## 1 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, [PERSON] [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 state-dependent 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], [PERSON] designed an event-triggered control mechanism determined by a dynamic differential equation. Subsequently, in [32], [PERSON] 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], [PERSON] 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-self-triggered 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 [PERSON] et al. (2013). Robust synchronization of multi-agent systems with uncertain Lur'e-type nonlinear dynamics was studied in [PERSON] 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.