## 3 Control Protocol Design and Stability Analysis  
In this section, we discuss the situation where we have consensus with exogenous disturbance, which means that \(\sigma=1\). In order to realize the prescribed-time bipartite output consensus of multi-agent systems of the undirected topology (1) with state and disturbance observers, the control protocol is proposed as  
\[u_{i}(t)= -\alpha(k_{1}+k_{2}\frac{\hat{\mu}(t)}{\mu(t)})\sum_{j\in N_{i}}a_ {ij}(\widehat{x}_{i}(t)-s_{ij}\widehat{x}_{j}(t))-\widehat{\omega}_{i}(t), \tag{10}\]  
with the state observer expressed as  
\[\hat{\hat{x}}_{i}(t)=u_{i}(t)+f(\widehat{x}_{i}(t))+\widehat{\omega}_{i}(t)-G _{1}(y_{i}(t)-C\widehat{x}_{i}(t)), \tag{11}\]  
and the disturbance observer expressed as  
\[\hat{\hat{d}}_{i}(t)=H\widehat{\hat{d}}_{i}(t)-G_{2}(\omega_{i}(t)-K\widehat{ \hat{d}_{i}}(t)), \tag{12}\]  
where \(k_{1}>\frac{2+p^{2}}{\lambda_{\min}(CC^{1})}\); \(k_{2}>0\); \(\widehat{\omega}_{i}(t)=K\widehat{\hat{d}_{i}}(t)\) denotes the estimated value of \(\omega_{i}(t)\); \(\widehat{\hat{d}_{i}}(t)\) is the estimated value of \(d_{i}(t)\); and \(G_{1}\) and \(G_{2}\) are observer gain matrices to be designed, respectively. **Remark 2**. : _The design of protocol (10) can achieve bipartite consensus including cooperation and competition without requiring a topological graph to be a balanced graph. Traditional bipartite consensus protocols typically require the communication topology to be a balanced graph (i.e., the in degree equals the out degree for each node) [29] to counteract the sign effects of competitive relationships. Here, by introducing the sign matrix as \(S=\text{diag}\{1,\ldots,1,-1,\ldots,-1\}\), this study directly encodes competitive interactions into the control protocol, eliminating the need for a balanced topology._  
**Theorem 1**. : _Under Assumptions 1 and 2, for any predefined \(T>0\), a prescribed-time bipartite output consensus can be achieved for multi-agent system (1) with \(\sigma=1\) under distributed control protocol (10) if the following apply:_  
_(1) Control parameter \(\alpha\geqslant\frac{\lambda_{\max}(CC^{\mathrm{T}})+\lambda_{\min}(CC^{ \mathrm{T}})}{2\lambda_{2}(L)}\);_  
_(2)_ \(K^{\mathrm{T}}KH+H^{\mathrm{T}}K^{\mathrm{T}}K\leqslant K^{\mathrm{T}}\)_(3) Observer gain matrix_ \(G_{1}=-(k_{1}+k_{2}\frac{\hat{\mu}(t)}{\mu(t)})C^{\mathrm{T}}\)_;_  
_(4) Observer gain matrix_  
\[G_{2}=-\frac{(k_{1}+k_{2}\frac{\hat{\mu}(t)}{\mu(t)})\lambda_{\min}(CC^{ \mathrm{T}})+\lambda_{\max}^{2}(L)\lambda_{\max}(C^{\mathrm{T}}C)}{2\lambda_{ \min}(KK^{\mathrm{T}})}K^{\mathrm{T}}.\]  
Proof. : We convert system (1) into the following compact forms under control protocol (10) with state observer (11):  
\[\left\{ \begin{aligned} \dot{X}(t)=&-\alpha(k_{1}+k_{2} \frac{\hat{\mu}(t)}{\mu(t)})(S^{-1}LS\otimes I_{n})\widehat{X}(t)\\ &+(I_{B+P}\otimes K)(d(t)-\widehat{d}(t))+F,\\ Y(t)=&(I_{B+P}\otimes C)X(t),\end{aligned} \right. \tag{13}\]  
with  
\[\dot{\widehat{X}}(t)= -\alpha(k_{1}+k_{2}\frac{\hat{\mu}(t)}{\mu(t)})(S^{-1}LS\otimes I _{n})\widehat{X}(t)+\widehat{F}\] \[-(I_{B+P}\otimes G_{1}C)(X(t)-\widehat{X}(t)). \tag{14}\]  
Above, \(S=diag\{1,1,\ldots,1,-1,\ldots,-1\}\), \(d(t)=(d_{1}^{\mathrm{T}}(t),d_{2}^{\mathrm{T}}(t),\ldots,d_{N+M}^{\mathrm{T}} (t))^{\mathrm{T}}\), \(\widehat{d}(t)=(\widehat{d}_{1}^{\mathrm{T}}(t),\widehat{d}_{2}^{\mathrm{T}} (t),\ldots,\widehat{d}_{N+M}^{\mathrm{T}}(t))^{\mathrm{T}}\), \(F=(f^{\mathrm{T}}(x_{1}(t)),f^{\mathrm{T}}(x_{2}(t)),\cdots,f^{\mathrm{T}}(x _{N+M}(t)))^{\mathrm{T}}\), and \(\widehat{F}=(f^{\mathrm{T}}\widehat{x}_{1}(t),f^{\mathrm{T}}\widehat{x}_{2}( t),\cdots,f^{\mathrm{T}}\widehat{x}_{N+M}(t)^{\mathrm{T}})\).