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Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays
Communicated by Jiahu Qin
Accepted Manuscript
Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays Fangcui Jiang, Bo Liu, Yongjun Wu, Yunru Zhu PII: DOI: Reference:
S0925-2312(17)31559-X 10.1016/j.neucom.2017.09.040 NEUCOM 18922
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
3 January 2017 11 June 2017 12 September 2017
Please cite this article as: Fangcui Jiang, Bo Liu, Yongjun Wu, Yunru Zhu, Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.09.040
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Highlights • The current work extends the existing results on impulsive consensus to the asynchronous setting.
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• The paper considers the case of multiple measurement time-delays as well. • In the technical contribution, the analysis method is entirely different from the analysis techniques used in some related existing works duo to the effects of both asynchronism of sampled information and impulsive nature of protocol.
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• The design of the protocol parameters is given by solving a feasible linear matrix inequality.
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Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays✩ Fangcui Jianga,b , Bo Liuc , Yongjun Wud , Yunru Zhue
School of Control Science and Engineering, Shandong University, Jinan 250061, China b School of Mathematics and Statistics, Shandong University, Weihai 264209, China c College of Science, North China University of Technology, Beijing 100144, China d School of Science, Tianjin Polytechnic University, Tianjin 300387, China e Center for Complex Systems, School of Mechano-electronic Engineering, Xidian University, Xian 710126, China
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a
Abstract
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This paper investigates the asynchronous consensus of second-order multiagent systems with directed networks and measurement time-delays via impulsive control. It is assumed that each agent receives the measurements of the information from its neighbouring agents and itself according to its own sampling clock, and different agents can have independent sampling times. The controller of agent is designed in impulsive framework and changes agent’s state instantaneously. By transforming the consensus problem into static consensus and using common Lyapunov function, it is proved that there exist protocol parameters ensuring consensus. Also, the design of the parameters is given in a specific form. Finally, simulation example is drawn to illustrate the effectiveness of the theoretical results.
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Keywords: Multi-agent systems, Second-order, Asynchronous consensus, Measurement time-delays, Impulsive control
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This work was supported by National Natural Science Foundation of China (No. 61304163, No. 61473337 and No. 61304049), Shandong Provincial Natural Science Foundation, China (No. ZR2013FQ008) and Independent Innovation Foundation of Shandong University (No. 2013ZRQP006). ∗ Corresponding author. Email addresses: [email protected] (Fangcui Jiang), [email protected] (Bo Liu), [email protected] (Yongjun Wu), [email protected] (Yunru Zhu) Preprint submitted to Elsevier
September 21, 2017
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1. Introduction
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In recent years, distributed coordination control of multi-agent systems has achieved momentous research findings in many scientific fields. Among the numerous branches of this subject, consensus problem is one of the fundamental topics and with the practical applications in autonomous formation flight, cooperative control of unmanned air vehicles, coordinated tracking and containment control of multi-agent systems [1, 2, 3, 4, 5]. Roughly speaking, the consensus problem of multiple dynamic agents aims at using simple local nearest neighbor rules so that certain states of interest eventually reach a common desired quantities. Researchers in the control community have studied the consensus problems from various perspectives, and obtained many significant and interesting results, such as average consensus of first-order agents [6, 7], consensus behavior of discrete-time dynamic agents with general topologies [8, 9, 10], static or dynamic consensus for second-order agents [11, 12, 13], finite-time consensus to improve convergence rate [14, 15], consensus analysis with transmission time-delays [16], group consensus or consensus tracking [17, 18, 19, 20], observer-based robust consensus of general linear dynamics with noises [21], consensus of switched multi-agent systems or switching disconnected topology [22, 23], just to name a few. The majority of the above mentioned works studied the distributed consensus with the assumption that all local information among agents is exchanged continuously or synchronously at the same discrete times. However, in real networked systems and robotic applications, it is more practical and implementable to consider asynchronous versions of consensus [24, 25, 26, 27, 28, 29, 30, 31] in which each agent can independently obtain its needed local information at discrete times according to its own sampling clock, because of the sensing limitations of agents, the unreliability of transmission channels, the energy consumption and the environment factors. For continuous-time first-order multi-agent systems, Xiao and Wang in [24] proposed an asynchronous consensus protocol based upon the neighbouring agents’ state information at independent discrete times and proved that the asynchronous consensus is reachable under the conditions that the union of the graph across any time interval with a given length contains a spanning tree and the outdated information coming from neighbors is with bounded time-varying delays. Another related work on this topic can be found in [25]. The asynchronous consensus based on sampled data for a number of agents with continuous-time second-order dynamics was considered in [26] for 3
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fixed topology and in [27] for time-dependent topology. In contrast to the first-order model, the analysis tool of nonnegative matrix theory employed in [24] is invalid to study the problem in [26] and [27], where the authors respectively made use of the stability theory of time-delayed discrete-time systems and the Schur stability of matrices. Also, the asynchronous consensus of discrete-time multi-agent systems was investigated extensively in [28, 29, 30, 31]. In this paper, the asynchronous consensus of continuous-time secondorder multi-agent systems with impulsive control and multiple measurement time-delays is studied. To the best of the authors’ knowledge, there are few works focusing on this topic, although the impulsive consensus has been addressed well in [13, 32, 33] and the references therein based on synchronous sampled-data. Specifically, [13] studied the synchronous consensus for the cases of periodic impulsive control and single time-delay, which is opposite to the current paper. (Note that the current paper studies the asynchronous consensus for the cases of aperiodic impulsive control and multiple timedelay.) In [32], the authors proposed two kinds of impulsive algorithms via periodic sampled information to solving the dynamic and static consensus for a group of continuous-time second-order agents with fixed topology, and obtained two necessary and sufficient conditions to ensure the consensus by utilizing Schur stability of matrices. Considering in real networks some partial states may not be obtained due to communication limitations or technology constraints, [33] established an aperiodic impulsive algorithm by using only position information for second-order multi-agent systems with switching topology and proved the consensus convergence according to the property of stochastic matrices. It is worth mentioning that the impulsive control is “an effective control technique in many practical applications, such as the orbit interception correction of orbiting objects, the population control of a kind of insects, the control of reaction process in a chemical reactor system, and the money supply in a financial system” and with “obvious advantages in less energy cost, fast transient, and easier to design” [13]. The contributions of this paper are threefold. Firstly, the current work extends the existing results on impulsive consensus to the asynchronous setting, and considers the case of multiple measurement time-delays as well. Secondly, in the technical contribution, the analysis method of this work is to transform the consensus problem into static consensus and then reduce only the order of position vector by a tree-type transformation but remain the order of velocity vector unchanged, duo to the effects of both asynchronism 4
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of sampled information and impulsive nature of protocol. This is entirely different from the analysis techniques used in [26, 27], where an invertible linear transformation was introduced to reduce the orders of both position vector and velocity vector. Moreover, the technical issue to analyse the convergence of the asynchronous setting is much more challenging than the synchronous case. Thirdly, the design of the protocol parameters is given by solving a feasible linear matrix inequality. The remainder of this paper is organized as follows. Some preliminaries on graph theory and problem formulation are given in Section 2 and Section 3. The convergence analysis and the main results are performed in Section 4. To illustrate the effectiveness of the theoretical finding, a numerical example is worked out in Section 5. At last, some concluding remarks are drawn in Section 6. Notations: We let R, N and N+ be the sets of real numbers, non-negative integers and positive integers, respectively. Rn is the n-dimensional Euclidean space. Rm×n is the set of m-by-n matrices. In ∈ Rn×n is an identity matrix. Sometimes we apply 0 to denote the zero matrix with appropriate dimension. 1n = [1 · · · 1]T ∈ Rn is a column vector with all elements equal to one. 2. Preliminaries on graph theory
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A weighted directed graph is denoted by G(Λ), which is consisted of vertex set W = {w1 , w2 , . . . , wN }, edge set F ⊂ W × W, and weighted adjacency matrix Λ = [akl ] ∈ RN ×N . The rows and columns of Λ are indexed by {w1 , w2 , . . . , wN }. In multi-agent systems, we use graph G(Λ) to describe the relationships of information transmission among agents. Specially, vertex wk represents the k-th agent of the system; an edge is denoted by ordered pair hwk , wl i, which means agent l can obtain the information of agent k; hwk , wl i ∈ F if and only if alk > 0 (otherwise, alk = 0), alk denotes the weight of the edge. If hwk , wl i ∈ F, then we say wk (or agent k) is a neighbor of wl (or agent l). In this paper, we assume (wk , wk ) 6∈ F and the element of F is not repeated. A path from wk to wl is defined as a string of edges (wk , w1 ), (w1 , w2 ), . . . , (wi , wl ) which are distinct in F. A graph is a directed tree, if it is a directed graph and there is a special vertex from which only one path can reach any other vertex. Given G(Λ), if there is F 0 ⊂ F such that W and F 0 compose a directed tree, then we say that G(Λ) contains a spanning tree.
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j=1,j6=k
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Laplacian matrix L = [bkl ] ∈ RN ×N of G(Λ) is defined as l 6= k −akl , N P bkl = . akj , l = k
Lemma 1. ([6, 9]) Given that G(Λ) is a graph with Laplacian matrix L. Then the algebraic multiplicity of the 0 eigenvalue is simple if and only if G(Λ) contains a spanning tree; L1N = 0; and the real parts of nonzero eigenvalues are positive. Define matrices
eT1 E
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E = [1N −1 − IN −1 ] ∈ R
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∈ RN ×N −1 ,
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= [1N F ], where e1 = [1 0 · · · 0]T ∈ RN . Then we have 0 eT1 LF N ×N −1 −1 , where U ∈ R is invertible, U = U , and U LU = 0 ELF L is Laplacian matrix of a graph. According to the property of similarity transformation of matrices, it follows that ELF has N − 1 eigenvalues of L. Hence Lemma 1 implies the following result.
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and U =
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Lemma 2. Given that G(Λ) is a graph with Laplacian matrix L, then the graph contains a spanning tree if and only if the real parts of all eigenvalues of ELF are positive, where E and F are defined as in (1).
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3. Model and problem formulations
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Consider a dynamical system of N autonomous agents, which are labeled 1 through N . Each agent is modeled by second-order dynamics p˙i (t)=vi (t), (2) v˙ i (t) =ui (t), t ≥ t0 , i = 1, 2, · · · , N, where pi (t) ∈ R, vi (t) ∈ R are the position and velocity states of agent i, respectively; ui (t) ∈ R is the control input or the protocol to be designed based on the information obtained by agent i; t0 ≥ 0 is the initial time of the dynamical system. 6
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Definition 1. Multi-agent system (2) is said to achieve consensus if for any initial states, limt→∞ [pi (t) − pj (t)] = 0 and limt→∞ [vi (t) − vj (t)] = 0, ∀i, j = 1, 2, · · · , N . Furthermore, if there exists p∗ , such that limt→∞ pi (t) = p∗ and limt→∞ vi (t) = 0, ∀i = 1, 2, · · · , N , then system (2) is said to achieve static consensus, where [p∗ , 0]T is called as the consensus state of the system. Because of the wide use of computer-based control and digital sensors, the agents in networks may only obtain sampled information from their neighbors at some sampling instants. Based on sampled information, [32] proposed the following impulsive algorithm
k=1
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ui (t) =
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where c1 , c2 > 0; aij is the weight of the edge from agent j to agent i; Ni is the set of neighbors of agent i; δ(·) is the Dirac impulsive function; the discrete time tk satisfies tk − tk−1 ≡ h. In the above algorithm, all agents sampled their required information at the same instants tk , k = 1, 2, · · · , and the sampling intervals were periodic. However, it is difficult for all agents to synchronously receive/measure the state information in the distributed control of some practical networks. Also the measured information may involve a certain degree of time-delay. Therefore, in this paper we mainly discuss the impulsive consensus of multi-agent system (2) with measurement time-delays based on asynchronous sampled information. This is a complete contrast to the synchronous sampling time studied in [13, 32, 33]. Specifically, we assume that agent i independently receives or measures its neighbors’ state information at times ti1 , ti2 , · · · , tis , · · · , i = 1, 2, · · · , N , which are determined by its own clock. For simplicity, let us denote the sampling instants by a real number sequence {tis }∞ s=1 . Note that by the concept of “asynchronous”, it is meant that for any different agents i and j, they may not measure their neighbors’ state at the same times, i.e., {tis }∞ s=1 6= {tjs }∞ . In addition, there may exist time-delays in the measured information s=1 of neighbors’ states. Hence different from the sampled controller proposed in [26, 27], we give the following impulsive consensus protocol ui (t) =
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aij (pi (t − τij ) − pj (t − τij )) δ(t − tis ), 7
(3)
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where c1 > 0 and c2 > 0 are the control gains; aij is the weight of the edge from agent j to agent i; Ni is the collection of neighbors of agent i; τij ≥ 0 is the measurement time-delay in relative position; δ(·) is the Dirac impulsive function with discontinuity points ti1 < ti2 < · · · < tis−1 < tis < · · · and tis → ∞ as s → ∞, ti1 > t0 ; tis , s ∈ N+ are the sampling times of agent i, also are the impulse times. For any i = 1, 2, · · · , N , it is assumed that {tis }∞ s=1 satisfies 0 < α ≤ tis+1 − tis ≤ β, s ∈ N,
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vi (tis
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where α and β are real numbers. Note that ti0 indicates the initial instant of the dynamical system of agent i rather than the first sampling instant. Seen from (2), it is obtained that ti0 = t0 for i = 1, 2, · · · , N . It’s known that the Dirac impulsive R ∞ function δ(·) is defined R ∞as δ(t) = 0 for t 6= 0, δ(t) → ∞ for t = 0 and −∞ δ(t)dt = 1. It satisfies −∞ f (t)δ(t − tis )dt = f (tis ). Then (3) implies that ui (t) = 0 for t 6= tis , s ∈ N+ . Moreover, for sufficiently small σ > 0, v˙ i (t)dt =
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Hence as σ → 0+ , one yields +
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) ∆vi (tis ):=vi (tis ) − vi (tis P i = −c1 vi (ts ) − c2 aij [pi (tis − τij ) − pj (tis − τij )],
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where vi (tis ) = limσ→0+ vi (tis − σ) and vi (tis ) = limσ→0+ vi (tis + σ) are, respectively, the left limit and the right limit of vi (t) at tis . Therefore, the closed-loop dynamics of agent i can be given by the following impulsive differential equations p˙i (t)=vi (t), v˙ i (t)=0, t ≥P t0 , t 6= tis , s ∈ N+ , (5) aij [pi (tis − τij ) − pj (tis − τij )], t = tis . ∆vi (tis )=−c1 vi (tis ) − c2 j∈Ni
In this impulsive control, we can assume vi (t) is left continuous at t = tis , − that is, vi (tis ) = vi (tis ), s ∈ N+ , and continuous at initial time t0 . 8
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Remark 1. In protocol (3), the agent i is required to receive the relative position information with measurement delay to its neighbors (i.e., pi (tis − τij ) − pj (tis − τij )) and its velocity information vi (tis ) at its own sampling times.
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Remark 2. Compared with the existing impulsive consensus algorithms in [13, 32, 33], the sampling times are independent of each other for different agents. That is to say the sampling sequence {tis } is independent of {tjs } for any agents i and j, i 6= j. Agents may not sample information at the same times. This asynchronous sampling setting is quite distinct from the synchronous sampling times therein. Remark 3. Compared with [26, 27], control input ui (t) can suddenly change the velocity of agent at the instants tis , i.e., ui (t) is only implemented at tis by impulsive actuator and causes a jump of the agent velocity. System (2) during sampling intervals is running without any control input. This is quite different from the sampled-data protocols therein, which are executed continuously during sampling interval.
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By recurring Definition 1, it is easily obtained that if system (2) with protocol (3) achieves static consensus, then the system achieves consensus. Moreover, system (5) can be rewritten as the following discrete-time impulsive system along time sequence {tis } + i i ) + (tis+1 − tis )vi (tis ), pi (tis+1 )=pi (tis+ vi (ts+1 ) =vi (ts ), (6) P i i i+ i − τ ) − p (t − τ )], ) − c a [p (t v (t ) =(1 − c )v (t ij ij j i 1 i 2 ij i s s s s j∈Ni
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with pi (tis − τij ) = pi (t0 ) for tis − τij ≤ t0 and vi (ti0 ) = vi (ti0 ) = vi (t0 ), i = 1, 2, · · · , N . Here, we need to point out two facts for the convergence of the system. The first fact is that the control gain c1 plays an important role for the consensus convergence, and 0 < c1 < 2 is also a necessary condition for system (2) with protocol (3) achieving consensus on the ground of Lemma 5 in [13]. The second fact is that from the same guideline of the proof of Theorem 2 in [32], we can derive the following lemma. Lemma 3. Suppose 0 < c1 < 2, then system (2) with protocol (3) achieving consensus is equivalent to the system achieving static consensus. 9
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4. Convergence analysis and main results
tis ≤ tk < tk+1 ≤ tis+1 .
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At first, we merge all the time sequences {tis }, {tis − τij } and the initial time {t0 } into a single ordered sequence, and relabel the sequence as t0 , t1 , · · · , tk , · · · . That is denoting the set {t0 , t1 , · · · , tk , · · · } = {t0 }∪{tis | i = 1, 2, · · · , N , s ∈ N+ } ∪ {tis − τij | tis − τij ≥ t0 , i = 1, 2, · · · , N , j ∈ Ni , s ∈ N+ }, which satisfies tk < tk+1 . Next, we will establish the closed-loop system (9) for the multi-agent system (5) along the time sequence {t0 , t1 , · · · , tk , · · · }. For any i = 1, 2, · · · , N and any k ∈ N, there exists s ∈ N such that (7)
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if Then according to system (5), it follows that if tk = tis , vi (tk + ) = + tk > tis , vi (tk + ) = vi (tk ) = vi (tis ). Hence for the time sequence {t0 , t1 , · · · , tk , · · · }, system (6) can be rewritten as ) + Tk vi (tk + ), pi (tk+1 )= pi (tk + vi (tk+1 ) = vi (tk ), k ∈ N, (8) P + i aij [pi (tis − τij ) − pj (tis − τij )], vi (tk ) = (1 − c1 )vi (ts ) − c2 j∈Ni
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where Tk = tk+1 − tk ; for k = 0, vi (t0 + ) = vi (t0 ), i = 1, 2, · · · , N . Let m e denoteTthe upper bound for the number of elements tk in the set {t0 , t1 , t2 , · · · } [tis , tis+1 ) for any i = 1, 2, · · · , N and any s ∈ N. Then Lemma 2 in [24] shows that m e can be taken as m e = N [(bβ/αc + 1)(N − 1) + ¯ 1][K(N − 1) + 1], where bβ/αc is the maximum integer which is not greater ¯ = bτmax /αc + 1 with τmax = than β/α, and K max {τij }. Denote i6=j,i,j=1,2,··· ,N
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¯ + 1)m. ¯ and k ≥ m, one has ti − τij ≥ tk−m+1 (this m = (K e For s ≥ K s conclusion can be deduced from the remarks after Lemma 2 of [24]). By recalling (7), we have that for any i = 1, 2, · · · , N and any k ∈ N, tk−m+1 ≤ tis − τij ≤ tis ≤ tk < tk+1 ≤ tis+1 .
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This means that for fixed k, tis and tis −τij take values in {tk , tk−1 , · · · , tk−m+1 }, i = 1, 2, · · · , N , j ∈ Ni . Here, we assume tk = t0 for k < 0. Consequently, if we let p(tk ) = [p1 (tk ), p2 (tk ), · · · , pN (tk )]T and v(tk ) = [v1 (tk ), v2 (tk ), · · · , vN (tk )]T , then system (8) can be written as p(tk+1 ) = p(tk ) + Tk v(tk + ), v(t )= v(t + ), k+1 k m−1 m−1 P P Lρ (tk )p(tk−ρ ) + (1 − c1 ) Bρ (tk )v(tk−ρ ), v(tk + ) = −c2 ρ=0
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where Lρ (tk ) ∈ RN ×N with the (i, j)-element being −aij , j ∈ Ni , 0, j ∈ Ni , 0, j 6∈ Ni , Lρ (tk )(i, j) = N P Lρ (tk )(i, l), j = i −
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B0 (tk )+B1 (tk )+· · ·+Bm−1 (tk ) = IN , and if there exists some agent i sampling information at time tk−ρ , i.e., tis = tk−ρ with ρ ∈ {0, 1, 2, · · · , m − 1}, then the i-th row of Bρ (tk ) is equal to the i-th row of matrix IN , while the ith rows of all the other matrices in {B0 (tk ), B1 (tk ), · · · , Bm−1 (tk )} are equal m−1 P to zeros. It should be mentioned that Lρ (tk ) = L and Lρ (tk )1N = 0 for ρ=0
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any ρ = 0, 1, · · · , m − 1, k ≥ m. Thus the dynamics of stacked vectors p(tk ) and v(tk ) can be described by the following discrete-time system with time-delays m−1 m−1 P P p(t ) =p(t ) − T c L (t )p(t ) + T (1 − c ) Bρ (tk )v(tk−ρ ), k+1 k k 2 ρ k k−ρ k 1 ρ=0
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(9) which is the closed-loop system of system (5) along the time sequence {t0 , t1 , · · · , tk , · · · }. Then system (2) with protocol (3) achieves consensus if and only if system (9) achieves consensus.
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Theorem 1. For system (2) with protocol (3), assume G is the directed topology which contains a panning tree, the sampling times of each agent satisfy (4), and there exists τ > 0 such that Tk in (8) satisfies Tk ≥ τ . Then for any τ > 0, system (2) with protocol (3) achieves consensus if there exists a positive definite matrix R such that IˆT RIˆ + IˆT RΣ(tk ) + ΣT (tk )RIˆ − R + ΣT (tk )RΣ(tk ) < 0
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with E and F be defined in (1).
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Iˆ1 0 0 0 0 I 0 N −1 N −1 Iˆ= Iˆ2 , Iˆ2 = , 0 0 , Iˆ1 = 0 0N 0 −IN 0 I(2N −1)(m−2) 0 Pm−1 Pm−1 Pm−1 − ρ=0 Mρ (tk ) Mρ (tk ) · · · Mρ (tk ) ρ=1 ρ=m−1 P Pm−1 Pm−1 , Σ(tk )= − m−1 ρ=0 Mρ (tk ) ρ=1 Mρ (tk ) · · · ρ=m−1 Mρ (tk ) 0 0 ··· 0 Tk c2 ELρ (tk )F −Tk (1 − c1 )EBρ (tk ) Mρ (tk )= , ρ = 0, 1, · · · , m − 1, c2 Lρ (tk )F −(1 − c1 )Bρ (tk )
Proof. We need only to prove that system (9) achieves consensus under the conditions given in Theorem 1. By recalling the matrices E and F in (1), let r(tk ) = Ep(tk ).
(11)
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Then p(tk ) = p1 (tk )1N + F r(tk ). (In literature [16], the transformation (11) is called as a tree-type transformation.) Applying (11) to system (9) and noting that Lρ (tk )1N = 0, one has m−1 P r(tk+1 ) =r(tk ) − Tk c2 ELρ (tk )F r(tk−ρ ) ρ=0 m−1 P +Tk (1 − c1 ) EBρ (tk )v(tk−ρ ), (12) ρ=0 m−1 m−1 P P Lρ (tk )F r(tk−ρ ) + (1 − c1 ) Bρ (tk )v(tk−ρ ). v(tk+1 )=−c2 ρ=0
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Denote x(tk ) = [rT (tk ), v T (tk )]T ∈ R2N −1 , then system (12) can be stacked as m−1 X x(tk+1 ) = Iˆ1 − M0 (tk ) x(tk ) − Mρ (tk )x(tk−ρ ), (13) ρ=1
where Iˆ1 and Mρ (tk ), ρ = 0, 1, · · · , m − 1 are given as in Theorem 1. It should be mentioned that transformation (11) implies r(tk ) = [p1 (tk ) − p2 (tk ), p1 (tk ) − p3 (tk ), · · · , p1 (tk ) − pN (tk )]T ∈ RN −1 . Hence, Lemma 3 indicates that system (9)achieves consensus if and only if limk→∞ x(tk ) = 0. 12
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Therefore, we next devote the effort to proving that time-delay system (13) is asymptotically stable. With some direct calculation, system (13) can be rewritten as
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T
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Mρ (tk ), i = 0, 1, · · · , m − 1.
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where
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x(tk+1 ) = Iˆ1 − H0 (tk ) x(tk ) + H1 (tk )[x(tk ) − x(tk−1 )] + · · · + Hm−2 (tk )[x(tk−m+3 ) − x(tk−m+2 )] +Hm−1 (tk )[x(tk−m+2 ) − x(tk−m+1 )],
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Let y(tk )=[x (tk ), x (tk ) − xT (tk−1 ), xT (tk−1 ) − xT (tk−2 ), · · · , xT (tk−m+2 ) −xT (tk−m+1 )]T . From (14), we obtain the following augmented system y(tk+1 ) = [Iˆ + Σ(tk )]y(tk )
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with Iˆ and Σ(tk ) being given in Theorem 1. Then system (13) is asymptotically stable if and only if system (16) is asymptotically stable. For system (16), consider the following common Lyapunov function
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each element being the same order infinitesimal of δ 2 . By calculation and denoting ΣT (tk )RΣ(tk ) = O(δ 2 ), one has
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ˆT RIˆ + IˆT RΣ(tk ) + ΣT (tk )RIˆ − R + ΣT (tk )RΣ(tk ) I Q11 (tk ) JH1 (tk ) · · · JHm−2 (tk ) JHm−1 (tk ) H T (tk )J T R3 − R2 1 . ... .. = + O(δ 2 ), T Hm−2 (tk )J T Rm − Rm−1 T Hm−1 (tk )J T −Rm
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where Q11 (tk ) = −(Iˆ1 R1 + Iˆ2 R2 )H0 (tk )−H0T (tk )(R1 Iˆ1 +R2 Iˆ2 )−R1 + Iˆ1 R1 Iˆ1 + Iˆ2 R2 Iˆ2 , J = Iˆ1 R1 + Iˆ2 R2 , H0 (tk ), H1 (tk ), · · · , Hm−1 (tk ) are given in (15). Next, due to the graph contains a spanning tree, from Lemma 2 one can yields that all the eigenvalues of ELF have positive real parts. This means that matrix −ELF is Hurwitz stable. Thus there exists a positive definite matrix W such that −[W (ELF ) + (ELF )T W ] < 0. Then one can take R1 = diag(W, 2c1 mIN ), R2 = mI2N −1 , R3 = (m − 1)I2N −1 , · · · , Rm−1 = 3I2N −1 , Rm = 2I2N −1 .
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+(ELF )T W ] + mδ [Tk W E +γm(LF )T ][Tk E T W +γm(LF )] < 0. Since Tk ≥ τ > 0 and −[W (ELF ) +(ELF )T W ] < 0, it is obtained that Q11 (tk ) < 0 holds for sufficiently small δ. Consequently, if we take R = diag(R1 , R2 , · · · , Rm ) with Ri , i = 1, 2, · · · , m given in (17), and c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0, then (10) holds for sufficiently small δ.
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Remark 4. Note that there is a concrete example in the existing results to satisfy the assumption Tk ≥ τ > 0 of Theorem 1, such as the scene of asynchronous periodic sampling, studied in [27]. Therein, the sampling times of each agent satisfy tis = t0 + shi and time delays τij = 0, where hi is the sampling period of agent i, independent of the others’ and hi = li h with h > 0 and li being positive integer. Then Tk ≥ min{m0 , m1 , · · · , mq−1 }h > 0, where q, m0 , m1 , · · · , mq−1 are determined only by li , and given in Page 1229 of [27]. Hence, for the case studied in [27], the assumption Tk ≥ τ > 0 of Theorem 1 holds.
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Remark 5. Seen from the proof of Theorem 1, the linear matrix inequality (10) holds for sufficiently small δ when we take the control gains c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0. In this case, Σ(tk ) can be written e k ) and the elements of Σ(t e k ) are independent of δ. Hence IˆT RIˆ + as δ Σ(t T T ˆ ˆ I RΣ(tk ) + Σ (tk )RI − R + ΣT (tk )RΣ(tk ) is a quadratic polynomial of δ with matrix coefficients, and hence is continuous with respect to δ. Then we can obtain the maximum value of δ such that (10) holds, which is denoted by δ ∗ , according to the following process (just as the method given in [26]): Step 1: In (10), let c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0. For a given γ > 0, we choose a sufficiently small δ1 ∈ (0, 1) such that inequality (10) with δ = δ1 holds. Step 2: Let δ2 = δ1 + σ, where σ > 0 is an admissible error. Then check the feasibility of inequality (10) with δ = δ2 . If it is solvable, then switch to next step; otherwise, stop and δ ∗ = δ1 . .. . Step k: Let δk = δk−1 + σ, and check the feasibility of inequality (10) with δ = δk . If it is solvable, then switch to next step; otherwise, stop and δ ∗ = δk−1 . .. . According to the proof of Theorem 1, it is known that the above process must stop after finite steps. 15
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Figure 1: Directed cycle.
5. Numerical example
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Consider system (2) of six agents with protocol (3). The topology among agents is described by the directed cycle in Figure 1. The elements of associated Laplacian matrix L are l11 = l33 = l55 = 2, l22 = l44 = l66 = 1, l21 = l43 = l65 = −1 and l32 = l54 = l16 = −2, and the others are zero. Assume the sampling times of agent 1, 4 and 6 are 0.2, 0.4, · · · , 0.2k, · · · ; the sampling times of agent 2, 3 and 5 are 0.3, 0.6, · · · , 0.3k, · · · , k ∈ N+ ; and measurement time-delays are τ16 = τ43 = τ65 = 0.1, τ21 = τ32 = τ54 = 0.2. By the proof of Theorem 1, there exists positive definite matrix R and control gains c1 , c2 such that (10) holds, that is, there are control gains c1 and c2 such that the system achieve consensus. Just as done in the proof, we take c1 and c2 as c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0, and let γ = 8. According to the process of Remark 5, we get the maximum value of δ being 0.1182. This means that the condition (10) holds (or say the system achieve consensus) when c1 = 1 − δ, c2 = δγ with 0 < δ < 0.1182 and γ = 8. Let δ = 0.1, then c1 = 0.9, c2 = 0.8. In the simulation, we choose c1 = 0.9, c2 = 0.8, and initial states r(0) = [ 1.5, −4, 0.5, −2.5, −1, 3 ]T , v(0) = [ −2.5, 0.9, −0.5, −1, 3.4, 2 ]T . Figure 2 shows the simulation result for the six agents. Therein, the subfigure (a) shows the trajectories of position which converge to a constant number as time evolves. The subfigure (b) presents the trajectories of velocity which converge to zero. These illustrate that system (2) achieves static consensus. To further investigate the effect of c1 on the consensus convergence, we respectively give the simulations for c1 = 1.99 and c1 = 2, in which c2 and initial states are the same as those in Figure 2. It is shown that the static consensus can be achieved when c1 = 1.99 (see Figure 3) but can not be achieved if c1 = 2 (see Figure 4). This verifies that 0 < c1 < 2 is a necessary condition for the consensus convergence of system (2) with protocol (3). 16
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6. Conclusions
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In this paper, the asynchronous consensus of multiple second-order agents with fixed topology and measurement time-delays has been studied. Based on the state measurements at some discrete times, an impulsive consensus protocol has been proposed. By the methods of reducing order and common Lyapunov function, it has been proved that there exist control gains such that the system achieves asynchronous consensus. Due to the asynchronism of sampled information and the impulsive nature of protocol, we have only reduced the order of position vector and remained the order of velocity vector unchanged. The future work will consider the impulsive consensus of multiple second-order agents with asynchronous sampling information and jointly connected topology. References [1] R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (1) (2007) 215-233.
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[2] W. Ren, Consensus strategies for cooperative control of vehicle formations, IET Control Theory Appl. 1 (2) (2007) 505-512.
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[3] H.S. Su, M.Z.Q. Chen, G.R. Chen, Robust Semi-global coordinated tracking of linear multi-agent systems with input saturation, Int. J. Robust Nonlinear Control 25 (14) (2015) 2375-2390.
[4] H.S. Su, G. Jia, M.Z.Q. Chen, Semi-global containment control of multiagent systems with intermittent input saturation, J. Frankl. Inst.-Eng. Appl. Math. 352 (9) (2015) 3504-3525.
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[5] J.H. Qin, Q.C. Ma, Y. Shi, L. Wang, Recent advances in consensus of multi-agent systems: A brief survey, IEEE Trans. Ind. Electron. 64 (6) (2017) 4972-4983. [6] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520-1533.
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[7] Y.J. Wu, L. Wang, Average consensus of continuous-time multi-agent systems with quantized communication, Int. J. Robust Nonlinear Control 24 (18) (2014) 3345-3371.
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[8] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control 48 (6) (2003) 988-1001.
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[9] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (1) (2005) 655-661.
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[10] W.J. Xiong, X.H. Yu, Y. Chen, J. Gao, Quantized iterative learning consensus tracking of digital networks with limited information communication, IEEE Trans. Neural Netw. Learn. Syst. 28 (6) (2017) 14731480. [11] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, Int. J. Robust Nonlinear Control 17 (10-11) (2007) 1002-1033.
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[12] G.M. Xie, L. Wang, Consensus control for a class of networks of dynamic agents, Int. J. Robust Nonlinear Control 17 (10-11) (2007) 941-959.
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[13] F.C. Jiang, D.M. Xie, B. Liu, Static consensus of second-order multiagent systems with impulsive algorithm and time-delays, Neurocomputing 223 (2017) 18-25. [14] F.C. Jiang, L. Wang, Finite-time weighted average consensus with respect to a monotonic function and its application, Syst. Control Lett. 60 (9) (2011) 718-725.
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[15] Y.S. Zheng, Y.R. Zhu, L. Wang, Finite-time consensus of multiple second-order dynamic agents without velocity measurements, Int. J. Syst. Sci. 45 (3) (2014) 579-588.
[16] Y.G. Sun, L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Trans. Autom. Control 54 (7) (2009) 1607-1613.
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[17] D.M. Xie, T. Liang, Second-order group consensus for multi-agent systems with time delays, Neurocomputing 153 (2015) 133-139.
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[18] Y.L. Gao, J.Y. Yu, J. L. Shao, M. Yu, Group consensus for secondorder discrete-time multi-agent systems with time-varying delays under switching topologies, Neurocomputing 207 (2016) 805-812.
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[19] Q. Cui, D. M. Xie, F.C. Jiang, Group consensus tracking control of second-order multi-agent systems with directed fixed topology, Neurocomputing 218 (2016) 286-295.
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[20] J.H. Qin, Q.C. Ma, W.X. Zheng, H.J. Gao, Y. Kang, Robust H-infinity group consensus for interacting clusters of integrator agents, IEEE Trans. Autom. Control, DOI: 10.1109/TAC.2017.2660240.
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[21] Z.M. Wang, H.S. Zhang, Observer-based robust consensus control for multi-agent systems with noises, Neurocomputing 207 (2016) 408-415. [22] Y.S. Zheng, L. Wang, Consensus of Switched Multiagent Systems, IEEE Trans. Circuits Syst. II-Express Briefs 63 (3) (2016) 314-318.
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[23] Y. Chen, W.W. Yu, S.L. Tan, H.H. Zhu, Synchronizing nonlinear complex networks via switching disconnected topology, Automatica 70 (8) (2016) 189-194.
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[24] F. Xiao, L. Wang, Asynchronous consensus in continuous-time multiagent systems with switching topology and time-varying delays, IEEE Trans. Autom. Control 53 (8) 2008 1804-1816. [25] J. Almeida, C. Silvestre, A. M. Pascoal, Continuous-time consensus with discrete-time communications, Syst. Control Lett. 61 (7) (2012) 788-796.
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[26] Y.P. Gao, L. Wang, Asynchronous consensus of continuous-time multiagent systems with intermittent measurements, Int. J. Control 83 (3) (2010) 552-562. [27] Y.P. Gao, L. Wang, Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology, IEEE Trans. Autom. Control 56 (5) (2011) 1226-1231.
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[28] M. Cao, A.S. Morse, B.D.O. Anderson, Agreeing asynchronously, IEEE Trans. Autom. Control 53 (8) (2008) 1826-1838.
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[29] L. Fang, P. J. Antsaklis, Asynchronous Consensus Protocols Using Nonlinear Paracontractions Theory, IEEE Trans. Autom. Control 53 (10) (2008) 2351-2355.
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[30] J.H. Qin, C.B. Yu, S. Hirche, Stationary Consensus of Asynchronous Discrete-Time Second-Order Multi-Agent Systems Under Switching Topology, IEEE Trans. Ind. Inform. 8 (4) (2012) 986-994.
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[31] B. Zhou, X.F. Liao, T.W. Huang, H.W. Wang, G. Chen, Constrained consensus of asynchronous discrete-time multi-agent systems with timevarying topology, Inf. Sci. 320 (2015) 223C234.
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[32] Z.H. Guan, Z.W. Liu, G. Feng, M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information, Automatica 48 (7) (2012) 1397-1404. [33] Z.W. Liu, Z.H. Guan, X. Shen, G. Feng, Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements, IEEE Trans. Autom. Control 57 (10) (2012) 2639-2643. 21
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Biography
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Fangcui Jiang received the Bachelor and Master degree in applied mathematics from Qufu Normal University, P.R. China in 2003 and 2006, and the Ph.D. degree in systems and control from Peking University in 2011. Now she is an assistant professor at School of Mathematics and Statistics, Shandong University, Weihai, and also studies as a Postdoctoral Research Fellow at School of Control Science and Engineering, Shandong University, Jinan, P.R. China. Her current research interests focus on networked systems, analysis and cooperative control of multi-agent systems.
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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.
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Yongjun Wu received the M.S. degree in operational research and cybernetics from the Tianjin University, Tianjin, China, in 2009, and the Ph.D. degree in general and fundamental mechanics from the Peking University, Beijing, China, in 2015. He is currently a lecturer with the Department of Applied Statistics, Tianjin Polytechnic University, Tianjin, China. His current research interest includes multi-agent systems, networked control systems and switched systems.
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Yunru Zhu was born in Xi’an, Shaanxi Province. She received the BS degree in Automatic from Xidian University in 2003, the MS degree from Huazhong University of Science and Technology in 2006, and PhD degree from Xidian University in 2015, respectively. Since 2006, she has been worked at the School of Mechano-electronic Engineering, Xidian University. Her current research interests are in the fields of coordination of multi-agent systems, quantized control and impulsive control.
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Accepted Manuscript
Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays Fangcui Jiang, Bo Liu, Yongjun Wu, Yunru Zhu PII: DOI: Reference:
S0925-2312(17)31559-X 10.1016/j.neucom.2017.09.040 NEUCOM 18922
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
3 January 2017 11 June 2017 12 September 2017
Please cite this article as: Fangcui Jiang, Bo Liu, Yongjun Wu, Yunru Zhu, Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.09.040
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Highlights • The current work extends the existing results on impulsive consensus to the asynchronous setting.
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• The paper considers the case of multiple measurement time-delays as well. • In the technical contribution, the analysis method is entirely different from the analysis techniques used in some related existing works duo to the effects of both asynchronism of sampled information and impulsive nature of protocol.
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• The design of the protocol parameters is given by solving a feasible linear matrix inequality.
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Asynchronous consensus of second-order multi-agent systems with impulsive control and measurement time-delays✩ Fangcui Jianga,b , Bo Liuc , Yongjun Wud , Yunru Zhue
School of Control Science and Engineering, Shandong University, Jinan 250061, China b School of Mathematics and Statistics, Shandong University, Weihai 264209, China c College of Science, North China University of Technology, Beijing 100144, China d School of Science, Tianjin Polytechnic University, Tianjin 300387, China e Center for Complex Systems, School of Mechano-electronic Engineering, Xidian University, Xian 710126, China
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This paper investigates the asynchronous consensus of second-order multiagent systems with directed networks and measurement time-delays via impulsive control. It is assumed that each agent receives the measurements of the information from its neighbouring agents and itself according to its own sampling clock, and different agents can have independent sampling times. The controller of agent is designed in impulsive framework and changes agent’s state instantaneously. By transforming the consensus problem into static consensus and using common Lyapunov function, it is proved that there exist protocol parameters ensuring consensus. Also, the design of the parameters is given in a specific form. Finally, simulation example is drawn to illustrate the effectiveness of the theoretical results.
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Keywords: Multi-agent systems, Second-order, Asynchronous consensus, Measurement time-delays, Impulsive control
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This work was supported by National Natural Science Foundation of China (No. 61304163, No. 61473337 and No. 61304049), Shandong Provincial Natural Science Foundation, China (No. ZR2013FQ008) and Independent Innovation Foundation of Shandong University (No. 2013ZRQP006). ∗ Corresponding author. Email addresses: [email protected] (Fangcui Jiang), [email protected] (Bo Liu), [email protected] (Yongjun Wu), [email protected] (Yunru Zhu) Preprint submitted to Elsevier
September 21, 2017
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1. Introduction
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In recent years, distributed coordination control of multi-agent systems has achieved momentous research findings in many scientific fields. Among the numerous branches of this subject, consensus problem is one of the fundamental topics and with the practical applications in autonomous formation flight, cooperative control of unmanned air vehicles, coordinated tracking and containment control of multi-agent systems [1, 2, 3, 4, 5]. Roughly speaking, the consensus problem of multiple dynamic agents aims at using simple local nearest neighbor rules so that certain states of interest eventually reach a common desired quantities. Researchers in the control community have studied the consensus problems from various perspectives, and obtained many significant and interesting results, such as average consensus of first-order agents [6, 7], consensus behavior of discrete-time dynamic agents with general topologies [8, 9, 10], static or dynamic consensus for second-order agents [11, 12, 13], finite-time consensus to improve convergence rate [14, 15], consensus analysis with transmission time-delays [16], group consensus or consensus tracking [17, 18, 19, 20], observer-based robust consensus of general linear dynamics with noises [21], consensus of switched multi-agent systems or switching disconnected topology [22, 23], just to name a few. The majority of the above mentioned works studied the distributed consensus with the assumption that all local information among agents is exchanged continuously or synchronously at the same discrete times. However, in real networked systems and robotic applications, it is more practical and implementable to consider asynchronous versions of consensus [24, 25, 26, 27, 28, 29, 30, 31] in which each agent can independently obtain its needed local information at discrete times according to its own sampling clock, because of the sensing limitations of agents, the unreliability of transmission channels, the energy consumption and the environment factors. For continuous-time first-order multi-agent systems, Xiao and Wang in [24] proposed an asynchronous consensus protocol based upon the neighbouring agents’ state information at independent discrete times and proved that the asynchronous consensus is reachable under the conditions that the union of the graph across any time interval with a given length contains a spanning tree and the outdated information coming from neighbors is with bounded time-varying delays. Another related work on this topic can be found in [25]. The asynchronous consensus based on sampled data for a number of agents with continuous-time second-order dynamics was considered in [26] for 3
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fixed topology and in [27] for time-dependent topology. In contrast to the first-order model, the analysis tool of nonnegative matrix theory employed in [24] is invalid to study the problem in [26] and [27], where the authors respectively made use of the stability theory of time-delayed discrete-time systems and the Schur stability of matrices. Also, the asynchronous consensus of discrete-time multi-agent systems was investigated extensively in [28, 29, 30, 31]. In this paper, the asynchronous consensus of continuous-time secondorder multi-agent systems with impulsive control and multiple measurement time-delays is studied. To the best of the authors’ knowledge, there are few works focusing on this topic, although the impulsive consensus has been addressed well in [13, 32, 33] and the references therein based on synchronous sampled-data. Specifically, [13] studied the synchronous consensus for the cases of periodic impulsive control and single time-delay, which is opposite to the current paper. (Note that the current paper studies the asynchronous consensus for the cases of aperiodic impulsive control and multiple timedelay.) In [32], the authors proposed two kinds of impulsive algorithms via periodic sampled information to solving the dynamic and static consensus for a group of continuous-time second-order agents with fixed topology, and obtained two necessary and sufficient conditions to ensure the consensus by utilizing Schur stability of matrices. Considering in real networks some partial states may not be obtained due to communication limitations or technology constraints, [33] established an aperiodic impulsive algorithm by using only position information for second-order multi-agent systems with switching topology and proved the consensus convergence according to the property of stochastic matrices. It is worth mentioning that the impulsive control is “an effective control technique in many practical applications, such as the orbit interception correction of orbiting objects, the population control of a kind of insects, the control of reaction process in a chemical reactor system, and the money supply in a financial system” and with “obvious advantages in less energy cost, fast transient, and easier to design” [13]. The contributions of this paper are threefold. Firstly, the current work extends the existing results on impulsive consensus to the asynchronous setting, and considers the case of multiple measurement time-delays as well. Secondly, in the technical contribution, the analysis method of this work is to transform the consensus problem into static consensus and then reduce only the order of position vector by a tree-type transformation but remain the order of velocity vector unchanged, duo to the effects of both asynchronism 4
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of sampled information and impulsive nature of protocol. This is entirely different from the analysis techniques used in [26, 27], where an invertible linear transformation was introduced to reduce the orders of both position vector and velocity vector. Moreover, the technical issue to analyse the convergence of the asynchronous setting is much more challenging than the synchronous case. Thirdly, the design of the protocol parameters is given by solving a feasible linear matrix inequality. The remainder of this paper is organized as follows. Some preliminaries on graph theory and problem formulation are given in Section 2 and Section 3. The convergence analysis and the main results are performed in Section 4. To illustrate the effectiveness of the theoretical finding, a numerical example is worked out in Section 5. At last, some concluding remarks are drawn in Section 6. Notations: We let R, N and N+ be the sets of real numbers, non-negative integers and positive integers, respectively. Rn is the n-dimensional Euclidean space. Rm×n is the set of m-by-n matrices. In ∈ Rn×n is an identity matrix. Sometimes we apply 0 to denote the zero matrix with appropriate dimension. 1n = [1 · · · 1]T ∈ Rn is a column vector with all elements equal to one. 2. Preliminaries on graph theory
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A weighted directed graph is denoted by G(Λ), which is consisted of vertex set W = {w1 , w2 , . . . , wN }, edge set F ⊂ W × W, and weighted adjacency matrix Λ = [akl ] ∈ RN ×N . The rows and columns of Λ are indexed by {w1 , w2 , . . . , wN }. In multi-agent systems, we use graph G(Λ) to describe the relationships of information transmission among agents. Specially, vertex wk represents the k-th agent of the system; an edge is denoted by ordered pair hwk , wl i, which means agent l can obtain the information of agent k; hwk , wl i ∈ F if and only if alk > 0 (otherwise, alk = 0), alk denotes the weight of the edge. If hwk , wl i ∈ F, then we say wk (or agent k) is a neighbor of wl (or agent l). In this paper, we assume (wk , wk ) 6∈ F and the element of F is not repeated. A path from wk to wl is defined as a string of edges (wk , w1 ), (w1 , w2 ), . . . , (wi , wl ) which are distinct in F. A graph is a directed tree, if it is a directed graph and there is a special vertex from which only one path can reach any other vertex. Given G(Λ), if there is F 0 ⊂ F such that W and F 0 compose a directed tree, then we say that G(Λ) contains a spanning tree.
5
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j=1,j6=k
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Laplacian matrix L = [bkl ] ∈ RN ×N of G(Λ) is defined as l 6= k −akl , N P bkl = . akj , l = k
Lemma 1. ([6, 9]) Given that G(Λ) is a graph with Laplacian matrix L. Then the algebraic multiplicity of the 0 eigenvalue is simple if and only if G(Λ) contains a spanning tree; L1N = 0; and the real parts of nonzero eigenvalues are positive. Define matrices
eT1 E
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E = [1N −1 − IN −1 ] ∈ R
N −1×N
, F =
0 −IN −1
∈ RN ×N −1 ,
(1)
= [1N F ], where e1 = [1 0 · · · 0]T ∈ RN . Then we have 0 eT1 LF N ×N −1 −1 , where U ∈ R is invertible, U = U , and U LU = 0 ELF L is Laplacian matrix of a graph. According to the property of similarity transformation of matrices, it follows that ELF has N − 1 eigenvalues of L. Hence Lemma 1 implies the following result.
ED
M
and U =
PT
Lemma 2. Given that G(Λ) is a graph with Laplacian matrix L, then the graph contains a spanning tree if and only if the real parts of all eigenvalues of ELF are positive, where E and F are defined as in (1).
CE
3. Model and problem formulations
AC
Consider a dynamical system of N autonomous agents, which are labeled 1 through N . Each agent is modeled by second-order dynamics p˙i (t)=vi (t), (2) v˙ i (t) =ui (t), t ≥ t0 , i = 1, 2, · · · , N, where pi (t) ∈ R, vi (t) ∈ R are the position and velocity states of agent i, respectively; ui (t) ∈ R is the control input or the protocol to be designed based on the information obtained by agent i; t0 ≥ 0 is the initial time of the dynamical system. 6
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Definition 1. Multi-agent system (2) is said to achieve consensus if for any initial states, limt→∞ [pi (t) − pj (t)] = 0 and limt→∞ [vi (t) − vj (t)] = 0, ∀i, j = 1, 2, · · · , N . Furthermore, if there exists p∗ , such that limt→∞ pi (t) = p∗ and limt→∞ vi (t) = 0, ∀i = 1, 2, · · · , N , then system (2) is said to achieve static consensus, where [p∗ , 0]T is called as the consensus state of the system. Because of the wide use of computer-based control and digital sensors, the agents in networks may only obtain sampled information from their neighbors at some sampling instants. Based on sampled information, [32] proposed the following impulsive algorithm
k=1
− c1 vi (t) − c2
X
i aij (pi (t) − pj (t)) δ(t − tk ).
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ui (t) =
+∞ h X
j∈Ni
AC
CE
PT
ED
M
where c1 , c2 > 0; aij is the weight of the edge from agent j to agent i; Ni is the set of neighbors of agent i; δ(·) is the Dirac impulsive function; the discrete time tk satisfies tk − tk−1 ≡ h. In the above algorithm, all agents sampled their required information at the same instants tk , k = 1, 2, · · · , and the sampling intervals were periodic. However, it is difficult for all agents to synchronously receive/measure the state information in the distributed control of some practical networks. Also the measured information may involve a certain degree of time-delay. Therefore, in this paper we mainly discuss the impulsive consensus of multi-agent system (2) with measurement time-delays based on asynchronous sampled information. This is a complete contrast to the synchronous sampling time studied in [13, 32, 33]. Specifically, we assume that agent i independently receives or measures its neighbors’ state information at times ti1 , ti2 , · · · , tis , · · · , i = 1, 2, · · · , N , which are determined by its own clock. For simplicity, let us denote the sampling instants by a real number sequence {tis }∞ s=1 . Note that by the concept of “asynchronous”, it is meant that for any different agents i and j, they may not measure their neighbors’ state at the same times, i.e., {tis }∞ s=1 6= {tjs }∞ . In addition, there may exist time-delays in the measured information s=1 of neighbors’ states. Hence different from the sampled controller proposed in [26, 27], we give the following impulsive consensus protocol ui (t) =
+∞ X s=1
− c1 vi (t) − c2
X
j∈Ni
aij (pi (t − τij ) − pj (t − τij )) δ(t − tis ), 7
(3)
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where c1 > 0 and c2 > 0 are the control gains; aij is the weight of the edge from agent j to agent i; Ni is the collection of neighbors of agent i; τij ≥ 0 is the measurement time-delay in relative position; δ(·) is the Dirac impulsive function with discontinuity points ti1 < ti2 < · · · < tis−1 < tis < · · · and tis → ∞ as s → ∞, ti1 > t0 ; tis , s ∈ N+ are the sampling times of agent i, also are the impulse times. For any i = 1, 2, · · · , N , it is assumed that {tis }∞ s=1 satisfies 0 < α ≤ tis+1 − tis ≤ β, s ∈ N,
(4)
+ σ) −
vi (tis
− σ) =
Z
tis +σ
M
vi (tis
AN US
where α and β are real numbers. Note that ti0 indicates the initial instant of the dynamical system of agent i rather than the first sampling instant. Seen from (2), it is obtained that ti0 = t0 for i = 1, 2, · · · , N . It’s known that the Dirac impulsive R ∞ function δ(·) is defined R ∞as δ(t) = 0 for t 6= 0, δ(t) → ∞ for t = 0 and −∞ δ(t)dt = 1. It satisfies −∞ f (t)δ(t − tis )dt = f (tis ). Then (3) implies that ui (t) = 0 for t 6= tis , s ∈ N+ . Moreover, for sufficiently small σ > 0, v˙ i (t)dt =
tis −σ
Z
tis +σ
ui (t)dt.
tis −σ
ED
Hence as σ → 0+ , one yields +
−
) ∆vi (tis ):=vi (tis ) − vi (tis P i = −c1 vi (ts ) − c2 aij [pi (tis − τij ) − pj (tis − τij )],
PT
j∈Ni
−
+
AC
CE
where vi (tis ) = limσ→0+ vi (tis − σ) and vi (tis ) = limσ→0+ vi (tis + σ) are, respectively, the left limit and the right limit of vi (t) at tis . Therefore, the closed-loop dynamics of agent i can be given by the following impulsive differential equations p˙i (t)=vi (t), v˙ i (t)=0, t ≥P t0 , t 6= tis , s ∈ N+ , (5) aij [pi (tis − τij ) − pj (tis − τij )], t = tis . ∆vi (tis )=−c1 vi (tis ) − c2 j∈Ni
In this impulsive control, we can assume vi (t) is left continuous at t = tis , − that is, vi (tis ) = vi (tis ), s ∈ N+ , and continuous at initial time t0 . 8
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Remark 1. In protocol (3), the agent i is required to receive the relative position information with measurement delay to its neighbors (i.e., pi (tis − τij ) − pj (tis − τij )) and its velocity information vi (tis ) at its own sampling times.
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Remark 2. Compared with the existing impulsive consensus algorithms in [13, 32, 33], the sampling times are independent of each other for different agents. That is to say the sampling sequence {tis } is independent of {tjs } for any agents i and j, i 6= j. Agents may not sample information at the same times. This asynchronous sampling setting is quite distinct from the synchronous sampling times therein. Remark 3. Compared with [26, 27], control input ui (t) can suddenly change the velocity of agent at the instants tis , i.e., ui (t) is only implemented at tis by impulsive actuator and causes a jump of the agent velocity. System (2) during sampling intervals is running without any control input. This is quite different from the sampled-data protocols therein, which are executed continuously during sampling interval.
PT
ED
M
By recurring Definition 1, it is easily obtained that if system (2) with protocol (3) achieves static consensus, then the system achieves consensus. Moreover, system (5) can be rewritten as the following discrete-time impulsive system along time sequence {tis } + i i ) + (tis+1 − tis )vi (tis ), pi (tis+1 )=pi (tis+ vi (ts+1 ) =vi (ts ), (6) P i i i+ i − τ ) − p (t − τ )], ) − c a [p (t v (t ) =(1 − c )v (t ij ij j i 1 i 2 ij i s s s s j∈Ni
+
AC
CE
with pi (tis − τij ) = pi (t0 ) for tis − τij ≤ t0 and vi (ti0 ) = vi (ti0 ) = vi (t0 ), i = 1, 2, · · · , N . Here, we need to point out two facts for the convergence of the system. The first fact is that the control gain c1 plays an important role for the consensus convergence, and 0 < c1 < 2 is also a necessary condition for system (2) with protocol (3) achieving consensus on the ground of Lemma 5 in [13]. The second fact is that from the same guideline of the proof of Theorem 2 in [32], we can derive the following lemma. Lemma 3. Suppose 0 < c1 < 2, then system (2) with protocol (3) achieving consensus is equivalent to the system achieving static consensus. 9
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4. Convergence analysis and main results
tis ≤ tk < tk+1 ≤ tis+1 .
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At first, we merge all the time sequences {tis }, {tis − τij } and the initial time {t0 } into a single ordered sequence, and relabel the sequence as t0 , t1 , · · · , tk , · · · . That is denoting the set {t0 , t1 , · · · , tk , · · · } = {t0 }∪{tis | i = 1, 2, · · · , N , s ∈ N+ } ∪ {tis − τij | tis − τij ≥ t0 , i = 1, 2, · · · , N , j ∈ Ni , s ∈ N+ }, which satisfies tk < tk+1 . Next, we will establish the closed-loop system (9) for the multi-agent system (5) along the time sequence {t0 , t1 , · · · , tk , · · · }. For any i = 1, 2, · · · , N and any k ∈ N, there exists s ∈ N such that (7)
+ vi (tis );
AN US
if Then according to system (5), it follows that if tk = tis , vi (tk + ) = + tk > tis , vi (tk + ) = vi (tk ) = vi (tis ). Hence for the time sequence {t0 , t1 , · · · , tk , · · · }, system (6) can be rewritten as ) + Tk vi (tk + ), pi (tk+1 )= pi (tk + vi (tk+1 ) = vi (tk ), k ∈ N, (8) P + i aij [pi (tis − τij ) − pj (tis − τij )], vi (tk ) = (1 − c1 )vi (ts ) − c2 j∈Ni
ED
M
where Tk = tk+1 − tk ; for k = 0, vi (t0 + ) = vi (t0 ), i = 1, 2, · · · , N . Let m e denoteTthe upper bound for the number of elements tk in the set {t0 , t1 , t2 , · · · } [tis , tis+1 ) for any i = 1, 2, · · · , N and any s ∈ N. Then Lemma 2 in [24] shows that m e can be taken as m e = N [(bβ/αc + 1)(N − 1) + ¯ 1][K(N − 1) + 1], where bβ/αc is the maximum integer which is not greater ¯ = bτmax /αc + 1 with τmax = than β/α, and K max {τij }. Denote i6=j,i,j=1,2,··· ,N
CE
PT
¯ + 1)m. ¯ and k ≥ m, one has ti − τij ≥ tk−m+1 (this m = (K e For s ≥ K s conclusion can be deduced from the remarks after Lemma 2 of [24]). By recalling (7), we have that for any i = 1, 2, · · · , N and any k ∈ N, tk−m+1 ≤ tis − τij ≤ tis ≤ tk < tk+1 ≤ tis+1 .
AC
This means that for fixed k, tis and tis −τij take values in {tk , tk−1 , · · · , tk−m+1 }, i = 1, 2, · · · , N , j ∈ Ni . Here, we assume tk = t0 for k < 0. Consequently, if we let p(tk ) = [p1 (tk ), p2 (tk ), · · · , pN (tk )]T and v(tk ) = [v1 (tk ), v2 (tk ), · · · , vN (tk )]T , then system (8) can be written as p(tk+1 ) = p(tk ) + Tk v(tk + ), v(t )= v(t + ), k+1 k m−1 m−1 P P Lρ (tk )p(tk−ρ ) + (1 − c1 ) Bρ (tk )v(tk−ρ ), v(tk + ) = −c2 ρ=0
ρ=0
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l=1,l6=i
defined as and tis − τij = tk−ρ and tis − τij 6= tk−ρ and j 6= i ;
CR IP T
where Lρ (tk ) ∈ RN ×N with the (i, j)-element being −aij , j ∈ Ni , 0, j ∈ Ni , 0, j 6∈ Ni , Lρ (tk )(i, j) = N P Lρ (tk )(i, l), j = i −
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B0 (tk )+B1 (tk )+· · ·+Bm−1 (tk ) = IN , and if there exists some agent i sampling information at time tk−ρ , i.e., tis = tk−ρ with ρ ∈ {0, 1, 2, · · · , m − 1}, then the i-th row of Bρ (tk ) is equal to the i-th row of matrix IN , while the ith rows of all the other matrices in {B0 (tk ), B1 (tk ), · · · , Bm−1 (tk )} are equal m−1 P to zeros. It should be mentioned that Lρ (tk ) = L and Lρ (tk )1N = 0 for ρ=0
M
any ρ = 0, 1, · · · , m − 1, k ≥ m. Thus the dynamics of stacked vectors p(tk ) and v(tk ) can be described by the following discrete-time system with time-delays m−1 m−1 P P p(t ) =p(t ) − T c L (t )p(t ) + T (1 − c ) Bρ (tk )v(tk−ρ ), k+1 k k 2 ρ k k−ρ k 1 ρ=0
ρ=0
m−1 m−1 P P Lρ (tk )p(tk−ρ ) + (1 − c1 ) Bρ (tk )v(tk−ρ ), v(tk+1 )=−c2 ρ=0
ED
ρ=0
PT
(9) which is the closed-loop system of system (5) along the time sequence {t0 , t1 , · · · , tk , · · · }. Then system (2) with protocol (3) achieves consensus if and only if system (9) achieves consensus.
AC
CE
Theorem 1. For system (2) with protocol (3), assume G is the directed topology which contains a panning tree, the sampling times of each agent satisfy (4), and there exists τ > 0 such that Tk in (8) satisfies Tk ≥ τ . Then for any τ > 0, system (2) with protocol (3) achieves consensus if there exists a positive definite matrix R such that IˆT RIˆ + IˆT RΣ(tk ) + ΣT (tk )RIˆ − R + ΣT (tk )RΣ(tk ) < 0
11
(10)
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holds, where
AN US
with E and F be defined in (1).
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Iˆ1 0 0 0 0 I 0 N −1 N −1 Iˆ= Iˆ2 , Iˆ2 = , 0 0 , Iˆ1 = 0 0N 0 −IN 0 I(2N −1)(m−2) 0 Pm−1 Pm−1 Pm−1 − ρ=0 Mρ (tk ) Mρ (tk ) · · · Mρ (tk ) ρ=1 ρ=m−1 P Pm−1 Pm−1 , Σ(tk )= − m−1 ρ=0 Mρ (tk ) ρ=1 Mρ (tk ) · · · ρ=m−1 Mρ (tk ) 0 0 ··· 0 Tk c2 ELρ (tk )F −Tk (1 − c1 )EBρ (tk ) Mρ (tk )= , ρ = 0, 1, · · · , m − 1, c2 Lρ (tk )F −(1 − c1 )Bρ (tk )
Proof. We need only to prove that system (9) achieves consensus under the conditions given in Theorem 1. By recalling the matrices E and F in (1), let r(tk ) = Ep(tk ).
(11)
PT
ED
M
Then p(tk ) = p1 (tk )1N + F r(tk ). (In literature [16], the transformation (11) is called as a tree-type transformation.) Applying (11) to system (9) and noting that Lρ (tk )1N = 0, one has m−1 P r(tk+1 ) =r(tk ) − Tk c2 ELρ (tk )F r(tk−ρ ) ρ=0 m−1 P +Tk (1 − c1 ) EBρ (tk )v(tk−ρ ), (12) ρ=0 m−1 m−1 P P Lρ (tk )F r(tk−ρ ) + (1 − c1 ) Bρ (tk )v(tk−ρ ). v(tk+1 )=−c2 ρ=0
CE
ρ=0
AC
Denote x(tk ) = [rT (tk ), v T (tk )]T ∈ R2N −1 , then system (12) can be stacked as m−1 X x(tk+1 ) = Iˆ1 − M0 (tk ) x(tk ) − Mρ (tk )x(tk−ρ ), (13) ρ=1
where Iˆ1 and Mρ (tk ), ρ = 0, 1, · · · , m − 1 are given as in Theorem 1. It should be mentioned that transformation (11) implies r(tk ) = [p1 (tk ) − p2 (tk ), p1 (tk ) − p3 (tk ), · · · , p1 (tk ) − pN (tk )]T ∈ RN −1 . Hence, Lemma 3 indicates that system (9)achieves consensus if and only if limk→∞ x(tk ) = 0. 12
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Therefore, we next devote the effort to proving that time-delay system (13) is asymptotically stable. With some direct calculation, system (13) can be rewritten as
Hi (tk ) =
m−1 X ρ=i
T
T
(14)
Mρ (tk ), i = 0, 1, · · · , m − 1.
(15)
AN US
where
CR IP T
x(tk+1 ) = Iˆ1 − H0 (tk ) x(tk ) + H1 (tk )[x(tk ) − x(tk−1 )] + · · · + Hm−2 (tk )[x(tk−m+3 ) − x(tk−m+2 )] +Hm−1 (tk )[x(tk−m+2 ) − x(tk−m+1 )],
(16)
Let y(tk )=[x (tk ), x (tk ) − xT (tk−1 ), xT (tk−1 ) − xT (tk−2 ), · · · , xT (tk−m+2 ) −xT (tk−m+1 )]T . From (14), we obtain the following augmented system y(tk+1 ) = [Iˆ + Σ(tk )]y(tk )
M
with Iˆ and Σ(tk ) being given in Theorem 1. Then system (13) is asymptotically stable if and only if system (16) is asymptotically stable. For system (16), consider the following common Lyapunov function
ED
V (tk ) = y T (tk )Ry(tk ). Then
PT
V (tk+1n ) − V (tk ) o T =y (tk ) [Iˆ + Σ(tk )]T R[Iˆ + Σ(tk )] − R y(tk ) =y T (tk ) IˆT RIˆ + IˆT RΣ(tk ) + ΣT (tk )RIˆ − R + ΣT (tk )RΣ(tk ) y(tk ).
AC
CE
According to the condition (10), there exists positive definite matrix R such that V (tk+1 ) − V (tk ) < 0. This implies that system (16) is asymptotically stable for any τ > 0. By summarizing the above discussion, one yields that system (2) with protocol (3) achieves consensus. At last, to improve the stringency of Theorem 1, we prove that there indeed exists a positive definite matrix R and control gains c1 , c2 such that (10) holds. Let R = diag(R1 , R2 , · · · , Rm ), with Ri ∈ R(2N −1)×(2N −1) , i = 1, 2, · · · , m, and take c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0. Introduce O(δ 2 ) to represent a matrix with 13
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each element being the same order infinitesimal of δ 2 . By calculation and denoting ΣT (tk )RΣ(tk ) = O(δ 2 ), one has
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ˆT RIˆ + IˆT RΣ(tk ) + ΣT (tk )RIˆ − R + ΣT (tk )RΣ(tk ) I Q11 (tk ) JH1 (tk ) · · · JHm−2 (tk ) JHm−1 (tk ) H T (tk )J T R3 − R2 1 . ... .. = + O(δ 2 ), T Hm−2 (tk )J T Rm − Rm−1 T Hm−1 (tk )J T −Rm
AN US
where Q11 (tk ) = −(Iˆ1 R1 + Iˆ2 R2 )H0 (tk )−H0T (tk )(R1 Iˆ1 +R2 Iˆ2 )−R1 + Iˆ1 R1 Iˆ1 + Iˆ2 R2 Iˆ2 , J = Iˆ1 R1 + Iˆ2 R2 , H0 (tk ), H1 (tk ), · · · , Hm−1 (tk ) are given in (15). Next, due to the graph contains a spanning tree, from Lemma 2 one can yields that all the eigenvalues of ELF have positive real parts. This means that matrix −ELF is Hurwitz stable. Thus there exists a positive definite matrix W such that −[W (ELF ) + (ELF )T W ] < 0. Then one can take R1 = diag(W, 2c1 mIN ), R2 = mI2N −1 , R3 = (m − 1)I2N −1 , · · · , Rm−1 = 3I2N −1 , Rm = 2I2N −1 .
(17)
ED
M
It is obvious that R = diag(R1 , R2 , · · · , Rm ) is a positive definite matrix. Also, −Tk δγ[W (ELF ) + (ELF )T W ] ∗ Q11 (tk ) = , (18) ∗T −mIN
AC
CE
PT
with ∗ = Tk δW E + δγm(LF )T . By Schur complement, Q11 (tk ) JH1 (tk ) · · · JHm−2 (tk ) JHm−1 (tk ) H T (tk )J T R3 − R2 1 .. .. . T . T Hm−2 (tk )J Rm − Rm−1 T T −Rm Hm−1 (tk )J Q11 (tk ) JH1 (tk ) · · · JHm−2 (tk ) JHm−1 (tk ) H T (tk )J T −I2N −1 1 .. .. = <0 . T . T Hm−2 (tk )J −I2N −1 T T Hm−1 (tk )J −2I2N −1 P 1 T T if and only if Q11 (tk ) + m−2 i=1 JHi (tk )Hi (tk )J + 2 JHm−1 (tk )Hm−1 (tk )J = 2 Q11 (tk ) + O(δ ) < 0. Moreover, Q11 (tk ) < 0 if and only if −Tk γ[W (ELF ) 14
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+(ELF )T W ] + mδ [Tk W E +γm(LF )T ][Tk E T W +γm(LF )] < 0. Since Tk ≥ τ > 0 and −[W (ELF ) +(ELF )T W ] < 0, it is obtained that Q11 (tk ) < 0 holds for sufficiently small δ. Consequently, if we take R = diag(R1 , R2 , · · · , Rm ) with Ri , i = 1, 2, · · · , m given in (17), and c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0, then (10) holds for sufficiently small δ.
AN US
Remark 4. Note that there is a concrete example in the existing results to satisfy the assumption Tk ≥ τ > 0 of Theorem 1, such as the scene of asynchronous periodic sampling, studied in [27]. Therein, the sampling times of each agent satisfy tis = t0 + shi and time delays τij = 0, where hi is the sampling period of agent i, independent of the others’ and hi = li h with h > 0 and li being positive integer. Then Tk ≥ min{m0 , m1 , · · · , mq−1 }h > 0, where q, m0 , m1 , · · · , mq−1 are determined only by li , and given in Page 1229 of [27]. Hence, for the case studied in [27], the assumption Tk ≥ τ > 0 of Theorem 1 holds.
AC
CE
PT
ED
M
Remark 5. Seen from the proof of Theorem 1, the linear matrix inequality (10) holds for sufficiently small δ when we take the control gains c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0. In this case, Σ(tk ) can be written e k ) and the elements of Σ(t e k ) are independent of δ. Hence IˆT RIˆ + as δ Σ(t T T ˆ ˆ I RΣ(tk ) + Σ (tk )RI − R + ΣT (tk )RΣ(tk ) is a quadratic polynomial of δ with matrix coefficients, and hence is continuous with respect to δ. Then we can obtain the maximum value of δ such that (10) holds, which is denoted by δ ∗ , according to the following process (just as the method given in [26]): Step 1: In (10), let c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0. For a given γ > 0, we choose a sufficiently small δ1 ∈ (0, 1) such that inequality (10) with δ = δ1 holds. Step 2: Let δ2 = δ1 + σ, where σ > 0 is an admissible error. Then check the feasibility of inequality (10) with δ = δ2 . If it is solvable, then switch to next step; otherwise, stop and δ ∗ = δ1 . .. . Step k: Let δk = δk−1 + σ, and check the feasibility of inequality (10) with δ = δk . If it is solvable, then switch to next step; otherwise, stop and δ ∗ = δk−1 . .. . According to the proof of Theorem 1, it is known that the above process must stop after finite steps. 15
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$ !
%
Figure 1: Directed cycle.
5. Numerical example
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" #
AC
CE
PT
ED
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Consider system (2) of six agents with protocol (3). The topology among agents is described by the directed cycle in Figure 1. The elements of associated Laplacian matrix L are l11 = l33 = l55 = 2, l22 = l44 = l66 = 1, l21 = l43 = l65 = −1 and l32 = l54 = l16 = −2, and the others are zero. Assume the sampling times of agent 1, 4 and 6 are 0.2, 0.4, · · · , 0.2k, · · · ; the sampling times of agent 2, 3 and 5 are 0.3, 0.6, · · · , 0.3k, · · · , k ∈ N+ ; and measurement time-delays are τ16 = τ43 = τ65 = 0.1, τ21 = τ32 = τ54 = 0.2. By the proof of Theorem 1, there exists positive definite matrix R and control gains c1 , c2 such that (10) holds, that is, there are control gains c1 and c2 such that the system achieve consensus. Just as done in the proof, we take c1 and c2 as c1 = 1 − δ, c2 = δγ with 0 < δ < 1 and γ > 0, and let γ = 8. According to the process of Remark 5, we get the maximum value of δ being 0.1182. This means that the condition (10) holds (or say the system achieve consensus) when c1 = 1 − δ, c2 = δγ with 0 < δ < 0.1182 and γ = 8. Let δ = 0.1, then c1 = 0.9, c2 = 0.8. In the simulation, we choose c1 = 0.9, c2 = 0.8, and initial states r(0) = [ 1.5, −4, 0.5, −2.5, −1, 3 ]T , v(0) = [ −2.5, 0.9, −0.5, −1, 3.4, 2 ]T . Figure 2 shows the simulation result for the six agents. Therein, the subfigure (a) shows the trajectories of position which converge to a constant number as time evolves. The subfigure (b) presents the trajectories of velocity which converge to zero. These illustrate that system (2) achieves static consensus. To further investigate the effect of c1 on the consensus convergence, we respectively give the simulations for c1 = 1.99 and c1 = 2, in which c2 and initial states are the same as those in Figure 2. It is shown that the static consensus can be achieved when c1 = 1.99 (see Figure 3) but can not be achieved if c1 = 2 (see Figure 4). This verifies that 0 < c1 < 2 is a necessary condition for the consensus convergence of system (2) with protocol (3). 16
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Figure 2: Trajectories of system (2) with protocol (3) and topology shown in Figure 1, where c1 = 0.9, c2 = 0.8. (a) and (b) repectively give the evolution of pi and vi .
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Figure 4: Trajectories of system (2) with protocol (3) and topology shown in Figure 1, where c1 = 2, c2 = 0.8. (a) and (b) repectively give the evolution of pi and vi .
6. Conclusions
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In this paper, the asynchronous consensus of multiple second-order agents with fixed topology and measurement time-delays has been studied. Based on the state measurements at some discrete times, an impulsive consensus protocol has been proposed. By the methods of reducing order and common Lyapunov function, it has been proved that there exist control gains such that the system achieves asynchronous consensus. Due to the asynchronism of sampled information and the impulsive nature of protocol, we have only reduced the order of position vector and remained the order of velocity vector unchanged. The future work will consider the impulsive consensus of multiple second-order agents with asynchronous sampling information and jointly connected topology. References [1] R. Olfati-Saber, J.A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (1) (2007) 215-233.
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[2] W. Ren, Consensus strategies for cooperative control of vehicle formations, IET Control Theory Appl. 1 (2) (2007) 505-512.
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[3] H.S. Su, M.Z.Q. Chen, G.R. Chen, Robust Semi-global coordinated tracking of linear multi-agent systems with input saturation, Int. J. Robust Nonlinear Control 25 (14) (2015) 2375-2390.
[4] H.S. Su, G. Jia, M.Z.Q. Chen, Semi-global containment control of multiagent systems with intermittent input saturation, J. Frankl. Inst.-Eng. Appl. Math. 352 (9) (2015) 3504-3525.
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[5] J.H. Qin, Q.C. Ma, Y. Shi, L. Wang, Recent advances in consensus of multi-agent systems: A brief survey, IEEE Trans. Ind. Electron. 64 (6) (2017) 4972-4983. [6] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520-1533.
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[7] Y.J. Wu, L. Wang, Average consensus of continuous-time multi-agent systems with quantized communication, Int. J. Robust Nonlinear Control 24 (18) (2014) 3345-3371.
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[8] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control 48 (6) (2003) 988-1001.
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[9] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (1) (2005) 655-661.
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[10] W.J. Xiong, X.H. Yu, Y. Chen, J. Gao, Quantized iterative learning consensus tracking of digital networks with limited information communication, IEEE Trans. Neural Netw. Learn. Syst. 28 (6) (2017) 14731480. [11] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, Int. J. Robust Nonlinear Control 17 (10-11) (2007) 1002-1033.
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[12] G.M. Xie, L. Wang, Consensus control for a class of networks of dynamic agents, Int. J. Robust Nonlinear Control 17 (10-11) (2007) 941-959.
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[13] F.C. Jiang, D.M. Xie, B. Liu, Static consensus of second-order multiagent systems with impulsive algorithm and time-delays, Neurocomputing 223 (2017) 18-25. [14] F.C. Jiang, L. Wang, Finite-time weighted average consensus with respect to a monotonic function and its application, Syst. Control Lett. 60 (9) (2011) 718-725.
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[15] Y.S. Zheng, Y.R. Zhu, L. Wang, Finite-time consensus of multiple second-order dynamic agents without velocity measurements, Int. J. Syst. Sci. 45 (3) (2014) 579-588.
[16] Y.G. Sun, L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Trans. Autom. Control 54 (7) (2009) 1607-1613.
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[17] D.M. Xie, T. Liang, Second-order group consensus for multi-agent systems with time delays, Neurocomputing 153 (2015) 133-139.
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[18] Y.L. Gao, J.Y. Yu, J. L. Shao, M. Yu, Group consensus for secondorder discrete-time multi-agent systems with time-varying delays under switching topologies, Neurocomputing 207 (2016) 805-812.
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[19] Q. Cui, D. M. Xie, F.C. Jiang, Group consensus tracking control of second-order multi-agent systems with directed fixed topology, Neurocomputing 218 (2016) 286-295.
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[20] J.H. Qin, Q.C. Ma, W.X. Zheng, H.J. Gao, Y. Kang, Robust H-infinity group consensus for interacting clusters of integrator agents, IEEE Trans. Autom. Control, DOI: 10.1109/TAC.2017.2660240.
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[21] Z.M. Wang, H.S. Zhang, Observer-based robust consensus control for multi-agent systems with noises, Neurocomputing 207 (2016) 408-415. [22] Y.S. Zheng, L. Wang, Consensus of Switched Multiagent Systems, IEEE Trans. Circuits Syst. II-Express Briefs 63 (3) (2016) 314-318.
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[23] Y. Chen, W.W. Yu, S.L. Tan, H.H. Zhu, Synchronizing nonlinear complex networks via switching disconnected topology, Automatica 70 (8) (2016) 189-194.
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[24] F. Xiao, L. Wang, Asynchronous consensus in continuous-time multiagent systems with switching topology and time-varying delays, IEEE Trans. Autom. Control 53 (8) 2008 1804-1816. [25] J. Almeida, C. Silvestre, A. M. Pascoal, Continuous-time consensus with discrete-time communications, Syst. Control Lett. 61 (7) (2012) 788-796.
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[26] Y.P. Gao, L. Wang, Asynchronous consensus of continuous-time multiagent systems with intermittent measurements, Int. J. Control 83 (3) (2010) 552-562. [27] Y.P. Gao, L. Wang, Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology, IEEE Trans. Autom. Control 56 (5) (2011) 1226-1231.
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[28] M. Cao, A.S. Morse, B.D.O. Anderson, Agreeing asynchronously, IEEE Trans. Autom. Control 53 (8) (2008) 1826-1838.
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[29] L. Fang, P. J. Antsaklis, Asynchronous Consensus Protocols Using Nonlinear Paracontractions Theory, IEEE Trans. Autom. Control 53 (10) (2008) 2351-2355.
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[30] J.H. Qin, C.B. Yu, S. Hirche, Stationary Consensus of Asynchronous Discrete-Time Second-Order Multi-Agent Systems Under Switching Topology, IEEE Trans. Ind. Inform. 8 (4) (2012) 986-994.
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[31] B. Zhou, X.F. Liao, T.W. Huang, H.W. Wang, G. Chen, Constrained consensus of asynchronous discrete-time multi-agent systems with timevarying topology, Inf. Sci. 320 (2015) 223C234.
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[32] Z.H. Guan, Z.W. Liu, G. Feng, M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information, Automatica 48 (7) (2012) 1397-1404. [33] Z.W. Liu, Z.H. Guan, X. Shen, G. Feng, Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements, IEEE Trans. Autom. Control 57 (10) (2012) 2639-2643. 21
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Biography
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Fangcui Jiang received the Bachelor and Master degree in applied mathematics from Qufu Normal University, P.R. China in 2003 and 2006, and the Ph.D. degree in systems and control from Peking University in 2011. Now she is an assistant professor at School of Mathematics and Statistics, Shandong University, Weihai, and also studies as a Postdoctoral Research Fellow at School of Control Science and Engineering, Shandong University, Jinan, P.R. China. Her current research interests focus on networked systems, analysis and cooperative control of multi-agent systems.
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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.
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Yongjun Wu received the M.S. degree in operational research and cybernetics from the Tianjin University, Tianjin, China, in 2009, and the Ph.D. degree in general and fundamental mechanics from the Peking University, Beijing, China, in 2015. He is currently a lecturer with the Department of Applied Statistics, Tianjin Polytechnic University, Tianjin, China. His current research interest includes multi-agent systems, networked control systems and switched systems.
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Yunru Zhu was born in Xi’an, Shaanxi Province. She received the BS degree in Automatic from Xidian University in 2003, the MS degree from Huazhong University of Science and Technology in 2006, and PhD degree from Xidian University in 2015, respectively. Since 2006, she has been worked at the School of Mechano-electronic Engineering, Xidian University. Her current research interests are in the fields of coordination of multi-agent systems, quantized control and impulsive control.
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