H-inﬁnity bipartite consensus of multi-agent systems with external disturbance and probabilistic actuator faults in signed networks

: The H-inﬁnity bipartite consensus problem is addressed for a class of linear multi-agent systems with external disturbance, where the positive and negative links are allowed in communication topology. A novel event-triggered communication scheme is presented to save limited network resources, which dependents on information from neighboring agents at event-triggered instants, the given event-triggered condition is detected only at discrete sampling times, thus Zeno behavior can be excluded, two types of event-triggered matrices have been introduced in our event-triggered communication scheme, which can further reduce the sampled-data transmission compared with some existed results. Considering the probabilistic actuator faults, the reliable controller is designed based on sampled-data, then a new distribution-based fault model is constructed by using coordinate transform. Some H-inﬁnity bipartite consensus criteria can be derived by the Lyapunov stability theory and algebraic graph theory, at the same time, the feedback matrices and event-triggered matrices can be obtained by solving some linear matrix inequalities. Finally, a numerical example is employed to show the validity and advantage of the proposed transmission scheme.


Introduction
Recently, the distributed consensus control for multi-agent systems has received compelling attentions owing to its potential applications [1][2][3]. The main objective is to design distributed control protocol such that all agents can achieve an agreement. Generally speaking, the consensus problems can be divided into three types, leader-following consensus problem, which means a leader is designated and all the followers track the leader. Containment control is a consensus-like tracking problem with multiple leaders, the control target is to let the followers states finally converge to the In practical multi-agent systems, each agent is equipped with micro-processor, it can collect information from neighboring agents, therefore, it is desired to implement actuating the controller updates on a digital platform [20], the scheduling can be done in a time-triggered scheme and an event-triggered scheme, where time-triggered scheme with a fixed sampling period, which should be selected to guarantee a desired performance under worst conditions, this kind of triggered method will lead to send many unnecessary sampling data to neighboring agents. Considering the limited network resources, the event-triggered scheme is an effective method to improve the resources utilization while ensuring a satisfactory performance [21,22]. Event-triggered bipartite leader-following consensus problems have been studied in [23][24][25][26], event-triggering condition is allowed to be intermittently examined at its own sampling instants in [23], a novel distributed protocol based on independent sampling period is presented. In [24], a novel event-triggered function is designed, where the continuous communication can be avoided and the Zeno-behavior can be excluded. In [25], the distributed static bipartite event-triggered controller and the fully distributed adaptive bipartite event-triggered controller are designed respectively, where the communication in both controller updating and triggered condition monitoring is intermittent, thus the communication resources can be greatly saved. In [26], by using pinning control strategies, a novel eventtriggered controller is designed to achieve bipartite consensus. The refereances [27,28] discuss fully distributed bipartite output consensus of heterogeneous multi-agent systems, the fully distributed bipartite compensator and dynamic event-triggered mechanism are proposed in [27], where the proposed compensator is independent of any global information of the network topology. The reference [28] discusses the distributed bipartite finite-time event-triggered output consensus for heterogeneous linear multi-agent systems under directed signed communication topology, where a novel distributed bipartite compensator with intermittent communication mechanism is proposed in finite time. In [29], distributed event-triggered control strategy is proposed to guarantee the semiglobal bipartite consensus. The reference [30] presents an even-triggered control framework to achieve bipartite consensus for networked Euler-Lagrange systems with external disturbance. In [31], event-triggered bipartite consensus is addressed under DoS attacks. Bipartite consensus of multiagent systems with switching communication topologies is addressed in [32], where event-triggered conditions are piecewise functions depending on the variable couplings and topologies. However, the above schemes still exist some room for improvement. Inspired by [33][34][35], the next transmission instant is determined by where Φ 1 and Φ 2 are two symmetric positive definite matrices, some comparisons are given in Section 4, it can be seen that our event-triggered scheme can lead to larger sampling period. The lower bound event interval is the sampling period h, which is strictly larger than zero, therefore, Zeno behavior is excluded in our proposed scheme. Furthermore, it is well known that actuators often suffer from faults inevitably in practical applications, which may degrade the performance of even cause instability [36,37]. In response to requirements for the reliability of multi-agent systems, some considerable works have been done on designing control schemes to accommodate actuator faults [38][39][40][41][42]. The leader-following consensus problem for multi-agent systems with actuator faults [38,39], actuator faults are time-varying [38], for two cases of the leader without control input and the leaders unknown bounded input, the distributed event-triggered fault-tolerant protocol is proposed to reach leader-following consensus. For a class of multi-agent systems against actuator faults and some perturbed factors as state-dependent uncertainties in [39], the distributed controllers is constructed to ensure leader-following consensus based on adaptive updates. In [40][41][42], the adaptive techniques are adopted to achieve the consensus of multiagent systems with actuator faults. By the sliding mode control method, the reference [40] investigates the consensus tracking problem of second-order nonlinear multi-agent systems with disturbance. In [41], for a class of high-order nonlinear systems with intermittent actuator faults, a novel distributed adaptive controller is proposed to solve the output consensus tracking problem. In [42], for the actuator faults of partial loss of effectiveness fault and biased fault, a neural network-based adaptive consensus protocol is developed based on the Lyapunov analysis. In addition, the disturbance is inevitable and may degrade the performance in practical applications, H-infinity bipartite consensus problem is studied of nonlinear multi-agent systems with bounded disturbances in [43], where an performance criterion from bounded disturbances to the consensus error has been provided. In [44], the bipartite consensus controller is designed for cooperation-competition multi-agent systems with round-robin protocols such that the predetermined H-infinity bipartite consensus is satisfied over a given finite horizon. But in the above references, the event-triggered scheme and probabilistic actuator faults have not considered, to the best of the author's knowledge, no result has been reported for H-infinity bipartite consensus of multi-agent systems with event-triggered scheme and probabilistic actuator faults in signed networks, which motivates the present study.
Motivated by the aforementioned discussion, this paper is devoted to investigate H-infinity bipartite consensus for a class of linear multi-agent systems with external disturbance in signed networks. The main contributions can be summarized as follows: (1) A novel concept of H-infinity bipartite consensus is introduced to quantify against the exogenous disturbance. (2) The proposed event-triggered scheme can ensure the desired performance while further reducing the frequency of data transmission, Zeno behavior is excluded owing to the lower bound of inter event interval is the sampling period. (3) The generalized random variables are introduced to describe probabilistic actuator failures, which represent the case of complete measurement failure , partial failure and complete normal of the actuator, but also including the case of the measured value of the actuator bigger than the real value.
The remainder of this paper is organized as follows: In Section 2, some preliminaries are introduced. In Section 3, main results are presented for H-infinity bipartite consensus. A numerical simulation is provided to verify the theoretical results in Section 4. Some conclusions and future works are provided in Section 5.
Notation R n denotes the n-dimensional Euclidean space and R n×m is a set of real n × m matrices. Notation . is the Euclidean norms of vector and matrix. If A is positive definite (or semi-definite), denote A > 0 (or A ≥ 0). Suppose that matrix M ∈ R n×n is a real symmetric matrix, λ max (M) and λ min (M) denote the maximum and minimum eigenvalues of matrix M, respectively. The notation ⊗ means the Kronecker product. diag{· · · } stands for a block-diagonal matrix. The superscript 1 n (0 n ) indicates the n-dimensional column vector with each entry being 1(0). sgn(·) is the standard sign function. E(·) stands the mathematical expectation of random variable.

Problem formulation and preliminaries
The communication topology can be described as a signed graph g = (v, ε, A), where v = (v 1 , v 2 , . . . , v N ) denotes a set of vertices, ε ⊆ v × v is a set of edges, A = (a i j ) N×N denotes a weighted adjacency matrix representing the communication topology, where the entry a i j of the matrix A can be allowed to negative and positive, a i j > 0 indicating node i and node j are cooperative relationship, if a i j < 0 indicating node i and node j are competitive relationship, otherwise, a i j = 0 means node i and node j have no information communication. The Laplacian matrix of the signed graph g can be defined as L = (l i j ) N×N , where l ii = k=1,k i |a ik |, and l i j = −a i j (i j). A signed graph g is called structurally balance if the node set v can be divided into two sub-networks V 1 and V 2 , which satisfy V 1 ∪ V 2 = v and V 1 ∩ V 2 = Φ, a i j ≥ 0 holds for any i, j ∈ V p , p ∈ {1, 2} and a i j < 0 holds for all i ∈ V q , j ∈ V r , where q r, otherwise, the signed graph g is called structurally unbalance. Consider a group of N identical agents with general linear dynamics, in which the dynamics of agent i is described as follow: where x i (t) ∈ R n , u i (t) ∈ R n and w i (t) ∈ R q represent the system state, control input and external disturbance of the ith agent, respectively. A ∈ R n×n , B ∈ R n×m and D ∈ R n×q are some known constant matrices with compatible dimensions.
In what follows, some key lemmas are summarized, which play the important roles to derive the main results. Lemma 1. [11] Suppose that signed network g is structurally balance, if and only if there is a gauge transformation W such that W T AW =Ā has all nonnegative entries, where a gauge transformation is a change of orthant order by W = diag{w 1 , w 2 , . . . , w N } with w i ∈ {1, −1}, A = (a i j ) N×N andĀ = (|a i j |) N×N are the adjacency matrices of the signed and unsigned networks, respectively. In addition, W provides a partition, that is is absolutely continuous function satisfying x(a) = 0 and M ∈ R n×n is a positive definite matrix, then the following wirtinger's inequality holds: It is assumed that there exists a positive definite matrix M ∈ R n×n and a vector function x(t) : (a, b) → R n , the following integrations are well defined, then For any two real vector functions y 1 (t) ∈ R n and y 2 (t) ∈ R n , a positive definite matrix R ∈ R n×n and any matrix S ∈ R n×n , then The following linear matrix inequality where S 11 = S T 11 , S 12 = S T 21 , S 22 = S T 22 are some constant matrices with appropriate dimensions, the following matrix inequalities hold: For reducing the communication burden among the agents, and considering the unreliable channel from control to actuator, the sampled control algorithm is designed as follow: where c > 0 is the coupling strength, Ξ = diag{Ξ 1 , Ξ 2 , . . . , Ξ m } with Ξ i being m unrelated random variables, taking on the interval [0, θ] with θ ≥ 1. K ∈ R m×n is the feedback matrix to be designed. t k h (k = 0, 1, 2, . . .) is the release times, t 0 h = 0, {t 0 , t 1 , . . .} ⊂ {0, 1, 2, . . .}, the data will be updated until next event-triggered instant t = t k+1 h. Remark 1. Inspired by some related literatures [49,50], the random variable Ξ is used to describe the actuator fault. For the case of Ξ = 0, it means complete failure of the ith actuator. For Ξ ∈ (0, 1), it means partial failure of the ith actuator. For the case of Ξ = 1, it means the ith actuator is in good working condition. For Ξ > 1, it means the data distortion with the measurement of the ith actuator.
Based on the definition of Laplacian matrix, Eq (2.6) can be rewritten aṡ Considering limited bandwidth of the communication network, to further reduce the data transmission among the agents, the event-triggered communication scheme is proposed for system (2.1), which is utilized to determine whether or not the current sampled data should be transmitted to neighboring agents. Supposed that the latest transmitted data is x(t k h), then for i = 2, 3, . . . , N, the next transmission instant t k+1 h can be determined by i are some positive definite event-triggered matrices to be designed, σ i > 0 is event-triggered parameter, t k+1 h − t k h denotes the release period which corresponds to the sampling period given by the event-triggered scheme.
Remark 2. Noted that the sampled state x(t k h) satisfying the inequality (2.8) will be sent to the neighbour agents, some of the sampled states that violate (2.8) will not be transmitted. Obviously, the communication burden of the network is reduced, the corresponding transmission energy can also be saved.   Letx In views of (2.9) and e i (t) =x i (t) −x 1 (t) (i = 2, 3, . . . , N), one haṡ Inspired by [34,35], the interval [t k h, t k+1 h) can be expressed as the union of several subintervals as follow: (2.12) Applying (2.12) into (2.10), one haṡ Rewrite (2.13) in compact matrix form, we can geṫ where Definition 1. For signed networks g, H-infinity bipartite consensus can be achieved for coupled linear system (2.1), if for given a weighting positive-definite matrix S ∈ R n×n , a performance index γ > 0 and the initial disagreement vector e(0), its solution satisfies

Main result
In this section, H-infinity bipartite leaderless consensus problem is investigated under eventtriggered scheme, supposed that the topology is directed signed graph and structurally balance.
14) can be formulated as follow: Combining (2.8) and (2.12), for t ∈ [t k h, t k+1 h), the event-triggered scheme can be rewritten as From the above analysis, it can be known from (3.2) that the current sampled data e i (t k h + lh) will not be sent, which will be employed in the following consensus analysis.
Corollary 1. Suppose that signed network is directed and structurally balance, for given sampled period h > 0, coupled strength c > 0, feedback matrix K ∈ R n×n , and a disturbance attenuation γ > 0, H-infinity bipartite consensus can be achieved in system (2.1) under the control algorithm (2.5) , if there exist some n × n positive definite matrices P > 0, Q > 0, R > 0, Ω > 0 and matrix U ∈ R n×n , such that the following matrix inequalities hold: Remark 5. When σ i → 0 + , it can be concluded that (3.3) can not be satisfied, the data of each sampling time will be transmitted to the neighbor agents, that is, event-triggered scheme reduces to time-triggered scheme, thus bipartite leaderless consensus can be derived based on the sampled control algorithm.
In the following, we will design the reliable protocol to achieve H-infinity bipartite consensus for multi-agent system (2.1) with probabilistic actuator faults.
Case 2: Event-triggered matrices Φ (1) = Φ (2) Suppose that the dynamics of linear multi-agent systems and the related parameters are the same with Case 1. By using Theorem 2, the event-triggered matrix Φ i (i = 2, 3, . . . , 7) can be obtained as follows: Taking t ∈ [0, 20), the simulation results show that only 106 sampled data are sent out, which take 5.3% of the sampled signals. Moreover, it can be computed that our event-triggered scheme can obtain a average sampling period of 0.1887, the release instants and release intervals are illustrated in Figure 5. The curve of x i j (t) (i = 1, 2, . . . , 7; j = 1, 2, 3) are presented in Figure 6, which can be seen H ∞ bipartite consensus is achieved. Figure 5. The release instants and release intervals under the event-triggered scheme for event-triggered matrices Φ (1) = Φ (2) . Figure 6. The trajectories of the state x i j (t)(i = 1, 2, . . . , 6; j = 1, 2, 3) for H ∞ bipartite consensus for event-triggered matrices Φ (1) = Φ (2) . Remark 6. It can be shown from Case 1 that the sampled times are 38 for the event-triggered matrices Φ (1) Φ (2) , for Case 2, the sampled times are 106 for the event-triggered matrices Φ (1) = Φ (2) , obviously, our event-triggered scheme can further reduce the sampled data transmission. From the perspective of average sampling period, in Case 1, the average sampling period is 0.4444, and the average sampling period is 0.1887 in Case 2, which can be seen from Figures 3 and 5, that is, our results are less conservative.

Conclusions
In this paper, a novel event-triggered communication scheme has been proposed to study H-infinity bipartite consensus with external disturbance in the signed networks. Compared with some existing event-triggered results, the proposed event-triggered communication scheme has the advantage to achieve a better performance while saving the limited network resources. A more generalized random variable is introduced to describe probabilistic actuator failure. Some H-infinity bipartite consensus criteria can be derived, the feedback matrices and event-triggered matrices can be obtained by solving some linear matrix inequalities. Notice that the above results are based on the given event-triggered parameters, it would be interesting to further investigate adaptive event-triggered communication scheme, where the triggered parameters can be adjusted with respect to the dynamic errors. The related results can also be extended to the cases of nonlinear multi-agent systems and stochastic multi-agent systems and so on.