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Sampled containment control for multi-agent systems with nonlinear dynamics
Author’s Accepted Manuscript Sampled containment control for multi-agent systems with nonlinear dynamics Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu, Wenshan Hu www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)31035-9 http://dx.doi.org/10.1016/j.neucom.2016.09.020 NEUCOM17549
To appear in: Neurocomputing Received date: 8 May 2016 Revised date: 1 August 2016 Accepted date: 12 September 2016 Cite this article as: Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu and Wenshan Hu, Sampled containment control for multi-agent systems with nonlinear dynamics, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.09.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Sampled containment control for multi-agent systems with nonlinear dynamics Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu*, Wenshan Hu**
Abstract In this paper, the sampled containment control scheme is proposed to study leader-following consensus problem in second-order systems. Compared to existing investigations, we develop the containment control to the sampled-data case for sake of the movements of the agents and the limited capacities of the communication. Furthermore, it is noted that the proposed algorithm is discussed for nonlinear dynamics which is much more challenging rather than linear ones. By using the LMIs and Lyapunov method, some useful sufficient conditions are presented. Finally the numerical examples are included to validate the effectiveness of proposed algorithms. Keywords: Containment control, sampled-data, consensus, nonlinear dynamics.
I. I NTRODUCTION Due to the wide application of the digital processor, the research of the sampled-data problems [1] [2] [3] [4] [5] [6] [7] [8] has attracted much attention. The digital processor can only process discrete data and obtain the discrete control signal which can be acquired as a continuous signal by a zero-order holder. In these sampled-data systems, the control signals are piecewise continuous. The effective method to analyze and deal with the sampled-data systems is the inputdelay method. That is, the systems are viewed as continuous systems with a time-varying delay This work was supported in part by the National Natural Science Foundation of China under Grants 61304152 and 6374034. Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu, and Wenshan Hu are with the Department of Automation, Wuhan University, Wuhan, 430072, P. R. China. Zhenhua Wang is with China Ship Development and Design Center (CSDDC), Wuhan, 430064, P. R. China. *Corresponding author. Email: [email protected] **Corresponding author. Email: [email protected]
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and the differential of the delay is equal to one. There are many literatures using the input delay method to dispose the sampled-data systems. Wu et al. [9] used the Lyapuov approach and the convex combination technique to investigate the synchronization of the sampled-data systems in complex dynamical networks. Wang et al. [10] studied the synchronization of the sampleddata systems with discrete-time communications. A piecewise Lyapunov function was proposed and the synchronization criterion was derived with LMIs. Fridman et al. [11] introduced the time-delay and sampled-data systems with linear dynamics. The basic analysis of sampled-data control and time-delay approach was obtained. Zarch et al. [12] considered the second-order dynamics in the non-periodic sampled-data systems. Xiao et al. [13] studied the consensus of multiple double integrators with arbitrary sampling in sampled-data systems and got a necessary and sufficient condition and a sufficient condition for the uniform and non-uniform sampleddata systems, respectively. Wen et al. [14] adopted the delay approach to dispose the consensus problems in the sampled-data systems with nonlinear dynamics. In the past years, consensus problems [15] [16] [17] [18] [19] in multi-agent systems have attracted a wide range of attention because their potential and practical applications in many aspects. One of the hot researches among the consensus problems is the consensus with one leader, called leader-following consensus. In the leader-following consensus systems, the leaders state is independent of all the other agents and agents of the followers are controlled to converge to the state of the only leader. The leader-following consensus of multi-agent systems with linear dynamics under fixed and switching topologies was considered in [20]. Using the Riccati inequality and Lyapunov approach, the consensus protocol with the undirected communications among the agents was obtained. While Liu et al. [21] investigated the consensus problems of multi-agent systems with directed topologies and the dynamics of the agents were nonlinear. Both the first order and second order systems under the fixed and switching topologies were considered. The above literatures can just apply to the cases with one leader. In the real applications, we may need to deal with the situations where there exist multiple leader agents. Recently, the leader-following consensus with multiple leaders, called containment control problems, thrives. Liu et al. [22] studied the containment problems of linear dynamics in the multi-agent systems and some basic lemmas were obtained. Xu et al. [23] focused on the containment for linear multi-agent systems with exogenous disturbances. Then the study was developed to nonlinear systems which was much more challenging. Wang et al. [24] extended the containment control September 13, 2016
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for nonlinear dynamics with uncertain and the robust control protocols were derived. In this paper, we mainly study the containment control problems in second-order systems. Unlike existing investigations, we develop the containment control to the sampled-data case for sake of the the movements of the agents and the limited capacities of the communication. Furthermore, a much more challenging nonlinear dynamics is considered. Using the LMIs and Lyapunov methods, we present some useful sufficient conditions. The rest of the paper is organized as follows. In Sec. II, the preliminaries and notations in graph theory are introduced. In Sec. III, the main theorem is presented and the proof is given. In Sec. IV, the numerical examples are given to validate the theoretical effectiveness. Finally, conclusions are drawn in Sec. V. II. P RELIMINARIES A. Notions In this part, some definitions and notions are introduced. Let G = {V, E, A} be a directed graph consisting of the set of agents V = {1, 2, . . . N}, the set of edges E ∈ V × V, and a weighted adjacency matrix A = [aij ], i, j = 1, 2, ..., n. If (j, i) ∈ E, then aij > 0, otherwise aij = 0 and the diagonal entries of A are zero, i.e., aii = 0. Ni denotes the set of the neighbors of agents i, where Ni = {j ∈ V : (j, i) ∈ E}. A directed path from agent i to agent j is a sequence of edges (i, s1 ), (s1 , s2 ), ..., (sk , j), where (i, s1 ), (s1 , s2 ), ..., (sk , j) ∈ E. A directed tree is a directed graph, where there exists a root connecting with any other agents by exactly one directed path. A directed forest is a directed graph consisting of one or more directed trees, but none of the nodes are in common. The Laplacian matrix of graph G is denoted as: L = [lij ] ∈ Rn×n , lii = nj=1 aij , and lij = −aij , for i = j Notation: N+ denotes the set of positive integers. · is the norm of the vector. B. Containment control of multi-agent systems with nonlinear dynamics Consider a multi-agent system of M + N agents, where there exist M ∈ N + leaders and N ∈ N+ followers. The dynamics of the followers are: ⎧ ⎨ x˙ (t) = v (t) i i ⎩ v˙ i (t) = f (xi (t), vi (t), t) + ui (t) September 13, 2016
(2.1)
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where i = 1, 2, ..., N, xi (t), vi (t) ∈ Rn are the position and the velocity of the agent i. u i (t) ∈ Rn is the control input. f (xi (t), vi (t), t) is the nonlinear dynamic of the agent i. The dynamics of the leaders are: ⎧ ⎨ x˙ (t) = v (t) j i ⎩ v˙ j (t) = f (xj (t), vj (t), t)
(2.2)
where j = N + 1, N + 2, ..., N + M. We use the following sampled-data control algorithm to achieve containment: ui (t) = −α aij (xi (tk ) − xj (tk )) − β aij (vi (tk ) − vj (tk )), t ∈ [tk , tk+1 ) j∈Ni
j∈Ni
(2.3)
where α > 0, β > 0 are the feedback gain parameters. aij is the element of the adjacency matrix associated with the communication topology G. N i is the set of the neighbors of agent i. It is assumed that the information transmission only occurs at discrete instants {t k |+∞ k=1 }, which satisfy 0 < t1 < t2 < · · · < tk−1 < tk < · · ·, called sampling instants. For briefness, we let n = 1. The case n > 1 can be analyzed with the property of the Kronecker. III. M AIN R ESULTS In this part, we consider the nonlinear dynamics of the multi-agent systems with sampleddata. We make three assumptions about the dynamics of the networks and the communication between the agents in preparation for introducing the main results. Assumption 1: The sampled-data instants t k satisfy the following condition: 0 ≤ t k+1 −tk ≤ η, where t = 1, 2... and η is a constant. Assumption 2: The communication topology among the agents in the multi-agent system (1) and (2) has a directed spanning forest. Assumption 3: There exists a constant ρ for any given constant numbers μ i ≥ 0, i = 1, ..., M, where M i=1 μi = 1, the nonlinear function satisfies the following condition:
2
2 M M M x− μi f (xi , vi , t) ≤ ρ1/2 μi xi + v − μi vi f (x, v, t) − i=1
i=1
i=1
where ∀x, v, xi , vi , i = 1, 2, ..., M. Remark 1: If the Assumption 2 is satisfied, ⎛ the Laplacian ⎞ matrix associated with the commuL11 L12 ⎠, where L11 ∈ RN ×N , L12 ∈ RN ×M . nication topology can be written as: L = ⎝ 0 0 September 13, 2016
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Using the sampled-data control algorithm (2.3) in system (2.1), the controlled system can be obtained: ⎧ ⎨ x˙ (t) = v (t) i i ⎩ v˙ i (t) = f (xi (t), vi (t), t) − α
j∈Ni
aij (xi (tk ) − xj (tk )) − β
j∈Ni
aij (vi (tk ) − vj (tk )) (3.4)
where i = 1, 2, ..., N. Let wi (t) = [xTi (t), viT (t)]T , T WF (t) = [w1T (t), w2T (t), ..., wN (t)]T , T T T T WL (t) = [wN +1 (t), wN +2 (t), ..., wN +M (t)] ,
FF (t) = [f (x1 , v1 , t), f (x2 , v2 , t), ..., f (xN , vN , t)] FL (t) = [f (xN +1 , vN +1 , t), f (xN +2 , vN +2 , t), ..., f (xN +M , vN +M , t)] The above equalities (3.4) ⎛ ⎡ 0 ˙ F (t) = ⎝IN ⊗ ⎣ W 0 ⎛
and (2.2) can be rewritten as: ⎤⎞ ⎡ ⎤ 1 0 ⎦⎠ WF (t) + FF (t) ⊗ ⎣ ⎦ 0 1 ⎡ ⎤⎞ ⎛ ⎡ 0 0 ⎦⎠ WF (tk ) − β ⎝L11 ⊗ ⎣ −α ⎝L11 ⊗ ⎣ 1 0 ⎛ ⎡ ⎤⎞ ⎛ ⎡ 0 0 ⎦⎠ WL (tk ) − β ⎝L12 ⊗ ⎣ −α ⎝L12 ⊗ ⎣ 1 0 ⎛ ⎡ ⎤⎞ ⎡ ⎤ 0 1 0 ˙ L (t) = ⎝IM ⊗ ⎣ ⎦⎠ WL (t) + FL (t) ⊗ ⎣ ⎦ . W 0 0 1
Let T (t) = WF (t) +
0 0 0 1 0 0 0 1
⎤⎞ ⎦⎠ WF (tk ) ⎤⎞
(3.5)
⎦⎠ WL (tk ) ,
L−1 11 L12 ⊗ I2 WL
According to the equalities (3.5), we can obtain the following error system: ⎛
⎡
T˙ (t) = ⎝IN ⊗ ⎣ ⎛
⎡
− ⎝L11 ⊗ ⎣
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0 1 0 0 0 0 α β
⎤⎞
⎡
⎦⎠ T (t) + FF (t) + L−1 ⎣ 11 L12 FL (t) ⊗ ⎤⎞
0 1
⎤ ⎦ (3.6)
⎦⎠ T (tk )
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Note that τ (t) = t − tk for t ∈ [tk , tk+1 ) and τ˙ (t) = 1 except at the communication instants tk , k = 1, 2, .... The system (3.7) can be converted to a time-varying delay system: ⎛
⎡
⎤⎞
⎡
⎤
0 ⎦⎠ T (t) + FF (t) + L−1 ⎣ ⎦ 11 L12 FL (t) ⊗ 0 0 1 ⎛ ⎡ ⎤⎞ (3.7) 0 0 ⎦⎠ T (t − τ (t)) , t ∈ [tk , tk+1) − ⎝L11 ⊗ ⎣ α β ⎛ ⎡ ⎤⎞ ⎛ ⎡ ⎤⎞ 0 1 0 0 ⎦⎠ , B = − ⎝L11 ⊗ ⎣ ⎦⎠ , For the simple representations, let A = ⎝IN ⊗ ⎣ 0 0 α β ⎡ ⎤ 0 ⎣ ⎦. The system (3.7) can be rewritten as: F (t) = FF (t) + L−1 11 L12 FL (t) ⊗ 1 T˙ (t) = ⎝IN ⊗ ⎣
0 1
T˙ (t) = AT (t) + BT (t − τ (t)) + F (t)
(3.8)
Because L11 is positive definite, there exists a invertible matrix U such that U −1 L11 U = diag(λ1 , λ2 , ..., λN ) = Λ, the above system can be rewritten as: ⎛ ⎡ ⎤⎞ ⎛ ⎡ ⎤⎞ 0 1 0 0 ⎦⎠ X (t) − ⎝Λ ⊗ ⎣ ⎦⎠ X (t − τ (t)) + F (t) X˙ (t) = ⎝IN ⊗ ⎣ 0 0 α β where (U −1 ⊗ I2 ) T (t) = X (t), (U −1 ⊗ I2 ) F (t) = F (t). Then we would like to present the main results in following theorem. Theorem 1: Assume the assumptions 1, 2 and 3 are satisfied. For given control gain parameters α, β and constant η, the multi-agent systems (2.1) using the control algorithm (2.3) can achieve containment control if there exist symmetry positive definite matrixes P1,P2,P3,P4, matrix Q1,Q2 with appropriate dimensions and the following two LMIs are satisfied: ⎡ ⎤ ηΞ2 + Ξ3 + Ξ4 ηQ1 ⎣ ⎦<0 ∗ −η (P3 + P4 ) ⎤ ⎡ η (Ξ1 + Ξ2 ) + Ξ3 + Ξ4 ηQ2 ⎦<0 ⎣ ∗ −ηP3
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(3.9)
(3.10)
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⎡ ⎢ ⎢ ⎢ where Ξ1 = ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤
AT P4 A AT P4 AT P4 B 0 P4
∗ ∗ P1
∗
−I
∗
∗
AT P3 A AT P3 AT P3 B 0
⎤
⎥ ⎥ ⎢ ⎥ ⎢ ∗ P 0 ⎥ P B 0 3 3 ⎥ ⎥ ⎢ ⎥, Ξ2 = ⎢ ⎥ , Ξ3 = T T ⎥ ⎢ ∗ B P4 B 0 ⎥ ∗ ∗ B P B 0 3 ⎦ ⎦ ⎣ ∗ ∗ 0 ∗ ∗ ∗ 0 ⎤ P1 B 0 ⎥ T 0 0 ⎥ ⎥ , Ξ = + ⎥ 4 Q1 0 Q1 − Q2 Q2 Q1 0 Q1 − Q2 Q2 0 0 ⎥ ⎦ ∗ −P2 following Lyapunov function:
∗
2P1 A + P2 + ρI
⎡
∗ ∗ Proof: Consider the
P4 B
V (t, T (t)) = V1 (t, T (t)) + V2 (t, T (t)) + V3 (t, T (t)) + V4 (t, T (t)) , V1 (t, T (t)) = T T (t) P1 T (t) , t V2 (t, T (t)) = T T (t) P2 T (t) ds, t−η 0 t
V3 (t, T (t)) =
−η
t+θ
T˙ T (t) P3 T˙ dsdθ,
V4 (t, T (t)) = (η − (t − tk ))
t tk
T˙ T (s) P4 T˙ (s) ds,
+ − = 0 and V t ≤ V tk . And we can prove (t) ≥ 0, we can get V It is obvious that V4 t+ 4 4 4 k k for V˙ (t) < 0 when t ∈ (tk , tk+1 ) in the following. For t ∈ (tk , tk+1 ), the derivative of the V (t, T (t)) along the system (3.8) yields: V˙ 1 (t, T ) = 2T T (t) P1 T˙ (t) , V˙ 2 (t, T ) = T T (t) P2 T (t) − T T (t − η) P2 T (t − η) , t T V˙ 3 (t, T ) = η T˙ (t) P3 T˙ (t) − T˙ T (t) P3 T˙ (t) ds ˙T
= η T (t) P3 T˙ (t) − V˙ 4 (t, T ) = −
t tk
t−η t
tk
˙T
T (s) P3 T˙ (s) ds −
tk t−η
T˙ T (s) P3 T˙ (s) ds,
T˙ T (s) P4 T˙ (s) ds + (η − (t − tk )) T˙ T (s) P4 T˙ (s) ,
T Define ω (t) = T T (t) , F T (t) , T T (tk ) , T T (t − η) . For any matrices Q1 , Q2 with appropriate dimensions,
t ˙ 2ω (t) Q1 T (t) − T (tk ) − T (s) ds = 0 T
tk
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2ω (t) Q2 T (tk ) − T (t − η) − T
tk
t−η
˙ T (s) ds = 0
we can obtain the following inequalities: t − T˙ T (s) (P3 + P4 ) T˙ (s) ds ≤ (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t)−2ω T (t) Q1 (T (t) − T (tk )) tk
−
tk
t−η
T˙ T (s) P3 T˙ (s) ds ≤ (η − (t − tk )) ω T (t) Q2 P3−1 QT2 ω (t)−2ω T (t) Q2 (T (tk ) − T (t − η))
According to the assumption 3, the following result can be obtained: F T (t) · F (t) ≤ ρT (t)2 = ρT T (t) T (t) From the above equalities and inequalities, it is obvious that: V˙ (t) = 2T T P1 T˙ + T T (t) P2 T (t) − T T (t − η) P2 T (t − η) t t T +η T˙ T P3 T˙ − T˙ (s) (P3 + P4 ) T˙ (s) ds − k T˙ T (s) P3 T˙ ds tk
+ (η − (t − tk )) T˙ T (s) P4 T˙ (s)
t−η
≤ 2T T (t) P1 (AT (t) + BT (tk ) + F (t)) + T T (t) P2 T (t) − T T (t − η) P2 T (t − η) +η(AT (t) + BT (tk ) + F (t))T P3 (AT (t) + BT (tk ) + F (t)) + (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t) − 2ω T (t) Q1 (T (t) − T (tk )) + (η − (t − tk )) ω T (t) Q2 P3−1 QT2 ω (t) − 2ω T (t) Q2 (T (tk ) − T (t − η)) + (η − (t − tk )) (AT (t) + BT (tk ) + F (t))T P4 (AT (t) + BT (tk ) + F (t)) +ρT T (t) T (t) − F T (t) · F (t)
= (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t) + (η − (t − tk )) ω T (t) Q2 P3−1QT2 + Ξ1 ω (t) +ω T (t) (ηΞ2 + Ξ3 + Ξ4 ) ω (t) ⎤ ⎤ ⎡ ⎡ AT P4 A AT P4 AT P4 B 0 AT P3 A AT P3 AT P3 B 0 ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ∗ P4 ∗ P3 P4 B 0 ⎥ P3 B 0 ⎥ ⎥ ⎥ ⎢ ⎢ where Ξ1 = ⎢ ⎥, Ξ2 = ⎢ ⎥ , Ξ3 = T T ⎥ ⎥ ⎢ ⎢ ∗ ∗ B ∗ ∗ B P B 0 P B 0 4 3 ⎦ ⎦ ⎣ ⎣ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ⎤ ⎡ 2P1 A + P2 + ρI P1 P1 B 0 ⎥ ⎢ ⎢ T ∗ −I 0 0 ⎥ ⎥ ⎢ ⎥, Ξ4 = Q1 0 Q1 − Q2 Q2 + Q1 0 Q1 − Q2 Q2 ⎢ ⎥ ⎢ ∗ ∗ 0 0 ⎦ ⎣ ∗ ∗ ∗ −P2 ˙ To make sure V (t) < 0, t ∈ (tk , tk+1 ), the following inequality should be satisfied: (t − tk ) Q1 (P3 + P4 )−1 QT1 + (η − (t − tk )) Q2 P3−1 QT2 + Ξ1 + (ηΞ2 + Ξ3 + Ξ4 ) < 0 September 13, 2016
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Using the convex combination method, the above inequality can be equivalently written: ηΞ2 + Ξ3 + Ξ4 + ηQ1 (P3 + P4 )−1 QT1 < 0, ηΞ2 + Ξ3 + Ξ4 + η Q2 P3−1 QT2 + Ξ1 < 0 ⎤ ⎡ ηQ1 ηΞ2 + Ξ3 + Ξ4 ⎦ < 0, ⎣ ∗ −η (P3 + P4 ) ⎤ ⎡ η (Ξ1 + Ξ2 ) + Ξ3 + Ξ4 ηQ2 ⎦<0 ⎣ ∗ −ηP3 Remark 2: For given systems and the feedback gain parameters α, β, we can use the conditions obtained in the Theorem 1 to find the maximum value of the sample period η by solving the maximization optimize problem: max {η},s.t. (3.9), (3.10).
IV. N UMERICAL S IMULATIONS In this part, some simulations are given to illustrate the effective results of the former analysis. Example 1: Consider a multi-agent system of 6 agents, (2 leaders and 4 followers) and the communication topology among the agents are shown in the Fig.1. The nonlinear dynamic function of the agents is f (x, v, t) = x + v sin(t). It is obvious that f (x, v, t) satisfies the condition in Assumption 3 and let ρ = 2. To achieve containment, the maximum of sampleddata interval η is 0.01. We let the feedback gain parameters α = 2, β = 2 and the initial values of the agents are random. The simulation results of the positions and velocities of the agents are shown in the Fig.2. It is easy to see that the final states of the followers converge into the convex hull of the states of the leaders. Example 2: Consider a multi-agent system of 6 agents, (2 leaders and 4 followers) and the communication topology among the agents are shown in the Fig.1. The nonlinear dynamic function of the agents is f (x, v, t) = x sin(t) + vcos(t). It is obvious that f (x, v, t) satisfies the condition in Assumption 3 and let ρ = 2. To achieve containment, the maximum of sampled-data interval η is 0.01. We let the feedback gain parameters α = 2, β = 3 and the initial values of the agents are random. The simulation results of the positions and velocities of the agents are shown in the Fig.3. It is easy to see that the final states of the followers converge into the convex hull of the states of the leaders. September 13, 2016
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Fig. 1: The fixed communication topology among the agents
70 follower1 follower2 follower3 follower4 leader5 leader6
60 50
x(t)
40 30 20 10 0 −10
0
0.5
1
1.5
2
2.5
2
2.5
time t
100 follower1 follower2 follower3 follower4 leader5 leader6
80
v(t)
60
40
20
0
−20
0
0.5
1
1.5 time t
Fig. 2: Trajectory of position and velocity of Example 1.
V. C ONCLUSIONS In this technique note, the sampled containment control scheme is proposed to investigate the leader-following consensus problem in second-order multi-agent systems. Considering the practical systems in real-world applications, the dynamics in our paper is considered in a nonlinear case, which is definitely more challenging. Compared to existing investigations, we develop the containment control to the sampled data case for sake of the the movements of September 13, 2016
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25 20 15 10
x(t)
5 0 follower1 follower2 follower3 follower4 leader5 leader6
−5 −10 −15 −20
0
1
2
3 time t
4
5
6
2
3 time t
4
5
6
20
10
0
v(t)
−10
−20 follower1 follower2 follower3 follower4 leader5 leader6
−30
−40
−50
0
1
Fig. 3: Trajectory of position and velocity of Example 2.
the agents and the limited capacities of the communication. By adopting Lyapunov and LMIs methods, some sufficient conditions are obtained for the results. Finally, the numerical examples are given to illustrate the results. R EFERENCES [1] Bo Shen, Zidong Wang, and Xiaohui Liu. Sampled-data synchronization control of dynamical networks with stochastic sampling. Automatic Control, IEEE Transactions on, 57(10):2644–2650, 2012. [2] Zidong Wang, Biao Huang, and Peijun Huo. Sampled-data filtering with error covariance assignment. Signal Processing, IEEE Transactions on, 49(3):666–670, 2001. [3] Bo Shen, Zidong Wang, and Xiaohui Liu. A stochastic sampled-data approach to distributed filtering in sensor networks. Circuits and Systems I: Regular Papers, IEEE Transactions on, 58(9):2237–2246, 2011. [4] Weiyuan Zhang, Junmin Li, Keyi Xing, and Chenyang Ding. Synchronization for distributed parameter nns with mixed delays via sampled-data control. Neurocomputing, 175:265–277, 2016. [5] R Rakkiyappan and N Sakthivel. Pinning sampled-data control for synchronization of complex networks with probabilistic time-varying delays using quadratic convex approach. Neurocomputing, 162:26–40, 2015. [6] Chao Ma and Qingshuang Zeng. Distributed formation control of 6-dof autonomous underwater vehicles networked by sampled-data information under directed topology. Neurocomputing, 154:33–40, 2015. [7] Bo Shen, Zidong Wang, Jinling Liang, and Xiaohui Liu. Sampled-data h filtering for stochastic genetic regulatory networks. International Journal of Robust and Nonlinear Control, 21(15):1759–1777, 2011. September 13, 2016
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[8] Zheng-Guang Wu, Peng Shi, Hongye Su, and Jian Chu. Sampled-data synchronization of chaotic lur’e systems with time delays. Neural Networks and Learning Systems, IEEE Transactions on, 24(3):410–421, 2013. [9] Zheng-Guang Wu, Ju H Park, Hongye Su, Bo Song, and Jian Chu. Exponential synchronization for complex dynamical networks with sampled-data. Journal of the Franklin Institute, 349(9):2735–2749, 2012. [10] Yan-Wu Wang, Jiang-Wen Xiao, Changyun Wen, and Zhi-Hong Guan. Synchronization of continuous dynamical networks with discrete-time communications. Neural Networks, IEEE Transactions on, 22(12):1979–1986, 2011. [11] Emilia Fridman, Alexandre Seuret, and Jean-Pierre Richard. Robust sampled-data stabilization of linear systems: an input delay approach. Automatica, 40(8):1441–1446, 2004. [12] Mehran Zareh, Dimos V Dimarogonas, Mauro Franceschelli, Karl H Johansson, and Carla Seatzu. Consensus in multiagent systems with second-order dynamics and non-periodic sampled-data exchange. In Emerging Technology and Factory Automation (ETFA), 2014 IEEE, pages 1–8. IEEE, 2014. [13] Feng Xiao and Tongwen Chen. Sampled-data consensus for multiple double integrators with arbitrary sampling. Automatic Control, IEEE Transactions on, 57(12):3230–3235, 2012. [14] Guanghui Wen, Zhisheng Duan, Wenwu Yu, and Guanrong Chen. Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach. International Journal of Robust and Nonlinear Control, 23(6):602–619, 2013. [15] Zidong Wang, Derui Ding, Hongli Dong, and Huisheng Shu. H consensus control for multi-agent systems with missing measurements: the finite-horizon case. Systems & Control Letters, 62(10):827–836, 2013. [16] Bo Shen, Zidong Wang, and Huisheng Shu. Nonlinear stochastic systems with incomplete information: filtering and control. Springer Science & Business Media, 2013. [17] Hongjun Chu, Yunze Cai, and Weidong Zhang. Consensus tracking for multi-agent systems with directed graph via distributed adaptive protocol. Neurocomputing, 166:8–13, 2015. [18] Xiaogang Yang, Jianxiang Xi, Jinying Wu, and Bailong Yang. Consensus transformation for multi-agent systems with topology variances and time-varying delays. Neurocomputing, 168:1059–1064, 2015. [19] Yuan Fan and Jianyu Yang. Average consensus of multi-agent systems with self-triggered controllers. Neurocomputing, 2015. [20] Wei Ni and Daizhan Cheng. Leader-following consensus of multi-agent systems under fixed and switching topologies. Systems & Control Letters, 59(3):209–217, 2010. [21] Kaien Liu, Guangming Xie, Wei Ren, and Long Wang. Consensus for multi-agent systems with inherent nonlinear dynamics under directed topologies. Systems & Control Letters, 62(2):152–162, 2013. [22] Huiyang Liu, Guangming Xie, and Long Wang. Containment of linear multi-agent systems under general interaction topologies. Systems & Control Letters, 61(4):528–534, 2012. [23] Chengjie Xu, Ying Zheng, Housheng Su, and Hong-bin Zeng. Containment for linear multi-agent systems with exogenous disturbances. Neurocomputing, 160:206–212, 2015. [24] Ping Wang and Yingmin Jia. Robust h containment control for uncertain multi-agent systems with inherent nonlinear dynamics. International Journal of Systems Science, pages 1–11, 2014.
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S0925-2312(16)31035-9 http://dx.doi.org/10.1016/j.neucom.2016.09.020 NEUCOM17549
To appear in: Neurocomputing Received date: 8 May 2016 Revised date: 1 August 2016 Accepted date: 12 September 2016 Cite this article as: Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu and Wenshan Hu, Sampled containment control for multi-agent systems with nonlinear dynamics, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.09.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Sampled containment control for multi-agent systems with nonlinear dynamics Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu*, Wenshan Hu**
Abstract In this paper, the sampled containment control scheme is proposed to study leader-following consensus problem in second-order systems. Compared to existing investigations, we develop the containment control to the sampled-data case for sake of the movements of the agents and the limited capacities of the communication. Furthermore, it is noted that the proposed algorithm is discussed for nonlinear dynamics which is much more challenging rather than linear ones. By using the LMIs and Lyapunov method, some useful sufficient conditions are presented. Finally the numerical examples are included to validate the effectiveness of proposed algorithms. Keywords: Containment control, sampled-data, consensus, nonlinear dynamics.
I. I NTRODUCTION Due to the wide application of the digital processor, the research of the sampled-data problems [1] [2] [3] [4] [5] [6] [7] [8] has attracted much attention. The digital processor can only process discrete data and obtain the discrete control signal which can be acquired as a continuous signal by a zero-order holder. In these sampled-data systems, the control signals are piecewise continuous. The effective method to analyze and deal with the sampled-data systems is the inputdelay method. That is, the systems are viewed as continuous systems with a time-varying delay This work was supported in part by the National Natural Science Foundation of China under Grants 61304152 and 6374034. Yan Wang, Hong Zhou, Zhenhua Wang, Zhiwei Liu, and Wenshan Hu are with the Department of Automation, Wuhan University, Wuhan, 430072, P. R. China. Zhenhua Wang is with China Ship Development and Design Center (CSDDC), Wuhan, 430064, P. R. China. *Corresponding author. Email: [email protected] **Corresponding author. Email: [email protected]
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and the differential of the delay is equal to one. There are many literatures using the input delay method to dispose the sampled-data systems. Wu et al. [9] used the Lyapuov approach and the convex combination technique to investigate the synchronization of the sampled-data systems in complex dynamical networks. Wang et al. [10] studied the synchronization of the sampleddata systems with discrete-time communications. A piecewise Lyapunov function was proposed and the synchronization criterion was derived with LMIs. Fridman et al. [11] introduced the time-delay and sampled-data systems with linear dynamics. The basic analysis of sampled-data control and time-delay approach was obtained. Zarch et al. [12] considered the second-order dynamics in the non-periodic sampled-data systems. Xiao et al. [13] studied the consensus of multiple double integrators with arbitrary sampling in sampled-data systems and got a necessary and sufficient condition and a sufficient condition for the uniform and non-uniform sampleddata systems, respectively. Wen et al. [14] adopted the delay approach to dispose the consensus problems in the sampled-data systems with nonlinear dynamics. In the past years, consensus problems [15] [16] [17] [18] [19] in multi-agent systems have attracted a wide range of attention because their potential and practical applications in many aspects. One of the hot researches among the consensus problems is the consensus with one leader, called leader-following consensus. In the leader-following consensus systems, the leaders state is independent of all the other agents and agents of the followers are controlled to converge to the state of the only leader. The leader-following consensus of multi-agent systems with linear dynamics under fixed and switching topologies was considered in [20]. Using the Riccati inequality and Lyapunov approach, the consensus protocol with the undirected communications among the agents was obtained. While Liu et al. [21] investigated the consensus problems of multi-agent systems with directed topologies and the dynamics of the agents were nonlinear. Both the first order and second order systems under the fixed and switching topologies were considered. The above literatures can just apply to the cases with one leader. In the real applications, we may need to deal with the situations where there exist multiple leader agents. Recently, the leader-following consensus with multiple leaders, called containment control problems, thrives. Liu et al. [22] studied the containment problems of linear dynamics in the multi-agent systems and some basic lemmas were obtained. Xu et al. [23] focused on the containment for linear multi-agent systems with exogenous disturbances. Then the study was developed to nonlinear systems which was much more challenging. Wang et al. [24] extended the containment control September 13, 2016
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for nonlinear dynamics with uncertain and the robust control protocols were derived. In this paper, we mainly study the containment control problems in second-order systems. Unlike existing investigations, we develop the containment control to the sampled-data case for sake of the the movements of the agents and the limited capacities of the communication. Furthermore, a much more challenging nonlinear dynamics is considered. Using the LMIs and Lyapunov methods, we present some useful sufficient conditions. The rest of the paper is organized as follows. In Sec. II, the preliminaries and notations in graph theory are introduced. In Sec. III, the main theorem is presented and the proof is given. In Sec. IV, the numerical examples are given to validate the theoretical effectiveness. Finally, conclusions are drawn in Sec. V. II. P RELIMINARIES A. Notions In this part, some definitions and notions are introduced. Let G = {V, E, A} be a directed graph consisting of the set of agents V = {1, 2, . . . N}, the set of edges E ∈ V × V, and a weighted adjacency matrix A = [aij ], i, j = 1, 2, ..., n. If (j, i) ∈ E, then aij > 0, otherwise aij = 0 and the diagonal entries of A are zero, i.e., aii = 0. Ni denotes the set of the neighbors of agents i, where Ni = {j ∈ V : (j, i) ∈ E}. A directed path from agent i to agent j is a sequence of edges (i, s1 ), (s1 , s2 ), ..., (sk , j), where (i, s1 ), (s1 , s2 ), ..., (sk , j) ∈ E. A directed tree is a directed graph, where there exists a root connecting with any other agents by exactly one directed path. A directed forest is a directed graph consisting of one or more directed trees, but none of the nodes are in common. The Laplacian matrix of graph G is denoted as: L = [lij ] ∈ Rn×n , lii = nj=1 aij , and lij = −aij , for i = j Notation: N+ denotes the set of positive integers. · is the norm of the vector. B. Containment control of multi-agent systems with nonlinear dynamics Consider a multi-agent system of M + N agents, where there exist M ∈ N + leaders and N ∈ N+ followers. The dynamics of the followers are: ⎧ ⎨ x˙ (t) = v (t) i i ⎩ v˙ i (t) = f (xi (t), vi (t), t) + ui (t) September 13, 2016
(2.1)
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where i = 1, 2, ..., N, xi (t), vi (t) ∈ Rn are the position and the velocity of the agent i. u i (t) ∈ Rn is the control input. f (xi (t), vi (t), t) is the nonlinear dynamic of the agent i. The dynamics of the leaders are: ⎧ ⎨ x˙ (t) = v (t) j i ⎩ v˙ j (t) = f (xj (t), vj (t), t)
(2.2)
where j = N + 1, N + 2, ..., N + M. We use the following sampled-data control algorithm to achieve containment: ui (t) = −α aij (xi (tk ) − xj (tk )) − β aij (vi (tk ) − vj (tk )), t ∈ [tk , tk+1 ) j∈Ni
j∈Ni
(2.3)
where α > 0, β > 0 are the feedback gain parameters. aij is the element of the adjacency matrix associated with the communication topology G. N i is the set of the neighbors of agent i. It is assumed that the information transmission only occurs at discrete instants {t k |+∞ k=1 }, which satisfy 0 < t1 < t2 < · · · < tk−1 < tk < · · ·, called sampling instants. For briefness, we let n = 1. The case n > 1 can be analyzed with the property of the Kronecker. III. M AIN R ESULTS In this part, we consider the nonlinear dynamics of the multi-agent systems with sampleddata. We make three assumptions about the dynamics of the networks and the communication between the agents in preparation for introducing the main results. Assumption 1: The sampled-data instants t k satisfy the following condition: 0 ≤ t k+1 −tk ≤ η, where t = 1, 2... and η is a constant. Assumption 2: The communication topology among the agents in the multi-agent system (1) and (2) has a directed spanning forest. Assumption 3: There exists a constant ρ for any given constant numbers μ i ≥ 0, i = 1, ..., M, where M i=1 μi = 1, the nonlinear function satisfies the following condition:
2
2 M M M x− μi f (xi , vi , t) ≤ ρ1/2 μi xi + v − μi vi f (x, v, t) − i=1
i=1
i=1
where ∀x, v, xi , vi , i = 1, 2, ..., M. Remark 1: If the Assumption 2 is satisfied, ⎛ the Laplacian ⎞ matrix associated with the commuL11 L12 ⎠, where L11 ∈ RN ×N , L12 ∈ RN ×M . nication topology can be written as: L = ⎝ 0 0 September 13, 2016
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Using the sampled-data control algorithm (2.3) in system (2.1), the controlled system can be obtained: ⎧ ⎨ x˙ (t) = v (t) i i ⎩ v˙ i (t) = f (xi (t), vi (t), t) − α
j∈Ni
aij (xi (tk ) − xj (tk )) − β
j∈Ni
aij (vi (tk ) − vj (tk )) (3.4)
where i = 1, 2, ..., N. Let wi (t) = [xTi (t), viT (t)]T , T WF (t) = [w1T (t), w2T (t), ..., wN (t)]T , T T T T WL (t) = [wN +1 (t), wN +2 (t), ..., wN +M (t)] ,
FF (t) = [f (x1 , v1 , t), f (x2 , v2 , t), ..., f (xN , vN , t)] FL (t) = [f (xN +1 , vN +1 , t), f (xN +2 , vN +2 , t), ..., f (xN +M , vN +M , t)] The above equalities (3.4) ⎛ ⎡ 0 ˙ F (t) = ⎝IN ⊗ ⎣ W 0 ⎛
and (2.2) can be rewritten as: ⎤⎞ ⎡ ⎤ 1 0 ⎦⎠ WF (t) + FF (t) ⊗ ⎣ ⎦ 0 1 ⎡ ⎤⎞ ⎛ ⎡ 0 0 ⎦⎠ WF (tk ) − β ⎝L11 ⊗ ⎣ −α ⎝L11 ⊗ ⎣ 1 0 ⎛ ⎡ ⎤⎞ ⎛ ⎡ 0 0 ⎦⎠ WL (tk ) − β ⎝L12 ⊗ ⎣ −α ⎝L12 ⊗ ⎣ 1 0 ⎛ ⎡ ⎤⎞ ⎡ ⎤ 0 1 0 ˙ L (t) = ⎝IM ⊗ ⎣ ⎦⎠ WL (t) + FL (t) ⊗ ⎣ ⎦ . W 0 0 1
Let T (t) = WF (t) +
0 0 0 1 0 0 0 1
⎤⎞ ⎦⎠ WF (tk ) ⎤⎞
(3.5)
⎦⎠ WL (tk ) ,
L−1 11 L12 ⊗ I2 WL
According to the equalities (3.5), we can obtain the following error system: ⎛
⎡
T˙ (t) = ⎝IN ⊗ ⎣ ⎛
⎡
− ⎝L11 ⊗ ⎣
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0 1 0 0 0 0 α β
⎤⎞
⎡
⎦⎠ T (t) + FF (t) + L−1 ⎣ 11 L12 FL (t) ⊗ ⎤⎞
0 1
⎤ ⎦ (3.6)
⎦⎠ T (tk )
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Note that τ (t) = t − tk for t ∈ [tk , tk+1 ) and τ˙ (t) = 1 except at the communication instants tk , k = 1, 2, .... The system (3.7) can be converted to a time-varying delay system: ⎛
⎡
⎤⎞
⎡
⎤
0 ⎦⎠ T (t) + FF (t) + L−1 ⎣ ⎦ 11 L12 FL (t) ⊗ 0 0 1 ⎛ ⎡ ⎤⎞ (3.7) 0 0 ⎦⎠ T (t − τ (t)) , t ∈ [tk , tk+1) − ⎝L11 ⊗ ⎣ α β ⎛ ⎡ ⎤⎞ ⎛ ⎡ ⎤⎞ 0 1 0 0 ⎦⎠ , B = − ⎝L11 ⊗ ⎣ ⎦⎠ , For the simple representations, let A = ⎝IN ⊗ ⎣ 0 0 α β ⎡ ⎤ 0 ⎣ ⎦. The system (3.7) can be rewritten as: F (t) = FF (t) + L−1 11 L12 FL (t) ⊗ 1 T˙ (t) = ⎝IN ⊗ ⎣
0 1
T˙ (t) = AT (t) + BT (t − τ (t)) + F (t)
(3.8)
Because L11 is positive definite, there exists a invertible matrix U such that U −1 L11 U = diag(λ1 , λ2 , ..., λN ) = Λ, the above system can be rewritten as: ⎛ ⎡ ⎤⎞ ⎛ ⎡ ⎤⎞ 0 1 0 0 ⎦⎠ X (t) − ⎝Λ ⊗ ⎣ ⎦⎠ X (t − τ (t)) + F (t) X˙ (t) = ⎝IN ⊗ ⎣ 0 0 α β where (U −1 ⊗ I2 ) T (t) = X (t), (U −1 ⊗ I2 ) F (t) = F (t). Then we would like to present the main results in following theorem. Theorem 1: Assume the assumptions 1, 2 and 3 are satisfied. For given control gain parameters α, β and constant η, the multi-agent systems (2.1) using the control algorithm (2.3) can achieve containment control if there exist symmetry positive definite matrixes P1,P2,P3,P4, matrix Q1,Q2 with appropriate dimensions and the following two LMIs are satisfied: ⎡ ⎤ ηΞ2 + Ξ3 + Ξ4 ηQ1 ⎣ ⎦<0 ∗ −η (P3 + P4 ) ⎤ ⎡ η (Ξ1 + Ξ2 ) + Ξ3 + Ξ4 ηQ2 ⎦<0 ⎣ ∗ −ηP3
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(3.9)
(3.10)
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⎡ ⎢ ⎢ ⎢ where Ξ1 = ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤
AT P4 A AT P4 AT P4 B 0 P4
∗ ∗ P1
∗
−I
∗
∗
AT P3 A AT P3 AT P3 B 0
⎤
⎥ ⎥ ⎢ ⎥ ⎢ ∗ P 0 ⎥ P B 0 3 3 ⎥ ⎥ ⎢ ⎥, Ξ2 = ⎢ ⎥ , Ξ3 = T T ⎥ ⎢ ∗ B P4 B 0 ⎥ ∗ ∗ B P B 0 3 ⎦ ⎦ ⎣ ∗ ∗ 0 ∗ ∗ ∗ 0 ⎤ P1 B 0 ⎥ T 0 0 ⎥ ⎥ , Ξ = + ⎥ 4 Q1 0 Q1 − Q2 Q2 Q1 0 Q1 − Q2 Q2 0 0 ⎥ ⎦ ∗ −P2 following Lyapunov function:
∗
2P1 A + P2 + ρI
⎡
∗ ∗ Proof: Consider the
P4 B
V (t, T (t)) = V1 (t, T (t)) + V2 (t, T (t)) + V3 (t, T (t)) + V4 (t, T (t)) , V1 (t, T (t)) = T T (t) P1 T (t) , t V2 (t, T (t)) = T T (t) P2 T (t) ds, t−η 0 t
V3 (t, T (t)) =
−η
t+θ
T˙ T (t) P3 T˙ dsdθ,
V4 (t, T (t)) = (η − (t − tk ))
t tk
T˙ T (s) P4 T˙ (s) ds,
+ − = 0 and V t ≤ V tk . And we can prove (t) ≥ 0, we can get V It is obvious that V4 t+ 4 4 4 k k for V˙ (t) < 0 when t ∈ (tk , tk+1 ) in the following. For t ∈ (tk , tk+1 ), the derivative of the V (t, T (t)) along the system (3.8) yields: V˙ 1 (t, T ) = 2T T (t) P1 T˙ (t) , V˙ 2 (t, T ) = T T (t) P2 T (t) − T T (t − η) P2 T (t − η) , t T V˙ 3 (t, T ) = η T˙ (t) P3 T˙ (t) − T˙ T (t) P3 T˙ (t) ds ˙T
= η T (t) P3 T˙ (t) − V˙ 4 (t, T ) = −
t tk
t−η t
tk
˙T
T (s) P3 T˙ (s) ds −
tk t−η
T˙ T (s) P3 T˙ (s) ds,
T˙ T (s) P4 T˙ (s) ds + (η − (t − tk )) T˙ T (s) P4 T˙ (s) ,
T Define ω (t) = T T (t) , F T (t) , T T (tk ) , T T (t − η) . For any matrices Q1 , Q2 with appropriate dimensions,
t ˙ 2ω (t) Q1 T (t) − T (tk ) − T (s) ds = 0 T
tk
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2ω (t) Q2 T (tk ) − T (t − η) − T
tk
t−η
˙ T (s) ds = 0
we can obtain the following inequalities: t − T˙ T (s) (P3 + P4 ) T˙ (s) ds ≤ (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t)−2ω T (t) Q1 (T (t) − T (tk )) tk
−
tk
t−η
T˙ T (s) P3 T˙ (s) ds ≤ (η − (t − tk )) ω T (t) Q2 P3−1 QT2 ω (t)−2ω T (t) Q2 (T (tk ) − T (t − η))
According to the assumption 3, the following result can be obtained: F T (t) · F (t) ≤ ρT (t)2 = ρT T (t) T (t) From the above equalities and inequalities, it is obvious that: V˙ (t) = 2T T P1 T˙ + T T (t) P2 T (t) − T T (t − η) P2 T (t − η) t t T +η T˙ T P3 T˙ − T˙ (s) (P3 + P4 ) T˙ (s) ds − k T˙ T (s) P3 T˙ ds tk
+ (η − (t − tk )) T˙ T (s) P4 T˙ (s)
t−η
≤ 2T T (t) P1 (AT (t) + BT (tk ) + F (t)) + T T (t) P2 T (t) − T T (t − η) P2 T (t − η) +η(AT (t) + BT (tk ) + F (t))T P3 (AT (t) + BT (tk ) + F (t)) + (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t) − 2ω T (t) Q1 (T (t) − T (tk )) + (η − (t − tk )) ω T (t) Q2 P3−1 QT2 ω (t) − 2ω T (t) Q2 (T (tk ) − T (t − η)) + (η − (t − tk )) (AT (t) + BT (tk ) + F (t))T P4 (AT (t) + BT (tk ) + F (t)) +ρT T (t) T (t) − F T (t) · F (t)
= (t − tk ) ω T (t) Q1 (P3 + P4 )−1 QT1 ω (t) + (η − (t − tk )) ω T (t) Q2 P3−1QT2 + Ξ1 ω (t) +ω T (t) (ηΞ2 + Ξ3 + Ξ4 ) ω (t) ⎤ ⎤ ⎡ ⎡ AT P4 A AT P4 AT P4 B 0 AT P3 A AT P3 AT P3 B 0 ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ∗ P4 ∗ P3 P4 B 0 ⎥ P3 B 0 ⎥ ⎥ ⎥ ⎢ ⎢ where Ξ1 = ⎢ ⎥, Ξ2 = ⎢ ⎥ , Ξ3 = T T ⎥ ⎥ ⎢ ⎢ ∗ ∗ B ∗ ∗ B P B 0 P B 0 4 3 ⎦ ⎦ ⎣ ⎣ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ⎤ ⎡ 2P1 A + P2 + ρI P1 P1 B 0 ⎥ ⎢ ⎢ T ∗ −I 0 0 ⎥ ⎥ ⎢ ⎥, Ξ4 = Q1 0 Q1 − Q2 Q2 + Q1 0 Q1 − Q2 Q2 ⎢ ⎥ ⎢ ∗ ∗ 0 0 ⎦ ⎣ ∗ ∗ ∗ −P2 ˙ To make sure V (t) < 0, t ∈ (tk , tk+1 ), the following inequality should be satisfied: (t − tk ) Q1 (P3 + P4 )−1 QT1 + (η − (t − tk )) Q2 P3−1 QT2 + Ξ1 + (ηΞ2 + Ξ3 + Ξ4 ) < 0 September 13, 2016
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Using the convex combination method, the above inequality can be equivalently written: ηΞ2 + Ξ3 + Ξ4 + ηQ1 (P3 + P4 )−1 QT1 < 0, ηΞ2 + Ξ3 + Ξ4 + η Q2 P3−1 QT2 + Ξ1 < 0 ⎤ ⎡ ηQ1 ηΞ2 + Ξ3 + Ξ4 ⎦ < 0, ⎣ ∗ −η (P3 + P4 ) ⎤ ⎡ η (Ξ1 + Ξ2 ) + Ξ3 + Ξ4 ηQ2 ⎦<0 ⎣ ∗ −ηP3 Remark 2: For given systems and the feedback gain parameters α, β, we can use the conditions obtained in the Theorem 1 to find the maximum value of the sample period η by solving the maximization optimize problem: max {η},s.t. (3.9), (3.10).
IV. N UMERICAL S IMULATIONS In this part, some simulations are given to illustrate the effective results of the former analysis. Example 1: Consider a multi-agent system of 6 agents, (2 leaders and 4 followers) and the communication topology among the agents are shown in the Fig.1. The nonlinear dynamic function of the agents is f (x, v, t) = x + v sin(t). It is obvious that f (x, v, t) satisfies the condition in Assumption 3 and let ρ = 2. To achieve containment, the maximum of sampleddata interval η is 0.01. We let the feedback gain parameters α = 2, β = 2 and the initial values of the agents are random. The simulation results of the positions and velocities of the agents are shown in the Fig.2. It is easy to see that the final states of the followers converge into the convex hull of the states of the leaders. Example 2: Consider a multi-agent system of 6 agents, (2 leaders and 4 followers) and the communication topology among the agents are shown in the Fig.1. The nonlinear dynamic function of the agents is f (x, v, t) = x sin(t) + vcos(t). It is obvious that f (x, v, t) satisfies the condition in Assumption 3 and let ρ = 2. To achieve containment, the maximum of sampled-data interval η is 0.01. We let the feedback gain parameters α = 2, β = 3 and the initial values of the agents are random. The simulation results of the positions and velocities of the agents are shown in the Fig.3. It is easy to see that the final states of the followers converge into the convex hull of the states of the leaders. September 13, 2016
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Fig. 1: The fixed communication topology among the agents
70 follower1 follower2 follower3 follower4 leader5 leader6
60 50
x(t)
40 30 20 10 0 −10
0
0.5
1
1.5
2
2.5
2
2.5
time t
100 follower1 follower2 follower3 follower4 leader5 leader6
80
v(t)
60
40
20
0
−20
0
0.5
1
1.5 time t
Fig. 2: Trajectory of position and velocity of Example 1.
V. C ONCLUSIONS In this technique note, the sampled containment control scheme is proposed to investigate the leader-following consensus problem in second-order multi-agent systems. Considering the practical systems in real-world applications, the dynamics in our paper is considered in a nonlinear case, which is definitely more challenging. Compared to existing investigations, we develop the containment control to the sampled data case for sake of the the movements of September 13, 2016
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25 20 15 10
x(t)
5 0 follower1 follower2 follower3 follower4 leader5 leader6
−5 −10 −15 −20
0
1
2
3 time t
4
5
6
2
3 time t
4
5
6
20
10
0
v(t)
−10
−20 follower1 follower2 follower3 follower4 leader5 leader6
−30
−40
−50
0
1
Fig. 3: Trajectory of position and velocity of Example 2.
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