###### Abstract  
The appointed-time leader-following consensus problem for second-order multi-agent systems with external disturbance on directed graphs is addressed. A distributed controller based on the cumulative position difference and cumulative velocity difference is proposed, which does not require prior knowledge of external disturbances. It is shown that the proposed controller guarantees the prescribed performance of the controlled systems, namely, to keep the cumulative position difference within a predefined boundary envelope. Furthermore, by employing a novel performance function, it is ensured that the position tracking error converges to an arbitrarily small expected region within the appointed time. Different from most existing finite/fixed-time control methods, here the settling time and the convergence region can both be predefined, which are also independent of initial conditions and the system parameters. Finally, the theoretical results are verified by simulations. l [applied]Applied Sciences  
A][PERSON], Z][PERSON] and [PERSON]  
[a]Check for updates  
[a][PERSON], H, [PERSON], Z]Z; [PERSON], W.  
[a]Applied-Time Leader-Following Consensus for Second-Order Multi-Agent Systems with Prescribed Performance Guarantees. _Appl. Sci._**2023**, _13_, 5937. [[https://doi.org/10.3390/appJ13105937](https://doi.org/10.3390/appJ13105937)]([https://doi.org/10.3390/appJ13105937](https://doi.org/10.3390/appJ13105937))  
[a]Accademic Editors: [PERSON] and [PERSON]  
[a]Received: 24 March 2023  
[a]Received: 4 May 2023  
[a]Accepted: 9 May 2023  
[a]Published: 11 May 2023  
[a]Applied-Time Leader-Following Consensus for Second-Order Multi-Agent Systems with Prescribed Performance Guarantees  
such as strict feedback systems, non-strict-feedback systems and multi-degree-of-freedom robot systems, which can be found in [14; 15; 16; 17; 18; 19]. Moreover, PPC has also been applied in MASs. In [20; 21], PPC is applied to linear MASs with guaranteed prescribed performance to address the average consensus on undirected graphs. Subsequently, scholars began to study leader-following consensus in MASs under directed graphs to ensure prescribed performance [22; 23; 24; 25; 26]. [PERSON] and [PERSON] designed a decentralized control law based on the Lyapunov theory to achieve leader-following consensus with guaranteed prescribed performance, for uncertain nonlinear MASs in [22]. PPC-based controllers were designed based on a backstepping method to address the consensus tracking problem for nonlinear MASs in [23; 24; 25; 26]. Considering the agents' state may be unmeasured, observers were designed to estimate state variables in [23; 25; 26]. However, applying the backstepping method may lead to the computational complexity explosion problem because of the repeated differentiation of virtual controllers at each step. In fact, we need a controller with lower complexity; in other words, we need the generation of control signals that only requires simple calculations. When analyzing and dealing with consensus problems, in addition to ensuring that the control variable can be predefined with maximum allowable overshoot and minimum convergence rate, whether the consensus can be achieved in a finite-time is also an important issue. Because in some scenarios, such as military drone operations, it is necessary to ensure that the agents achieve formation consensus at the appointed time. Therefore, this topic has also attracted the attention and research of many scholars. The distributed finite-time consensus protocol was firstly proposed by [PERSON] [27]. Subsequently, various finite-time controllers for MASs have been designed in the current works [28; 29; 30]. However, in these studies, the convergence time depends on the initial states of agents, which means that if the initial states are not known a priori, and the convergence time for the consensus can not be estimated.