## 3 Main Results
### Design of Event-Triggered Control Protocol
\begin{cases}\|E_{i}(t)\|^{2}-r_{1i}\|\overline{\varphi}_{i}(t)\|^{2 },&i\in I_{m}\\ \|E_{i}(t)\|^{2}-r_{2i}\|\overline{\varphi}_{i}(t)+\overline{\xi}_{i}(t))\|^{2 },&i\in I_{N-m}\end{cases}  \tag{8}\]  
where \(r_{1i},r_{2i}>0\) are the constants, and the measurement error \(E_{i}(t)\) of the \(i\)th agent is given by  
\[E_{i}(t)= \begin{cases}k_{1}\overline{\varphi}_{i}(t_{k}^{i})+\overline{ \varphi}_{i}(t_{k}^{i})^{\frac{p}{q}}-k_{1}\overline{\varphi}_{i}(t)-\overline{ \varphi}_{i}(t)^{\frac{p}{q}},&i\in I_{m}\\ (c+k_{2})(\overline{\psi}_{i}(t_{k}^{i})+\overline{\xi}_{i}(t_{k}^{i}))+( \overline{\psi}_{i}(t_{k}^{i})+\overline{\xi}_{i}(t_{k}^{i}))^{\frac{p}{q}}\\ -(c+k_{2})(\overline{\psi}_{i}(t)+\overline{\xi}_{i}(t))-(\overline{\psi}_{i}(t )+\overline{\xi}_{i}(t))^{\frac{p}{q}},&i\in I_{N-m}\end{cases}  \tag{9}\]  
**Remark 3**. : _Utilizing Lemma 1 enables the conversion of the signed digraph associated with system (1) into an unsigned digraph, thereby transforming the bipartite consensus problem into a common consensus problem._  
Combined with the control protocol (7), measurement error (9), and error term (6), the error dynamical system of system (1) is described by  
\[\left\{ \begin{aligned} \overleftarrow{\overline{\varphi}}_{i}(t)=& -\sum\limits_{j=1}^{m}\overline{l}_{ij}(w_{j}f_{j}(x_{j}(t),t)+w_{j}u_{j}(t)) -b_{i}(w_{i}f_{i}(x_{i}(t),t)+w_{i}u_{i}(t))\\ &-\sum\limits_{j=m+1}^{N}\overline{l}_{ij}w_{i}v_{j}(t),\\ \left\{\overleftarrow{\overline{\psi}}_{i}(t)=& \overline{\xi}_{i}(t),\\ \overleftarrow{\overline{\xi}}(t)=&-\sum\limits_{j=m +1}^{N}\overline{l}_{ij}(w_{j}g_{j}(x_{j}(t),v_{j}(t),t)+w_{j}u_{j}(t))-b_{i} (w_{i}\delta_{i}(x_{i}(t),v_{i}(t),t)\quad i\in I_{N-m}\\ &+w_{i}u_{i}(t)),\end{aligned} \right. \tag{10}\]  
where \(\overline{l}_{ij}\) is defined as \(\overline{l}_{ij}=|a_{ij}|(i\
eq j)\), \(\overline{l}_{ii}=\sum_{k=1,k\
eq i}^{N}|a_{ik}|(i,j=1,2,\ldots,N)\), and \(\tilde{l}_{ij}\) is defined as \(\tilde{l}_{ij}=|a_{ij}|(i,j\geq m+1,i\
eq j)\), \(\tilde{l}_{ii}=\sum_{k=m+1,k\
eq i}^{N}|a_{ik}|\); otherwise, \(\tilde{l}_{ij}=0\)\((i,j=1,2,\ldots,N)\). **Remark 4**. : _The matrix \(\overline{L}=(\overline{l}_{ij})_{N\times N}\) is the Laplacian matrix of the unsigned digraph \(\overline{\mathcal{A}}\) defined in Lemma 1, which is a zero-row-sum Laplacian matrix. The matrix \(\tilde{L}=(\tilde{l}_{ij})_{N\times N}\) is a block-diagonal matrix composed of blocks \(0_{m}\) and \(L_{2}\), where \(L_{2}\) is the Laplacian matrix of the unsigned digraph \(\bar{\mathcal{A}}=(|a_{ij}|)_{(N-m)\times(N-m)}(i,j=m+1,m+2,\ldots,N)\), meaning that \(\bar{\mathcal{A}}\) is the adjacency matrix of the unsigned communication network for all second-order multi-agent systems._  
For the purpose of facilitating the subsequent theoretical analysis, let  
\[ \begin{aligned} &\overline{\varphi}(t)=(\overline{\varphi}_{1}(t), \overline{\varphi}_{2}(t),\ldots,\overline{\varphi}_{m}(t),0,0,\ldots,0)^{ \mathrm{T}},\\ &\overline{\psi}(t)=(0,0,\ldots,0,\overline{\psi}_{m+1}(t), \overline{\psi}_{m+2}(t),\ldots,\overline{\psi}_{N}(t))^{\mathrm{T}},\\ &\overline{\xi}(t)=(0,0,\ldots,0,\overline{\xi}_{m+1}(t), \overline{\xi}_{m+2}(t),\ldots,\overline{\xi}_{N}(t))^{\mathrm{T}},\\ & F(x(t),t)=[f_{1}(x_{1}(t),t),f_{2}(x_{2}(t),t),\ldots,f_{N}(x_{N }(t),t),0,0,\ldots,0]^{\mathrm{T}},\\ & G(x(t),v(t),t)=[0,0,\ldots,0,g_{m+1}(x_{m+1}(t),v_{m+1}(t),t),g _{m+2}(x_{m+2}(t),v_{m+2}(t),t),\\ &\qquad\qquad\qquad\ldots,g_{N}(x_{N}(t),v_{N}(t),t)]^{\mathrm{T}},\\ & E^{1}(t)=(E_{1}(t),E_{2}(t),\ldots,E_{m}(t),0,0,\ldots,0)^{ \mathrm{T}},\\ & E^{2}(t)=(0,0,\ldots,0,E_{m+1}(t),E_{m+2}(t),\ldots,E_{N}(t))^{ \mathrm{T}},\\ & v(t)=(0,0,\ldots,0,v_{m+1}(t),v_{m+2}(t),\ldots,v_{N}(t))^{ \mathrm{T}}.\end{aligned} \]