###### Abstract  
In this study, we developed a framework for achieving prescribed-time bipartite output consensus in nonlinear multi-agent systems by introducing a distributed observer approach. We propose a novel control protocol that utilizes a bipartite projective parameter and allows the settling time to be predetermined within a physically feasible range, independent of the system parameters and initial conditions. By using a time-varying scaling function (\(\mu(t)\)), the prescribed time may not depend on initial conditions and system parameters and covers more complex, nonlinear multi-agent systems. State and disturbance observers are co-designed to estimate unmeasurable states and reject external perturbations. Sufficient conditions for scenarios regarding the presence of exogenous disturbances are established by using algebraic graph theory, nonlinear system theory, Lyapunov stability, and matrix theory. Finally, numerical simulations verify the effectiveness of the designed protocol and the correctness of the proposed theorem. p +
Footnote †: : 2025: 15, 3620  
group. The bipartite consensus theory presented in [17] was utilized to develop a novel multi-agent secure image communication system. The leading agents and following agents responsible for image transmission can produce consistent encryption and decryption keystreams. The experimental results demonstrate that the newly constructed multi-agent secure image communication system exhibits outstanding performance in terms of statistical properties, correlation, key sensitivity, resistance to differential attacks, and information entropy analysis. In the early stages of research on consensus in multi-agent systems, more attention was paid to the asymptotic stability of consensus. With the advancement of contemporary society, scholars have increasingly focused on finite-time consensus in MASs due to its potential to achieve rapid convergence within a specified time frame [18; 19; 20; 21]. Nevertheless, the finite-time consensus presented in the abovementioned literature studies exhibits certain limitations, as the designated finite times are intricately dependent on the initial conditions and parameter configurations of the multi-agent system. In response to these limitations, the concept of fixed-time consensus, which operates independently of initial conditions, thereby enabling the system to reach consensus within a predetermined fixed time, was introduced [22; 23; 24]. However, it is essential to recognize that fixed time is still constrained by the influence of system parameters and, consequently, may not always be set arbitrarily. To address the aforementioned shortcomings, the concept of arbitrary prescribed time has been proposed. This framework offers the advantage of allowing the convergence time of MASs to be explicitly predetermined, independently of both the initial conditions and system parameters [25]. Furthermore, the authors of [25] introduced a novel time-varying scalar function designed for control systems that ensures that system stability can be achieved within the designated preset time. The measurement of system state information is largely influenced by the precision of instruments and external interference. There may be significant errors between the measured data and the actual state. Thus, designing a proper observer-based protocol is a promising approach to solve the formation control problem. The observer-based method has been successfully applied across various fields [26], and numerous researchers have utilized observers for linear systems, yielding favorable results. However, more in-depth research is needed in the area of nonlinear systems. The operation of multi-agent system is inevitably affected by the external environment. Interference signals generated by external systems can significantly impact the consensus of multi-agent systems [27]. Therefore, it is essential to consider the effects of exogenous disturbances when addressing the consensus issues of multi-agent systems.