Integral-Type Edge-Event- and Edge-Self-Triggered Synchronization to Multi-Agent Systems with Lur’e Nonlinear Dynamics

: This paper proposes event- and self-triggered control strategies to achieve distributed synchronization for multiple Lur’e systems with unknown static nonlinearities. Firstly, the integral-type edge-event-triggered function is designed here without Zeno behaviors. Compared to the traditional event-triggered schemes, the considered algorithm has the advantages of reducing controller update frequency and sensor energy consumption. Then, the integral-type self-triggered is further investigated, which implements discontinuous monitoring and discontinuous agent listening. Finally, numerical simulations veriﬁed the effectiveness and superiority of our policies.


Introduction
In recent years, distributed control of multi-agent systems has emerged as an essential topic in the field of control theory [1][2][3][4][5][6][7][8][9][10][11][12], and it has been applied in more and more fields such as the coordinated attitude of multiple spacecraft, formation cooperative control, multi-drone coordination, and multiple robotic arms coordination. The main concern on distributed control is how to coordinate all individuals to achieve synchronization and consensus control under information interaction [1,2,5,6]. Generally, sampled-data control is applied to the studies on synchronization and consensus analysis in actual digital computers, among which the most common solution is periodic sampling based on a time-triggered mechanism. However, consensus analysis and performance requirements typically result in conservative choices of the sampling period (i.e., minimum), which consume excessive system resources.
In contrast to specific time intervals, the other is event-triggered sampling control, whose sampling instants are determined by preset triggering conditions [13][14][15][16][17][18][19][20][21][22][23][24][25]. Here, these events are triggered at time points when the norm of a certain measurement error becomes large to the norm of a designed function as time evolves. Earlier studies on event-triggered control include a conference paper on event-triggered control of PID controllers [13]. However, the application and development of event-triggered control theories have not made a good progress for a long time. This is because Zeno behavior often appears in event-triggered control systems, and its related theoretical problems have not been well resolved. Until 2007, Tabuada [14] strictly proved that there is a minimum time-bound between the interval of any two consecutive events to avoid Zeno behaviors, in a class of nonlinear feedback control systems based on state-dependent event-triggered control. Furthermore, self-triggered control strategies were developed in [26][27][28][29][30], in which each agent predicts its next triggering time at any event instant; that is, continuous or periodic event detections can be avoided. Such schemes aim to replace the resource consumption of all sensors with an increase in computational complexity.
The above event-triggered and self-triggered conditions are all designed to be statedependent or decaying exponential function-dependent, which demands that the measurement error needs to meet these conditions at any time. To relax such conservative settings, in [31], Girard designed an event-triggered control mechanism determined by a dynamic differential equation. Subsequently, in [32], Mousavi et al. proposed an integral event-triggered condition, which requires that the measurement error only needs to be less than the given threshold function over a certain integration interval. Then, in [33,34], Zhang et al. extend the integral-type method to first-order linear multi-agent systems. The integral-type event-triggered control relaxed the requirement for measurement error and also relaxed the requirement for decreasing Lyapunov functions, further reducing the controller updates.
In this paper, we study the design problem of integral-type edge-event-and edge-selftriggered control for Lur'e uncertain nonlinear systems with unknown static nonlinearities. First, the research on synchronization of Lur'e systems has greatly promoted the development of control theory [35][36][37][38][39]. Recently, distributed synchronization of multi-agent systems with Lur'e node dynamics was investigated in Wen et al. (2013). Robust synchronization of multi-agent systems with uncertain Lur'e-type nonlinear dynamics was studied in Zhao et al. (2013). Then, most distributed event-triggered control policies require each agent to gather its own information and then communicate with its neighbors to achieve global control goals. In actual scenarios, the accurate measurement of self-state information is often difficult to achieve. For instance, the individual's own position, velocity, and other information often need to be obtained with the help of GPS or inertial sensors using of sophisticated algorithmic measurements. Here, it may be easier for agents to implement coordinated control by using relative state information. Therefore, another state measurement scheme is applied. The relative state information between itself and its neighbors, termed edge state information, is measured by means of relative state sensors such as Lidar or laser speedometer mounted on each individual to achieve the control goal. Quite a number of studies on synchronization and consensus analysis for distributed edge-event-triggered control have been represented [40][41][42].
Hinted by these observations, the integral-type edge-event-and edge-self-triggered control strategies are developed for Lur'e uncertain nonlinear systems in this paper. In the proposed event-triggered strategy, an integral-type triggering function is designed to reduce the controller updates. Moreover, we accomplish the edge-self-triggered control on the above system to avoiding continuous monitoring, in which each controller only updates at certain sampling instants and can effectively reduce controller burden and sensor energy consumption. The main contributions of this work are stated as follows: 1. The dominant motive of this work is to design integral-type edge-event-and edgeself-triggered control strategies for Lur'e uncertain nonlinear systems to seek novel scheduling policies of active sensors. Here, only relative states are employed, while absolute state information is uninvolved. These combined edge-based control policies can provide another approach, where absolute state information is not easily available [16,17,19,20,23]. 2. A distributed integral-type edge-event-triggered control algorithm is designed in the proposed control strategies, which relaxes the setting of the measurement error in the traditional event-triggered strategy [13][14][15][16][17][19][20][21][22][23]25] that needs to meet the trigger conditions at all times. The application of Barbalat's Lemma in proof guarantees the convergence. Compared to the studies on distributed integral-type event-triggered control in [33,34], our results on edge states related to each agent are evaluated asynchronously, based on nonlinear dynamic models. 3. The self-triggering condition does not require continuous or periodic detection of measurement error information, which avoids the periodic monitoring of the sensor [13][14][15][16][17][19][20][21][22][23]25,32,33], further reducing the number of sensor measurement samples and then saving system resources.
The remainder of this paper is organized as follows. Some preliminaries and the problem formulation are given in Section 2. The main results are presented in Sections 3 and 4. Section 5 validates the obtained theoretical results using numerical simulations. Finally, concluding remarks close the paper in Section 6.

Preliminaries on Graph Theory
Consider an undirected graph G with N edges and N nodes, which is denoted by Then the weighted adjacency matrix A ∈ R N×N is used to describe the topology of system (2), in which self-loops and multiple information links are not allowed, where a ij > 0 if (v i , v j ) ∈ E , while a ij = 0, otherwise. The graph Laplacian of G is defined as L = {l ij } ∈ R N×N , in which l ij = −a ij if i = j, and l ij = ∑ j∈N i a ij otherwise. We now introduce several important matrices in the graph theory. The matrix which relates the nodes to the edges is called incidence matrix, denoted by H = {h ki } ∈ R N ×N [23]. By choosing arbitrary orientation for all edges for an undirected graph, the entries of its incidence matrix are defined as h ki = 1 if the kth edge sinks at node i, or h ki = −1 if the kth edge leaves node i, or h ki = 0 otherwise. Therefore, the corresponding Laplacian matrix is represented as L = H T H [4,43].
For the edge ε p ∈ E with the orientation from agents be the corresponding edge state. We say that system (2) reaches the state synchronization if and only if lim t→∞ r p (t) = 0, p = 1, 2, · · · , N , for any initial values r p (t 0 ). The following assumption and lemma are presented: Definition 1. [44] Let S 1 , S 2 ∈ R s×s be real symmetric matrices such that S 1 is positive semidefinite and S 1 − S 2 is positive definite, i.e., 0 ≤ S 1 < S 2 . Then φ(·) is called sector bounded within the sector [S 1 , for all y ∈ R m . Assumption 1. The interaction topology G of system (2) is connected.
Proof. By (1), we have Then we derived that we just need to find the positive constant η that satisfies that where D 1 = η I + (S 1 S 2 + S 2 S 1 ). The above matrix S is semi-positive definite if and only if its contract matrixŜ is semi-positive definite.
based on the fact that D 2 , −η I are invertible matrices. The following, we take Therefore, there's an η here that makes φ(y i ) − φ(y j ) 2 ≤ η y i − y j 2 true. This completes the proof.

Problem Formulation
We consider a event-triggered control of Lur'e systems.
where x i (t) ∈ R n is the state vector of the ith agent, u i (t) ∈ R p is the control input, and y i (t) ∈ R m is the output. Let A, B, B 1 , and C be given constant matrices with appropriate dimensions. The function φ(y i (t)) ∈ R p is defined as follows.
To this end, the focal issue is how to design a distributed protocol u i according to an edge-event-triggered rule, which will be detailed in the next section.

Integral-Type Edge-Event-Triggered Policy
Here, the control strategy will be introduced. The designed policy on combined relative state measurements is given as follows: where with j ∈ N i represent the relative states between v i and all neighboring agents v j with j ∈ N i . We introduce the following integral-type triggering condition. Define the state measurement error between the sampled state and the current state by Each triggering instant t i k is determined by where Here, k i , β and α are positive constants to be designed. (3), the closed-loop system can be represented by where The following theorem presents some sufficient conditions for achieving synchronism tracking of system (2) with integral event-triggered information.
Theorem 1. For a given positive real constant a, η > λ m (S 1 S 2 + S 2 S 1 ), γ = 2k max 1−2k max , k max < 1 2 , if there exists a symmetric positive definite X ∈ R n×n , a matrix Y ∈ R p×n and positive real constants c, such that the linear matrix inequality is negative definite matrix, whereĀ = I N ⊗ (XA T + AX),B = −c[HH T ⊗ (BY + Y T B T )], X = I n − 2X, then the system (7) with K = YX −1 consensus tracking for sector bounded within the sector [I N ⊗ S 1 , I N ⊗ S 2 ] under the integral-type edge-event-triggering condition (6). Furthermore, no Zeno behavior occurs for all t > t 0 .
Proof. Choose the following Lyapunov function: where P is a positive-definite matrix with P = X −1 . Then, taking the time derivative of V(t) along the trajectory of (7) yields where we use the fact that −R T [2(H ⊗ PBK)]e ≤ 1 a R T (HH T ⊗ PBKK T B T P)]R + ae T e for any positive real constant a to derive the above inequality. However, a has to be fixed due to the product of a and X.
In addition, it follows from the event condition (6) that Noting that k max = max i∈{1,2,...,N} k i < 1 2 and z(t) = (H T ⊗ I n )R(t). Then, integrating (10) from t 0 to t, one gets Similarly, by the fact that R T (2(H ⊗ PB 1 ))Φ(y) and then, based on Lemma 1, we get The following, we will prove that (I N ⊗ A T P) is a negative definite matrix, due to the fact that −X 2 ≤ I n − 2X and (I n − X) 2 = I n − 2X + X 2 ≥ 0. By premultipling and postmultipling the matrix inequality with diag(I N ⊗ P, I N ×n , I N×n , I N ×m , I N ⊗ P), along with K = YX −1 .
is negative definite, whereÂ = I N ⊗ A T P + PA,B = −c[HH T ⊗ (PBK + K T B T P)]. Then, using the Schur complement lemma, we obtain that whereĈ = c 2 a (HH T ⊗ PBKK T B T P). Similarly, we have that being equivalent toÂ +B +Ĉ + I N ⊗ PB 1 B T 1 P + aγ(HH T ⊗ I n ) + η C 2 I N ×n < 0. i.e., (I N ⊗ A T P) + (I N ⊗ PA) − c(HH T ⊗ PBK) − c(HH T ⊗ K T B T P) + c 2 a (HH T ⊗ PBKK T B T P) + aγ(HH T ⊗ I n ) + (I N ⊗ PB 1 B T 1 P) + η C 2 I N ×n < 0. Therefore, the above matrix is negative definite, with defined as −P. Then where λ min (P) is the minimum eigenvalue of the matrixP. Inequality (19) yields based on the fact that V(t) ≥ 0, e −αt > 0, As a result, t t 0 R(s) 2 ds has an upper bound. Furthermore, we implies that V(t) is bounded, and hence, V (t) is also bounded. Thus, we can further have that Therefore, the synchronization is achieved. Next, we show that no Zone behavior occurs. If there is a lower bound on the interevent intervals of any agent in the integral-type event condition (6), Zeno-free behavior can be assured. Let us consider the derivative of e i (t) over the interval [t i k , t i k+1 ).
where t j k(t) denotes the latest triggering time instant of agent j for t < t i k , with Then, the derivative of e i (t) 2 satisfies Using the comparison principle, one can further obtain that On the other hand, when triggering condition (6) is satisfied, the next event time t i k+1 is obtained from f i (t) = 0.
The sufficient condition is obtained to guarantee Accordingly, with Equation (25), one obtains Letting τ i = t − t i k and combining with the condition (26), one gets In order to show τ i > 0, we proceed by contradiction. Suppose that τ i < 0, the left side of (27) is negative, whereas the right side of (27) is positive. Furthermore, if τ i = 0, the left side of (27) is zero, which is not equal to k i z i (t i k ) 2 . Equation (27) implicitly defines τ i as a positive lower bound between any two consecutive event instants. Hence, no Zeno behavior occurs. The proof is completed.

Remark 2.
For the triggering condition (6), only the integral of e i (t) 2 from each triggering instant t i k to the time t, needs to be less than a designed variable in time intervals, and e i (t) 2 does not need to be less than a given variable at any time. That is, the proposed mechanism relaxes the requirement in traditional event-triggered control [13][14][15][16][17][19][20][21][22][23]25]. Then, we design the following nonintegral-type event condition where {t i k } k ∈Z ≥0 represents the triggering instant sequence of agent v i . Under the system (2) and control protocol (3), it can achieve synchronization asymptotically, and its minimum inter-event interval is less than Theorem 1.
βe −αs ds in the proposed triggering function, which is different from the results on single nonlinear systems. Therefore, the application of Barbalat's Lemma can guarantee the convergence results.

Integral-Type Edge-Self-Triggered Policy
In the event-triggered formulation, it becomes apparent that continuous monitoring of the measurement error norm e(t) is required to check condition (6). In the following self-trigged control, this requirement is relaxed. Specifically, the next time t i k+1 at which control law is updated is predetermined at the previous event time t i k and no state or error measurement is required in between the control updates. Define an increasing positive sequence k i 0 , k i 1 , · · · , k iq , taking into account the effects of neighboring agent updates, with t i k = t k i 0 and t k iq ≤ t i k+1 . By the inequality (22), the measurement error e i (t) satisfies where and A = A . Solving such inequality in t ∈ [t k i q , t k i q+1 ], q = 0, 1, · · · ,q − 1, we have where M i ], q = 0, 1, · · · ,q − 1. Defined a variable τ i k = τ i k (t) = t − t i k , we can now compute By (6) and (34), the self-triggered function is as follows: For each i = 1, 2, · · · , N the self-triggered ruling defines the next update time as following: if there is a τ i k > 0 such thatf (τ i k ) = 0, the next update time t i k+1 take place at most τ i k time units after t i k . Agent v i also checks the condition (35) when its neighbors updated. Otherwise, if the inequalityf (τ i k ) < 0 holds for all τ i k > 0, then agent v i waits until the next update of the control law of one of its neighbors to recheck its condition.
According to the above analysis, we give Algorithm 1 as follows: Algorithm 1 Integral-Type Edge-Self-Triggered Control Algorithm Step 1. Set the algorithm execution time T . At initial time t = t 0 , let k = 0, t i k = t 0 , q = 0 and k i q = 0. Let z i (t) = z i (t 0 ); update u i (t 0 ) = −cBKz i (t 0 ); compute Ξ i k i q and M i k i q by (31) and (33); compute maximum allowable value τ i k by (35) such thatf (τ i k ) = 0.
Step 2. When 0 < t ≤ T , check the events related to agents v i , v j ; perform the following steps: i. If any agent updated before t i k + τ i k , let q = q + 1; recompute Ξ i k i q and M i k i q by (31) and (33), such thatf (τ i k ) = 0; ii. If there is no agent updated before t i k + τ i k , the event related to agent v j ,

Theorem 2.
Under the assumptions and conditions in Theorem 1, the system (2) with the control law (3) and Algorithm 1 can achieve synchronization for any initial condition in R N . Furthermore, no Zeno behavior occurs for all t > t 0 .

Remark 5.
The designed distributed integral-type edge-event-triggered policy still requires continuous monitoring of measurement errors. The self-triggered control can solve this problem. That is, the equipped sensors only perform intermittent sampling, which can further reduce the number of sensor measurement samples and then save system resources. Based on the self-triggered policies [26][27][28], a distributed integral-type edge-self-triggered algorithm is proposed in this paper. This algorithm, in addition to intermittent updates, schedules sensors measurements intermittently.

Simulations
In this section, we use an example to illustrate the effectiveness of the results developed in above sections. We consider Chua's circuit, which is described by the following system: , belonging to the sector [0, 2], represents the change in resistance vs. Choosing the parameters as h = 0.25, g = 1, m 1 = 3, and m 2 = −1. Then, we can rewrite (36) in the form of a Lur'e system with control φ(y) = |y + 1| − |y − 1|. Its Lur'e-type nonlinearity satisfies the sector bounded condition φ(y)(φ(y) − 2y) ≤ 0 with S 1 = 0 and S 2 = 2. Taking (37) as each agent dynamics, a network of five such agents is shown in Figure 1. The interconnection topology is undirected and connected. and c = 1.0004, respectively. Take k i = 0.1 * 0.01i < 1 2 , β = 0.11, and α = 1.85. Here, all the above parameters satisfy the conditions in Theorem 1. The results of the nonintegraltype edge-event-triggered policy (under condition (29)), the integral-type edge-eventtriggered policy (under condition (5)) and the integral-type edge-self-triggered policy (under Algorithm 1) are shown in Figures 2, 3 and 4, respectively. Tables 1 and 2 are the event numbers and average sampling intervals under the triggering mechanisms (5), (29), and Algorithm 1. Obviously, compared with the nonintegral-type event-triggered control in (29), the integral-type edge-event-triggered control can reduce sampling frequency, thereby reducing system resource consumption. Moreover, the integral-type edge-self-triggered control avoids continuous monitoring of all measurement errors, which further reduces sensor resource consumption at the expense of increased sampling frequency.

Conclusions
In this paper, the synchronization problem of multi-agent systems with Lur'e nonlinear dynamics has been studied via integral-type edge-event-and edge-self-triggered policies. In event-triggered control, the provided LMI condition ensures the convergence of the considered system. Additionally, Zeno-free triggering has also been strictly proven. This study relaxes the conditions of the traditional event-triggered mechanism, thereby resulting in lower sampling frequencies. Besides, the integral-type self-triggering is further designed to avoid continuous monitoring required in integral-type edge-event-triggered control, thereby saving sensor resources. Further studies involve directed graphs, prescribed performance control, and reinforcement learning.