants. In practice, the coupling strengths and feedback gains are bounded generally.
Lemma 6. For any real differentiable vector function xðtÞ∈Rn and any n n constant matrix W ¼ W T 4 0, we have the following inequality: Z t T Z t Z t xðsÞ ds W xðsÞ ds ≤τ xT ðsÞWxðsÞ ds; t≥0; t−τðtÞ
t−τðtÞ
where 0 ≤τðtÞ ≤τ. Proof. Z t t−τðtÞ
Z
Z
t
¼
Note that Assumption 4 can be satisfied by many well-known systems, such as the Lü system, the chaotic Chua circuit, the Lorenz system and so on. Proposition 1 (Zorich [24] The Weierstrass maximum-value theorem). A function that is continuous on a closed interval is bounded on that interval. Moreover, there is a point in the interval where the function assumes its maximum value and a point where it assumes its minimal value. Lemma 1 (Heterogeneous Nonlinear Dynamics Inequality (HNDI)). The vector field f i : Rn -Rn ði ¼ 0; 1; …; NÞ satisfies ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ω2 ∥pi ðtÞ−p0 ðtÞ∥; ∀pi ðtÞ; p0 ðtÞ∈Rn for the constants ω1 40 and ω2 4 0. Proof. From Assumption 2, we can have ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f i ðp0 ðtÞÞ þ f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f i ðp0 ðtÞÞ þ ðpi ðtÞ−p0 ðtÞÞT ½f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ
t−τðtÞ
□
T Z xðsÞ ds W
t−τðtÞ
Assumption 4 (Hu et al. [20]). The virtual leader moves in a bounded region consistently in sense that there exists a compact set C ¼ Cðq0 ; p0 Þ∈Rn Rn , and the virtual leader starting with ðq0 ; p0 Þ is always in C.
291
xðsÞ ds
t
t−τðtÞ
t
1
1
½W 2 xðsÞT ½W 2 xðrÞ ds dr
t−τðtÞ
Z Z t 1 1 1 t 2½W 2 xðsÞT ½W 2 xðrÞ ds dr 2 t−τðtÞ t−τðtÞ Z Z t 1 t ½xT ðsÞWxðsÞ þ xT ðrÞWxðrÞ ds dr ≤ 2 t−τðtÞ t−τðtÞ Z t ¼ τðtÞ xT ðsÞWxðsÞ ds ¼
Z ≤τ
t−τðtÞ
t
xT ðsÞWxðsÞ ds:
t−τðtÞ
□
This completes the proof.
Definition 1. System (1) with the virtual leader described as Eq. (2) is said to achieve second-order consensus if lim ∥qi ðtÞ−q0 ðtÞ∥ ¼ 0;
t-∞
lim ∥pi ðtÞ−p0 ðtÞ∥ ¼ 0;
t-∞
i ¼ 1; 2; …; N;
hold for any initial condition Xð0Þ ¼ ðqT ð0Þ; pT ð0ÞÞT ∈R2N , where qð0Þ ¼ ðqT1 ð0Þ; qT2 ð0Þ; …; qTN ð0ÞÞT , pð0Þ ¼ ðpT1 ð0Þ; pT2 ð0Þ; …; pTN ð0ÞÞT . 3. Theoretical analysis and main results In this section, we will give detailed analysis for the adaptive second-order consensus of multi-agent systems from two cases, that is, the weighted coupling configuration matrix A is symmetric and asymmetric, respectively.
≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ðpi ðtÞ−p0 ðtÞÞT ½f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ðpi ðtÞ−p0 ðtÞÞT ½ðf i −f 0 Þðp0 ðtÞÞ:
3.1. The weighted coupling configuration matrix A is symmetric
By Assumption 4 and Proposition 1, we know that the value of ðf i −f 0 Þðp0 ðtÞÞ is bounded since each f i ði ¼ 0; 1; 2; …; NÞ is continuous function. Therefore,
Firstly, we investigate the adaptive consensus of multi-agent systems described by Eq. (1) with the weighted coupling configuration matrix A is symmetric in detail.
ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ω2 ∥pi ðtÞ−p0 ðtÞ∥: □
Theorem 1. Consider system (1) with a virtual leader as (2) which is steered by the adaptive strategies (4)–(7) under Assumptions1–4, suppose that the network is undirected and connected and the weighted coupling configuration matrix A is symmetric, then the position and velocity of each agent can converge to those of the virtual leader when at least one agent is controlled by (3).
Lemma 2 (Yu [23]). For any vectors x; y∈Rn and positive definite matrix G∈Rnn , the following matrix inequality holds: 2xT y ≤xT Gx þ yT G−1 y: Lemma 3 (Su and Zhang [8]). Suppose that a and b are vectors, then for any positive-definite matrix E, the following inequality holds: −2aT b ≤ inf faT Ea þ b E−1 bg: T
E40
Lemma 4 (Su et al. [18]). If A ¼ ðaij Þ∈RNN is a symmetric irreducible matrix with aii ¼ −∑N j ¼ 1;j≠i aij , and the matrix E ¼ diagfe; 0; …; 0g with e 4 0, then all eigenvalues of the matrix A−E are negative. Lemma 5 (Godsil and Royle [26]). The matrix A of an undirected graph G is irreducible if and only if the undirected graph is connected.
Proof. Let q~ i ðtÞ ¼ qi ðtÞ−q0 ðtÞ; p~ i ðtÞ ¼ pi ðtÞ−p0 ðtÞ;
q~ i ðt−τðtÞÞ ¼ qi ðt−τðtÞÞ−q0 ðtÞ; p~ i ðt−τðtÞÞ ¼ pi ðt−τðtÞÞ−p0 ðtÞ;
we have 8 > q~_ i ðtÞ ¼ q_ i ðtÞ−q_ 0 ðtÞ ¼ pi ðtÞ−p0 ðtÞ ¼ p~ i ðtÞ > > > > > p~_ i ðtÞ ¼ p_ i ðtÞ−p_ 0 ðtÞ > > > > < ¼ f ðp ðtÞÞ−f ðp ðtÞÞ þ ∑ α ðtÞa ðq~ ðt−τðtÞÞ−q~ ðt−τðtÞÞ ij ij j i i 0 0 i j∈N i > > > > þ ∑ βij ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞ > > > j∈Ni > > > : −h c ðtÞq~ ðt−τðtÞÞ−h d ðtÞp~ ðt−τðtÞÞ: i i
i
i i
i
ð8Þ
292
B. Liu et al. / Neurocomputing 118 (2013) 289–300
~ ~ In order to obtain the desired result, i.e., qðtÞ-0 and pðtÞ-0 as t-∞, we choose
and ∥p~ i ðtÞ∥ ≤ ∥p~ i ðt−τðtÞÞ∥ þ ‖
1 N 1 N T V 1 ðtÞ ¼ ∑ p~ Ti ðtÞp~ i ðtÞ þ ∑ q~ i ðtÞq~ i ðtÞ: 2i¼1 2i¼1 Taking the derivative of V 1 ðtÞ, there appears the terms of adaptive coefficients and time-varying delays according to Eq. (8). To offset these terms, we select ðαij ðtÞ−2αij −mÞ2 þ ðβij ðtÞ−2βij −mÞ2 4kij i ¼ 1 j∈Ni
Z
t
t−τðtÞ
p~_ ðsÞ ds‖ ≤∥p~ i ðt−τðtÞÞ∥ þ
Z
t t−τðtÞ
‖p~_ ðsÞ‖ds:
In addition, Lemma 6 demonstrates that we can have Z t T Z t Z t _ ds; _ _ ds ≤τ xðsÞds xðsÞ x_ T ðsÞxðsÞ t−τðtÞ
t−τðtÞ
t−τðtÞ
N
V 2 ðtÞ ¼ ∑ ∑
thus, Z p~ Ti ðt Þp~ i ðt Þ ¼ p~ i ðt−τðt ÞÞ þ
3 3 2 2 N ðci ðtÞ− ci −mÞ þ ðdi ðtÞ− di −mÞ 2 2 þ ∑ ; 2ki i¼1 N
"
V 3 ðtÞ ¼ τ ∑
i¼1
Z
t
! 2ω1 þ
∑ aij ðαij þ βij Þ
#
N
t−τ
T ðs−t þ τÞp~_ i ðsÞp~_ i ðsÞ ds þ ω2 ∑ N
þτ ∑
i¼1
Z
i¼1
t t−τ
t−τðt Þ
Z
t
Z þ
t−τ
T ðs−t þ τÞq~_ i ðsÞq~_ i ðsÞ ds:
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ; where m 4 0 is sufficiently large. By Lemmas 1 and 2, the derivative of V 1 ðtÞ satisfies N
V_ 1 ðtÞ ¼ ∑ p~ Ti ðtÞff i ðpi ðtÞÞ−f ðp0 ðtÞÞ þ ∑ αij ðtÞaij ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞ
T Z p~_ i ðsÞ ds
t t−τðt Þ
t−τðt Þ
≤ p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
p~_ i ðsÞ ds :
p~ i ðt−τðt ÞÞ þ t
t−τðt Þ
Z
t
t−τðt Þ
p~_ i ðsÞ ds
p~_ i ðsÞ ds
Z
t
þ
t−τðt Þ
T Z p~_ i ðsÞ ds
t
t−τðt Þ
p~_ i ðsÞ ds :
Therefore, p~ Ti ðt Þp~ i ðt Þ ≤2p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ Z t T Z t p~_ i ðsÞds p~_ i ðsÞ ds þ2 t−τðt Þ
j∈N i
þ ∑ βij ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞ−hi ci ðtÞq~ i ðt−τðtÞÞ−hi di ðtÞp~ i ðt−τðtÞÞg
T
From Lemma 2, choosing G ¼ I N and we have Z t p~_ i ðsÞ ds 2p~ Ti ðt−τðt ÞÞ
Thus, the global Lyapunov function is chosen as
i¼1
t
t−τðt Þ
ðs−t þ τÞ∥p~_ i ðsÞ∥ ds
p~_ i ðsÞ ds
Z ¼ p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ þ 2p~ Ti ðt−τðt ÞÞ
þ hi ðci þ di Þ þ 1
j∈Ni
t
t−τðt Þ
¼ 2p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
Z
t
þ 2τ
t−τðt Þ
j∈N i
T p~_ i ðsÞp~_ i ðsÞ ds;
i.e.,
N
T þ ∑ q~ i ðtÞp~ i ðtÞ
∥p~ i ðt Þ∥2 ≤2∥p~ i ðt−τðt ÞÞ∥2 þ 2τ
i¼1
Z
t t−τðt Þ
N
∥p~_ i ðsÞ∥2 ds:
≤ ∑ p~ Ti ðtÞ½f i ðpi ðtÞÞ−f ðp0 ðtÞÞ i¼1
þ
1 N ∑ ∑ αij ðtÞaij p~ Ti ðtÞp~ i ðtÞ 2 i ¼ 1 j∈Ni
Similarly,
1 N þ ∑ ∑ αij ðtÞaij ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞT ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞ 2 i ¼ 1 j∈Ni þ þ þ þ
1 N ∑ ∑ β ðtÞaij p~ Ti ðtÞp~ i ðtÞ 2 i ¼ 1 j∈Ni ij
N
1 ∑ ∑ β ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞÞT ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞÞ 2 i ¼ 1 j∈Ni ij 1 N 1 N T ∑ h c ðtÞp~ Ti ðtÞp~ i ðtÞ þ ∑ hi ci ðtÞq~ i ðt−τðtÞÞq~ i ðt−τðtÞÞ 2i¼1 i i 2i¼1 N
1 1 ∑ h d ðtÞp~ Ti ðtÞp~ i ðtÞ þ ∑ hi di ðtÞp~ Ti ðt−τðtÞÞp~ i ðt−τðtÞÞ 2i¼1 i i 2i¼1
1 N T 1 N þ ∑ q~ i ðtÞq~ i ðtÞ þ ∑ p~ Ti ðtÞp~ i ðtÞ: 2i¼1 2i¼1
V_ 1 ðt Þ ≤ ∑
Using Leibniz–Newton formula as Z t _ ds; xðsÞ xðtÞ−xðt−τðtÞÞ ¼ t−τðtÞ
i¼1
t t−τðtÞ
p_~ ðsÞ ds; T
t−τðt Þ
∥q~_ i ðsÞ∥2 ds:
# ! ∑ aij αij þ βij þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2
2ω1 þ
j∈N i
þ ∑ ω2 ∥p~ i ðt−τðt ÞÞ∥ i¼1 " N
þτ ∑
i¼1
Z
t t−τðt Þ N
i¼1
2ω1 þ
N
∑ aij αij þ βij
i¼1
!
# þ hi ðci þ di Þ þ 1
j∈N i
T p~_ i ðsÞp~_ i ðsÞ ds
Z
t t−τðt Þ
∥p~_ i ðsÞ∥ ds
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ τ
Z
t t−τðt Þ
∥q~_ i ðsÞ∥2 ds
þ
T 1 N ∑ ∑ α ðt Þaij q~ j ðt−τðt ÞÞ−q~ i ðt−τðt ÞÞ q~ j ðt−τðt ÞÞ−q~ i ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij
þ
T 1 N ∑ ∑ βij ðt Þaij p~ j ðt−τðt ÞÞ−p~ i ðt−τðt ÞÞ p~ j ðt−τðt ÞÞ−p~ i ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni
we can have Z
t
N
þ ∑ ω2
p~ Ti ðtÞ ¼ p~ Ti ðt−τðtÞÞ þ
Z
Under Assumption 3 "
N
N
∥q~ i ðt Þ∥2 ≤2∥q~ i ðt−τðt ÞÞ∥2 þ 2τ
B. Liu et al. / Neurocomputing 118 (2013) 289–300
þ
1 N 1 N T ∑ hi ci ðt Þq~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ hi di ðt Þp~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ: 2i¼1 2i¼1
Differentiating V 2 ðt Þ, T 1 N q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ V_ 2 ðt Þ ¼ ∑ ∑ αij ðt Þ−2αij −m aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni T 1 N ∑ ∑ αij ðt Þ−2αij −m aij q~_ i ðt Þ−q~_ j ðt Þ q~_ i ðt Þ−q~_ j ðt Þ 2 i ¼ 1 j∈Ni T 1 N þ ∑ ∑ βij ðt Þ−2βij −m aij p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni
þ
T 1 N þ ∑ ∑ βij ðt Þ−2βij −m aij p~_ i ðt Þ−p~_ j ðt Þ p~_ i ðt Þ−p~_ j ðt Þ 2 i ¼ 1 j∈Ni N N T 3 3 T þ ∑ ci ðt Þ− ci −m hi q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ ci ðt Þ− ci −m hi q~_ i ðt Þq~_ i ðt Þ 2 2 i¼1 i¼1
N
þ ∑
i¼1
d dt
N T 3 3 T di ðt Þ− di −m hi p~ i ðt−τðt ÞÞp~ i ðt−τðt ÞÞ þ ∑ di ðt Þ− di −m hi p~_ i ðt Þp~_ i ðt Þ: 2 2 i¼1
Based on the Leibniz derivative rule Z t Z f ðτ; t Þ dτ ¼ f ðbðt Þ; t Þb_ ðt Þ−f ðaðt Þ; t Þa_ ðt Þ þ t−τij ðt Þ
the derivative of V 3 ðt Þ yields "
N
V_ 3 ðt Þ ¼ τ2 ∑
i¼1
j∈N i
"
N
2ω1 þ
−τ ∑
i¼1
∑ aij αij þ βij
2ω1 þ
∑ aij αij þ βij j∈N i
N
N
i¼1
i¼1
þτω2 ∑ ∥p~ i ðt Þ∥−ω2 ∑
Z
t t−τ
!
#Z þ hi ðci þ di Þ þ 1
t t−τ
N
d f ðτ; t Þ dτ; dt t−τij ðt Þ
∥p~_ i ðsÞ∥ ds þ τ2 ∑
i¼1
T p~_ i ðsÞp~_ i ðsÞ ds
T q~_ i ðt Þq~_ i ðt Þ−τ
N
∑
i¼1
Z
2
3 ∥p~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 6 ∥p 2 ðt−τðt ÞÞ∥ 7 7 þm½∥p~ 1 ðt−τðt ÞÞ∥; ∥p~ 2 ðt−τðt ÞÞ∥; …; ∥p~ N ðt−τðt ÞÞ∥ðA−H Þ6 6 7 ⋮ 4 5 ∥p~ N ðt−τðt ÞÞ∥ "
N
þτ2 ∑
∑ aij αij þ βij
2ω1 þ
i¼1
t−τ
T q~_ i ðsÞq~_ i ðsÞ
#
N
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2 þ τω2 ∑ ∥p~_ i ðt Þ∥ i¼1
j∈N i
3 2 _ ∥p~ 1 ðt Þ∥ 7 6 _ N 6 ∥p~ 2 ðt Þ∥ 7 7 þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þm½∥p~_ 1 ðt Þ∥; ∥p~_ 2 ðt Þ∥; …; ∥p~_ N ðt Þ∥ðA−HÞ6 7 6 ⋮ 5 i¼1 4 ∥p~_ N ðt Þ∥
2
3 ∥q~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 6 ∥q 2 ðt−τðt ÞÞ∥ 7 7 þm½∥q~ 1 ðt−τðt ÞÞ∥; ∥q~ 2 ðt−τðt ÞÞ∥; …; ∥q~ N ðt−τðt ÞÞ∥ðA−H Þ6 6 7 ⋮ 4 5 ∥q~ N ðt−τðt ÞÞ∥ 0
3 2 _ ∥q~ 1 ðt Þ∥ B 7 6 _ B _ 7 6 ~ T B∥q~ 1 ðt Þ∥; ∥q~_ 2 ðt Þ∥; …; ∥q~_ N ðt Þ∥ðA−H Þ6 ∥q 2 ðt Þ∥ 7: þτ2 ∑ q~_ i ðt Þq~_ i ðt Þ þ mB 7 6 ⋮ i¼1 @ 5 4 _ ~ ∥q N ðt Þ∥ N
ij
i¼1
t
!
From the conditions of Theorem 1, we know that the matrix H is a diagonal matrix with at least one element equaling to 1. From Lemma 5, one can easily get that A is irreducible because system (1) is connected. Since A is symmetric, all eigenvalues of A−H are negative from Lemma 4. Let λ1 be the maximum eigenvalue of A−H and λ1 o 0. Then " # ! N _ þ hi ðci þ di Þ þ 1 ∥p~ ðt−τðt ÞÞ∥2 V ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ β
t
# T þ hi ðci þ di Þ þ 1 p~_ i ðt Þp~_ i ðt Þ
!
293
ds:
i
j∈Ni
N
þ ω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ i¼1
N
By Assumption 3, V_ ðt Þ ¼ V_ 1 ðt Þ þ V_ 2 ðt Þ þ V_ 3 ðt Þ " # ! N ≤ ∑ 2ω1 þ ∑ aij αij þ βij þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 i¼1
j∈N i
T N 1 p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ − m ∑ ∑ aij p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni hi p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
N
"
i¼1
∑ aij αij þ βij
2ω1 þ
∑
i¼1
∑ aij αij þ βij
2ω1 þ
!
# þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
j∈Ni
N
þmλ1 ∑ ∥p~_ i ðt Þ∥2 i¼1
N
!
#
N
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2 þ τω2 ∑ ∥p~_ i ðt Þ∥ i¼1
j∈N i
N
N
N
i¼1
i¼1
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ mλ1 ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ τ2 ∑ ∥q~_ i ðt Þ∥2 i¼1
T N N T 1 p~_ i ðt Þ−p~_ j ðt Þ −m ∑ hi p~_ i ðt Þp~_ i ðt Þ − m ∑ ∑ aij p~_ i ðt Þ−p~_ j ðt Þ 2 i ¼ 1 j∈Ni i¼1 þτ2 ∑
"
i¼1
i¼1
i¼1
þτ
2
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥
−m ∑
i¼1
N
þτω2 ∑ ∥p~_ i ðt Þ∥
N
N
þmλ1 ∑ ∥p~ i ðt−τðt ÞÞ∥2
N
þmλ1 ∑ ∥q~_ i ðt Þ∥2 :
ð9Þ
i¼1
Since λ1 o 0 and m 4 0 is sufficiently large, V_ ðt Þ ≤0:
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 i¼1
T N 1 q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ − m ∑ ∑ aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni N
−m ∑ hi q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ T
Define the level set of V(t) in the space of the positions and velocities of agents Ω ¼ f q~ i ; p~ i jV ðt Þ ≤c; c 4 0g:
i¼1
T N N T 1 q~_ i ðt Þ−q~_ j ðt Þ −m ∑ hi q~_ i ðt Þq~_ i ðt Þ − m ∑ ∑ aij q~_ i ðt Þ−q~_ j ðt Þ 2 i ¼ 1 j∈Ni i¼1 " ! N N T þτ2 ∑ q~_ ðt Þq~_ ðt Þ ¼ ∑ 2ω þ ∑ a α þ β i¼1
i
i
i¼1
1
ij
ij
ij
j∈Ni
N
þhi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 þ ω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ i¼1
Since G is connected and V ðt Þ ≤c, on the one hand, p~ Ti ðt Þp~ i ðt Þ ≤2c pffiffiffiffiffi T and then ∥p~ i ∥ ≤ 2c. On the other hand, q~ i ðt Þq~ i ðt Þ ≤2c and then pffiffiffiffiffi ∥q~ i ∥ ≤ 2c. So, ∥q~ i ðt Þ∥ and ∥p~ i ðt Þ∥ are both bounded. Thus, Ω is bounded. Since ∥q~ i ðt Þ∥ and ∥p~ i ðt Þ∥ are bounded, the set Ω is closed by the continuity of V ðt Þ ≤c. Therefore, the set Ω is compact.
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B. Liu et al. / Neurocomputing 118 (2013) 289–300
Consequently, we can apply LaSalle Invariance Principle and the corresponding trajectories will converge to the largest invariant set of ~ p~ ÞjV_ ðt Þ ¼ 0g: S ¼ fðq;
And define the same Lyapunov function as Theorem 1, we can obtain " # ! N _ V ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ β þ hi ðci þ di Þ þ 1 ∥p~ ðt−τðt ÞÞ∥2 ij
i¼1
i
j∈Ni
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥
From (9), V_ ðt Þ ¼ 0 if and only if q~ i ðt−τðt ÞÞ ¼ 0;
q~_ i ðt Þ ¼ 0;
p~ i ðt−τðt ÞÞ ¼ 0;
p~_ i ðt Þ ¼ 0; i ¼ 1; 2; …; N:
i¼1
þm½∥p~ 1 ðt−τðt ÞÞ∥; ∥p~ 2 ðt−τðt ÞÞ∥; …; ∥p~ N ðt−τðt ÞÞ∥ 2
Based on (8), we can gain that V_ ðt Þ ¼ 0 if and only if q~ i ðt−τðt ÞÞ ¼ 0; p~ i ðt Þ ¼ p~ i ðt−τðt ÞÞ ¼ 0;
i ¼ 1; 2; …; N:
ð10Þ
Since lim p~ i ðt−τðt ÞÞ ¼ p~ i ðt Þ;
lim p~ i ðt−τðt ÞÞ ¼ p~ i ðt Þ;
i ¼ 1; 2; …; N;
hence, according to Eq. (10), we can obtain lim q~ i ðt Þ ¼ 0;
t-∞
lim p~ i ðt Þ ¼ 0;
t-∞
i ¼ 1; 2; …; N;
which imply that the position and the velocity of each agent can converge to those of the virtual leader respectively. This completes the proof. □ 3.2. The weighted coupling configuration matrix A is asymmetric In the following, we are interested in discussing the consensus of system (1) with the weighted coupling configuration matrix A is asymmetric. Firstly, we give the following assumption. Assumption 5 (Liu et al. [25]). The weights satisfy the balance condition: ∑ aij ¼ ∑ aji ; j∈N i
∀i:
j∈N i
Theorem 2. Consider system (1) with a virtual leader as (2) which is steered by the adaptive strategies (4)–(7) under Assumptions 1–5, suppose that the network is undirected and connected and the weighted coupling configuration matrix A is asymmetric, then the position and velocity of each agent can converge to those of the virtual leader even when only one agent is controlled by (3). Proof. Under Assumption 5, we have −
þ
1 N 1 N T T ∑ ∑ aij q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ ∑ aij q~ i ðt−τðt ÞÞq~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 2 i ¼ 1 j∈Ni N
!
j∈Ni
! A þ AT _ _ _ _ ~ ~ ~ ~ −H þτω2 ∑ ∥p i ðt Þ∥ þ m½∥p 1 ðt Þ∥; ∥p 2 ðt Þ∥; …; ∥p N ðt Þ∥ 2 i¼1 2 _ 3 ∥p~ 1 ðt Þ∥ 6 _ 7 N 6 ∥p~ 2 ðt Þ∥ 7 6 7 þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 6 7 ⋮ 4 5 i¼1 ∥p~_ N ðt Þ∥ ! A þ AT −H þm½∥q~ 1 ðt−τðt ÞÞ∥; ∥q~ 2 ðt−τðt ÞÞ∥; …; ∥q~ N ðt−τðt ÞÞ∥ 2 2 3 ∥q~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 N 6 ∥q 2 ðt−τðt ÞÞ∥ 7 6 7 þ τ2 ∑ q~_ T ðt Þq~_ i ðt Þ þ m½∥q~_ 1 ðt Þ∥; ∥q~_ 2 ðt Þ∥; …; ∥q~_ N ðt Þ∥ i 6 7 ⋮ i¼1 4 5 ∥q~ N ðt−τðt ÞÞ∥ 2 _ 3 ∥q~ ðt Þ∥ !6 1 7 6 ∥q~_ 2 ðt Þ∥ 7 A þ AT 7: −H 6 6 7 2 ⋮ 4 5 ∥q~_ N ðt Þ∥ Even though matrix A is asymmetric, matrix A þ AT =2 is symmetric. Thus, all eigenvalues of matrix A þ AT =2−H are negative from Lemma 2. Let λ2 ð o 0Þ be the maximum eigenvalue of A þ AT =2−H . Therefore, " # ! N þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 V_ ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ βij i¼1
j∈Ni N
T 1 N ∑ ∑ a q~ ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij i ¼−
∑ aij αij þ βij
!
N
t-∞ t-∞
3 ∥p~ 1 ðt−τðt ÞÞ∥ " 6 ~ 7 N 6 ∥p 2 ðt−τðt ÞÞ∥ 7 6 7 þ τ2 ∑ 2ω1 þ 6 7 ⋮ i¼1 4 5 ∥p~ N ðt−τðt ÞÞ∥
þhi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
A þ AT −H 2
N
1 1 T T ∑ ∑ a q~ ðt−τðt ÞÞq~ i ðt−τðt ÞÞ− ∑ ∑ aij q~ j ðt−τðt ÞÞq~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij j 2 i ¼ 1 j∈Ni
T 1 N q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ − ∑ ∑ aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 1 T 1 N T ¼ q~ ðt−τðt ÞÞAq~ ðt−τðt ÞÞ þ ∑ ∑ aij q~ j ðt−τðt ÞÞq~ i ðt−τðt ÞÞ 2 2 i ¼ 1 j∈Ni ! 1 N A þ AT T T q~ ðt−τðt ÞÞ: − ∑ ∑ aij q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ ¼ q~ ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 2
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ þ mλ2 ∑ ∥p~ i ðt−τðt ÞÞ∥2 i¼1
i¼1
"
N
þτ2 ∑
i¼1
2ω1 þ
∑ aij αij þ βij
#
!
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
j∈Ni
N
N
i¼1
i¼1
þτω2 ∑ ∥p~_ i ðt Þ∥ þ mλ2 ∑ ∥p~_ i ðt Þ∥2 N
N
N
i¼1
i¼1
i¼1
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ mλ2 ∑ ∥q~ i ðt−τðÞÞ∥2 þ τ2 ∑ ∥q~_ i ðt Þ∥2 N
þmλ2 ∑ ∥q~_ i ðt Þ∥2 : i¼1
Since m 4 0 is sufficiently large, we can get V_ ðt Þ ≤0:
B. Liu et al. / Neurocomputing 118 (2013) 289–300
295
30 20
y axis
2500
0 −100 −40
y axis
10 100
2000
−20
1500 −20
−30 60
1000 0
20
40
x axis
500 60
80
0 −10
40 20
z axis
0
0 −20
z axis
−10
0
10
20
x axis
Fig. 1. The position and velocity of the virtual leader. (a) The position of the virtual leader and (b) the velocity of the virtual leader. Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
5
10
15
20
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
5
10
15
20
0
5
time
time
10
15
20
time
Fig. 2. Position convergence of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Velocity convergence(x axis)
Velocity convergence(y axis)
10
Velocity convergence(z axis)
10
0 −5 −10
6
piz− p0z, i=1,2,3.
piy− p0y, i=1,2,3.
pix − p0x, i=1,2,3.
8 5
4 2 0 −2 −4 −6
−15
−8 0
5
10
15
20
0
5
time
10
15
12 10 8 6 4 2 0 −2 −4 −6 −8
20
0
time
5
10
15
20
time
Fig. 3. Velocity convergence of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Similar to Theorem 1, the position and velocity of each agent described by (1) under Assumption 5 can converge to those of the virtual leader. This completes the proof. □ Remark 4. It is obvious that the results of both Theorems 1 and 2 hold if the time delay of the agents is time-invariant.
Remark 5. Notice also that the results keep conservatism since the upper bound of the time-varying delays τ is necessary in the theoretical analysis. Moreover, comparing with the adaptive strategies of the second-order consensus algorithms without time delay in [18–20], the decentralized adaptive strategies in this paper need more agents' information, such as the delayed information and the derivative information.
4. Numerical example This section presents some numerical simulations to illustrate the adaptive consensus of second-order multi-agent systems with time-varying delays. The dynamics of the virtual leader is
described by a Lorenz system 8 q_ 0 ðt Þ ¼ p0 ðt Þ > > 8 > > > > < > > 10 p0y −p0x < > p_ 0 ðt Þ ¼ 28p0x −p0x p0z −p0y ; > > > > > > > : : p0x p0y −8p0z
ð11Þ
3
where q0 ¼ ½q0x ; q0y ; q0z T and p0 ¼ ½p0x ; p0y ; p0z T . The position and the velocity of the virtual leader are presented as Fig. 1. Example 1. Consider a multi-agent system with the undirected topology described as the following symmetric matrix 2
−0:2 6 0:1 A¼4 0:0
0:1 −0:2 0:1
0:0
3
0:1 7 5: −0:2
ð12Þ
with that Consistent therequirements of the theory, we choose f 1 p1 ðt Þ ¼ sin 2 p1 ðt Þ , f 2 p2 ðt Þ ¼ tanh p1 ðt Þ and f 3 p3 ðt Þ ¼ sin p3 ðt Þ satisfying Assumption 2, as well as let kij ¼ 1 and ki ¼ 0:1. Here, only agent 3 is controlled by control input (3). In Figs. 2 and 3, we have shown that the adaptive second-order consensus of such system contains three agents with a timeinvariant delay and symmetric weighted coupling configuration
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B. Liu et al. / Neurocomputing 118 (2013) 289–300
Example 2. Consider a multi-agent system with the undirected topology described as the following asymmetric matrix: 2 3 −0:1976 0:1976 0 6 0:1677 −0:3623 0:1946 7 A¼4 ð13Þ 5: 0 0:3218 −0:3218
matrix A described as Eq. (12), for simplicity. Fig. 2 shows the positions' errors of the multi-agent system with time delay τ ¼ 0:14, while Fig. 4 shows the velocities' errors of the multiagent system with the same time delay. Fig. 4 presents the adaptive coupling strengths of position αij ðt Þ; ði; j ¼ 1; 2; 3Þ and the velocity βij ðt Þ; ði; j ¼ 1; 2; 3Þ when the time delay is τ ¼ 0:14. Every adaptive coupling strength tends to a constant. Similarly, Fig. 5 demonstrates the feedback gains of position and velocity with time delay τ ¼ 0:14, respectively. We can find that the feedback gains of position and velocity also tend to constants, which implies Assumption 3 is reasonable. Figs. 6-9 show the convergence of the positions' errors and the velocities' errors, adaptive coupling strengths and adaptive feedback gains with symmetric matrix A as well as τ ¼ 0:18. From these figures, we also find that the position and the velocity of each agent can converge to those of the virtual leader, at the same time, all the adaptive coupling strengths and the adaptive feedback gains tend to be constant. Note that from Figs. 2–9, it is easily to see that the larger the time delay is, the slower convergent speed is.
The selection of all parameters is the same as Example 1 and only the agent 3 is steered by control input (3). Figs. 10 and 11 depict the adaptive second-order consensus of a multi-agent system with time delay τ ¼ 0:05 and asymmetric weighted coupling configuration matrix A described as Eq. (13). These two figures show that both the positions and the velocities of all agents can converge to those of the virtual leader. Fig. 12 displays the variable trend of the adaptive coupling strengths of the positions and the velocities, which can reach to be constant eventually. While, Fig. 13 shows the adaptive feedback gains of the positions and the velocities respectively, which can also reach to be constant eventually.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
9
25
8 20
7
βij, i,j=1,2,3.
αij, i,j=1,2,3.
6 5 4 3 2
15 10 5
1 0
0 0
5
10
15
20
0
5
10
time
15
20
time
Fig. 4. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Adaptive feedback gain of position
Adaptive feedback gain of velocity
8
14
7
12
6
10
5
d3
c3
8 4
6 3 4
2
2
1 0
0 0
5
10
15
20
0
5
10
time
15
20
time
Fig. 5. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
10
20
30
time
40
50
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
10
20
30
time
40
50
0
10
20
30
40
time
Fig. 6. Position convergence of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
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Velocity convergence(x axis)
297
Velocity convergence(y axis)
10
Velocity convergence(z axis)
10 8
0 −5 −10
6
piz − p0z, i=1,2,3.
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
5
4 2 0 −2 −4 −6
−15
−8 0
10
20
30
40
50
0
10
20
time
30
40
12 10 8 6 4 2 0 −2 −4 −6 −8
50
0
10
20
time
30
40
50
time
Fig. 7. Velocity convergence of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity 90
12
80 10 70
βij, i,j=1,2,3.
αij, i,j=1,2,3.
8 6 4
60 50 40 30 20
2 10 0
0 0
10
20
30
40
50
0
10
20
time
30
40
50
time
Fig. 8. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric. Adaptive feedback gain of position
Adaptive feedback gain of velocity
9
40
8
35
7
30
6
25
c3
d3
5 20
4 15
3 2
10
1
5
0
0 0
10
20
30
40
50
0
10
20
time
30
40
50
time
Fig. 9. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
5
10
15
20
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
5
time
10
time
15
20
0
5
10
15
20
time
Fig. 10. Position convergence of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Figs. 14–17 present the convergence of the positions' errors, velocities' errors, adaptive coupling strengths and adaptive feedback gains, respectively, when the weighted coupling configuration matrix A is asymmetric and the time delay is τ ¼ 0:1. These four figures demonstrate that the positions and the velocities of all agents converge to those of the virtual leader, in which all the
adaptive parameters tend to be constant. Similarly, the larger the time delay is, the longer the convergent time is. From Figs. 2–17, we know that the delayed multi-agent system with the adaptive strategies (4)–(7) and symmetric or asymmetric weighted coupling configuration matrix A, can reach consensus via the second-order consensus algorithm (1).
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Velocity convergence(y axis)
Velocity convergence(z axis) 10
8
8
8
6
6
6
4 2 0 −2
4 2 0 −2 −4
−4
−6
−6
−8 0
5
10
15
piz − p0z, i=1,2,3.
10
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
Velocity convergence(x axis) 10
4 2 0 −2 −4 −6 −8
20
0
5
10
time
15
20
0
5
10
time
15
20
time
Fig. 11. Velocity convergence of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
25
50 45
20
40
βij, i,j=1,2,3.
αij, i,j=1,2,3.
35 15
10
30 25 20 15
5
10 5
0
0 0
5
10
15
20
0
5
time
10
15
20
time
Fig. 12. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Adaptive feedback gain of position
Adaptive feedback gain of velocity
8
10
7
9 8
6
7 6
d3
c3
5 4
5 4
3
3
2
2
1
1
0
0 0
5
10
15
20
0
5
time
10
15
20
time
Fig. 13. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Position convergence(x axis)
Position convergence(y axis) qiy− q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5 4 3 2 1 0 −1 −2 0
10
20
30
40
50
60
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2
qiz − q0z, i=1,2,3.
6
0 −2 −4 −6 −8 −10 −12
0
10
20
time
30
time
40
50
60
0
10
20
30
40
50
60
time
Fig. 14. Position convergence of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric.
5. Conclusion In this paper, we have investigated second-order consensus of multi-agent systems with time-varying delays. Differing from the current works, all agents and the virtual leader are described by
heterogeneous nonlinear intrinsic dynamics. By introducing local decentralized adaptive strategies to the coupling strengths and feedback gains of both the position and velocity, we have proved that the position and velocity of each agent can converge to those of the virtual leader respectively, when the network is connected
B. Liu et al. / Neurocomputing 118 (2013) 289–300
Velocity convergence(y axis)
0
10
20
30
40
50
Velocity convergence(z axis)
10 8 6 4 2 0 −2 −4 −6 −8
60
piz − p0z, i=1,2,3.
10 8 6 4 2 0 −2 −4 −6 −8
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
Velocity convergence(x axis)
299
0
10
20
time
30
40
50
10 8 6 4 2 0 −2 −4 −6 −8
60
0
10
20
time
30
40
50
60
time
Fig. 15. Velocity convergence of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric. Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
30
80 70
25
βij, i,j=1,2,3.
αij, i,j=1,2,3.
60 20 15 10
50 40 30 20
5
10
0
0 0
10
20
30
40
50
60
0
10
20
time
30
40
50
60
time
Fig. 16. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric. Adaptive feedback gain of position
Adaptive feedback gain of velocity 9
8
8
7
7
6
6
5
5
c3
d3
9
4
4
3
3
2
2
1
1
0
0 0
10
20
30
40
50
60
0
10
time
20
30
40
50
60
time
Fig. 17. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric.
and at least one agent is steered by the controller, even though the weighted coupling configuration matrix is asymmetric. Finally, some simulation results are showed to demonstrate the theoretical results. Future work will be to further optimize the proposed algorithm, and apply it to the filtering problems in sensor networks.
National High Technology Research and Development Program of China 863 (No.2012AA112401), the Research Fund for the Doctoral Program of Higher Education (RFDP) under Grant No. 20100142120023, the Natural Science Foundation of Hubei Province of China under Grant No. 2011CDB042. References
Acknowledgements This work was supported in part by the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, the National Natural Science Foundation of China under Grant (No. 61104140, No. 61203150, No. 61170113, No. 61174116), the Beijing Natural Science Foundation Program (1102016, 4122019), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201108055), Science and Technology Development Plan Project of Beijing Education Commission (No. KM201310009011)), the
[1] W. Yang, H. Shi, Sensor selection schemes for consensus based distributed estimation over energy constrained wireless sensor networks, Neurocomputing 87 (2012) 132–137. [2] B. Shen, Z. Wang, Y. Hung, Distributed H∞ consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case, Automatica 46 (10) (2010) 1682–1688. [3] H. Dong, Z. Wang, H. Gao, Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts, IEEE Trans. Signal Process. 60 (6) (2012) 3164–3173. [4] B. Shen, Z. Wang, X. Liu, A stochastic sampled-data approach to distributed H∞ filtering in sensor networks, IEEE Trans. Circuits Syst.-I: Regular Pap. 58 (9) (2011) 2237–2246. [5] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (2005) 655–661.
300
B. Liu et al. / Neurocomputing 118 (2013) 289–300
[6] Y. Sun, L. Wang, G. Xie, Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Syst. Control Lett. 57 (2008) 175–183. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 53 (2008) 1503–1509. [8] H. Su, W. Zhang, Second-order consensus of multiple agents with coupling delay, Commun. Theor. Phys. 51 (1) (2009) 101–109. [9] P. Lin, Y. Jia, Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica 45 (2009) 2154–2158. [10] Z. Tang, T. Huang, J. Shao, J. Hu, Consensus of second-order multi-agent systems with nonuniform time-varying delays, Neurocomputing 97 (2012) 410–414. [11] C. Liu, F. Liu, Consensus problem of second-order multi-agent systems with time-varying communication delay and switching topology, J. Syst. Eng. Electron. 22 (2011) 672–678. [12] X. Lu, F. Austin, S. Chen, Formation control for second-order multi-agent systems with time-varying delays under directed topology, Commun. Nonlinear Sci. Numer. Simulation 17 (2012) 1382–1391. [13] F. Jiang, L. Wang, Y. Jia, Consensus in leaderless networks of high-orderintegrator agents, in: Proceeding of the 2009 American Control Conference, 2009, pp. 4458–4463. [14] X. Wang, G. Chen, Pinning control of scale-free dynamical networks, Physica A 310 (2002) 521–531. [15] H. Su, X. Wang, Z. Lin, Flocking of multi-agents with a virtual leader, IEEE Trans. Autom. Control 54 (2) (2009) 293–307. [16] Z. Hou, et al., Decentralized robust adaptive control for the multiagent system consensus problem using neural networks, IEEE Trans. Syst. Man Cybern.-Part B: Cybern. 39 (2009) 636–647. [17] P. Delellis, M. Bernardo, F. Garofalo, Synchronization of complex networks through local adaptive coupling, Chaos 18 (2008) 037110. [18] H. Su, Z. Rong, M. Chen, X. Wang, G. Chen, H. Wang, Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks, IEEE Trans. Cybern. 43 (1) (2013) 394–399. [19] H. Su, G. Chen, X. Wang, Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica 47 (2011) 368–375. [20] Y. Hu, H. Su, J. Lam, Adaptive consensus with a virtual leader of multiple agents governed by local Lipschitz nonlinearity, Int. J. Robust Nonlinear Control 23 (2013) 978–990. [21] O. Faydasicok, S. Arik, A new robust stability criterion for dynamical neural networks with multiple time delays, Neurocomputing 99 (2013) 290–297. [22] J. Zhang, M. Li, B. Liu, Synchronization of a complex network for neighboring agents with time-varying delays, Commun. Computer Inf. Sci. 225 (2011) 73–81. [23] W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays, Nonlinear Dyn. 48 (2007) 165–174. [24] V. Zorich, Mathematical Analysis I, Springer, New York, 2004. [25] B. Liu, T. Chu, L. Wang, Stability and oscillation of swarm with interaction time delays, in: Proceeding of the 2007 American Control Conference, 2007, pp. 4600–4605. [26] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001.
Bo Liu was born in 1977. She received the Ph.D. degree in Dynamics and Control from Peking University in 2007. She was a visiting research fellow at the City University of Hong Kong in 2009 and is currently an Associate Professor in North China University of Technology. Her research interests include swarm dynamics, networked systems, collective behavior and coordinate control of multi-agent systems.
Xiaoling Wang was born in 1987. She received the B.S. degree in mathematics and applied mathematics from Anqing Normal University, Anqing, China, in 2010, and is currently pursuing the M.S. degree in applied mathematics at North China University of Technology, Beijing, China. Her research interests are in the fields of networked systems and collective intelligence.
Housheng Su received the B.S. degree in automatic control and the M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. Since January 2010, he has been an associate professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
Yanping Gao received the B.S. and M.S. degrees in Applied Mathematics from Ocean University of China in 2003 and 2006, respectively, and the Ph.D. degree from Peking University, Beijing, in 2011. She is currently working as a lecturer at Beijing Technology and Business University. Her current research interests are in the fields of multi-agent systems, consensus problems, and sampled-data control.
Li Wang was born in 1978. He graduated from BeiHang University and achieved Ph.D degree in 2006. Currently, he is an Associate Professor of North China University of Technology. He won Beijing New Star of Science and Technology Award in 2011. His main research field is intelligent transportation system, intelligent traffic system modeling and control, traffic signal control, dynamic traffic simulation. He has chaired more than 10 national and Beijing local government projects like 863 project, Olympic special project and project founded by Beijing Government Education Committee. In 2011, he was rewarded by 3rd Prize of Beijing Science and Technology and 1st Prize of aLian Science and Technology Progress. He has published 20 papers on international traffic journals and three patents.
Contents lists available at SciVerse ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Adaptive second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays Bo Liu a, Xiaoling Wang a, Housheng Su b,n, Yanping Gao c, Li Wang d a
College of Science, North China University of Technology, Beijing 100144, PR China Department of Control Science and Engineering, Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Huazhong University of Science and Technology, Wuhan 430074, PR China c College of Computer and Information Engineering, Beijing Technology and Business University, Beijing 100048, PR China d Beijing Key Lab of Urban Intelligent Traffic Control Technology, North China University of Technology, Beijing 100144, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2012 Received in revised form 18 January 2013 Accepted 12 February 2013 Communicated by H. Dong Available online 3 May 2013
This paper investigates second-order consensus of multi-agent systems with heterogeneous nonlinear dynamics and time-varying delays by introducing novel decentralized adaptive strategies to both the coupling strengths and the feedback gains. Based on the Lyapunov stability theory, it is proved that the position and the velocity of each agent can converge to those of the virtual leader respectively, when the network is connected and at least one agent is informed, without any global information of the multiagent system. Finally, some numerical simulations are presented to verify the theoretical results. & 2013 Elsevier B.V. All rights reserved.
Keywords: Adaptive control Second-order consensus Pinning control Time-varying delays Virtual leader
1. Introduction The consensus of multi-agent systems is ubiquitous in nature, such as flocking of birds, schooling of fish, etc. Nowadays, the consensus problem of multi-agent systems is an important developing direction of control theory due to the broad application of the consensus in control engineering and sensor networks [1–4]. Consensus problems are investigated for single-integrator kinematics [5,6], double-integrator dynamics [7–12] and high-orderintegrator dynamics [13], respectively. So far, many techniques, such as the pinning control and the adaptive control have been brought to drive the system to achieve consensus. The pinning control, signifying that only a small fraction of agents can receive information from the desired state, is a valid method to study the consensus of multi-agent systems. Wang and Chen [14] compared the effectiveness of specifically and randomly pinning schemes for the stabilization problem of the scale-free dynamical networks. In [15], the authors investigated the flocking of multi-agent systems with a virtual leader by virtue of pinning control and found that an uninformed agent can track the virtual leader if it can receive information from the informed
n
Corresponding author. Tel.: +86 27 87540210. E-mail address: [email protected] (H. Su).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.02.038
agents directly or indirectly from time to time. The adaptive strategy is usually used for the coordinated control algorithms when the global information of complex dynamical networks is unavailable. In [16], a robust adaptive control approach was proposed to solve the consensus problem of multi-agent systems. Two local adaptive strategies, that is, vertex-based strategy and edge-based strategy, were discussed for the synchronization of complex networks in [17]. In [18], the authors discussed the adaptive synchronization of first-order complex dynamical networks, while second-order consensus was investigated in [19,20] by introducing adaptive strategies to both the coupling strengths and feedback gains. Time delays exist widely in practice due to the jams of channel, noise, etc. Therefore, we should pay attention to the influence of the time delays when the consensus of multi-agent systems is considered. Average consensus of dynamical agents with timevarying delays was studied in both fixed and switching networks by applying a linear matrix inequality method [6]. The stability of a class of neural networks with multiple time delays was solved in [21]. In [8], Su et al. presented two kinds of algorithms to investigate the second-order consensus of multi-agent systems with identical time delay in fixed directed networks. Refs. [9,10] studied the second-order consensus of multi-agent systems with nonuniform time-delays, and Ref. [11] provided the consensus criteria in the form of linear matrix inequality (LMI) for the
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multi-agent systems with time-varying delays under both static interconnection topology and switching topologies, respectively. Motivated by the works above, in this paper, we focus on addressing the second-order consensus of multi-agent systems with nonlinear dynamics and time-varying delays via pinning control. The main contributions of the paper are threefold: (1) The nonlinear intrinsic dynamics of all the agents and the virtual leader are heterogeneous, (2) the position and the velocity of the agents are both with time-varying delays, and (3) novel decentralized adaptive strategies are introduced to solve the second-order consensus of multi-agent systems. For the bounded time-varying delays, we investigate the second-order consensus of two cases that the weighted coupling configuration matrix is symmetric and the weighted coupling configuration matrix is asymmetric by adding novel adaptive decentralized strategies to both the coupling strengths and feedback gains, and obtain that the position and the velocity of each agent can converge to those of the virtual leader respectively when the network is connected even though only one agent is informed, without requiring global information of the multi-agent system which is needed in Refs. [6,8], and calculating any LMI as Ref. [22]. The rest of the paper is organized as follows. Section 2 states the second-order model investigated and introduces some preliminaries to solve the second-order consensus. Section 3 provides theoretical analysis to the second-order consensus of multi-agent systems with nonlinear dynamics and time-varying delays. Section 4 presents the numerical simulations to the main results. Section 5 concludes this paper. Notation: Throughout this paper, the notation AT means the transpose of the matrix A, ∥ ∥ is the Euclidean norm, H ¼ diagfh1 ; h2 ; …; hN g.
2. Preliminaries and problem statement Consider a multi-agent system consisting of N agents and each agent moves according to the following dynamics: 8_ q i ðtÞ ¼ pi ðtÞ > > >_ > > p ðtÞ ¼ f i ðpi ðtÞÞ þ ∑ αij ðtÞaij ðqj ðt−τðtÞÞ−qi ðt−τðtÞÞ > < i j∈N i ð1Þ þ ∑ βij ðtÞaij ðpj ðt−τðtÞÞ−pi ðt−τðtÞÞ > > > j∈Ni > > > : þui ðtÞ; i ¼ 1; …; N; where qi ðtÞ ¼ ðqi1 ðtÞ; qi2 ðtÞ; …; qin ðtÞÞT ∈Rn (i ¼ 1; 2; …; N) describes the position vector of the i-th agent, and pi ðtÞ ¼ ðpi1 ðtÞ; pi2 ðtÞ; …; pin ðtÞÞT ∈Rn (i ¼ 1; 2; …; N) is its velocity vector. f i : Rn -Rn is a continuous function and f i ðpi ðtÞÞ describes the intrinsic dynamics of the ith agent, τðtÞ represents time-varying delays in transmission between agent i and agent j. αij ðtÞ and βij ðtÞ denote the coupling strengths of the position and the velocity between agent i and agent j, respectively. The matrix A ¼ ðaij Þ∈RNN with aii ¼ −∑N j ¼ 1;j≠i aij represents the weighted coupling configuration of system (1) and aij is the weight of the edge which connects agent i and agent j. G(A) describes the topology structure of G. If there exists communication channel between agent i and agent j, then aij 4 0, otherwise, aij ¼ 0. Ni is the neighbor set of agent i. The virtual leader regulates itself according to the following dynamics: ( q_ 0 ðtÞ ¼ p0 ðtÞ ; ð2Þ p_ 0 ðtÞ ¼ f 0 ðp0 ðtÞÞ where q0 ðtÞ and p0 ðtÞ are the position and velocity of the virtual leader, f 0 : Rn -Rn is a continuous function and f 0 ðp0 ðtÞÞ is the intrinsic dynamics of the virtual leader. Note that this virtual
leader is an external command (an external signal or a desired destination) to the system. The control input is designed as ui ðtÞ ¼ −hi ci ðtÞðqi ðt−τðtÞÞ−q0 ðtÞÞ−hi di ðtÞðpi ðt−τðtÞÞ−p0 ðtÞÞ;
i ¼ 1; …; N; ð3Þ
where hi is a binary value, if the ith agent is controlled, then hi ¼ 1, otherwise hi ¼ 0. ci(t) and di(t) are the feedback gains of position and velocity. Based on (1) and (3), the adaptive strategies on the coupling strengths of position and velocity are designed respectively as α_ ij ðtÞ ¼ aij kij fðqi ðt−τðtÞÞ−qj ðt−τðtÞÞÞT ðqi ðt−τðtÞÞ−qj ðt−τðtÞÞÞ þðq_ i ðtÞ−q_ j ðtÞÞT ðq_ i ðtÞ−q_ j ðtÞÞg;
ð4Þ
and β_ ij ðtÞ ¼ aij kij fðpi ðt−τðtÞÞ−pj ðt−τðtÞÞÞT ðpi ðt−τij ðtÞÞ−pj ðt−τij ðtÞÞÞ þðp_ i ðtÞ−p_ j ðtÞÞT ðp_ i ðtÞ−p_ j ðtÞÞg;
ð5Þ
where kij 4 0 is the weight of αij ðtÞ and βij ðtÞ with αij ð0Þ≥0 and βij ð0Þ≥0. The adaptive strategies on the feedback gains of position and velocity are designed respectively as c_ i ðtÞ ¼ hi ki ½ðqi ðt−τðtÞÞ−q0 ðtÞÞT ðqi ðt−τðtÞÞ−q0 ðtÞÞ þðq_ i ðtÞ−q_ 0 ðtÞÞT ðq_ i ðtÞ−q_ 0 ðtÞÞ;
ð6Þ
and d_ i ðtÞ ¼ hi ki ½ðpi ðt−τðtÞÞ−p0 ðtÞÞT ðpi ðt−τðtÞÞ−p0 ðtÞÞ þðp_ i ðtÞ−p_ 0 ðtÞÞT ðp_ i ðtÞ−p_ 0 ðtÞÞ;
ð7Þ
where ki 4 0 is the weight The weighted coupling and the velocity of system 2 a11 α11 a12 α12 ⋯ 6 a21 α21 a22 α22 ⋯ 6 B¼6 ⋮ ⋱ 4 ⋮ aN1 αN1
aN2 αN2
of ci(t) and di(t), ci ð0Þ≥0 and di ð0Þ≥0. configuration matrices of the position (1) are given respectively as follows: 3 a1N α1N a2N α2N 7 7 NN 7∈R ⋮ 5 ⋯ aNN αNN
with aii αii ¼ −∑N j ¼ 1;j≠i aij αij ; 2 a11 β11 a12 β12 ⋯ a1N β1N 6 6 a21 β21 a22 β22 ⋯ a2N β2N C¼6 6 ⋮ ⋮ ⋱ ⋮ 4 aN1 βN1
aN2 βN2
⋯
aNN βNN
3 7 7 NN 7∈R 7 5
with aii βii ¼ −∑N j ¼ 1;j≠i aij βij . Remark 1. Eq. (1) describes a second-order consensus algorithm for the multi-agent system with time-varying delays, and each agent uses the delayed information to update its position and velocity. Remark 2. In Eq. (1), the time-varying delays are in the presence of both the position and the velocity because the sensors have the broadcast delays, and this is different from [12] in which only the position has time-varying delays. To end this section, some necessary assumptions, propositions, definitions and lemmas are stated. Assumption 1. 0 ≤ τðtÞ ≤τ, t≥0, where τ is a positive constant. Remark 3. Differing from the existed results, such as [22], which required the derivative of the time delay satisfying 0 ≤ τ_ ðtÞ ≤1, in this paper, any information about the derivative of the time delay τðtÞ need not be known.
B. Liu et al. / Neurocomputing 118 (2013) 289–300
Assumption 2 (Su et al. [19]). The vector field f i : Rn -Rn ði ¼ 1; 2; …; NÞ satisfies ðx−yÞT ½f i ðxÞ−f i ðyÞ ≤ðx−yÞT Δðx−yÞ;
∀x; y∈Rn ;
i ¼ 0; 1; 2; …; N;
for some positive definite diagonal constant matrix Δ. Assumption 3. The coupling strengths and feedback gains of the position and the velocity are bounded, that is, ∥αij ðtÞ∥ ≤ αij ;
∥βij ðtÞ∥ ≤βij ;
∥ci ðtÞ∥ ≤ci ;
∥di ðtÞ∥ ≤di ;
where αij ; βij ; ci ; di , are positive constants. In practice, the coupling strengths and feedback gains are bounded generally.
Lemma 6. For any real differentiable vector function xðtÞ∈Rn and any n n constant matrix W ¼ W T 4 0, we have the following inequality: Z t T Z t Z t xðsÞ ds W xðsÞ ds ≤τ xT ðsÞWxðsÞ ds; t≥0; t−τðtÞ
t−τðtÞ
where 0 ≤τðtÞ ≤τ. Proof. Z t t−τðtÞ
Z
Z
t
¼
Note that Assumption 4 can be satisfied by many well-known systems, such as the Lü system, the chaotic Chua circuit, the Lorenz system and so on. Proposition 1 (Zorich [24] The Weierstrass maximum-value theorem). A function that is continuous on a closed interval is bounded on that interval. Moreover, there is a point in the interval where the function assumes its maximum value and a point where it assumes its minimal value. Lemma 1 (Heterogeneous Nonlinear Dynamics Inequality (HNDI)). The vector field f i : Rn -Rn ði ¼ 0; 1; …; NÞ satisfies ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ω2 ∥pi ðtÞ−p0 ðtÞ∥; ∀pi ðtÞ; p0 ðtÞ∈Rn for the constants ω1 40 and ω2 4 0. Proof. From Assumption 2, we can have ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f i ðp0 ðtÞÞ þ f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f i ðp0 ðtÞÞ þ ðpi ðtÞ−p0 ðtÞÞT ½f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ
t−τðtÞ
□
T Z xðsÞ ds W
t−τðtÞ
Assumption 4 (Hu et al. [20]). The virtual leader moves in a bounded region consistently in sense that there exists a compact set C ¼ Cðq0 ; p0 Þ∈Rn Rn , and the virtual leader starting with ðq0 ; p0 Þ is always in C.
291
xðsÞ ds
t
t−τðtÞ
t
1
1
½W 2 xðsÞT ½W 2 xðrÞ ds dr
t−τðtÞ
Z Z t 1 1 1 t 2½W 2 xðsÞT ½W 2 xðrÞ ds dr 2 t−τðtÞ t−τðtÞ Z Z t 1 t ½xT ðsÞWxðsÞ þ xT ðrÞWxðrÞ ds dr ≤ 2 t−τðtÞ t−τðtÞ Z t ¼ τðtÞ xT ðsÞWxðsÞ ds ¼
Z ≤τ
t−τðtÞ
t
xT ðsÞWxðsÞ ds:
t−τðtÞ
□
This completes the proof.
Definition 1. System (1) with the virtual leader described as Eq. (2) is said to achieve second-order consensus if lim ∥qi ðtÞ−q0 ðtÞ∥ ¼ 0;
t-∞
lim ∥pi ðtÞ−p0 ðtÞ∥ ¼ 0;
t-∞
i ¼ 1; 2; …; N;
hold for any initial condition Xð0Þ ¼ ðqT ð0Þ; pT ð0ÞÞT ∈R2N , where qð0Þ ¼ ðqT1 ð0Þ; qT2 ð0Þ; …; qTN ð0ÞÞT , pð0Þ ¼ ðpT1 ð0Þ; pT2 ð0Þ; …; pTN ð0ÞÞT . 3. Theoretical analysis and main results In this section, we will give detailed analysis for the adaptive second-order consensus of multi-agent systems from two cases, that is, the weighted coupling configuration matrix A is symmetric and asymmetric, respectively.
≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ðpi ðtÞ−p0 ðtÞÞT ½f i ðp0 ðtÞÞ−f 0 ðp0 ðtÞÞ ¼ ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ðpi ðtÞ−p0 ðtÞÞT ½ðf i −f 0 Þðp0 ðtÞÞ:
3.1. The weighted coupling configuration matrix A is symmetric
By Assumption 4 and Proposition 1, we know that the value of ðf i −f 0 Þðp0 ðtÞÞ is bounded since each f i ði ¼ 0; 1; 2; …; NÞ is continuous function. Therefore,
Firstly, we investigate the adaptive consensus of multi-agent systems described by Eq. (1) with the weighted coupling configuration matrix A is symmetric in detail.
ðpi ðtÞ−p0 ðtÞÞT ½f i ðpi ðtÞÞ−f 0 ðp0 ðtÞÞ ≤ω1 ∥pi ðtÞ−p0 ðtÞ∥2 þ ω2 ∥pi ðtÞ−p0 ðtÞ∥: □
Theorem 1. Consider system (1) with a virtual leader as (2) which is steered by the adaptive strategies (4)–(7) under Assumptions1–4, suppose that the network is undirected and connected and the weighted coupling configuration matrix A is symmetric, then the position and velocity of each agent can converge to those of the virtual leader when at least one agent is controlled by (3).
Lemma 2 (Yu [23]). For any vectors x; y∈Rn and positive definite matrix G∈Rnn , the following matrix inequality holds: 2xT y ≤xT Gx þ yT G−1 y: Lemma 3 (Su and Zhang [8]). Suppose that a and b are vectors, then for any positive-definite matrix E, the following inequality holds: −2aT b ≤ inf faT Ea þ b E−1 bg: T
E40
Lemma 4 (Su et al. [18]). If A ¼ ðaij Þ∈RNN is a symmetric irreducible matrix with aii ¼ −∑N j ¼ 1;j≠i aij , and the matrix E ¼ diagfe; 0; …; 0g with e 4 0, then all eigenvalues of the matrix A−E are negative. Lemma 5 (Godsil and Royle [26]). The matrix A of an undirected graph G is irreducible if and only if the undirected graph is connected.
Proof. Let q~ i ðtÞ ¼ qi ðtÞ−q0 ðtÞ; p~ i ðtÞ ¼ pi ðtÞ−p0 ðtÞ;
q~ i ðt−τðtÞÞ ¼ qi ðt−τðtÞÞ−q0 ðtÞ; p~ i ðt−τðtÞÞ ¼ pi ðt−τðtÞÞ−p0 ðtÞ;
we have 8 > q~_ i ðtÞ ¼ q_ i ðtÞ−q_ 0 ðtÞ ¼ pi ðtÞ−p0 ðtÞ ¼ p~ i ðtÞ > > > > > p~_ i ðtÞ ¼ p_ i ðtÞ−p_ 0 ðtÞ > > > > < ¼ f ðp ðtÞÞ−f ðp ðtÞÞ þ ∑ α ðtÞa ðq~ ðt−τðtÞÞ−q~ ðt−τðtÞÞ ij ij j i i 0 0 i j∈N i > > > > þ ∑ βij ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞ > > > j∈Ni > > > : −h c ðtÞq~ ðt−τðtÞÞ−h d ðtÞp~ ðt−τðtÞÞ: i i
i
i i
i
ð8Þ
292
B. Liu et al. / Neurocomputing 118 (2013) 289–300
~ ~ In order to obtain the desired result, i.e., qðtÞ-0 and pðtÞ-0 as t-∞, we choose
and ∥p~ i ðtÞ∥ ≤ ∥p~ i ðt−τðtÞÞ∥ þ ‖
1 N 1 N T V 1 ðtÞ ¼ ∑ p~ Ti ðtÞp~ i ðtÞ þ ∑ q~ i ðtÞq~ i ðtÞ: 2i¼1 2i¼1 Taking the derivative of V 1 ðtÞ, there appears the terms of adaptive coefficients and time-varying delays according to Eq. (8). To offset these terms, we select ðαij ðtÞ−2αij −mÞ2 þ ðβij ðtÞ−2βij −mÞ2 4kij i ¼ 1 j∈Ni
Z
t
t−τðtÞ
p~_ ðsÞ ds‖ ≤∥p~ i ðt−τðtÞÞ∥ þ
Z
t t−τðtÞ
‖p~_ ðsÞ‖ds:
In addition, Lemma 6 demonstrates that we can have Z t T Z t Z t _ ds; _ _ ds ≤τ xðsÞds xðsÞ x_ T ðsÞxðsÞ t−τðtÞ
t−τðtÞ
t−τðtÞ
N
V 2 ðtÞ ¼ ∑ ∑
thus, Z p~ Ti ðt Þp~ i ðt Þ ¼ p~ i ðt−τðt ÞÞ þ
3 3 2 2 N ðci ðtÞ− ci −mÞ þ ðdi ðtÞ− di −mÞ 2 2 þ ∑ ; 2ki i¼1 N
"
V 3 ðtÞ ¼ τ ∑
i¼1
Z
t
! 2ω1 þ
∑ aij ðαij þ βij Þ
#
N
t−τ
T ðs−t þ τÞp~_ i ðsÞp~_ i ðsÞ ds þ ω2 ∑ N
þτ ∑
i¼1
Z
i¼1
t t−τ
t−τðt Þ
Z
t
Z þ
t−τ
T ðs−t þ τÞq~_ i ðsÞq~_ i ðsÞ ds:
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ; where m 4 0 is sufficiently large. By Lemmas 1 and 2, the derivative of V 1 ðtÞ satisfies N
V_ 1 ðtÞ ¼ ∑ p~ Ti ðtÞff i ðpi ðtÞÞ−f ðp0 ðtÞÞ þ ∑ αij ðtÞaij ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞ
T Z p~_ i ðsÞ ds
t t−τðt Þ
t−τðt Þ
≤ p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
p~_ i ðsÞ ds :
p~ i ðt−τðt ÞÞ þ t
t−τðt Þ
Z
t
t−τðt Þ
p~_ i ðsÞ ds
p~_ i ðsÞ ds
Z
t
þ
t−τðt Þ
T Z p~_ i ðsÞ ds
t
t−τðt Þ
p~_ i ðsÞ ds :
Therefore, p~ Ti ðt Þp~ i ðt Þ ≤2p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ Z t T Z t p~_ i ðsÞds p~_ i ðsÞ ds þ2 t−τðt Þ
j∈N i
þ ∑ βij ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞ−hi ci ðtÞq~ i ðt−τðtÞÞ−hi di ðtÞp~ i ðt−τðtÞÞg
T
From Lemma 2, choosing G ¼ I N and we have Z t p~_ i ðsÞ ds 2p~ Ti ðt−τðt ÞÞ
Thus, the global Lyapunov function is chosen as
i¼1
t
t−τðt Þ
ðs−t þ τÞ∥p~_ i ðsÞ∥ ds
p~_ i ðsÞ ds
Z ¼ p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ þ 2p~ Ti ðt−τðt ÞÞ
þ hi ðci þ di Þ þ 1
j∈Ni
t
t−τðt Þ
¼ 2p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
Z
t
þ 2τ
t−τðt Þ
j∈N i
T p~_ i ðsÞp~_ i ðsÞ ds;
i.e.,
N
T þ ∑ q~ i ðtÞp~ i ðtÞ
∥p~ i ðt Þ∥2 ≤2∥p~ i ðt−τðt ÞÞ∥2 þ 2τ
i¼1
Z
t t−τðt Þ
N
∥p~_ i ðsÞ∥2 ds:
≤ ∑ p~ Ti ðtÞ½f i ðpi ðtÞÞ−f ðp0 ðtÞÞ i¼1
þ
1 N ∑ ∑ αij ðtÞaij p~ Ti ðtÞp~ i ðtÞ 2 i ¼ 1 j∈Ni
Similarly,
1 N þ ∑ ∑ αij ðtÞaij ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞT ðq~ j ðt−τðtÞÞ−q~ i ðt−τðtÞÞÞ 2 i ¼ 1 j∈Ni þ þ þ þ
1 N ∑ ∑ β ðtÞaij p~ Ti ðtÞp~ i ðtÞ 2 i ¼ 1 j∈Ni ij
N
1 ∑ ∑ β ðtÞaij ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞÞT ðp~ j ðt−τðtÞÞ−p~ i ðt−τðtÞÞÞ 2 i ¼ 1 j∈Ni ij 1 N 1 N T ∑ h c ðtÞp~ Ti ðtÞp~ i ðtÞ þ ∑ hi ci ðtÞq~ i ðt−τðtÞÞq~ i ðt−τðtÞÞ 2i¼1 i i 2i¼1 N
1 1 ∑ h d ðtÞp~ Ti ðtÞp~ i ðtÞ þ ∑ hi di ðtÞp~ Ti ðt−τðtÞÞp~ i ðt−τðtÞÞ 2i¼1 i i 2i¼1
1 N T 1 N þ ∑ q~ i ðtÞq~ i ðtÞ þ ∑ p~ Ti ðtÞp~ i ðtÞ: 2i¼1 2i¼1
V_ 1 ðt Þ ≤ ∑
Using Leibniz–Newton formula as Z t _ ds; xðsÞ xðtÞ−xðt−τðtÞÞ ¼ t−τðtÞ
i¼1
t t−τðtÞ
p_~ ðsÞ ds; T
t−τðt Þ
∥q~_ i ðsÞ∥2 ds:
# ! ∑ aij αij þ βij þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2
2ω1 þ
j∈N i
þ ∑ ω2 ∥p~ i ðt−τðt ÞÞ∥ i¼1 " N
þτ ∑
i¼1
Z
t t−τðt Þ N
i¼1
2ω1 þ
N
∑ aij αij þ βij
i¼1
!
# þ hi ðci þ di Þ þ 1
j∈N i
T p~_ i ðsÞp~_ i ðsÞ ds
Z
t t−τðt Þ
∥p~_ i ðsÞ∥ ds
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ τ
Z
t t−τðt Þ
∥q~_ i ðsÞ∥2 ds
þ
T 1 N ∑ ∑ α ðt Þaij q~ j ðt−τðt ÞÞ−q~ i ðt−τðt ÞÞ q~ j ðt−τðt ÞÞ−q~ i ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij
þ
T 1 N ∑ ∑ βij ðt Þaij p~ j ðt−τðt ÞÞ−p~ i ðt−τðt ÞÞ p~ j ðt−τðt ÞÞ−p~ i ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni
we can have Z
t
N
þ ∑ ω2
p~ Ti ðtÞ ¼ p~ Ti ðt−τðtÞÞ þ
Z
Under Assumption 3 "
N
N
∥q~ i ðt Þ∥2 ≤2∥q~ i ðt−τðt ÞÞ∥2 þ 2τ
B. Liu et al. / Neurocomputing 118 (2013) 289–300
þ
1 N 1 N T ∑ hi ci ðt Þq~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ hi di ðt Þp~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ: 2i¼1 2i¼1
Differentiating V 2 ðt Þ, T 1 N q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ V_ 2 ðt Þ ¼ ∑ ∑ αij ðt Þ−2αij −m aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni T 1 N ∑ ∑ αij ðt Þ−2αij −m aij q~_ i ðt Þ−q~_ j ðt Þ q~_ i ðt Þ−q~_ j ðt Þ 2 i ¼ 1 j∈Ni T 1 N þ ∑ ∑ βij ðt Þ−2βij −m aij p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni
þ
T 1 N þ ∑ ∑ βij ðt Þ−2βij −m aij p~_ i ðt Þ−p~_ j ðt Þ p~_ i ðt Þ−p~_ j ðt Þ 2 i ¼ 1 j∈Ni N N T 3 3 T þ ∑ ci ðt Þ− ci −m hi q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ ci ðt Þ− ci −m hi q~_ i ðt Þq~_ i ðt Þ 2 2 i¼1 i¼1
N
þ ∑
i¼1
d dt
N T 3 3 T di ðt Þ− di −m hi p~ i ðt−τðt ÞÞp~ i ðt−τðt ÞÞ þ ∑ di ðt Þ− di −m hi p~_ i ðt Þp~_ i ðt Þ: 2 2 i¼1
Based on the Leibniz derivative rule Z t Z f ðτ; t Þ dτ ¼ f ðbðt Þ; t Þb_ ðt Þ−f ðaðt Þ; t Þa_ ðt Þ þ t−τij ðt Þ
the derivative of V 3 ðt Þ yields "
N
V_ 3 ðt Þ ¼ τ2 ∑
i¼1
j∈N i
"
N
2ω1 þ
−τ ∑
i¼1
∑ aij αij þ βij
2ω1 þ
∑ aij αij þ βij j∈N i
N
N
i¼1
i¼1
þτω2 ∑ ∥p~ i ðt Þ∥−ω2 ∑
Z
t t−τ
!
#Z þ hi ðci þ di Þ þ 1
t t−τ
N
d f ðτ; t Þ dτ; dt t−τij ðt Þ
∥p~_ i ðsÞ∥ ds þ τ2 ∑
i¼1
T p~_ i ðsÞp~_ i ðsÞ ds
T q~_ i ðt Þq~_ i ðt Þ−τ
N
∑
i¼1
Z
2
3 ∥p~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 6 ∥p 2 ðt−τðt ÞÞ∥ 7 7 þm½∥p~ 1 ðt−τðt ÞÞ∥; ∥p~ 2 ðt−τðt ÞÞ∥; …; ∥p~ N ðt−τðt ÞÞ∥ðA−H Þ6 6 7 ⋮ 4 5 ∥p~ N ðt−τðt ÞÞ∥ "
N
þτ2 ∑
∑ aij αij þ βij
2ω1 þ
i¼1
t−τ
T q~_ i ðsÞq~_ i ðsÞ
#
N
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2 þ τω2 ∑ ∥p~_ i ðt Þ∥ i¼1
j∈N i
3 2 _ ∥p~ 1 ðt Þ∥ 7 6 _ N 6 ∥p~ 2 ðt Þ∥ 7 7 þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þm½∥p~_ 1 ðt Þ∥; ∥p~_ 2 ðt Þ∥; …; ∥p~_ N ðt Þ∥ðA−HÞ6 7 6 ⋮ 5 i¼1 4 ∥p~_ N ðt Þ∥
2
3 ∥q~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 6 ∥q 2 ðt−τðt ÞÞ∥ 7 7 þm½∥q~ 1 ðt−τðt ÞÞ∥; ∥q~ 2 ðt−τðt ÞÞ∥; …; ∥q~ N ðt−τðt ÞÞ∥ðA−H Þ6 6 7 ⋮ 4 5 ∥q~ N ðt−τðt ÞÞ∥ 0
3 2 _ ∥q~ 1 ðt Þ∥ B 7 6 _ B _ 7 6 ~ T B∥q~ 1 ðt Þ∥; ∥q~_ 2 ðt Þ∥; …; ∥q~_ N ðt Þ∥ðA−H Þ6 ∥q 2 ðt Þ∥ 7: þτ2 ∑ q~_ i ðt Þq~_ i ðt Þ þ mB 7 6 ⋮ i¼1 @ 5 4 _ ~ ∥q N ðt Þ∥ N
ij
i¼1
t
!
From the conditions of Theorem 1, we know that the matrix H is a diagonal matrix with at least one element equaling to 1. From Lemma 5, one can easily get that A is irreducible because system (1) is connected. Since A is symmetric, all eigenvalues of A−H are negative from Lemma 4. Let λ1 be the maximum eigenvalue of A−H and λ1 o 0. Then " # ! N _ þ hi ðci þ di Þ þ 1 ∥p~ ðt−τðt ÞÞ∥2 V ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ β
t
# T þ hi ðci þ di Þ þ 1 p~_ i ðt Þp~_ i ðt Þ
!
293
ds:
i
j∈Ni
N
þ ω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ i¼1
N
By Assumption 3, V_ ðt Þ ¼ V_ 1 ðt Þ þ V_ 2 ðt Þ þ V_ 3 ðt Þ " # ! N ≤ ∑ 2ω1 þ ∑ aij αij þ βij þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 i¼1
j∈N i
T N 1 p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ − m ∑ ∑ aij p~ i ðt−τðt ÞÞ−p~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni hi p~ Ti ðt−τðt ÞÞp~ i ðt−τðt ÞÞ
N
"
i¼1
∑ aij αij þ βij
2ω1 þ
∑
i¼1
∑ aij αij þ βij
2ω1 þ
!
# þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
j∈Ni
N
þmλ1 ∑ ∥p~_ i ðt Þ∥2 i¼1
N
!
#
N
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2 þ τω2 ∑ ∥p~_ i ðt Þ∥ i¼1
j∈N i
N
N
N
i¼1
i¼1
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ mλ1 ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ τ2 ∑ ∥q~_ i ðt Þ∥2 i¼1
T N N T 1 p~_ i ðt Þ−p~_ j ðt Þ −m ∑ hi p~_ i ðt Þp~_ i ðt Þ − m ∑ ∑ aij p~_ i ðt Þ−p~_ j ðt Þ 2 i ¼ 1 j∈Ni i¼1 þτ2 ∑
"
i¼1
i¼1
i¼1
þτ
2
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥
−m ∑
i¼1
N
þτω2 ∑ ∥p~_ i ðt Þ∥
N
N
þmλ1 ∑ ∥p~ i ðt−τðt ÞÞ∥2
N
þmλ1 ∑ ∥q~_ i ðt Þ∥2 :
ð9Þ
i¼1
Since λ1 o 0 and m 4 0 is sufficiently large, V_ ðt Þ ≤0:
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 i¼1
T N 1 q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ − m ∑ ∑ aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni N
−m ∑ hi q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ T
Define the level set of V(t) in the space of the positions and velocities of agents Ω ¼ f q~ i ; p~ i jV ðt Þ ≤c; c 4 0g:
i¼1
T N N T 1 q~_ i ðt Þ−q~_ j ðt Þ −m ∑ hi q~_ i ðt Þq~_ i ðt Þ − m ∑ ∑ aij q~_ i ðt Þ−q~_ j ðt Þ 2 i ¼ 1 j∈Ni i¼1 " ! N N T þτ2 ∑ q~_ ðt Þq~_ ðt Þ ¼ ∑ 2ω þ ∑ a α þ β i¼1
i
i
i¼1
1
ij
ij
ij
j∈Ni
N
þhi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 þ ω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ i¼1
Since G is connected and V ðt Þ ≤c, on the one hand, p~ Ti ðt Þp~ i ðt Þ ≤2c pffiffiffiffiffi T and then ∥p~ i ∥ ≤ 2c. On the other hand, q~ i ðt Þq~ i ðt Þ ≤2c and then pffiffiffiffiffi ∥q~ i ∥ ≤ 2c. So, ∥q~ i ðt Þ∥ and ∥p~ i ðt Þ∥ are both bounded. Thus, Ω is bounded. Since ∥q~ i ðt Þ∥ and ∥p~ i ðt Þ∥ are bounded, the set Ω is closed by the continuity of V ðt Þ ≤c. Therefore, the set Ω is compact.
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Consequently, we can apply LaSalle Invariance Principle and the corresponding trajectories will converge to the largest invariant set of ~ p~ ÞjV_ ðt Þ ¼ 0g: S ¼ fðq;
And define the same Lyapunov function as Theorem 1, we can obtain " # ! N _ V ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ β þ hi ðci þ di Þ þ 1 ∥p~ ðt−τðt ÞÞ∥2 ij
i¼1
i
j∈Ni
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥
From (9), V_ ðt Þ ¼ 0 if and only if q~ i ðt−τðt ÞÞ ¼ 0;
q~_ i ðt Þ ¼ 0;
p~ i ðt−τðt ÞÞ ¼ 0;
p~_ i ðt Þ ¼ 0; i ¼ 1; 2; …; N:
i¼1
þm½∥p~ 1 ðt−τðt ÞÞ∥; ∥p~ 2 ðt−τðt ÞÞ∥; …; ∥p~ N ðt−τðt ÞÞ∥ 2
Based on (8), we can gain that V_ ðt Þ ¼ 0 if and only if q~ i ðt−τðt ÞÞ ¼ 0; p~ i ðt Þ ¼ p~ i ðt−τðt ÞÞ ¼ 0;
i ¼ 1; 2; …; N:
ð10Þ
Since lim p~ i ðt−τðt ÞÞ ¼ p~ i ðt Þ;
lim p~ i ðt−τðt ÞÞ ¼ p~ i ðt Þ;
i ¼ 1; 2; …; N;
hence, according to Eq. (10), we can obtain lim q~ i ðt Þ ¼ 0;
t-∞
lim p~ i ðt Þ ¼ 0;
t-∞
i ¼ 1; 2; …; N;
which imply that the position and the velocity of each agent can converge to those of the virtual leader respectively. This completes the proof. □ 3.2. The weighted coupling configuration matrix A is asymmetric In the following, we are interested in discussing the consensus of system (1) with the weighted coupling configuration matrix A is asymmetric. Firstly, we give the following assumption. Assumption 5 (Liu et al. [25]). The weights satisfy the balance condition: ∑ aij ¼ ∑ aji ; j∈N i
∀i:
j∈N i
Theorem 2. Consider system (1) with a virtual leader as (2) which is steered by the adaptive strategies (4)–(7) under Assumptions 1–5, suppose that the network is undirected and connected and the weighted coupling configuration matrix A is asymmetric, then the position and velocity of each agent can converge to those of the virtual leader even when only one agent is controlled by (3). Proof. Under Assumption 5, we have −
þ
1 N 1 N T T ∑ ∑ aij q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ þ ∑ ∑ aij q~ i ðt−τðt ÞÞq~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 2 i ¼ 1 j∈Ni N
!
j∈Ni
! A þ AT _ _ _ _ ~ ~ ~ ~ −H þτω2 ∑ ∥p i ðt Þ∥ þ m½∥p 1 ðt Þ∥; ∥p 2 ðt Þ∥; …; ∥p N ðt Þ∥ 2 i¼1 2 _ 3 ∥p~ 1 ðt Þ∥ 6 _ 7 N 6 ∥p~ 2 ðt Þ∥ 7 6 7 þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 6 7 ⋮ 4 5 i¼1 ∥p~_ N ðt Þ∥ ! A þ AT −H þm½∥q~ 1 ðt−τðt ÞÞ∥; ∥q~ 2 ðt−τðt ÞÞ∥; …; ∥q~ N ðt−τðt ÞÞ∥ 2 2 3 ∥q~ 1 ðt−τðt ÞÞ∥ 6 ~ 7 N 6 ∥q 2 ðt−τðt ÞÞ∥ 7 6 7 þ τ2 ∑ q~_ T ðt Þq~_ i ðt Þ þ m½∥q~_ 1 ðt Þ∥; ∥q~_ 2 ðt Þ∥; …; ∥q~_ N ðt Þ∥ i 6 7 ⋮ i¼1 4 5 ∥q~ N ðt−τðt ÞÞ∥ 2 _ 3 ∥q~ ðt Þ∥ !6 1 7 6 ∥q~_ 2 ðt Þ∥ 7 A þ AT 7: −H 6 6 7 2 ⋮ 4 5 ∥q~_ N ðt Þ∥ Even though matrix A is asymmetric, matrix A þ AT =2 is symmetric. Thus, all eigenvalues of matrix A þ AT =2−H are negative from Lemma 2. Let λ2 ð o 0Þ be the maximum eigenvalue of A þ AT =2−H . Therefore, " # ! N þ hi ðci þ di Þ þ 1 ∥p~ i ðt−τðt ÞÞ∥2 V_ ðt Þ ≤ ∑ 2ω1 þ ∑ aij αij þ βij i¼1
j∈Ni N
T 1 N ∑ ∑ a q~ ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij i ¼−
∑ aij αij þ βij
!
N
t-∞ t-∞
3 ∥p~ 1 ðt−τðt ÞÞ∥ " 6 ~ 7 N 6 ∥p 2 ðt−τðt ÞÞ∥ 7 6 7 þ τ2 ∑ 2ω1 þ 6 7 ⋮ i¼1 4 5 ∥p~ N ðt−τðt ÞÞ∥
þhi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
A þ AT −H 2
N
1 1 T T ∑ ∑ a q~ ðt−τðt ÞÞq~ i ðt−τðt ÞÞ− ∑ ∑ aij q~ j ðt−τðt ÞÞq~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni ij j 2 i ¼ 1 j∈Ni
T 1 N q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ − ∑ ∑ aij q~ i ðt−τðt ÞÞ−q~ j ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 1 T 1 N T ¼ q~ ðt−τðt ÞÞAq~ ðt−τðt ÞÞ þ ∑ ∑ aij q~ j ðt−τðt ÞÞq~ i ðt−τðt ÞÞ 2 2 i ¼ 1 j∈Ni ! 1 N A þ AT T T q~ ðt−τðt ÞÞ: − ∑ ∑ aij q~ i ðt−τðt ÞÞq~ i ðt−τðt ÞÞ ¼ q~ ðt−τðt ÞÞ 2 i ¼ 1 j∈Ni 2
N
þω2 ∑ ∥p~ i ðt−τðt ÞÞ∥ þ mλ2 ∑ ∥p~ i ðt−τðt ÞÞ∥2 i¼1
i¼1
"
N
þτ2 ∑
i¼1
2ω1 þ
∑ aij αij þ βij
#
!
þ hi ðci þ di Þ þ 1 ∥p~_ i ðt Þ∥2
j∈Ni
N
N
i¼1
i¼1
þτω2 ∑ ∥p~_ i ðt Þ∥ þ mλ2 ∑ ∥p~_ i ðt Þ∥2 N
N
N
i¼1
i¼1
i¼1
þ ∑ ∥q~ i ðt−τðt ÞÞ∥2 þ mλ2 ∑ ∥q~ i ðt−τðÞÞ∥2 þ τ2 ∑ ∥q~_ i ðt Þ∥2 N
þmλ2 ∑ ∥q~_ i ðt Þ∥2 : i¼1
Since m 4 0 is sufficiently large, we can get V_ ðt Þ ≤0:
B. Liu et al. / Neurocomputing 118 (2013) 289–300
295
30 20
y axis
2500
0 −100 −40
y axis
10 100
2000
−20
1500 −20
−30 60
1000 0
20
40
x axis
500 60
80
0 −10
40 20
z axis
0
0 −20
z axis
−10
0
10
20
x axis
Fig. 1. The position and velocity of the virtual leader. (a) The position of the virtual leader and (b) the velocity of the virtual leader. Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
5
10
15
20
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
5
10
15
20
0
5
time
time
10
15
20
time
Fig. 2. Position convergence of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Velocity convergence(x axis)
Velocity convergence(y axis)
10
Velocity convergence(z axis)
10
0 −5 −10
6
piz− p0z, i=1,2,3.
piy− p0y, i=1,2,3.
pix − p0x, i=1,2,3.
8 5
4 2 0 −2 −4 −6
−15
−8 0
5
10
15
20
0
5
time
10
15
12 10 8 6 4 2 0 −2 −4 −6 −8
20
0
time
5
10
15
20
time
Fig. 3. Velocity convergence of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Similar to Theorem 1, the position and velocity of each agent described by (1) under Assumption 5 can converge to those of the virtual leader. This completes the proof. □ Remark 4. It is obvious that the results of both Theorems 1 and 2 hold if the time delay of the agents is time-invariant.
Remark 5. Notice also that the results keep conservatism since the upper bound of the time-varying delays τ is necessary in the theoretical analysis. Moreover, comparing with the adaptive strategies of the second-order consensus algorithms without time delay in [18–20], the decentralized adaptive strategies in this paper need more agents' information, such as the delayed information and the derivative information.
4. Numerical example This section presents some numerical simulations to illustrate the adaptive consensus of second-order multi-agent systems with time-varying delays. The dynamics of the virtual leader is
described by a Lorenz system 8 q_ 0 ðt Þ ¼ p0 ðt Þ > > 8 > > > > < > > 10 p0y −p0x < > p_ 0 ðt Þ ¼ 28p0x −p0x p0z −p0y ; > > > > > > > : : p0x p0y −8p0z
ð11Þ
3
where q0 ¼ ½q0x ; q0y ; q0z T and p0 ¼ ½p0x ; p0y ; p0z T . The position and the velocity of the virtual leader are presented as Fig. 1. Example 1. Consider a multi-agent system with the undirected topology described as the following symmetric matrix 2
−0:2 6 0:1 A¼4 0:0
0:1 −0:2 0:1
0:0
3
0:1 7 5: −0:2
ð12Þ
with that Consistent therequirements of the theory, we choose f 1 p1 ðt Þ ¼ sin 2 p1 ðt Þ , f 2 p2 ðt Þ ¼ tanh p1 ðt Þ and f 3 p3 ðt Þ ¼ sin p3 ðt Þ satisfying Assumption 2, as well as let kij ¼ 1 and ki ¼ 0:1. Here, only agent 3 is controlled by control input (3). In Figs. 2 and 3, we have shown that the adaptive second-order consensus of such system contains three agents with a timeinvariant delay and symmetric weighted coupling configuration
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Example 2. Consider a multi-agent system with the undirected topology described as the following asymmetric matrix: 2 3 −0:1976 0:1976 0 6 0:1677 −0:3623 0:1946 7 A¼4 ð13Þ 5: 0 0:3218 −0:3218
matrix A described as Eq. (12), for simplicity. Fig. 2 shows the positions' errors of the multi-agent system with time delay τ ¼ 0:14, while Fig. 4 shows the velocities' errors of the multiagent system with the same time delay. Fig. 4 presents the adaptive coupling strengths of position αij ðt Þ; ði; j ¼ 1; 2; 3Þ and the velocity βij ðt Þ; ði; j ¼ 1; 2; 3Þ when the time delay is τ ¼ 0:14. Every adaptive coupling strength tends to a constant. Similarly, Fig. 5 demonstrates the feedback gains of position and velocity with time delay τ ¼ 0:14, respectively. We can find that the feedback gains of position and velocity also tend to constants, which implies Assumption 3 is reasonable. Figs. 6-9 show the convergence of the positions' errors and the velocities' errors, adaptive coupling strengths and adaptive feedback gains with symmetric matrix A as well as τ ¼ 0:18. From these figures, we also find that the position and the velocity of each agent can converge to those of the virtual leader, at the same time, all the adaptive coupling strengths and the adaptive feedback gains tend to be constant. Note that from Figs. 2–9, it is easily to see that the larger the time delay is, the slower convergent speed is.
The selection of all parameters is the same as Example 1 and only the agent 3 is steered by control input (3). Figs. 10 and 11 depict the adaptive second-order consensus of a multi-agent system with time delay τ ¼ 0:05 and asymmetric weighted coupling configuration matrix A described as Eq. (13). These two figures show that both the positions and the velocities of all agents can converge to those of the virtual leader. Fig. 12 displays the variable trend of the adaptive coupling strengths of the positions and the velocities, which can reach to be constant eventually. While, Fig. 13 shows the adaptive feedback gains of the positions and the velocities respectively, which can also reach to be constant eventually.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
9
25
8 20
7
βij, i,j=1,2,3.
αij, i,j=1,2,3.
6 5 4 3 2
15 10 5
1 0
0 0
5
10
15
20
0
5
10
time
15
20
time
Fig. 4. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Adaptive feedback gain of position
Adaptive feedback gain of velocity
8
14
7
12
6
10
5
d3
c3
8 4
6 3 4
2
2
1 0
0 0
5
10
15
20
0
5
10
time
15
20
time
Fig. 5. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:14 when weighted coupling configuration matrix A is symmetric.
Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
10
20
30
time
40
50
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
10
20
30
time
40
50
0
10
20
30
40
time
Fig. 6. Position convergence of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
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B. Liu et al. / Neurocomputing 118 (2013) 289–300
Velocity convergence(x axis)
297
Velocity convergence(y axis)
10
Velocity convergence(z axis)
10 8
0 −5 −10
6
piz − p0z, i=1,2,3.
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
5
4 2 0 −2 −4 −6
−15
−8 0
10
20
30
40
50
0
10
20
time
30
40
12 10 8 6 4 2 0 −2 −4 −6 −8
50
0
10
20
time
30
40
50
time
Fig. 7. Velocity convergence of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity 90
12
80 10 70
βij, i,j=1,2,3.
αij, i,j=1,2,3.
8 6 4
60 50 40 30 20
2 10 0
0 0
10
20
30
40
50
0
10
20
time
30
40
50
time
Fig. 8. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric. Adaptive feedback gain of position
Adaptive feedback gain of velocity
9
40
8
35
7
30
6
25
c3
d3
5 20
4 15
3 2
10
1
5
0
0 0
10
20
30
40
50
0
10
20
time
30
40
50
time
Fig. 9. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:18 when weighted coupling configuration matrix A is symmetric.
Position convergence(x axis)
Position convergence(y axis)
4
qiy − q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5
3 2 1 0 −1 −2 0
5
10
15
20
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2 0
qiz − q0z, i=1,2,3.
6
−2 −4 −6 −8 −10 −12
0
5
time
10
time
15
20
0
5
10
15
20
time
Fig. 10. Position convergence of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Figs. 14–17 present the convergence of the positions' errors, velocities' errors, adaptive coupling strengths and adaptive feedback gains, respectively, when the weighted coupling configuration matrix A is asymmetric and the time delay is τ ¼ 0:1. These four figures demonstrate that the positions and the velocities of all agents converge to those of the virtual leader, in which all the
adaptive parameters tend to be constant. Similarly, the larger the time delay is, the longer the convergent time is. From Figs. 2–17, we know that the delayed multi-agent system with the adaptive strategies (4)–(7) and symmetric or asymmetric weighted coupling configuration matrix A, can reach consensus via the second-order consensus algorithm (1).
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Velocity convergence(y axis)
Velocity convergence(z axis) 10
8
8
8
6
6
6
4 2 0 −2
4 2 0 −2 −4
−4
−6
−6
−8 0
5
10
15
piz − p0z, i=1,2,3.
10
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
Velocity convergence(x axis) 10
4 2 0 −2 −4 −6 −8
20
0
5
10
time
15
20
0
5
10
time
15
20
time
Fig. 11. Velocity convergence of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
25
50 45
20
40
βij, i,j=1,2,3.
αij, i,j=1,2,3.
35 15
10
30 25 20 15
5
10 5
0
0 0
5
10
15
20
0
5
time
10
15
20
time
Fig. 12. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Adaptive feedback gain of position
Adaptive feedback gain of velocity
8
10
7
9 8
6
7 6
d3
c3
5 4
5 4
3
3
2
2
1
1
0
0 0
5
10
15
20
0
5
time
10
15
20
time
Fig. 13. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:05 when weighted coupling configuration matrix A is asymmetric.
Position convergence(x axis)
Position convergence(y axis) qiy− q0y, i=1,2,3.
qix − q0x, i=1,2,3.
5 4 3 2 1 0 −1 −2 0
10
20
30
40
50
60
Position convergence(z axis)
4 3 2 1 0 −1 −2 −3 −4 −5 −6
2
qiz − q0z, i=1,2,3.
6
0 −2 −4 −6 −8 −10 −12
0
10
20
time
30
time
40
50
60
0
10
20
30
40
50
60
time
Fig. 14. Position convergence of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric.
5. Conclusion In this paper, we have investigated second-order consensus of multi-agent systems with time-varying delays. Differing from the current works, all agents and the virtual leader are described by
heterogeneous nonlinear intrinsic dynamics. By introducing local decentralized adaptive strategies to the coupling strengths and feedback gains of both the position and velocity, we have proved that the position and velocity of each agent can converge to those of the virtual leader respectively, when the network is connected
B. Liu et al. / Neurocomputing 118 (2013) 289–300
Velocity convergence(y axis)
0
10
20
30
40
50
Velocity convergence(z axis)
10 8 6 4 2 0 −2 −4 −6 −8
60
piz − p0z, i=1,2,3.
10 8 6 4 2 0 −2 −4 −6 −8
piy − p0y, i=1,2,3.
pix − p0x, i=1,2,3.
Velocity convergence(x axis)
299
0
10
20
time
30
40
50
10 8 6 4 2 0 −2 −4 −6 −8
60
0
10
20
time
30
40
50
60
time
Fig. 15. Velocity convergence of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric. Adaptive coupling strengths of position
Adaptive coupling strengths of velocity
30
80 70
25
βij, i,j=1,2,3.
αij, i,j=1,2,3.
60 20 15 10
50 40 30 20
5
10
0
0 0
10
20
30
40
50
60
0
10
20
time
30
40
50
60
time
Fig. 16. The adaptive coupling strengths of the position and the velocity of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric. Adaptive feedback gain of position
Adaptive feedback gain of velocity 9
8
8
7
7
6
6
5
5
c3
d3
9
4
4
3
3
2
2
1
1
0
0 0
10
20
30
40
50
60
0
10
time
20
30
40
50
60
time
Fig. 17. The adaptive feedback gains of the position and the velocity of system (1) with time delay as τ ¼ 0:1 when weighted coupling configuration matrix A is asymmetric.
and at least one agent is steered by the controller, even though the weighted coupling configuration matrix is asymmetric. Finally, some simulation results are showed to demonstrate the theoretical results. Future work will be to further optimize the proposed algorithm, and apply it to the filtering problems in sensor networks.
National High Technology Research and Development Program of China 863 (No.2012AA112401), the Research Fund for the Doctoral Program of Higher Education (RFDP) under Grant No. 20100142120023, the Natural Science Foundation of Hubei Province of China under Grant No. 2011CDB042. References
Acknowledgements This work was supported in part by the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, the National Natural Science Foundation of China under Grant (No. 61104140, No. 61203150, No. 61170113, No. 61174116), the Beijing Natural Science Foundation Program (1102016, 4122019), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201108055), Science and Technology Development Plan Project of Beijing Education Commission (No. KM201310009011)), the
[1] W. Yang, H. Shi, Sensor selection schemes for consensus based distributed estimation over energy constrained wireless sensor networks, Neurocomputing 87 (2012) 132–137. [2] B. Shen, Z. Wang, Y. Hung, Distributed H∞ consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case, Automatica 46 (10) (2010) 1682–1688. [3] H. Dong, Z. Wang, H. Gao, Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts, IEEE Trans. Signal Process. 60 (6) (2012) 3164–3173. [4] B. Shen, Z. Wang, X. Liu, A stochastic sampled-data approach to distributed H∞ filtering in sensor networks, IEEE Trans. Circuits Syst.-I: Regular Pap. 58 (9) (2011) 2237–2246. [5] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (2005) 655–661.
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B. Liu et al. / Neurocomputing 118 (2013) 289–300
[6] Y. Sun, L. Wang, G. Xie, Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Syst. Control Lett. 57 (2008) 175–183. [7] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 53 (2008) 1503–1509. [8] H. Su, W. Zhang, Second-order consensus of multiple agents with coupling delay, Commun. Theor. Phys. 51 (1) (2009) 101–109. [9] P. Lin, Y. Jia, Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica 45 (2009) 2154–2158. [10] Z. Tang, T. Huang, J. Shao, J. Hu, Consensus of second-order multi-agent systems with nonuniform time-varying delays, Neurocomputing 97 (2012) 410–414. [11] C. Liu, F. Liu, Consensus problem of second-order multi-agent systems with time-varying communication delay and switching topology, J. Syst. Eng. Electron. 22 (2011) 672–678. [12] X. Lu, F. Austin, S. Chen, Formation control for second-order multi-agent systems with time-varying delays under directed topology, Commun. Nonlinear Sci. Numer. Simulation 17 (2012) 1382–1391. [13] F. Jiang, L. Wang, Y. Jia, Consensus in leaderless networks of high-orderintegrator agents, in: Proceeding of the 2009 American Control Conference, 2009, pp. 4458–4463. [14] X. Wang, G. Chen, Pinning control of scale-free dynamical networks, Physica A 310 (2002) 521–531. [15] H. Su, X. Wang, Z. Lin, Flocking of multi-agents with a virtual leader, IEEE Trans. Autom. Control 54 (2) (2009) 293–307. [16] Z. Hou, et al., Decentralized robust adaptive control for the multiagent system consensus problem using neural networks, IEEE Trans. Syst. Man Cybern.-Part B: Cybern. 39 (2009) 636–647. [17] P. Delellis, M. Bernardo, F. Garofalo, Synchronization of complex networks through local adaptive coupling, Chaos 18 (2008) 037110. [18] H. Su, Z. Rong, M. Chen, X. Wang, G. Chen, H. Wang, Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks, IEEE Trans. Cybern. 43 (1) (2013) 394–399. [19] H. Su, G. Chen, X. Wang, Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica 47 (2011) 368–375. [20] Y. Hu, H. Su, J. Lam, Adaptive consensus with a virtual leader of multiple agents governed by local Lipschitz nonlinearity, Int. J. Robust Nonlinear Control 23 (2013) 978–990. [21] O. Faydasicok, S. Arik, A new robust stability criterion for dynamical neural networks with multiple time delays, Neurocomputing 99 (2013) 290–297. [22] J. Zhang, M. Li, B. Liu, Synchronization of a complex network for neighboring agents with time-varying delays, Commun. Computer Inf. Sci. 225 (2011) 73–81. [23] W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays, Nonlinear Dyn. 48 (2007) 165–174. [24] V. Zorich, Mathematical Analysis I, Springer, New York, 2004. [25] B. Liu, T. Chu, L. Wang, Stability and oscillation of swarm with interaction time delays, in: Proceeding of the 2007 American Control Conference, 2007, pp. 4600–4605. [26] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001.
Bo Liu was born in 1977. She received the Ph.D. degree in Dynamics and Control from Peking University in 2007. She was a visiting research fellow at the City University of Hong Kong in 2009 and is currently an Associate Professor in North China University of Technology. Her research interests include swarm dynamics, networked systems, collective behavior and coordinate control of multi-agent systems.
Xiaoling Wang was born in 1987. She received the B.S. degree in mathematics and applied mathematics from Anqing Normal University, Anqing, China, in 2010, and is currently pursuing the M.S. degree in applied mathematics at North China University of Technology, Beijing, China. Her research interests are in the fields of networked systems and collective intelligence.
Housheng Su received the B.S. degree in automatic control and the M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. Since January 2010, he has been an associate professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
Yanping Gao received the B.S. and M.S. degrees in Applied Mathematics from Ocean University of China in 2003 and 2006, respectively, and the Ph.D. degree from Peking University, Beijing, in 2011. She is currently working as a lecturer at Beijing Technology and Business University. Her current research interests are in the fields of multi-agent systems, consensus problems, and sampled-data control.
Li Wang was born in 1978. He graduated from BeiHang University and achieved Ph.D degree in 2006. Currently, he is an Associate Professor of North China University of Technology. He won Beijing New Star of Science and Technology Award in 2011. His main research field is intelligent transportation system, intelligent traffic system modeling and control, traffic signal control, dynamic traffic simulation. He has chaired more than 10 national and Beijing local government projects like 863 project, Olympic special project and project founded by Beijing Government Education Committee. In 2011, he was rewarded by 3rd Prize of Beijing Science and Technology and 1st Prize of aLian Science and Technology Progress. He has published 20 papers on international traffic journals and three patents.