Synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control

Communicated by Prof. Liu Xiwei

Accepted Manuscript

Synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control Mihua Ma, Jianping Cai PII: DOI: Reference:

S0925-2312(18)30779-3 10.1016/j.neucom.2018.06.043 NEUCOM 19721

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Neurocomputing

Received date: Revised date: Accepted date:

11 October 2017 6 April 2018 3 June 2018

Please cite this article as: Mihua Ma, Jianping Cai, Synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.06.043

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Synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control Mihua Ma, Jianping Cai

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School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, PR China Email: [email protected]

Abstract

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Synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control was developed in this paper. By applying linear feedback injections to a fraction of agents at discontinuous time, all the agents described by Lagrangian systems can be regulated to follow a synchronization state. Based on aperiodically intermittent pinning control, some simple yet general synchronization criteria are derived. Compared with some existing works on control problem of Lagrangian networks, the distinctive advantages of the proposed controllers here include: (i) discontinuous-time control input; (ii) only a fraction of agents to be controlled; (iii) independence on the knowledge of system models. As a direct application, the results are illustrated by a Lagrangian network composing of six two-link revolute manipulators. Subsequently, numerical simulations with different kinds of pinning schemes demonstrate the effectiveness of the proposed control strategy.

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Keywords: Lagrangian networks, Aperiodically intermittent control, Pinning control, Synchronization criteria, Revolute manipulators 1. Introduction

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The design of control strategy to synchronize a group of interconnected dynamical systems has become a rather significant topic [1-3]. In recent years, there have been increasing interest in the study of synchronization and control of networked mechanical systems described by Lagrangian dynamics, which are called as Lagrangian networks [4-21]. This is mainly because the classical Lagrangian equations can well formulate a lot of mechanical systems, including robotic manipulators, autonomous vehicles, flying spacecrafts, walking robots and so on. What’s more, Lagrangian networks possess extensive applications in many engineering fields, especially involving complex and integrated production processes, such as coordination of multiple manipulators, formation of mobile sensor networks and flying spacecrafts. This has resulted in a large number of works on this topic. For example, some nonlinear control strategies are introduced to study the synchronization of multiple Lagrangian systems with application to tethered formation flight spacecraft [4]. Contraction analysis method is used to study the synchronization problem of Lagrangian networks [5]. PD-type control law with robustness property is introduced to address Preprint submitted to Neurocomputing

the coordinating problem of Lagrangian systems [6-7]. Distributed adaptive controlled synchronization scheme is designed for Lagrangian networks [8-13]. Passivity based synchronization for networked robots described by Lagrangian systems is investigated [14-16]. In recent years, pinning control strategy is developed to discuss the synchronization problem of Lagrangian networks [17-18], while impulsive control strategy is given to deal with the same problem [19-20]. Lately, intermittent control is introduced to study the synchronization of master-slave Lagrangian systems [21]. There are many interesting results on synchronization and control problems of Lagrangian systems cited above. However, it seems that few works considered synchronization of Lagrangian networks via aperiodically intermittent pinning control. In fact, aperiodically intermittent pinning control has been widely used to study control problem of dynamical networks in engineering fields [22-26]. The distinctive advantages of the aperiodically intermittent pinning control consisting of aperiodically intermittent control and pinning control are twofold. On one hand, aperiodically intermittent control as a discontinuous control is more effective and lower cost than the continuous control. On the June 27, 2018

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other hand, pinning control is a powerful technique because it can be easily realized by applying local feedback injections to a small fraction of nodes on a largesize network. Generally speaking, directly controlling every node in a dynamical network with a large number of agents might be impossible but might also not be necessary. Although pinning control strategy is developed to discuss the synchronization problem of Lagrangian networks [17-18], it is desirable to synchronize the Lagrangian networks via aperiodically intermittent control instead of continuous-time control. It seems that most of the existing works were concentrated on the continuous control [5-18]. Despite the synchronization of Lagrangian systems discussed by intermittent control [21], only two coupled master-slave Lagrangian systems were considered. Besides, the intermittent control strategy studied in Ref. [21] is periodic. Obviously, it is more better to propose synchronization via aperiodically intermittent control since it is more widespread and interesting in applications. To the best of our knowledge, so far little attention has been paid to synchronization problem of Lagrangian networks via aperiodically intermittent pinning control. Therefore, as an interesting and challenging topic, this motivates our present research. Motivated by the aforementioned comments, the main objective of this paper is to investigate synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control. Based on the aperiodically intermittent pinning control, some simple yet generic criteria are presented to make all the states of Lagrangian networks synchronize to a desired synchronization state. Subsequently, the results are illustrated by a Lagrangian network composing of six two-link revolute manipulators in detail, and simulation results are finally given to show the effectiveness of the proposed theoretical results. The control strategy designed here further extend the ideas and techniques presented in previous corresponding results. First, different from previous works on Lagrangian networks [4-16], we apply linear feedback control to a fraction of agents, which means that the proposed control strategy does not require the prerequisite knowledge of system models, such as the inertial matrix, the Coriolis and centrifugal torques, and the gravitational torque. Second, compared with Refs.[4-18], the controller proposed here is not continuous-time control input. Besides, in contrast to the synchronization of master-slave Lagrangian systems via intermittent control [21] , the intermittent control developed here is aperiodic, and it is used to investigate the synchronization of networked Lagrangian systems.

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The rest of this paper is organized as follows. In Sec. 2, some preliminaries are presented. In Sec. 3, model description and aperiodically intermittent pinning control problem formulation are depicted. In Sec. 4, some simple yet general synchronization criteria are derived. In Sec. 5, examples and simulations are given to illustrate the effectiveness of the proposed control strategy. Finally, Sec. 6 presents a brief conclusion to this paper. 2. Preliminaries

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2.1. Notations Some mathematical notations will be used throughout this paper. Rn×n represents a set of all n × n real matrices. Let In ∈ Rn×n be the n-order identity matrix, 0n×n ∈ Rn×n be the n-order matrix with all zeros, diag(γ1 , γ2 , · · · , γn ) ∈ Rn×n be the diagonal matrix with diagonal entries γi (i = 1, 2, · · · , n), Rn be a set of n × 1 real vectors, 0n ∈ Rn be the vector with all zeros. For x ∈ Rn , xT stands for its transpose, andqthe norm of Pn 2 x = (x1 , x2 , ..., xn )T is defined as k x k= i=1 xi . For A ∈ Rn×n , AT stands for its transpose, and eigenvalues def of A are defined as λmin (A) = λ1 (A) ≤ λ2 (A) ≤ · · · ≤ def n×n λn (A) = λmax is defined p (A). The norm of A ∈ R as k A k= λmax (AT A), and the symmetric part of A is denoted by AS = (A + AT )/2. Write matrix A > 0 (A < 0) if A is real symmetric and positive (negative) definite. The symbol ⊗ denotes the Kronecker product of two matrices.

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2.2. Graph theory Information exchange among agents in a network can be conveniently represented in a graph. Let G = (V, E, G) denote a directed graph of order N (N ≥ 2) with a set of nodes V = {1, 2, · · · , N}, a set of edges E ⊆ V × V, and a weighted adjacency matrix G = [gi j ] ∈ RN×N . A directed edge of G denoted by εi j = (i, j) means that node i has access to node j, i.e., node i can receive information from node j. gi j is defined by the rule that gi j > 0 if and only if εi j ∈ E. Otherwise gi j = 0. We usually assume that gii = 0 for all i ∈ V. The Laplacian matrix LG = [li j ] ∈ RN×N with P respect to G is defined as lii = Nj=1, j,i gi j and li j = −gi j P where i , j [12-13]. Let degin (i) = Nj=1, j,i gi j and PN degout (i) = j=1, j,i g ji be the in-degree and out-degree of node i, respectively. degout (i) − degin (i) represents the degree-difference for the ith node [17, 27]. A directed path from node j to node i is a sequence of edges ( j, i1 ), (i1 , i2 ), ..., (il , i) in the directed graph G with distinct nodes. A directed graph has a directed spanning 2

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(i) MIn ≤ M(q) ≤ MIn , k C(q, q) ˙ k≤ kC1 k q˙ k and k g(q) k≤ G; (ii) k M(x)z − M(y)z k≤ k M k x − y kk z k and k g(x) − g(y) k≤ kg k x − y k; (iii)k C(x, z)w − C(y, v)w k≤ kC1 k z − v kk w k +kC2 k z kk x − y kk w k.

tree if there exists at least one node called root which has a directed path to all the other nodes[27]. 2.3. Instrumental lemmas Lemma 1 [28]. If X and Y are real matrices with appropriate dimensions, then there exists ε > 0 such that

3. Problem statement

These properties are exhibited for a wide class of robotic manipulator with revolute joints. One way to compute the corresponding constants k M , kC1 , kC2 and kg can be seen from the literature [29]. The analysis in this paper is focused on robotic manipulators with revolute joints, which are described by Lagrangian dynamics (2) with k q(t) ˙ k bounded. In fact, intermittent control is hard to be realized if the velocity q˙ of Lagrangian system is unbounded in the rest time.

3.1. Lagrangian dynamics Following [4], the n degree-of-freedom robotic manipulator is expressed by the following Lagrangian system

3.2. Lagrangian network On the basis of Lagrangian structure of system (2), we consider a Lagrangian network consisting of N robotic agents as

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1 X T Y + Y T X ≤ εX T X + Y T Y. ε

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Lemma 2 [27]. If B and D are N × N Hermitian matrices, then one has λi (B) + λ1 (D) ≤ λi (B + D) ≤ λi (B) + λN (D), i = 1, 2, ..., N.

M(q)q¨ + C(q, q) ˙ q˙ + g(q) = τ,

(1)

n

M(qi )q¨ i + C(qi , q˙ i )q˙ i + g(qi ) + λq˙ i XN gi j [c1 (t)(q j − qi ) + c2 (t)(q˙ j − q˙ i )] + ui (t), (3) =

where q ∈ R is the vector of generalized coordinate, q˙ ∈ Rn is the vector of generalized velocity, M(q) : Rn → Rn×n is a symmetric positive definite inertia matrix, C(q, q) ˙ q˙ ∈ Rn is the vector function containing coriolis and centrifugal forces, g(q) is the vector of gravitational force, and τ ∈ Rn is the vector of generalized forces acting on the system. If we take τ = −λq˙ with λ > 0, then the Lagrangian system (1) is rewritten as

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j=1

(2)

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M(q)q¨ + C(q, q) ˙ q˙ + g(q) + λq˙ = 0.

For the above Lagrangian system (2), we have the following Lemma according to the literature [21].

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Lemma 3 [21]. If λ > 0 is taken, then k q(t) ˙ k in system (2) will be bounded for all t ≥ 0 and any initial value. Namely, for given λ > 0 and any initial value of system (2), there always exists constant H > 0 such that k q(t) ˙ k≤ H for all t ≥ 0.

=

M(qi )q¨ i + C(qi , q˙ i )q˙ i + g(qi ) + λq˙ i XN li j [c1 (t)q j + c2 (t)q˙ j ] + ui (t), − j=1

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where LG = [li j ] ∈ RN×N is the corresponding Laplacian matrix of the digraph G = (V, E, G) defined in subsection 2.2. Generally speaking, the Laplacian matrix LG of a diagraph is asymmetric. We will design the coupling strengths c1 (t) and c2 (t), and the controllers ui (t) (i = 1, 2, ..., N) to make the Lagrangian network (4) synchronize to a desired trajectory q0 (t). This means that limt→+∞ k qi (t) − q0 (t) k= 0 and limt→+∞ k q˙ i (t) − q˙ 0 (t) k= limt→+∞ k vi (t) − v0 (t) k= 0 for given any initial value of the Lagrangian network (4). We assume that the node system (2) has a unique solution with respect to a given initial condition. Then

In addition, the Lagrangian systems used to describe the dynamic behavior of robotic manipulators with revolute joints exhibit certain fundamental properties due to their Lagrangian dynamic structure [11-13, 29].

˙ Property 1. The matrix M(q) − 2C(q, q) ˙ is skew˙ symmetric, implying that xT [ M(q) − 2C(q, q)]x ˙ = 0 for arbitrary x ∈ Rn . Property 2. There exist positive constants M, M, G, k M , kC1 , kC2 and kg such that for all q, x, y, z, w, v ∈ Rn ,

where qi = (qi1 , qi2 , ..., qin )T ∈ Rn and q˙ i = vi = (vi1 , vi2 , ..., vin )T ∈ Rn (i = 1, 2, ..., N) are the vectors of generalized coordinates and velocities, respectively. gi j characterizes the interaction of the generalized coordinates and velocities between robot i and robot j, and it is the entry of the adjacency matrix G = [gi j ] ∈ RN×N associated with the digraph G = (V, E, G). c1 (t) > 0 and c2 (t) > 0 are the coupling strengths with respect to the generalized coordinates and velocities, respectively. ui (t) is the control input acting on robot i. The Lagrangian network (3) can be rewritten as

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the desired trajectory q0 (t) will be taken as a solution of the node system (2) satisfying   M(q0 )q¨ 0 + C(q0 , q˙ 0 )q˙ 0 + g(q0 ) + λq˙ 0 = 0,    T q (5)  0 (0) = (q01 , q02 , ..., q0n ) ,    v (0) = (v , v , ..., v )T . 0

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02

pinned. The aperiodically intermittent pinning control ui (t) is designed as   −ki [γq˜ i (t) + v˜ i (t)] − γλq˜ i (t), 1 ≤ i ≤ l,      t2m ≤ t ≤ t2m+1 ,    (8) ui (t) =    0, l + 1 ≤ i ≤ N, t2m ≤ t ≤ t2m+1 ,       0, 1 ≤ i ≤ N, t
0n

According to Lemma 3 and the inequalities in item (i) of property 2, k q˙ 0 (t) k and k q¨ 0 (t) k in system (5) are bounded. This implies that there exist constants ζ > 0 and η > 0 such that k q˙ 0 (t) k≤ ζ and k q¨ 0 (t) k≤ η. System (5) is usually viewed as the master system. Of course, system (5) also can be referred to as the dynamics of a virtual leader, and N robotic agents can be seen as N followers. Taking the state errors as q˜ i (t) = qi (t) − q0 (t) and v˜ i (t) = q˙ i (t)− q˙ 0 (t) = vi (t)−v0 (t), i = 1, 2, ..., N, the error dynamical system can be derived from the Lagrangian network (4) and the mast system (5) as   q˙˜ i = v˜ i ,       M(qi )v˙˜ i = −C(qi , q˙ i )˜vi − λ˜vi − ∆i (6)      P   − N li j [c1 (t)q˜ j + c2 (t)˜v j ] + ui (t),

2m+1

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where m = 0, 1, 2, ..., γ > 0 and ki > 0 (i = 1, 2, ..., l) are constants of control gains to be designed. The time span [t2m , t2m+1 ] is called the control time or work time, while (t2m+1 , t2m+2 ) is called the rest time. Letting α = min{t2m+1 − t2m , m = 0, 1, 2, ...} and β = max{t2m+2 − t2m+1 , m = 0, 1, 2, ...}, the values of α and β should be taken to satisfy α > 0 and 0 ≤ β < +∞, which will be designed later. In addition, the coupling strengths c1 (t) and c2 (t) are designed as

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j=1

c1 (t) = γc2 (t)

and

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k [M(qi ) − M(q0 )]q¨ 0 + [C(qi , q˙ i ) − C(q0 ,

+kg k q˜ i k

H1 k q˜ i k +H2 k v˜ i k,

    c2 , t2m ≤ t ≤ t2m+1 , c2 (t) =    0, t2m+1 < t < t2m+2 ,

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(10)

where c2 is a constant to be determined, and m = 0, 1, 2, .... It is worth noting that c1 (t) > 0 and c2 (t) > 0 if and only if t ∈ [t2m , t2m+1 ]. Otherwise c1 (t) = 0 and c2 (t) = 0. Remark 1. Due to the inherent strong nonlinearity of Lagrangian system, the controllers designed in many results on synchronization and control problems of Lagrangian networks were continuous-time control input or relied on the knowledge of system models [418]. The designed controllers (8) are only activated in each work time [t2m , t2m+1 ] and is off in the rest time (t2m+1 , t2m+2 ). In addition, the controllers show that the linear feedback injections are only added on partial agents of the network, which means that the presented controllers do not require the prerequisite knowledge of system models. Thus, the designed control strategy here is essentially different from the control input mentioned in Refs.[4-21].

q˙ 0 )]q˙ 0 + g(qi ) − g(q0 ) k k M η k q˜ i k +kC1 ζ k q˙˜ i k +kC2 ζ 2 k q˜ i k

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≤

M

where ∆i = [M(qi )−M(q0 )]q¨ 0 +[C(qi , q˙ i )−C(q0 , q˙ 0 )]q˙ 0 + g(qi ) − g(q0 ). By using the inequalities in items (ii) and (iii) of property 2, together with the bounds k q˙ 0 (t) k and k q¨ 0 (t) k, we can obtain that k ∆i k =

2m+2

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where H1 = k M η + kC2 ζ 2 + kg and H2 = kC1 ζ. The problem of synchronization in the Lagrangian network (4) is corresponding to stability analysis of error system around the origin. We will design the aperiodically intermittent pinning control to make that limt→+∞ k q˜ i (t) k= 0 and limt→+∞ k v˜ i (t) k= 0 for given any initial value of the Lagrangian network (4).

Remark 2. Let T > 0 and 0 < σ < 1. If t2m+1 − t2m = σT and t2m+2 − t2m+1 = (1 − σ)T , m = 0, 1, 2, ..., then the periodically intermittent pinning control scheme with the control period T and the rate of control duration σ is recovered.

3.3. Aperiodically intermittent pinning control scheme Now we shall address the aperiodically intermittent pinning control scheme for the Lagrangian system (4). In order to realize synchronization between the Lagrangian network (4) and the master system (5), some controllers are needed to add on partial agents of the network. Without loss of generality, we can rearrange the order of all agents and select the first l nodes to be

In what follows, we rewrite the error dynamical system (6). Due to the inherent strong nonlinearity of Lagrangian system, the study of pinning synchronization 4

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to a desired trajectory q0 (t) described by the master system (5). Based on the error dynamical systems (12) and (13), the values of α, β, the coupling strength c2 , the control gains γ and ki (i = 1, 2, ..., l) will be designed. We define two positive numbers as follows H3

H4

=

=

2 ε1 (γ2 kC1 H 2 + 1 + γ2 H22 + H1 ) + ε2 γ4 M 2

2

+γM + H2 − λ,

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H1 + 3 1 + , 2ε1 2ε2

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in networked Lagrangian systems by utilizing aperiodically intermittent is very challenging. Thus, we introduce reference trajectory si (t) = γq˜ i (t) + v˜ i (t) (i = 1, 2, ..., N) to deal with the strong nonlinearity of Lagrangian system. In the following, we will consider two cases according to t in different time intervals. In the case with t ∈ [t2m , t2m+1 ], substituting v˜ i = si − γq˜ i , ui (t), c1 (t) and c2 (t) into the error dynamical system (6), the error dynamical system can be rewritten as   q˙˜ i = si − γq˜ i , 1 ≤ i ≤ N,         M(qi ) s˙i = [γM(qi ) − C(qi , q˙ i ) − (ki + λ)In ]si         +[γC(qi , q˙ i ) − γ2 M(qi )]q˜ i − ∆i      P  −c2 Nj=1 li j s j , 1 ≤ i ≤ l, (11)         M(qi ) s˙i = [γM(qi ) − C(qi , q˙ i ) − λIn ]si         +[γC(qi , q˙ i ) − γ2 M(qi ) + λγIn ]q˜ i      P   −∆i − c2 Nj=1 li j s j , l + 1 ≤ i ≤ N.

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where ε1 > 0 and ε2 > 0. It can be seen that H3 is monotonic increasing function for ε1 and ε2 , and H4 is monotonic decreasing one. So ε1 and ε2 can be chosen properly to make the values of H3 and H4 be suitable and optimal in specific applications. Now the synchronization criteria are derived in the following. Theorem 1. The Lagrangian network (4) can synchronize to the master system (5) by using the aperiodically intermittent pinning controllers ui (t) (i = 1, 2, ..., l) if, given for ξ1 > 0, the values of α, β, the coupling strength c2 , the control gains γ and ki (i = 1, 2, ..., l) are designed as

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Let q˜ = (q˜ T1 , q˜ T2 , · · · , q˜ TN )T , s = (sT1 , sT2 , · · · , sTN )T , M(q) = diag(M(q1 ), M(q2 ), · · · , M(qN )), C(q, q˙ ) = diag(C(q1 , q˙ 1 ), C(q2 , q˙ 2 ), · · · , C(qN , q˙ N )), ∆ = (∆T1 , ∆T2 , · · ·, ∆TN )T , and K = diag(k1 , k2 , · · · , kl , 0, · · · , 0). Thus, system (11) can be rewritten in the vector form as

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 ˙ q˜ = s − γq, ˜           M(q) s˙ = [γM(q) − C(q, q˙ ) − (K + c2 LG + λIN ) (12)     ⊗In ]s + [γC(q, q˙ ) − γ2 M(q)       +γλΛ ⊗ In ]q˜ − ∆, ! 0l×l 0N−l where Λ = ∈ RN×N . 0N−l IN−l In the case with t ∈ (t2m+1 , t2m+2 ), we have ui (t) = 0, i = 1, 2, ..., N, c1 (t) = 0 and c2 (t) = 0. In this case, the error dynamical system in the vector form can be obtained as  ˙    q˜ = s, (13)    M(q) s˙ = [−C(q, q˙ ) − λINn ]s − ∆, In fact, the above error dynamical system in this case can be easily derived from system (12) with γ = 0, K = 0 and c2 = 0. 4. Synchronization criteria

(A1)

λmin (K + c2 LGS − H3 IN −

(A2)

γ≥

(A3)

ξ1 α − ξ2 β > 0,

where

Mξ1 γλ1 Λ) ≥ ; 2 2

ξ1 /2 + H4 ; 1 − λ/21

ξ2 = max{

2H2 − 2λ + 2 (H1 + 1) H1 + 1 , }, (16) M 2

λ , 2 > 0. 2 Proof. In order to analyze the stability of the error dynamical systems (12) and (13) around the origin, we choose the quadratic Lyapunov function as

with 1 >

1 1 T s M(q)s + q˜ T q. ˜ (17) 2 2 We first calculate the derivative of V(t) with respect to time t ∈ [t2m , t2m+1 ], along the trajectory of the error dynamical system (12). By using inequality (7), Lemma 1, property 1 and the inequalities in item (i) of property 2, one can obtain that V(t) =

˙ V(t)

In this section, we will derive some synchronization criteria to make the Lagrangian network (4) synchronize

= 5

sT M(q) s˙ +

1 T ˙ s M(q)s + q˜ T q˙˜ 2

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=

sT [γM(q) − C(q, q˙ ) − (K + c2 LG + λIN ) ⊗ In ]s 2

T

T

+s [γC(q, q˙ ) − γ M(q) + γλΛ ⊗ In ]q˜ − s ∆(t) 1 ˙ + q˜ T (s − γq) ˜ + sT M(q)s 2 T

s [γMIN − K − c2 LG − λIN ) ⊗ In ]s

i=1

sT [−K − c2 LGS + γMIN + H2 IN − λIN

γλ γλ1 Λ) ⊗ In ]s + q˜ T [(−γIN + ) ⊗ In ]q˜ 2 21

−sT [K + c2 LGS − H3 IN − −[(1 −

γλ1 Λ) ⊗ In ]s 2

λ )γ − H4 )]q˜ T q. ˜ 21

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Thus, for given ξ1 > 0, it can obtain from conditions (A1) and (A2) that

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˙ ≤ − 1 ξ1 (M k s k2 + k q˜ k2 ) ≤ −ξ1 V(t), V(t) 2

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for t ∈ [t2m , t2m+1 ]. Subsequently, we consider the case with t ∈ (t2m+1 , t2m+2 ). In this case, the controller is off in the rest time. Thus, the derivative of V(t) with respect to time t ∈ (t2m+1 , t2m+2 ) is obtained along the trajectory of the error dynamical system (13). By the use of inequality (7), Lemma 1 and property 1, we have

=

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1 ˙ s M(q) s˙ + sT M(q)s + q˜ T q˙˜ 2

≤ ≤ ≤

Xm−1 +

Z

r=0

t2r+2

t2r+1

t2r

ξ2 dt −

Z

t

t2m

ξ1 dt



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Noting +∞, if t − Rt2m → +∞,  P tR → R t then we have Pm−1 t2r+2 m−1 t2r+1 − r=0 t ξ1 dt + r=0 t ξ2 dt − t ξ1 dt → −∞. 2r 2r+1 2m Thus, limt→+∞ V(t) = 0 can be easily obtained. If t − t2m < +∞, the following inequalities can be obtained  Xm−1 Z t2r+1 V(t) ≤ V(0) exp − ξ1 dt

≤

T

t2r+2

t2r+1

ξ2 dt

h

t2r



i V(0) exp − m(ξ1 α − ξ2 β) ,

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where α = min{t2r+1 − t2r , r = 0, 1, 2, ..., m − 1} and β = max{t2r+2 − t2r+1 , r = 0, 1, 2, ..., m − 1} are used. According to condition (A3), it can be easily obtained that −m(ξ1 α − ξ2 β) → −∞ as t → +∞. This is because t → +∞ implies m → +∞. Thus, we can get limt→+∞ V(t) = 0. In the case with t ∈ (t2m+1 , t2m+2 ), we can obtain  Xm Z t2r+1 V(t) ≤ V(0) exp − ξ1 dt

sT [−C(q, q˙ ) − λINn ]s − sT ∆(t)

1 ˙ + sT M(q)s + q˜ T s 2

=

Xm−1 +

Z

r=0

AC

=

r=0

r=0

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˙ V(t)

(21)

where m = 0, 1, 2, ..., ξ1 > 0 and ξ2 > 0. Next, we will show from system (22) that limt→+∞ V(t) = 0, if condition (A3) is satisfied. It is noted that there exists a positive integer m such that t ∈ [t2m , t2m+1 ] or t ∈ (t2m+1 , t2m+2 ) for any t ∈ [0, +∞). Thus, we will discuss two cases according to t in different time intervals. The following proof is similar to that of Theorem 1 in Ref. [30]. In the case with t ∈ [t2m , t2m+1 ], we can obtain  Xm−1 Z t2r+1 V(t) ≤ V(0) exp − ξ1 dt

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+(γkC1 H + γ2 M + 1 + γH2 + H1 ) k s kk q˜ k ≤

Therefore, it can be obtained from expression (16) that

for t ∈ (t2m+1 , t2m+2 ). From inequalities (19) and (21), one can obtain ( ˙ ≤ −ξ1 V(t), t2m ≤ t ≤ t2m+1 , V(t) (22) ˙ ≤ ξ2 V(t), t2m+1 < t < t2m+2 , V(t)

−γq˜ T q˜ + (γkC1 H + γ2 M + 1) k s kk q˜ k XN XN k s i k k ∆i k k si k k q˜ i k + +γλ +

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˙ ≤ 1 ξ2 (M k s k2 + k q˜ k2 ) ≤ ξ2 V(t), V(t) 2

i=l+1

≤

H1 + 1 k q˜ k2 . 22

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≤

+

−λsT s − sT ∆(t) + q˜ T s

XN −λ k s k2 + k q˜ kk s k + k s i k k ∆i k i=1

r=0

(H2 − λ) k s k2 +(H1 + 1) k q˜ kk s k 2 (H1 + 1) [H2 − λ + ] k s k2 2

Xm−1 + r=0

6

Z

t2r+2

t2r+1

t2r

ξ2 dt +

Z

t

t2m+1

ξ2 dt



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 Xm Z V(0) exp − r=0

Xm +

r=0

≤

Z

t2r+2

t2r+1

h

ξ2 dt

t2r+1

t2r

· · · ≤ λ¯ N be the eigenvalues of the matrix −K − c2 LGS . It can be obtained from Lemma 2 that  S S ¯    −ki − c2 λN (LG ) ≤ λi ≤ −ki − c2 λ1 (LG ), i = 1, 2, ..., l,    −c λ (LS ) ≤ λ¯ ≤ −c λ (LS ), i = l + 1, l + 2, ..., N, 2 N G i 2 1 G

ξ1 dt



i V(0) exp − (m + 1)(ξ1 α − ξ2 β) .

and

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−k − c2 λi (LGS ) ≤ λ¯ i ≤ −c2 λi (LGS ), i = 1, 2, ..., N.

Therefore, one can obtain that −(m + 1)(ξ1 α − ξ2 β) → −∞ as t → +∞ according to condition (A3), which results in limt→+∞ V(t) = 0. Hence, we have shown from system (22) and condition (A3) that V(t) → 0 as t → +∞. This implies that k s(t) k→ 0 and k q(t) ˜ k→ 0 as t → +∞. It should be noted that v˜ (t) = s(t) − γq(t) ˜ for t ∈ [t2m , t2m+1 ] and v˜ (t) = s(t) for t ∈ (t2m+1 , t2m+2 ). Thus, we can also get k v˜ (t) k→ 0 as t → +∞. As a result, if the conditions in Theorem 1 are satisfied, then we can get k q(t) ˜ k→ 0 and k v˜ (t) k→ 0 as t → +∞. Therefore, the Lagrangian network (4) can synchronize to the master system (5) under the aperiodically intermittent pinning controllers. This completes the proof.

Since −K − c2 LGS is a real symmetric matrix, there exists an orthogonal matrix P satisfying −K − c2 LGS = PT diag(λ¯ 1 , λ¯ 2 , ..., λ¯ N )P. Therefore, one can get that

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≤

γλ1 Λ)z 2 (Pz)T diag(λ¯ 1 + H3 , λ¯ 2 + H3 , ..., λ¯ l + H3 , λ¯ l+1 + H3 γλ1 γλ1 + , ..., λ¯ N + H3 + )(Pz) 2 2 (Pz)T diag(−k1 − c2 λ1 (LGS ) + H3 , −k2 − c2 λ1 (LGS ) zT (−K − c2 LGS + H3 IN +

=

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≤

M

Remark 3. Conditions in Theorem 1 is often called as synchronization criteria. 1 and 2 are introduced to make the values of the control gains be optimal. In specific applications, we first take the value of 2 properly to make the value of ξ2 be optimal from expression (16). Then ξ1 can be properly chosen according to the value of ξ2 , and the values of α and β can be obtained from condition (A3). If we further take a value of 1 to satλ isfy 1 > , the value of γ can be got from condition 2 (A2). Finally, the values of ki (i = 1, 2, ..., l) and c2 are taken to satisfy condition (A1).

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Remark 5. It can be seen from inequality (27) that the minimum number of the pinned agents l can be determined by λl+1 (LGS ) > 0. We do not have to assume that the interaction diagraph of the Lagrangian network contains a directed spanning tree. However, the virtual leader and followers should form a directed spanning tree. Motivated by the pinned-node selection scheme in [27], we can pick all agents with zero in-degrees as pinned agent and pin the remaining agents in descending order according to their degree-differences. Numerical examples in Section 5 show that the bigger the degree-difference of the pinned agent is, the easier it is to pin the considered Lagrangian network to a synchronization state.

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Remark 4. It is not difficult to calculate the values of ki (i = 1, 2, ..., l) and c2 satisfying condition (A1) with the aid of computer. However, it is necessary to point out that the values of ki (i = 1, 2, ..., l) and c2 should be taken to satisfy

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ki > H3 − c2 λ1 (LGS )

(26)

and

c2 λl+1 (LGS ) > H3 +

γλ1 . 2

(27)

In fact, it can be derived from condition (A1) that −K − 1 c2 LGS + H3 IN + γλ 2 Λ < 0. In the following, we can 1 prove that −K − c2 LGS + H3 IN + γλ 2 Λ < 0 if inequalities (26) and (27) are true. Equivalently, we will show that 1 N zT (−K−c2 LGS +H3 IN + γλ 2 Λ)z < 0 for any vector z ∈ R . ¯ ¯ To do this, let k = max{k1 , k2 , ..., kl }, and λ1 ≤ λ2 ≤

+H3 , ..., −kl − c2 λ1 (LGS ) + H3 , −c2 λl+1 (LGS ) + H3 γλ1 γλ1 , ..., −c2 λN (LGS ) + H3 + )(Pz). + 2 2 If inequalities (26) and (27) are true, we have −ki − c2 λ1 (LGS ) + H3 < 0(i = 1, 2, .., l) and −c2 λi (LGS ) + H3 + γλ1 2 < 0 (i = l + 1, l + 2, ..., N) . Then, one can obtain 1 T z (−K −c2 LGS +H3 IN + γλ 2 Λ)z < 0. This implies that the values of ki (i = 1, 2, ..., l) and c2 satisfying inequalities 1 (26) and (27) can ensure −K − c2 LGS + H3 IN + γλ 2 Λ < 0, γλ1 S i.e., K + c2 LG − H3 IN − 2 Λ > 0.

Remark 6. It is important to point out that the conditions of Theorem 1 are the sufficient conditions but not necessary. Thus, the Lagrangian network (4) may synchronize to a desired trajectory described by the master system (5) under the intermittent pinning controllers, although conditions in Theorem 1 are not satisfied. This will be further illustrated through numerical examples in Section 5. 7

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5. Application examples and simulation results In this section, we utilize the proposed aperiodically intermittent pinning control strategy to study the synchronization of the Lagrangian network consisting of six two-link revolute manipulators. The mechanical structure of the two-link revolute manipulator is shown in Fig.1, whose dynamic is described by the Lagrangian equation (1) [4].

3

2

6

5

1

4

Fig.2. Network structure of six manipulators.

     LG =    

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From Fig.2, the Laplacian matrix LG can be shown as 0 0 0 −1 −1 −1

0 1 −1 0 0 −1

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Fig.1 shows that the vector of generalized coordinates of agent i is denoted by qi = (qi1 , qi2 )T ∈ R2 , and the masses of each link are denoted by m1 and m2 , respectively. For the below link, we denote its link length, the distance from the previous joint to the center of mass of link, and moment of inertia as l1 , lc1 and I1 , respectively. As for the upper link, its corresponding amounts are denoted by l2 , lc2 and I2 , respectively. Then the dynamic of the two-link revolute manipulator described by Lagrangian equation can be referred to Ref.[4]. In the following simulations, the physical parameters of the robot manipulators are taken as m1 = 2 kg, m2 = 1.5 kg, l1 = 0.3 m, l2 = 0.4 m, lc1 = 0.1 m, lc2 = 0.1 m, I1 = 0.06 kg m2 and I2 = 0.08kg m2 . By calculating, one can obtain M = 0.46, M = 0.04, k M = 0.36, kC1 = 0.18, kC2 = 0.36 and kg = 15.68. In addition, the initial values of the master system (5) and the Lagrangian network (4) are taken arbitrarily as q0 (0) = (0.5π, 0.02π)T , q˙ 0 (0) = (0, 0)T , qi (0) = (−7 + 2i, −5 + 2i)T and q˙ i (0) = (0, 0)T , i = 1, 2, ..., 6. Taking λ = 8, we will obtain that the bounds of k q˙ k, k q˙ 0 k and k q¨ 0 k are H = 0.9, ζ = 1 and η = 1.35, respectively. According to the above data, we have H1 = 16.53 and H2 = 0.18. The interaction diagraph of the Lagrangian network consisting of six two-link revolute manipulators is shown in Fig. 2, which does not have a directed spanning tree. We have rearranged the order of the agents in descending order according to their degree-differences. The degree-differences of the six agents are equal to 3, 1, 0, 0, −1, −3, respectively.

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0 0 0 1 0 −1

0 0 0 0 1 0

0 0 0 0 0 3

      .   

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The eigenvalues of the symmetric part of LG are λ1 (LGS ) = −0.4851, λ2 (LGS ) = −0.0317, λ3 (LGS ) = 0.9464, λ4 (LGS ) = 1.3626, λ5 (LGS ) = 1.9145, λ6 (LGS ) = 3.2933. Now we consider the synchronization criteria proposed in Theorem 1. Taking 2 = 0.935, ξ2 = 18.75 is obtained from expression (16). Choosing ξ1 = 6, we can further obtain from criteria (A3) that 6α − 18.75β > 0. Based on the data, we let the aperiodically intermittent control exist on the following time span in the next simulations [0, 0.4] ∪ [0.45, 0.9] ∪ [1, 1.4] ∪ [1.45, 1.9] ∪ [2, 2.4] ∪ [2.45, 2.9] ∪ [3, 3.4] ∪ [3.45, 3.9] ∪ ... So, we have α = min{t2m+1 − t2m , m = 0, 1, 2, ...} = 0.4 and β = max{t2m+2 − t2m+1 , m = 0, 1, 2, ...} = 0.1 satisfying 6α − 18.75β > 0. This implies that condition (A3) in Theorem 1 is satisfied. In addition, taking ε1 = 10, ε2 = 1 and 1 = 25, γ ≥ 5.33 can be got from expression (15) and criteria (A2). For γ = 5.33, one can get H3 = 176.02 from expression (14). If five agents are allowed to be pinned, the coupling strength c2 > 215.29 can be obtained from expression (27). Taking c2 = 352, ki > 346.78 (i = 1, 2, ..., 5) are obtained from expression (26). We further choose ki = 485 (i = 1, 2, ..., 5) to satMξ1 1 isfy λmin (K + c2 LGS − H3 IN − γλ 2 Λ) = 1.5 ≥ 2 = 1.38. This implies that criteria (A1) in Theorem 1 is satisfied. Based on the above data, the simulation results by pinning the first five agents are presented in Fig.3. Fig.3 shows that limt→+∞ k q˜ i (t) k= 0 and limt→+∞ k v˜ i (t) k= 0, i = 1, 2, ..., 6. This means that the Lagrangian network (4) consisting of six two-link revolute manipulators can synchronize to the desired trajectory described by the master system (5) under the aperiodically intermittent pinning controllers. Therefore, our present control strategy is effective and feasible.

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Fig.1. The two-link revolute manipulator model for the robotic agents.

0 −1 1 0 0 0

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1.5

||vi(t)−v0(t)|| (i=1,2,3,4,5,6)

||qi(t)−q0(t)|| (i=1,2,3,4,5,6)

8

1

0.5

7 6 5 4 3 2 1

0

0.5

1

1.5

2

2.5

0

3

1

t

4

5

6

7

8

9

10

Fig.4. The synchronization errors of the six manipulators by pinning the first two agents with k1 = k2 = 100, c2 = 10, γ = 5.33, α = 0.4 and β = 0.1.

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||vi(t)−v0(t)|| (i=1,2,3,4,5,6)

3

t

14

10 8 6 4 2 0 0.5

1

1.5

2

2.5

3

t

If we take agent 1 and agent 3 to be pinned, the Lagrangian network (4) consisting of six two-link revolute manipulators can also achieve synchronization. However, the quality of the pinning process in this case is worse than that in the case of pinning agent 1 and agent 2. In order to facilitate comparison, the qualities P of the pinning process are defined as E1 (t) = 6i=1 k P6 qi (t) − q0 (t) k and E2 (t) = i=1 k vi (t) − v0 (t) k. According to the definition, the evolutions of E1 (t) and E2 (t) are shown in Fig.5 with k1 = k2 = 100, c2 = 10, γ = 5.33, α = 0.4 and β = 0.1, which are taken as the same as those in Fig.4. Fig.5 shows that the quality of the pinning process in the case of pinning agent 1 and agent 2 is better than that in the case of pinning agent 1 and agent 3. That is because the degree-difference of agent 2 is bigger than that of agent 3. As a comparison, it implies that the bigger the degree-difference of the chosen pinned agent is, the easier it is to pin the controlled Lagrangian network to the synchronization state, just as stated in Remark 5.

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Fig.3. The synchronization errors of the six manipulators by pinning the first five agents with k1 = k2 = ... = k5 = 485, c2 = 352, γ = 5.33, α = 0.4 and β = 0.1.

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It is important to point out that conditions in Theorem 1 are the sufficient conditions but not necessary, as stated in Remark 6. Thus, even if the conditions in Theorem 1 are not satisfied, the Lagrangian network (4) can also achieve synchronization. To verify this, we only take two agents to be pinned in the following simulations. In fact, since λ2 (LGS ) < 0 and λ3 (LGS ) > 0, the minimum number of the pinned agents is l = 2 according to Remark 5. Note that agent 1 has zero in-degree, which implies that the first agent must be pinned. Of course, agent 2 or agent 3 should be pinned to make the virtual leader and followers form a directed spanning tree. In addition, the degree-difference of agent 2 is bigger than that of agent 3. Thus, we should pin the first two agents when l = 2 is taken. In this case, we take k1 = k2 = 100, c2 = 10. The other parameters are taken as the same as those in Fig.3, which are not taken to satisfy the conditions in Theorem 1. However, the Lagrangian network (4) can synchronize to the desired trajectory under the aperiodically intermittent pinning controllers, as shown in Fig.4. This also demonstrates that the present aperiodically intermittent pinning control strategy is effective and feasible.

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0.12 0.1

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E1 (t)

0.08 0.06 0.04 0.02 0

5

6

7

8

9

10

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t

0.45 0.4 0.35 0.3

E2 (t)

2

||qi(t)−q0(t)|| (i=1,2,3,4,5,6)

2

0.25 0.2 0.15

1.5

0.1 0.05

1

0 4

0.5

0

1

2

3

4

5

6

7

8

9

5

6

7 t

8

9

10

Fig.5. Evolutions of E1 (t) and E2 (t): Dot lines denote the case of pinning agent 1 and agent 2; Solid lines denote the case of pinning agent 1 and agent 3.

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6. Conclusion

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In this paper, synchronization of Lagrangian networks with a directed graph via aperiodically intermittent pinning control has been investigated. By using aperiodically intermittent pinning control, linear feedback injections are only applied to a fraction of agents in each work time instead of continuous time. The designed control strategy is not continuous-time control input and does not depend on the knowledge of system models. In addition, some simple yet general synchronization criteria are developed to make the Lagrangian networks with any initial condition achieve synchronization. The theoretical results are illustrated by a Lagrangian network consisting of six two-link revolute manipulators in detail. Simulation results show the effectiveness of the proposed control strategy. It is believed that the presented technique may provide a valuable insight into the underlying use of synchronization between multiple agents in practical designs and engineering applications. Acknowledgements

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This work was supported by the National Natural Science Foundation of China(No.61603174), the 2015th Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (No.B11540), the Natural Science Foundation of Zhangzhou (No.ZZ2016J33), and the Natural Science Foundation of Fujian (No.2017J05012).

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References References

[7] P.F. Hokayem, D.M. Stipanovic, M.W. Spong, Semiautonomous control of multiple networked Lagrangian systems, Int. J. Robust Nonlinear Control 19 (2009) 2040-2055. [8] C.C. Cheah, C. Liu, J.J.E. Slotine, Adaptive tracking control for robots with unknown kinematic and dynamic properties, Int. J. Robot. Res. 25 (2006) 283-296. [9] E. Nu˜no, R. Ortega, et al., Synchronization of networks of nonidentical Lagrange systems with uncertain parameters and communication delays, IEEE Trans. Autom. Conrol 56 (2011) 935941. [10] G. Chen, F.L. Lewis, Distributed adaptive tracking control for synchronization of unknown networked Lagrangian systems, IEEE Trans. Syst. Man. Cybern. B Cybern. 41 (2011) 805-816. [11] H. Min, F. Sun, S. Wang, H. Li, Distributed adaptive consensus algorithm for networked Euler-Lagrange systems, IET Control Theory Appl. 5 (2011) 145-154. [12] Z.Y. Meng, Z.L. Lin, W. Ren, Leader-follower swarm tracking for networked Lagrange systems, Syst. Control Lett. 61 (2012) 117-126. [13] J. Mei, W. Ren, G.F. Ma, Distributed containment control for Lagrangian networks with parametric uncertainties under a directed graph, Automatica 56 (2012) 653-659. [14] N. Chopra, M.W. Spong, Passivity-based control of multi-agent systems, In Advances in Robot Control, Springer, Berlin Heidelberg (2006). [15] Y.C. Liu, N. Chopra, Controlled synchronization of heterogeneous robotic manipulators in the task space, IEEE Trans. Robotics 28 (2012) 268-275. [16] H.L. Wang, Passivity based synchronization for networked robotic systems with uncertain kinematics and dynamics, Automatica 49 (2013) 755-761. [17] M.H. Ma, J.P. Cai, J. Zhou, Adaptive practical synchronization of Lagrangian networks with a directed graph via pinning control, IET Control Theory Appl. 9 (2015) 2157-2164. [18] M.H. Ma, J. Zhou, J.P. Cai, Pinning synchronization in networked Lagrangian systems, Asian J. Control 18 (2016) 569580. [19] X.J. Wu, J. Zhou, et al., Impulsive synchronization motion in networked open-loop multibody systems, Multibody Syst. Dyn. 30 (2013) 37-52. [20] M.H. Ma, J. Zhou, J.P. Cai, Impulsive practical tracking synchronization of networked uncertain Lagrangian systems without and with time-delays, Physica A 415 (2014) 116-132. [21] M.H. Ma, J.P. Cai, Synchronization of master-slave Lagrangian systems via intermittent control, Nonlinear Dyn. 89 (2017) 3948. [22] X. Liu, T. Chen, Synchronization of complex networks via aperiodically intermittent pinning control, IEEE Trans. Autom. Conrol 60 (2015) 3316-3321. [23] X. Liu, T. Chen, Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control, Neurocomputing 173 (2016) 759-767. [24] M. Liu, H. Jiang, C. Hu, Synchronization of hybrid-coupled delayed dynamical networks via aperiodically intermittent pinning control, J. Franklin I. 353 (2016) 2722-2742. [25] S. Cai, X. Lei, Z. Liu, Outer synchronization between two hybrid coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control, Neurocomputing 222 (2017) 26-35. [26] P. Zhou, S. Cai, Pinning synchronization of complex directed dynamical networks under decentralized adaptive strategy for aperiodically intermittent control, Nonlinear Dyn. 54 (2017) 113. [27] Q. Song, J. Cao, W. Yu, Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Syst.

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[1] Z. Wang, J. Cao, Z. Duan, et al., Synchronization of coupled Duffing-type oscillator dynamical networks, Neurocomputing 136 (2014) 162-169. [2] S. Das, M. Srivastava, A.Y.T. Leung, Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method, Nonlinear Dyn. 73 (2013) 2261-2272. [3] J.L. Wang, H.N. Wu, Adaptive output synchronization of complex delayed dynamical networks with output coupling, Neurocomputing 142 (2014) 174-181. [4] S.J. Chung, Nonlinear control and synchronization of multiple Lagrangian systems with application to tethered formation flight spacecraft, Dept. Aeronaut. Astronaut., Massachusetts Institute of Technology, Cambridge, MA (2007). [5] S.J. Chung, J.J.E. Slotine, Cooperative robot control and concurrent synchronization of Lagrangian systems, IEEE Trans. Robotics 25 (2009) 686-700. [6] Y. Su, D. Sun, L. Ren, J.K. Mills, Integration of saturated PI synchronous control and PD feedback for control of parallel manipulators, IEEE Trans. Robotics 22 (2006) 202-207.

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Control Lett. 59 (2010) 553-562. [28] W. Zhang, J. Huang, P. Wei, Weak synchronization of chaotic neural networks with parameter mismatch via periodically intermittent control, Appl. Math. Model. 35 (2011) 612-620. [29] R. Kelly, R. Salgado, PD control with computed feedforward of robot manipulators: a design procedure, IEEE Trans. Rob. Autom. 10 (1994) 566-571. [30] R.Z. Luo, Y.H. Zeng, The control and synchronization of a class of chaotic systems with a novel input, Chin. J. Phys. 54 (2016) 147-158.

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Biographies of all the authors

Mihua Ma received both the B.S. degree in applied mathematics and M. S. degree in fundamental mathematics from Minnan Normal University, Zhangzhou, China, in

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2005 and 2007, respectively, and Ph.D. degree in general and fundamental mechanics from Shanghai University, Shanghai, China, in 2015. She is currently an associate professor of Minnan Normal University, Zhangzhou, China. Her research interests include chaos control and synchronization of dynamical systems, analysis and control

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of complex networks, and coordinated control in networked multi-agent systems.

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Jianping Cai received both the B.S. degree in mathematics and M.S. degree in fundamental mathematics from Fujian Normal University, Fuzhou, China, in 1988

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and 1991, respectively, and Ph.D. degree in applied mathematics from Zhongshan University, Guangzhou, China, in 2004. He is currently a professor of Minnan Normal University, Zhangzhou, China. His research interests include chaos control and synchronization, chaotic secure communication, complex network and nonlinear vibration.