###### Abstract
To relax such a limitation, in [31; 32], the fixed-time control was realized, and the upper bound of the settling time can be determined by some state-independent parameter by using the terminal sliding mode control technology. The cost is that the controller will be discontinuous, and in this case, it is difficult to be effectively applied in practical engineering. Therefore, it is necessary to find other control schemes, so that the controller is continuous, making it suitable for a wide range of practical engineering applications. In addition, most of the PPC methods in the literature usually use traditional exponential performance functions to quantitatively describe the transient and steady-state performance of the controlled systems, and sometimes it is difficult to guarantee that the controlled variable achieves the expected accuracy in fixed time. Motivated by the above issues, we propose a PPC method to achieve leader-following consensus for the second-order MASs in this paper. The specified performance index is described as error constraints, and the constrained MAS is transformed into an unconstrained system by using the transformation error function. The controller of each agent only uses the position and velocity information of the neighbor agent and itself. The main contributions are given as follows. * Compared with the works [28; 29; 30; 33; 34], where the consensus error accuracy cannot be specified a priori and the overshoot was neglected, the proposed controller based on PPC methods in this paper can guarantee the transient performance constraint, which means that the cumulative position difference evolves within a predefined region and the position tracking error can converge to a predefined arbitrarily small residual set within the appointed-time. The appointed time in the paper can be chosen arbitrarily. * Compared with [22; 23; 24; 25; 26], we successfully drive the cumulative position difference to fall into the expected region in the appointed time by using a novel performance function. Furthermore, compared with the backstepping method in [23; 24; 25; 26], our proposed scheme has low complexity. * Compared with the work on sliding mode control-based fixed-time control [31; 32], we only need to adjust several design parameters and the proposed controller is continuous,which it is more applicable for practical engineering. * The control input of each agent is based on the position and velocity information of the neighbor agents, without the prior knowledge of the external disturbance of the dynamics of agents. The rest of this paper is arranged as follows. In Section 2, the basic knowledge of graph theory, the core idea of PPC design, and the description of the problem studied in this paper are introduced. The main results are presented in Section 3, which includes the proposed controller and the proof of leader-following appointed-time consensus with prescribed performance. Simulation examples are presented in Section 4. Some conclusions are drawn in Section 5. Notations: Throughout this paper, \(\mathbb{R}^{i}\) denotes the i-dimensional Euclidean space; \(|\cdot|\) is the absolute value of a real number; \(\|\cdot\|\) is the Euclidean norm of a vector; \(\lambda_{min}(\cdot)\) and \(\lambda_{max}(\cdot)\), respectively, denote the smallest eigenvalue of a matrix and the largest eigenvalue of a matrix; matrix \(K>0\) means \(K\) is positive definite. An M-matrix \(\Lambda\in R^{n\times n}\) can be expressed in the form \(\Lambda=sI-D\), where \(D=(d_{ij})_{n\times n}\) with \(d_{ij}\geq 0\) for \(\forall i,j\), and \(s>\rho(D)\), with \(\rho(D)\) being the spectral radius of matrix \(D\).