Event-based Consensus Tracking for Nonlinear Multi-Agent Systems under Semi-Markov Jump Topology

This paper studies the event-triggering leader-follower consensus with the strictly dissipative performance for nonlinear multi-agent systems (MASs) with semi-Markov changing topologies. First, a polynomial fuzzy model is established to describe the error nonlinear multi-agent system that is formed by one virtual leader and followers. Then, a new event-triggering transmission strategy is proposed to mitigate communication and computational load. By utilizing the event-triggering mechanism and modeling the switching topologies by semi-Markov process, a sampled-data based consensus protocol is designed. Compared with traditional Markov jump topologies, the transition rate is time-varying for semi-Markov switching topologies. By mode-dependent Lyapunov-Krasovskii functional, the sum of square based relaxed stabilization conditions for fuzzy MASs are obtained to guarantee event-triggering consensus with strict dissipativity in an even-square sense, i.e., the derived conditions take into account the joint effects of event-triggering control, semi-Markov jump topologies and external disturbance. An illustrative example is provided to verify the proposed consensus design schemes.


I. INTRODUCTION
Cooperative consensus of multi-agent systems (MASs) has received considerable attention owing to its wide applications, including flocking [1], formation control [2], [3]. The main purpose of consensus problems is to design a distributed controller (consensus protocol), which can guarantee that all agents can reach a common state by exchanging local information among neighboring agents via communication link. Various control schemes have been utilized, such as finite time control in [4], fault-tolerant control in [5], [6], adaptive control in [7]- [9] and optimal control in [11], [12].
The communication topologies among the agents may not be often fixed due to links interruption and new establishment partly stem from the communication equipment failures and disturbance. To describe the time-varying topology, a common method is that the switching topologies are modeled by the Markov process, which have been attracted a lot of concerns. For example, see [13]- [15]. However, in practice, Markov changing topologies have many limitations because the dwell time obeys exponential distribution and the transition rates are constant. Different from traditional Markov jump topologies, the dwell time of semi-Markov changing topologies obeys more general distribution, including Gaussian distribution and Weibull distribution. For semi-Markov switching topologies, the transition rates are time-varying and depend on the dwell time. Recently, fruitful results have been reported on semi-Markov switching topologies [16]- [18]. Hence, semi-Markov changing topology is one of the issues worth considering here.
Dissipativity theory is introduced in [19], which plays a key role in the analysis and synthesis of control systems. In practice, it is necessary to guarantee the dissipativity to reach the purpose of interference attenuation. The dissipativity is regarded as a generalization of the H ∞ performance, the passivity theory, and the bounded real lemma. The dissipative performance is discussed for a variety of dynamic systems [20], [21]. For instance, [21] studies the observedbased event-triggering sliding mode control with the strict dissipativity of the switched stochastic discrete system. Event-triggering control (ETC), as an effective scheme in saving communication resources and alleviating control updates, has gain remarkable attention. Different from the time-triggering scheme, data transmission and control updates are decided by an event-triggering condition. When the triggering condition is met, the event occurs. The central idea and challenge of ETC are to establish the time sequence of data transmission through a predefined event-triggering strategy which is different from the time series of traditional periodic control. For example, see [22] and the references therein. Recently, event-triggering consensus problems for MASs have attracted extensive attention. Rich results have been obtained [23]- [28]. For instance, the control problem of event-triggering consensus is discussed for linear MASs with changing topologies in [26].
Recently, the polynomial fuzzy model in [29] is introduced for modeling a nonlinear system by polynomial expression. The new fuzzy model can be viewed as a generalization of the T-S fuzzy model [30]- [32], which has attracted extensive attention. One may refer to [33]. To date, few results are reported on even-triggering consensus with strictly dissipative performance for polynomial fuzzy MASs under semi-Markov jump topologies.
Motivated by the above discussion, this paper investigates the even-triggering consensus with strict dissipativity of polynomial fuzzy MASs with semi-Markov changing topologies. The main contributions of this paper are summarized as follows: (i) Most existing results deal with the consensus problems of the nonlinear MASs by using the Lipschitz conditions, such as [9], [10]. A polynomial fuzzy model is established to describe the error nonlinear multi-agent system in this paper. Compared with [34], the fuzzy model here is simpler and without extra assumptions.
(ii) In [35], [36], the consensus problems of continuoustime communication are investigated for nonlinear MASs under changing topologies. [13] addresses the time-triggering consensus problems for nonlinear MASs under Markov switching topologies. Unlike [13], [35], [36], a sampled-data mode-dependent event-triggering transmission strategy is presented here to reduce communication and computational load. By using the event-triggering scheme and modeling the switching topologies by semi-Markov process, the modedependent event-triggering consensus protocols are designed.
(iii) Using mode-dependent Lyapunov-Krasovskii functional, relaxed stabilization conditions based on sum of square (SOS) [37] are obtained to assure event-triggering consensus with strict dissipativity in an even-square sense, i.e., the presented conditions take into account the joint effects of event-triggering communication, semi-Markov jump topologies and external disturbance.
The remainder of this paper is organized as follows: In Section 2, the related knowledge of graph theory is introduced and the problem formulation is given. In Section 3, the polynomial fuzzy model is built and the sample-data mode-dependent event-triggering transmission scheme is designed. In Section 4, event-triggering consensus protocols and the main results are presented. In Section 5, an illustrative example is provided. We conclude this paper in Section 6.
Notation: The symbol ⊗ denotes the Kronecker product. · is the Euclidean norm. I represents the identity matrix with appropriate dimensions. E{·} is the expectation operator. (Ω, F, P) denotes a probability space. Q > 0 means that the matrix Q is positive definite. The superscript T for matrix Q T denotes transpose of matrix Q. Sym(A) means A + A T . Σ 2 represents SOS.

II. PRELIMINARIES AND PROBLEM FORMULATION
Here, we introduce the related knowledge of graph theory and the problem formulation is presented.

A. GRAPH THEORY
Let G = (V, E, A) be a digraph generated by N follower agents, in which V = {1, ..., N } is a nonempty node set, E = {(i, j) : i, j ∈ V} denotes an edge set, and A = [a ij ] ∈ R N ×N represents a weighted adjacency matrix.
We denoteḠ as a digraph formed by one virtual leader labeled 0 and N follower agents marked 1 ∼ N .

C. PROBLEM FORMULATION
Here, we consider a leader-follower nonlinear multi-agent system formed by N followers and one virtual leader. Each agent's dynamics is described bẏ where x 0 ∈ R n is the state of virtual leader. x i ∈ R n is the state of agent i, and i = 1, . . . , N. f (x i ) ∈ R n is a polynomial vector in x i . u i ∈ R n is the control input. d p ∈ R n×q , and w i ∈ R q is the external disturbance. The error state is e i = x i − x 0 . Then, the error dynamics can be expressed aṡ Before going further, the following assumptions and concepts are given to obtain the main results. Assumption 1: EachḠ , γ(t) ∈ S, contains a directed spanning tree with the root of the virtual leader. Assumption 2: States of each agent are periodically sampled. The sampling period is synchronized by a clock. Definition 1: [36]: Given matrices Y ∈ R ι×q , X = X T ∈ R ι×ι , and Z=Z T ∈ R q×q with X ≤0 and Z >0, if for T * ≥ 0 and δ > 0, then (3) is called strictly (X , Y, Z)-δ-dissipative. Definition 2: Under the consensus protocol u i , (2) is called mean-square consensus if for any initial conditions x i (0), x 0 (0) ∈ R n . Remark 1: Inspired by [35], the definition of mean-square consensus in (5) for event-based MASs under semi-Markov jump topologies is presented.

A. POLYNOMIAL FUZZY MODEL
To describe system (3), a polynomial fuzzy model is established below: where The compact form of (6) iṡ where The function h p (θ i ) has the properties of

B. EVENT-TRIGGERING MECHANISM
To save communication resources, an event-triggering control strategy is presented for the system (2). To determined whether the sampled data is transmitted or not, the modedependent event-triggering condition for the ith agent is defined as where ρ i > 0 denotes the threshold, and Φ γ(t) > 0 is the weighting matrices to be designed later.
, where l = 1, 2, . . ., and h is the sampling period. Define where α is a scalar with α ∈ (0, 1]. E i (t i k + lh) denotes the measurement error formed by the last released state x i (t i k ) and the current state where m is an integer. Remark 2: If the condition (8) holds, the event is triggering, and then the sampled data is sent to its neighbors and controller. The event-triggering time sequence represents Zeno behavor does not happen. Remark 3: In (9), motivated by [32], α is introduced to smooth the input signal. If α = 1, the event-triggering mechanism will reduced to the conventional one as in [16], [38]. Compared with the conventional event-triggering mechanism, the event-triggering mechanism in (9) will reduce erroneous events induced by the abrupt changing of the output measurement.

IV. EVENT-TRIGGERING CONSENSUS DESIGN AND CONSENSUS CONDITIONS
Now, we consider the event-triggering dissipative consensus conditions for system (7) under semi-Markov changing topologies. Then, the derived results can be extended to a fixed topological case.

A. EVENT-TRIGGERING CONSENSUS PROTOCOL
Here, we first design a distributed event-triggering consensus protocol for system (7) under semi-Markov changing topologies.
Considering the controlled output z i , the augmented system of agent i iṡ where c zp ∈ R ι×n , d zp ∈ R ι×q , and z i ∈ R ι .

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An event-triggering consensus protocol for agent i is designed as follows: is the weight of information flow. If agent i can get the leader's information at γ(t), then d i = 1. Remark 4: Our purpose is to design the consensus protocol (11) to ensure that all agents can achieve agreement and alleviate the consumption of communication resources.

By Definition 1, (14) is strictly
Based on (17), one obtains Therefore, there exists a scalar > 0, such that By using Dynkin's formula, one has

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Similarly, one obtains Since Therefore, it follows from (31)-(33) that That is which indicates that lim T →∞ E e(σ) 2 = 0. According to Definition 2, all agents achieve consensus. This proof is completed. ∇∇∇ Remark 5: In Theorem 1, by polynomial Lyapunov-Krasovskii functional technique, SOS-based relaxed sufficient condition is presented to ensure that the polynomial fuzzy MASs can achieve even-square agreement with strictly dissipative performance under event-triggering control and semi-Markov switching topologies. Based on Theorem 1, we present the following approach to design the control gains.

C. CONSENSUS CONDITIONS UNDER SEMI-MARKOV SWITCHING TOPOLOGIES
Here, based on the obtained result in Theorem 2, consensus conditions for MASs under semi-Markov switching topologies is given.

V. ILLUSTRATIVE EXAMPLE
Consider a nonlinear multi-agent network, which the switching topologies are shown in Figure 1. Each agent's dynamics is described from [40] x where The polynomial fuzzy model is established as follows: , and x i3 ∈ [ ζ 1 3 , ζ 2 3 ], where ζ 1 1 = −24, ζ 2 1 = 24, ζ 1 2 = −32, ζ 2 2 = 32, ζ 1 3 = −46, and ζ 2 3 = 46.  The augmented fuzzy error system iṡ Without loss of generality, assume that the edges' weights of all communication topologies are 1. Figure 2 depicts the semi-Markov switching signal. Laplacian matrices are expressed as The external disturbance is w(t) = 1.5e −0.25t | cos t|. Let    The event-triggering instants of each agent are depicted in Figure 3, which indicates the amounts of sampling data transmitted are reduced. The state trajectories of each agent are shown in Figure 4. The error states are given in Figure 5. The simulation results show that all agents achieve consensus, which demonstrates the effectiveness of the presented design schemes.

VI. CONCLUSION
In this paper, the event-triggering consensus with strict dissipativity have been studied for fuzzy MASs with semi-Markov jump topologies and external disturbance. A new VOLUME 4, 2016  He is currently a Professor with the Graduate School of Science and Technology, Tokai University, Japan. His research interests include approximate reasoning, fuzzy reasoning, fuzzy system modelling and applications, neuro-fuzzy learning algorithms for system identification. He has published over 200 papers in journals and conferences. He has actively served in a number of journals. He is the Executive Editor of International Journal of Innovative Computing, Information and Control; Editor-in-Chief of International Journal of Biomedical Soft Computing and Human Sciences; and Editor-in-Chief of ICIC Express Letters. He is a member, the board of directors of Biomedical Fuzzy Systems Association. VOLUME 4, 2016