Exponential consensus of discrete-time non-linear multi-agent systems via relative state-dependent impulsive protocols

Accepted Manuscript Exponential consensus of discrete-time non-linear multi-agent systems via relative state-dependent impulsive protocols Yiyan Han, Chuandong Li, Zhigang Zeng, Hongfei Li

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S0893-6080(18)30239-9 https://doi.org/10.1016/j.neunet.2018.08.013 NN 4017

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Neural Networks

Received date : 28 April 2018 Revised date : 1 July 2018 Accepted date : 13 August 2018 Please cite this article as: Han, Y., Li, C., Zeng, Z., Li, H., Exponential consensus of discrete-time non-linear multi-agent systems via relative state-dependent impulsive protocols. Neural Networks (2018), https://doi.org/10.1016/j.neunet.2018.08.013 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 proof before it is published in its final 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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Exponential Consensus of Discrete-time Non-linear Multi-agent Systems via Relative State-dependent Impulsive Protocols Yiyan Hana , Chuandong Lia,∗, Zhigang Zengb , Hongfei Lia a Chongqing

Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China b School of Automation, Huazhong University of Science and Technology, and Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China

Abstract In this paper, we discuss the exponential consensus problem of discrete-time multi-agent systems with non-linear dynamics via relative state-dependent impulsive protocols. Impulsive protocols of which the impulsive instants are depended on the weighted relative states of any two agents are introduced for general discrete-time multi-agent systems. The analysis of such impulsive protocols are transformed into an investigation on reduced fixed-time impulsive protocols by constructing a map, which is achieved mainly by a derived B-equivalence method in discrete-time domain. Our main results indicate that the exponential consensus of the multi-agent systems via relative state-dependent impulsive protocols can be achieved if the reduced systems via fixed-time impulsive protocols can achieve exponential consensus, which need to satisfy suitable sufficient conditions. Numerical simulations are presented to support the theoretical results. Keywords: Relative state-dependent impulsive protocols, Discrete-time non-linear multi-agent systems, Exponential consensus 1. Introduction It is worth noting that multi-agent systems have drawn much attention due to the promising application. Many single agents which can represent different individuals with their own state and dynamics constitute a group as so-called multi-agent system. Multi-agent systems are tools to achieve goals that must be done by cooperation including formation, flocking, rendezvous and so on (Tanner, Jadbabaie, & Pappas, 2007; Zhou, Lu, & Lu, 2006; Lu, & Chen, 2005). The consensus problem, as a fundamental problem demanding the states finally to be same has been naturally introduced and has kept attracting researchers in last two decades. In a multi-agent system, each agent shares information with its neighbors to reach an agreement. Once the agreement is reached by every agent, the whole system achieves consensus. Existing researches mainly focused on the consensus problem (Jadbabaie, Lin, & Morse, 2003; Lu, Lu, Chen, & Lu, 2013; Chen, Lu, & Yu, 2011) based on integrator and first-order dynamics. Recently the consensus problem of second-order multiagent systems (Ren, 2007; Mei, Ren, & Chen, 2016; Sun, Austin, Lu, & Chen, 2011) with states which represent both position and velocity became more attractive, where distributed consensus problem was considered with both undirected and directed networks topologies. And because non-linear phenomena are everywhere in real-world, many studies (Rezaee, & ∗ Corresponding

author Email addresses: [email protected] (Yiyan Han), [email protected] (Chuandong Li), [email protected] (Zhigang Zeng), [email protected] (Hongfei Li) Preprint submitted to Neural Networks

Abdollahi, 2017; Li, Chen, Dong, & Xia, 2016; Fan, Chen, & Zhang, 2014; Li, 2016) were done with the non-linear dynamics of multi-agent systems. Hua, You and Guan (2017) studied the adaptive leader-following consensus of second-order multi-agent systems with time-varying non-linear dynamics by a fully distributed algorithm. And the consensus problem that is mainly concerned with the analysis of the second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching topologies was studied in Li, Liao, and Huang (2013). Shen and Shi (2016) studied the output consensus control problem of uncertain second-order nonlinear multiagent systems with unknown non-linear dead zone. In Wen et al.(2017), the consensus tracking problem was investigated by an observer-based protocol, which indicates that only the relative output measurements of neighboring agents are available for information exchange. However, in some specific circumstances, discrete-time dynamics are more suitable for analyzing real problems. There are also many researches (Han, & Li, 2018; Gao, Yu, Shao, & Yu, 2016; Liang, Zhang, Wang, & Wang, 2015) on the consensus problem of discrete-time multiagent systems with linear or non-linear dynamics. Moreover, impulsive control systems were paid more attention in recent years in that an evolution of a system may suddenly changes in real circumstances. To get closed to the reality, impulsive control method was introduced because its high efficiency to multi-agent systems with high-robustness and low-cost. Impulsive control had been widely applied to the synchronization and consensus problems (Lu, Ho, & Cao, 2010; Liu, Lu, & Chen, 2013; Morarescu, Martin, Girard, & Muller-Gueudin, 2016) with kinds of complex networks over June 30, 2018

last decade. Hu et al.(2013) investigated the consensus problem of multi-agent systems with each agent described by impulsive dynamics under directed communication networks topology. In Jiang and Bi (2010), by impulsive control method, several conditions are built for synchronization and consensus with switching topology were obtained. In Guan, Wu and Feng (2012), the convergence speed depending on the consensus algorithms via impulsive control was studied furthermore. Recently, the consensus problem of second-order multi-agent systems was also dealt with impulsive control, which may deserve more researches(Qian, Wu, Lu, & Lu, 2014; Ding, Yu, Liu, Guan, & Feng, 2013) due to its meaningful application. Wang and Yi (2015) solved the consensus problem of second-order multi-agent systems via impulsive control using only position information with communication delays. Static consensus of second-order multiagent systems was investigated in Jiang, Xie and Liu (2017). by using impulsive algorithm and the author further considered the case with communication delays. Many researches emphasized on the second-order multi-agent systems with linear or integrator dynamics, some focused on the non-linear dynamics case. In Liu et al. (2017), consensus criteria of second-order multi-agent systems were delivered based on so-called pulsemodulated intermittent control with non-linear velocity dynamics, which can be reduced to the consensus problem with sampled control. Li and Su (2016) proposed protocols which permit that all the feedbacks are only on velocity in communication intervals. If the interval were only a instant, the protocols would change into impulsive protocols. Seeing from previous researches, the consensus of multiagent systems by impulsive method demands more strict conditions than using continuous control methods. However, existing researches focused on continuous time multi-agent systems especially via impulsive protocols or algorithms, where the authors often gave complex conditions and the system required very small impulsive intervals. In addition, fixed-time impulsive protocols require the design of all impulsive instants before the systems commence, but for application we often need to control the multi-agent systems due to suitable instants which depend on the states. So the use of state-dependent impulsive protocols are considered as an advanced method to better deal with the consensus problem. As an attractive but difficult problem to be investigated, only several publications can be found. In Li, Li and Huang (2017), the periodicity and stability for variable-time impulsive neural networks were studied by a method called B-equivalence, where a comparison system was constructed to guarantee the purpose. Furthermore, statedependent impulses on synchronization of hybrid and switching Hopfield neural networks were also investigated by the same method in Zhang, Li and Huang (2017). Since the state-dependent impulsive control has been applied to neural networks, we naturally consider it in other application. Until now, no studies have been found about the consensus of discrete-time multi-agent systems by such state-dependent impulsive method, which urges us to try to investigate this interesting and promising issue. Notice that the B-equivalence has been only used for continuous time systems in reported researches, which leads to a challenge in discrete-time case. Furthermore,

we consider the state will no longer be the state of the agents own but the weighted relative states of any two agents. This may be utilized in more real circumstances because relative states contain more complex information. This paper aims at an investigation into the consensus of discrete-time multi-agent systems with non-linear dynamics by designing suitable relative state-dependent impulsive protocols. By constructing comparison systems, the analysis of the systems via relative state-dependent impulsive protocols is reduced to the case via fixed-time impulsive protocols, where the latter one has been well studied recently. Based on graph theory and impulsive control theory, several fundamental consensus conditions and criteria are obtained. The organization of this paper is describe as follows. Fundamental preliminaries of related graph theory are introduced in section 2. The consensus problem of the multi-agent systems via relative state-dependent impulsive protocols is transformed into a case of multi-agent systems via fixed-time impulsive protocols in section 3. The equivalence of both the multi-agent systems via two sorts of impulsive protocols is proved and some sufficient conditions for the exponential consensus in fixed topology is demonstrated in section 4. Numerical examples that strongly support the effectiveness of the theoretical results are presented in section 5. The conclusion and future work are discussed in section 6.

Notions: Throughout this paper, N and R represent the set

of natural numbers and the set of real numbers, respectively. For any square matrix A ∈ RN×N , AT denotes the transpose of A. IN denotes the N-dimensional identity matrix and 1N denotes the N-dimensional column vector in which all elements equal to one. For any vector x = (x1 , x2 , ..., xN )T kxk∞ = max(|x1 |, |x2 |, ...,q |xN |) represents the infinite norm of vector x and kxk =

x12 + x22 + · · · + x22 represents Eu-

clidean norm. And for any matrix A ∈ RN×N , kAk∞ = P P P max( Nj=1 |x1 j |, Nj=1 |x2 j |, ..., Nj=1 |xN j |) is the infinite norm of matrix A. 2. Preliminaries

2

2.1. Graph Theory A directed graph with N nodes is represented by G = (V, E, A), which consists a node set V = {1, 2, ...N} and a edge set E = {(i, j)|i, j ∈ V} ⊆ V × V. The set of neighbors of node i is denoted by Ni = { j ∈ V|(i, j) ∈ E, i , j} and |Ni | denotes the number of its neighbors. And A = [ai j ] ∈ RN×N is called the adjacency matrix of graph G, where aii = 0 and ai j ≥ 0, i , j, and ai j = 0 if and only if there is no edge between node i and node j. The out-degree of node i in a directP ed graph is defined as deg(i) = Nj=1 ai j . Let D be the outdegree matrix of graph G, which is a diagonal matrix with the out-degree of each node along the principal diagonal, that is, D = diag{deg(1), deg(2), ..., deg(N)}. A Laplacian L = [li j ]N×N of graph G is defined by   deg(i), i = j    −1, j ∈ Ni li j =     0, otherwise

which satisfies L = D − A. And by the definition, every rowsum of L is zero, which means there exists a right eigenvector of L associated with a unique zero eigenvalue. Its eigenvalues can be arranged such as 0 = |λ1 (L)| ≤ |λ2 (L)| ≤ |λ3 (L)| ≤ · · · ≤ |λN (L)|. From now on, all the topologies of multi-agent systems we discussed are directed graphs which are strongly connected.

In addition to that, multi-agent systems via fixed time impulsive protocols has been well studied already, where the impulsive instants must be predesigned. To avoid this inconvenience, protocols of which the impulsive instants depend on the states of agents seem to be reasonable. This consideration may make the behavior of multi-agent systems easier to control. Moreover, notice that the event-triggered scheme for multi-agent systems is also a state-dependent control method where the trigger instants are decided by the state of every agent. But this requires that the design of trigger conditions for each agent and usually the trigger conditions are in specific forms. Furthermore, the trigger errors should be calculated on predesigned event-checking period. And the most important of all, although event-trigger scheme dose not demand the real-time states of agents, it is still a method of continuous control. Enlightened by event-triggered scheme and based on the state-dependent impulsive method, we figure out relative state-dependent protocols to simplify this problem and reveals a more general relationship between the states and the impulsive instants. By such method, the communication instants are based on the sum of several weighted relative states of any two agents. To further investigate the speed of convergence of multiagent systems, that is, the speed to achieve consensus, we also introduce exponential consensus.

3. Problem Formulation 3.1. Multi-agent Systems via Relative State-dependent Protocols The simplest discrete-time multi-agent systems are described by xi (n + 1) = xi (n) + ui (n) (1) where the discrete time instants n satisfies n = 0, 1, 2, .... xi (n) ∈ R is the state of the ith agent at time instant n and ui (n) ∈ R is called protocol. In general, the protocol ui (n) for i = 1, 2, ..., N is usually described by X ui (n) = α ai j (x j (n) − xi (n)) (2) j∈Ni

where α ∈ R is a constant to be designed. Easily, the vector form of these protocols for i = 1, ..., N is described as u(n) = −αLx(n)

Definition 1. The multi-agent systems are said to achieve exponential consensus for any initial state, if there exist two positive constants W and ψ such that

(3)

where x(n) = (x1 (n), x2 (n), ..., xN (n))T and the uniform u(n) is called protocols. The general definition of consensus is defined by limn→+∞ |x j (n) − xi (n)|, i, j = 1, 2, ..., N. Then the above multi-agent system via such protocols can achieve consensus PN xi (0) if it is subject to suitable with an average value N1 i=1 conditions, where xi (0) is the initial state of the ith agent. And in recent years, multi-agent systems via impulsive protocols have attracted more investigation in that some communication environment makes it difficult to acquire continuous information from neighbors for any agent. By such impulsive method, the protocol ui (n) is as follows X ui (n) = δ(n − nh )(b ai j (x j (n) − xi (n))) (4)

|x j (n) − xi (n)| ≤ We−ψ(n−n0 )

where i, j = 1, 2, ..., N.

Based on above discussion, we construct the problem by discrete-time multi-agent systems with non-linear dynamics via relative state-dependent impulsive protocols in networks topology G, which can be described by xi (n + 1) = xi (n) + s f (xi , n) + ui (n),

i = 1, 2, ..., N

(5)

where constant s is step-size and f (xi , n) is a discrete-time nonlinear function. Protocol ui (n) is designed as X ui (n) = δ(n − rk (e x ))(b ai j (x j (n) − xi (n))) (6)

j∈Ni

j∈Ni

where the discrete impulsive instants nh satisfy 0 ≤ n0 < n1 < · · · < nh−1 < nh < · · · and limh→+∞ nh = +∞ with the impulsive intervals τh = nh+1 − nh for h ∈ N. ∆xi (nh ) = xi (nh + 1) − xi (nh ). The constant b is called impulsive gain. Notice that by using Dirac function δ(n) that has a property such that δ(n − a)H(n) = H(a), the states of all agents can only update at impulsive instants. It is noticed that the instant n0 is set to be initial instant and there is no impulsive effect at n0 . But the real-world circumstances are far more complicated than the above can describe. From previous researches, conditions related to the networks topology are often used to guarantee the consensus of most multi-agent systems with integrator or linear dynamics, which can be only applied to ideal cases. Actually, dynamics of multi-agent systems are often nonlinear. This definitely leads to more difficult analysis but can solve more realistic problems in engineering.

where b is a constant and rk (e x ) ∈ N is a function with P e x = e x (n) = | i,Nj=1 ci j (x j (n) − xi (n))| at kth communication for k ∈ N. Also, rk (·) characterizes the kth impulsive surface where the relative state e x (n) will meet to cause communication between agents. Thus, system (5) can be rewritten as a discretetime impulsive system such that ( xi (n + 1) = xi (n) + s f (xi , n), n , rk (e x ) (7) ∆xi (rk (e x )) = s f (xi , rk (e x )) + ui (rk (e x ))

It follows that such a multi-agent system composed of N agents can be described as ( x(n + 1) = x(n) + sF(x, n), n , rk (e x ) (8) x(rk (e x ) + 1) =(IN − bL)x(rk (e x )) + sF(x, rk (e x )) where F(x, n) = ( f (x1 , n), f (x2 , n), ..., f (xN , n))T .

3

Specially, there is z1 (n) = q1 γ1 (IN − 1N γ1T )x(n) ≡ 0 at any instant. Based on this, we could only analyze the systems for i = 2, 3, ..., N such that  ˜ n), n , rk (e x )  z (n + 1) = zi (n) + sqi γi F(x,    i  z (r (e ) + 1) =(1 − bq λ i k x i i (L))zi (rk (e x )) (11)      ˜ rk (e x )) + sqi γi F(x,

Remark 1. The innovation of our protocols are the communication instants decided by the relative states of any two agents. This demands detection on the agents’ states but the function rk (·) is unique for all. As an explanation, since the relative states are the key to measure consensus, they are naturally important information to control the whole multi-agent system. Remark 2. Consider a specific case that the state represents the position when all agents work together in a narrow region. Then controlling every two agents’ relative position is equivalent to control the admissible region, which can be considered as a real world application. The function rk (e x ) can describe the boundary so that once the limitation of the region is reached by agents, they must communicate immediately to converge to secure themselves. In addition, the weight ci j is designed to control the region of specifically chosen agents. For a more general case, we can consider a situation that the impulsive instants are dependent on any two agents relative state. If the relative state is smaller than a threshold, then there may not need a communication, which efficiently reduces the number of times of communication. If the relative state is too big, then the frequency of communication will be larger. This may be more intelligent for the control of multi-agent systems.

Thus, the stability of equivalent system (10) at origin indicates the consensus of system (5).

Remark 3. We notice that once the systems are with disturbance from the uncertainty of non-linear dynamics, the consensus problem becomes more complicated compared with linear and integrator cases. So for such multi-agent systems, we can no longer guarantee the consensus with an average value 1 PN i=1 xi (0) due to the conservativeness of non-linear control N PN system theory. In this paper, the consensus value is i=1 wi xi (n) instead with a dynamic value, which indicates the consensus value mostly depends on the networks topology and the realtime states. Furthermore, the trajectories and the final states of agents cannot be predicted unless one introduces the leaderfollowing scheme to control the traces. Despite the construction of equivalent systems, the relative state-dependent impulsive protocols still bring much difficulty for the investigation on the consensus problem. Here, we use a method on general state-dependent impulsive systems, which is called B-equivalence method. Since there is no reports about such method in discrete-time domain, so we introduce follows in details.

3.2. Error Systems and Equivalent Systems In this subsection, we construct error systems and their equivalent systems to help analyze the exponential consensus problem of the proposed multi-agent systems. Moreover, the relative state-dependent cases are analyzed by cases with a kind of variable-time impulsive systems. Here we assume that for a directed graph, its Laplacian is diagonalizable such that L = Γ−1GΓ, where G = diag(λ1 (L), λ2 (L), ..., λN (L)). And the invertible matrix Γ = (γ1 , γ2 , ..., γN )T ∈ CN×N where γi is the left eigenvector according to λi (L). It can be easily notice that γ1 can be chosen as a positive eigenvector satisfying γ1T L = 0, where γ1 = PN wi = 1. Other γi is (w1 , w2 , ..., wN )T ∈ RN×1 and satisfies i=1 chosen as an unit vector for i = 2, 3, ..., N. PN Denote yi (n) = xi (n)− xˆ(n), where xˆ(n) = i=1 wi xi (n). Obviously, the error |xi (n) − xˆ(n)| = |yi (n)| has the same convergence with |x j (n) − xi (n)| for n → +∞. Therefore, the error system of system (8) can be described by ( ˜ n), n , rk (e x ) y(n + 1) = y(n) + sF(x, (9) ˜ rk (e x )) y(rk (e x ) + 1) =(IN − bL)y(rk (e x )) + sF(x, ˜ n) = ( f (x1 , n) − where y(n) = (y1 (n), y2 (n), ..., yN (n))T and F(x, PN PN T i=1 wi f (xi , n), ..., f (xN , n) − i=1 wi f (xi , n)) . Furthermore, let z(n) = Λy(n), where z(n) = (z1 (n), z2 (n), ..., zN (n))T , Λ = QΓ and Q is a positive diagonal matrix Q = diag(q1 , q2 , ..., qN ) to be designed. Notice that matrix Λ is invertible and so ky(n)k ≤ kΛ−1 kkz(n)k, which means y(n) and z(n) have same stability at origin. Then the equivalent system of system (9) is described by ( ˜ n), n , rk (e x ) z(n + 1) = z(n) + sΛF(x, ˜ rk (e x )) (10) z(rk (e x ) + 1) = (IN − bΛL)z(rk (e x )) + sΛF(x,

3.3. Intersection with Impulsive Surfaces In this subsection, we discuss the intersection with impulsive surfaces. It indicates only when weighted relative states intersect the impulsive surfaces, information for communication can be delivered and received. As mentioned before, the state of system (10) via relative state-dependent impulsive control has discontinuities not at moments with fixed time instants, but at the moments depending on the impulsive surfaces and relative states. This is a significant difference from the systems via fixed-time impulsive control. Moreover, states with different initial conditions yield different impulsive instants in general. In this paper, note that since the impulsive instants are decided by the weighted relative state and impulsive surfaces that are denoted by Γk and characterzied by rk (·) for k ∈ N. This characterization presents additional difficulty in analysis, exceptionally the number of the impulsive instants on a same surface is not predictable. Then for a better situation, it is necessary to assume that the intersection between the relative state and every impulsive surface Γk happens exactly once. Here we present some assumptions and lemmas for later use. (A1) rk (·) = nk + τ˜ k (·), where τ˜ k (·) = [τk (·)] is the integer part of a continuous function τk (·).

4

(A2) There exists a positive constant ζ > 0 such that 0 ≤ τk (·) ≤ ζ.

(A3) There exists a sequence of integers {nk } such that 0 = n0 < n1 < · · · < nk < · · · and nk → ∞ for k → ∞ with two positive constants n and n satisfying n + ζ < nk+1 − nk ≤ n − ζ.

3.4. Comparison Systems To reduce the difficulty of solving the stability of system (10), now we introduce a type of fixed-time impulsive systems. In this subsection, we will construct a fixed-time impulsive system that can be regarded as the comparison system of system (10) and the relationship between these two is presented. Notice that a subsystem of system (10) without any impulsive effect is described as

(A4) The non-linear function satisfies | f (x, n) − f (v, n)| ≤ l f |x − v|

x, v ∈ R

where l f is a positive constant.

˜ n) z(n + 1) = z(n) + sΛF(x,

(A5) On every impulsive surface, there only exists an unique positive integer ξk such that rk (e x (ξk )) = ξk = nk + τ˜ k (e x (ξk )).

Now we consider following in interval n ∈ [n0 , ξ1 + 1]. Denote a state of system (15) by z s1 (n) = z(n, n0 , z s1 (n0 )) which has impulsive effect at ξ1 = n1 + τ˜ 1 (z(ξ1 )). And let zc1 (n) = z(n, ξ1 +1, zc1 (ξ1 +1)) be another state of it such that zc1 (ξ1 +1) = ˜ s1 , ξ1 ), where x s1 (n) z s1 (ξ1 + 1) = z s1 (ξ1 ) − bΛLz s1 (ξ1 ) + sΛF(x is the uniform state of agents according to z s1 (n). Similarly, the uniform state xc1 (n) is according to zc1 (n). Furthermore, we define zc1 (n) = z s1 (n) for n ∈ [n0 , n1 ]. Then it is obvious that zc1 (n) can be extended backward as a state of system (15) for n ∈ [n1 , ξ1 + 1]. Similar and so on, we can define two states for n ∈ [ξk−1 , nk ] such that zck (n) = z sk (n) and for n ∈ [nk , ξk +1] such that z sk (n) = z(n, nk , z sk (nk )) with zck (n) = z(n, ξk + 1, zck (ξk + 1)) and zck (ξk + ˜ sk , ξk ), where 1) = z sk (ξk + 1) = z sk (ξk ) − bΛLz sk (ξk ) + sΛF(x sk x (n) is the uniform state. Similarly, there is an uniform state xck (n). Obviously, zck (n) can be extended backward as a state of system (15) for n ∈ [nk , ξk + 1]. Based on all the definition of two states z sk (n) and z sk (n) for k ∈ N, we unify all of them in infinite instants interval so that we can easily have z s (n), zc (n), x s (n) and xc (n) for n ≥ n0 , which ˜ s , ξk ). satisfies zc (ξk +1) = z s (ξk +1) = z s (ξk )−bΛLz s (ξk )+sΛF(x Here we can construct a map to establish a relationship between zc (n) and z s (n) such that

Here we make an explanation about the assumptions through the following lemma. Lemma 1. (A1) and (A5) are feasible and under all assumptions, the weighted relative state of system (10) meets every impulsive surface at impulsive instant ξk for k ∈ N exactly once in turn, where ξk = nk + τ˜ k (e x (ξk )). Proof . Firstly, we proof the feasibility of (A1) and (A5). Actually under (A2), there may not exists a ξk and a nk satisfying ξk = nk + τ˜ k (e x (ξk )). But since τ˜ k (·) is upper bounded and ξk > ζ, there must exists a ξk and a nk = ξk − τ˜ k (e x (ξk )) ≥ 1. Here we might as well set nk = min{κ|κ = ξk − τ˜ k (e x (ξk )), ξk = [ζ] + 1, [ζ] + 2, ...} for k ∈ N and nk is subject to (A3). So there must exists a ξk such that rk (e x (ξk )) = ξk = nk + τ˜ k (e x (ξk )). Furthermore, under (A2) and (A3) we consider ξk+1 − ξk ≥ nk+1 − nk − ζ > n > 0

(12)

Θk (z, k)

and since τ˜ k (·) is bounded, there is ξk+1 − ξk = nk+1 + τ˜ k+1 (e x (ξk+1 )) − nk − τ˜ k (e x (ξk ))

=zc (nk + 1) − z s (nk )

(13)

=zc (ξk + 1) − sΛ

which easily leads to a fact that the impulsive interval Ik = ξk − ξk−1 is both lower and upper bounded such that n ≥ Ik ≥ n

(15)

ξk X

w=nk +1

˜ c , w) − z s (nk ) F(x

˜ s , ξk ) − z s (nk ) =z s (ξk ) − bΛLz s (ξk ) + sΛF(x

(14)

− sΛ

All the above indicate that the weighted relative states intersect each impulsive surface Γk in turn, i.e., ξ1 < ξ2 < · · · < ξk < · · · and on every impulsive surface there only exists one ξk , which guarantees the feasibility of (A5). Thus the weighted relative states of system (10) meet every impulsive surface at impulsive instant ξk exactly once in turn. Here the proof is complete.

=sΛ(

ξk X

(16)

˜ c , w) F(x

w=nk +1

ξk X

w=nk

˜ s , w) − F(x

ξk X

w=nk +1

˜ c , w)) − bΛLz s (ξk ) F(x

Thus, for k ∈ N and n ∈ [n0 , +∞], we notice that z s (n) can be a state of system (10). By such method, zc (n) can be extended as the state of a fixed-time impulsive system described by ( ˜ n), z(n + 1) =z(n) + sΛF(x, n , nk (17) z(nk + 1) = z(nk ) + Θk (z, k)

Remark 4. The assumptions also lead to a fact that nk+1 − ξk > 0 which is proved by nk+1 − ξk > nk + n + ζ − ξk > 0. Moreover, we obtain that n0 < n1 < ξ1 < n2 < ξ2 < · · · < ξk < nk+1 < ξk+1 < · · ·, which indicates all the instants nk and ξk are different from each other. It is convenient for constructing comparison systems in next subsection by such definition.

which is called a comparison system of system (10). Then we will prove that the asymptotical stability of (17) indicates the same stability of (10) in next section.

5

Proof . Based on (A4), consider the following

It is significant that a kind of state-dependent impulsive systems can be reduced to simpler fixed-time impulsive systems. Since the systems with fixed moments of impulses have been well investigated, it is beneficial to use them for our later results. But it is worth noting that the map Θk (z, k) is not only related to non-linear dynamics at instant nk , but also involves the information at future time instants, which still leaves us much difficulty to analyze the stability. In this paper, the key way to solve this problem is to estimate the map Θk (z, k).

| f (xi , n) − =|

N X j=1

≤ lf

4. Main Results

= lf

4.1. Comparison Consensus

w j |xi (n) − xˆ(n) − x j (n) + xˆ(n)|

(19)

j=1

≤ 2l f kΛ−1 k∞ kz(n)k∞

N X

wj

j=1

and it follows that ˜ n)k∞ ≤ 2l f kΛ−1 k∞ kz(n)k∞ kF(x,

(20)

Then we can have that h1 (q)h3 (q)

q=p0

˜ n)| |zi (n + 1)| ≤ |zi (n)| + sqi |γi F(x, √ ˜ n)k∞ ≤ |zi (n)| + sqi Nkγi kkF(x, √ ≤ |zi (n)| + 2sqi l f NkΛ−1 k∞ kz(n)k∞ √ ≤ (1 + 2sq0 l f NkΛ−1 k∞ )kz(n)k∞

(21)

kz(n + 1)k∞ ≤ lkz(n)k∞

(22)

and

0
Lemma 3. Let z s (n) be the state of system (10) and zc (n) be the state of comparison system (17) satisfying zc (nk + 1) = z s (nk ) + Θk (z, k) and zc (ξk + 1) = z s (ξk + 1) = z s (ξk ) − bΛLz s (ξk ) + ˜ s , ξk ) for k ∈ N. Denote z s (n) = zc (n) = z(n) for n < sΛF(x [nk + 1, ξk ]. Then for n ∈ [nk + 1, ξk ], there are two valid claims as follows

Easily, one obtains kz(n)k∞ ≤ ln−nk kz(nk )k∞

(23)

Then for n ∈ [nk + 1, ξk ], it makes

(i) kzc (nk + 1)k∞ ≤ gkz(nk )k∞

kz s (n)k∞ ≤ ln−nk kz s (nk )k∞

(ii) kzc (n) − z s (n))k∞ ≤ βkz(nk )k∞

and

kzc (n)k∞ ≤ ln−nk −1 kzc (nk + 1)k∞

= ln−nk −1 kz s (nk ) + Θk (z, k)k∞

where constants are as follows

(24)

(25)

Now we see at n = ξk , one has

q0 = max qi √

NkΛ−1 k∞

l0 = max ||1 − bqi λi (L)| + sλz qi l f i=2,...,N

N X

w j |xi (n) − x j (n)|

≤ l f |yi (n)| + ky(n)k∞

for p ≥ p0 and h2 (p) ≤ hb , where hb is an upper bound of h2 (p), the following inequity holds. Y h1 (p) ≤ hb (1 + h3 (q))

l = 1 + 2sq0 l f

j=1

j=1

Lemma 2. Let {h1 (p)}, {h2 (p)} and {h3 (p)} be three sequences of real numbers. Sequence {h3 (p)} satisfies that {h3 (p)} ≥ 0 for p ≥ p0 , where p0 is a positive real number. Then if

i=2,...,N

N X

N N X X ≤ l f ( w j |yi (n)| + w j |y j (n)|)

In this subsection, we will discuss that if the system (17) under fixed-time impulsive control is exponentially stable, then the system (10) under relative state-dependent impulsive control has the same stability. And notice both the systems can indicate the consensus of their according multi-agent systems, then we called this relationship comparison consensus. Here we give some lemmas for later use.

p−1 X

w j f (x j , n)|

j=1

w j ( f (xi , n) − f (x j , n))|

j=1

h1 (p) ≤ h2 (p) +

N X

√

|zci (ξk + 1)| = |zis (ξk + 1)|

N|

˜ c , ξk )| ≤ |1 − bqi λi (L)||zci (ξk )| + sqi |γi F(x √ ≤ (|1 − bqi λi (L)| + 2sqi l f NkΛ−1 k∞ )|zci (ξk )|

(18)

l0 l ζ (2 − lζ ) β = lζ (l + g)

g=

(26)

With z1 (n) ≡ 0, we immediately have 6

kzc (ξk + 1)k∞ = kz s (ξk + 1)k∞ ≤ l0 kz s (ξk )k∞

(27)

Therefore, based on (A2) and (A4), we estimate the state combined with map Θk (z, k) such that

thus claim (ii) holds. Here the proof of all claims above are complete.

kzc (nk + 1)k∞

Next we prove that if system (17) is exponentially stable, then system (10) is the same.

= kz s (nk ) + Θk (z, k)k∞ = kzc (ξk + 1) − sΛ

ξk X

w=nk +1

≤ l0 kz s (ξk )k∞ + 2sq0 l f

√

s

≤ l0 kz (nk )k∞ + l0 (l − 1) + (l − 1)kzc (nk + 1)k∞ s

Theorem 1. If there exist positive constants Wc , ψc such that

˜ c , w)k∞ F(x

NkΛ−1 k∞ ξX k −1

w=nk ξX k −1

ξk X

w=nk +1

kzc (n)k∞ ≤ Wc e−ψc (n−n0 )

kzc (w)k∞

then there must exist positive constants W s , ψ s satisfying kz s (n)k∞ ≤ W s e−ψs (n−n0 )

s

kz (w)k∞

which indicates the multi-agent system (5) can achieve exponential consensus.

(28)

lw−nk

Proof . First, we prove the equivalence of the stability of the both systems. For n ∈ [n0 , n1 ], it is obvious that

w=nk s

≤ l0 kz (nk )k∞ + l0 (l − 1)kz (nk )k∞

ξX k −1

lw−nk

kz s (n)k∞ = kzc (n)k∞

w=nk

then we have

(l − 1)(1 − lξk −nk ) c kz (nk + 1)k∞ 1−l (l − 1)(1 − lζ ) s ≤ l0 (1 + )kz (nk )k∞ 1−l (l − 1)(lζ − 1) c + kz (nk + 1)k∞ l−1 ζ s = l0 l kz (nk )k∞ + (lζ − 1)kzc (nk + 1)k∞ +

kz s (n1 )k∞ = kzc (n1 )k∞ ≤ Wc e−φc (n1 −n0 ) And for n ∈ [n1 + 1, ξ1 ], from Lemma 3, one has kz s (n)k∞ = kz s (n) − zc (n) + zc (n)k∞

≤ kzc (n) − z s (n)k∞ + kzc (n)k∞ ≤ βkzc (n1 )k∞ + Wc e−ψc (n−n0 )

So that by kz s (nk )k∞ = kzc (nk )k∞ , one has

l0 lζ kz s (nk )k∞ = gkz(nk )k∞ kz (nk + 1)k∞ ≤ (2 − lζ ) c

≤ Wc (e−ψc (n1 −n0 ) + e−ψc (n−n0 ) )

(29)

≤ Wc (1 + eψc (n−n1 ) )e−ψc (n−n0 )

Thus claim (i) holds. And we are going to prove that claim (ii) is also valid. Consider the term kzc (n) − z s (n)k∞

≤ Wc (1 + eψc ζ )e−ψc (n−n0 ) When n ∈ [ξ1 + 1, n2 ], obviously one has

kz s (n)k∞ = kzc (n)k∞ ≤ Wc e−ψc (n−n0 )

6 kzc (n)k + kz s (n)k∞

n−1 X

c

= kz (nk + 1) + sΛ + kz s (nk ) + sΛ

w=nk +1

n−1 X

w=nk

+s

n−1 X

kz s (n)k∞ ≤ Wc (1 + eψζ )e−ψ(n−n0 )

˜ s , w)k∞ F(x

Thus, there must be two constants W s = Wc (1 + eψc ζ ) and ψ s = ψc for n ≥ n0 satisfying

(30)

kz s (n)k∞ ≤ W s e−ψs (n−n0 )

˜ , w)k∞ + kΛF(x ˜ , w)k∞ ) (kΛF(x s

w=nk +1

= (1 + g + 2sq0 l f + 2sq0 l f

Then to sum up, when n ∈ [nk + 1, ξk ], similar to the case for n ∈ [n1 + 1, ξ1 ], one observes that

˜ c , w)k∞ F(x

˜ s , nk )k∞ ≤ kz s (nk )k + gkz s (nk )k + skΛF(x

√

√

c

Thus the proof is complete.

NkΛ−1 k∞ )kz s (nk )k∞

NkΛ−1 k∞

n−1 X

w=nk +1

Remark 5. The theorem establishes a relationship of consensus between the multi-agent systems via fixed-time impulsive protocols and state-dependent impulsive protocols. A map is constructed to keep the state of the comparison system (17) same as the system (10) except for n ∈ [nk , ξk ]. And for n ∈ [nk , ξk ], the states of both systems are not the same but the error between them is bounded by βkz(nk )k∞ . Besides, with ψ s = ψc , the rate of convergence dose not change due to our comparison method.

(kzc (w)k∞ + kz s (w)k∞ )

then from Lemma 2, we can easily have kzc (n) − z s (n)k∞ ≤ kzc (n)k∞ + kz s (n)k∞ = lζ (l + g)kz s (nk )k

(32)

(31)

= βkp(nk )k 7

4.2. Exponential Consensus In this subsection, we discuss the exponential stability of the comparison system (17) which indicates the system (10) is exponentially stable. Furthermore, based on Lemma 3, if the exponential stability of comparison system is guaranteed, the consensus problem of multi-agent system (5) can be solved by our proposed method. Through following theorem, we use the normal symbol z(n) to represent the state of system (17) for all time instants in analysis.

for n ∈ [nk−1 , nk ], claim (33) holds. Then we note that for n ∈ [nk , nk+1 ], there is kz(n)k∞ ≤ gln−nk −1 kz(nk )k∞

≤ gln−ζ−1 Me−µ(nk −n0 )

≤ Me−µ(n−n0 )

which indicates the validity of claim (33). Then for all n ≥ n0 , there is kz(n)k∞ ≤ Me−µ(n−n0 )

Theorem 2. The system (17) is exponentially stable such that kz(n)k∞ ≤ Me−µ(n−n0 )

Since kz(n)k∞ , ky(n)k∞ and |x j (n)−xi (n)| have same convergence to zero, the exponential consensus of system (5) can be solved with a constant M0 . Thus the proof is complete.

for any initial state kx(n0 )k∞ ≤ C0 if there exist positives constant µ and M with the following condition

Remark 6. In this paper, we discuss the exponential consensus of multi-agent system (5) instead of consensus. As we see, the exponential consensus can characterize the speed of convergence by constant µ. But to obtain the convergence speed of the multi-agent system, the method we used makes the result only sufficient and kind of conservative. To reduce the inaccuracy and conservativeness, the constant e could be replaced by any other positive constant which needs to be bigger than one.

gln−ζ−1 eµ(n−ζ) ≤ 1 satisfied. Furthermore, multi-agent system (5) can achieve exponential consensus if there exists a positive constant M0 such that |x j (n) − xi (n)| ≤ M0 e−µ(n−n0 ) Proof . In this proof, mathematical induction method will be used. For n ∈ [nk−1 + 1, nk ] and k ∈ N, we firstly claim that kz(n)k∞ ≤ Me

−µ(n−n0 )

Remark 7. Compared to other researches on state-dependent impulsive systems, we investigate the case of discrete-time system. Also, under the assumptions a complex problem called beating phenomena is directly avoided, which greatly reduced the conservativeness of the choice of f (xi , n) and τk (·). Furthermore, since it is apparent that rk (·) can be a function with respect to any variable, this state-dependent impulsive method seems meaningful for engineering applications. Except for our proposed method, this state-dependent impulsive method could be extended to more general case.

(33)

then show it holds by follows. For n ∈ [n0 , n1 ], obviously there is kz(n)k∞ ≤ ln−n0 kz(n0 )k∞

≤ ln−ζ e−µ(n−n0 ) eµ(n−n0 ) kz(n0 )k∞ ≤ ln−ζ eµ(n−ζ) e−µ(n−n0 ) kz(n0 )k∞

(34)

= Me−µ(n−n0 )

In this section, numerical examples are given to illustrate the theoretical results.

where constant M = C0 kΛk∞ ln−ζ eµ(n−ζ) . And for n ∈ [n1 +1, n2 ], we will show that the claim still holds. Assume not, then there at least exists a m1 ∈ [n1 + 1, n2 ] satisfying that kz(m1 )k∞ > Me−µ(m1 −n0 ) . Notice that kz(m1 )k∞ ≤ lm1 −n1 −1 kz(n1 + 1)k∞ ≤ glm1 −n1 −1 kz(n1 )k∞ ≤ gl

n−ζ−1

Example 1. We consider a multi-agent system composed of 4 agents and the non-linear function can be described as bellow f (xi , n) = 1.2| cos(xi )| − | arctan(n)| + 2.54 + 6.5 sin(n)

(35)

with constant l f = 1.2. The directed fixed topology is strongly connected and its Laplacian matrix is as following

−µ(n1 −n0 )

Me

    L =   

From the condition, we have kz(m1 )k∞ ≤ Me−µ(n1 −n0 ) e−µ(n−ζ)

≤ Me−µ(n1 −n0 ) e−µ(n2 −n1 )

(37)

≤ Me−µ(nk −n0 ) e−µ(n−ζ)

(36)

−µ(n2 −n0 )

= Me

With e−µ(m1 −n0 ) ≥ e−µ(n2 −n0 ) , this contradicts with the assumption kz(m1 )k∞ > Me−µ(m1 −n0 ) for any m1 ∈ [n1 + 1, n2 ]. So that claim (33) holds for n ∈ [n1 + 1, n2 ]. Similarly and so on, consider it for n ∈ [nk , nk+1 ] and k = 2, 3, ..., we will also prove that claim (33) holds. Assume that 8

2 0 −1 0 0

0 2 0 −1 0

−1 0 2 −1 −1

−1 −1 0 3 −1

0 −1 −1 −1 2

       

The eigenvalues of its according Laplacian are λ1 = λ2 = 1.6753, λ3 = 2, λ4 = 3.6624 + 0.5623i and λ5 3.6624 − 0.5623i. The eigenvector w can be calculated w = (0.1522, 0.087, 0.3043, 0.1739, 0.2826)T . The sum P the weights of relative states is chosen as i,Nj=1,i, j |ci j | = Choose the function characterizing the impulsive surfaces

0, = as of 1. as

with constant l f = 1. The directed fixed topology is strongly connected and its Laplacian matrix is as following         L =      

2 −1 −1 0 −1 0 −1 −1 0

0 4 0 −1 0 0 −1 −1 −1

0 0 3 0 0 −1 0 0 −1

−1 −1 0 3 0 −1 −1 0 −1

0 −1 −1 0 2 0 −1 −1 −1

0 0 0 −1 −1 3 −1 0 −1

0 0 −1 −1 0 0 6 −1 −1

0 −1 0 0 0 0 0 5 −1

−1 0 0 0 0 −1 −1 −1 7

              

Its eigenvalues are λ1 = 0, λ2 = 2.5807 + 0.4643i, λ3 = 2.5807−0.4643i, λ4 = 3.1349+0.8522i, λ5 = 3.1349−0.8522i, λ6 = 4.2236, λ7 = 5.9082 + 0.4616i, λ8 = 5.9082 − 0.4616i and λ9 = 7.5287. The eigenvector w can be calculated as w = (0.1990, 0.0843, 0.0722, 0.1862, 0.1537, 0.1538, 0.0585, 0.0294, 0.0630)T . The sum of the weights of relative states is P chosen as i,Nj=1,i, j |ci j | = 4. Here we introduce the switching impulsive surfaces of which dynamics could be described as ( 3.2 − 2 arccos(k · k∞ ), k = 1, 3, 5... τ(·) = 4.2 − 4.2sech(k · k∞ ), k = 2, 4, 6, ... with constant and ζ = 4.2. Choose the constants n = 0.8 and n = 9.2 with ζ such that nk+1 − nk = 5. Since we choose b = 0.22 and s = 0.012, it is easy to verify that the conditions in Theorem 2 hold with a diagonal matrix Q = diag(10, 0.5, 1.6, 1.6, 1.3, 1.3, 0.9, 0.9, 0.6). Moreover, we choose the initial states such that 10 ≤ |xi (n0 )| ≤ 30, then there exist constants g = 0.5805 and µ = 0.027 satisfying the conditions in Theorem 2. Fig.2 shows that the multi-agent system can achieve exponential consensus and Fig.3 shows the relative state-dependent impulsive intervals. It is worth noting that since e x (n) will converge to zero, the impulsive interval will finally converge to a fixed constant.

Fig. 1. States and error states of the agents in Example 1

5. Conclusions

τ(e x (n)) = 3.14 sin2 (2e x (n)) and obviously τ(·) < ζ = 3.15. Choose the constants n = 0.85 and n = 7.15 with ζ such that nk+1 − nk = 4. Since we choose b = 0.48 and s = 0.0044, it is easy to verify that the conditions in Theorem 2 hold with a diagonal matrix Q = diag(20, 1.2, 1.1, 0.55, 0.55). Moreover, the initial states such is chosen as |xi (n0 )| ≤ 1, then there exist constants g = 0.6489 and µ = 0.01 satisfying the conditions in Thm 3. Fig.1 shows that the multi-agent system can achieve exponential consensus.

In this paper, the exponential consensus of discrete-time multi-agent systems with non-linear dynamics via relative state-dependent impulsive protocols has been investigated. Protocols based on former researches has been designed to guarantee the exponential consensus with wider impulsive intervals. By constructing comparison systems, the problem has been transformed into a general consensus problem of multi-agent systems via fixed-time impulsive protocols. Sufficient conditions which guarantee the exponential consensus have been obtained by non-linear control system theory. Numerical simulations have demonstrated that our impulsive protocols are efficient to solve the consensus problem in fixed networks topology.

Example 2. Consider a hyperbolic tangent function as the dynamics of a multi-agent system, which is described by

Acknowledgment f (xi , n) =

e x(n) − e−x(n) e x(n) + e−x(n)

This study is supported by the Natural Science Foundation of China (No. 61374078, 61633011), Chongqing Research 9

Fig. 2. States and error states of the agents in Example 2

Fig. 3. The impulsive interval Ik = ξk+1 − ξk in Example 2

Program of Basic Research and Frontier Technology (No. cstc2015jcyjBX0052). 10

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