## Appendix A  
Introduce the definitions as follows:  
\[ \begin{array}{l}\vspace{0.2 cm}e_{1}=\sum\limits_{j=1}^{N}a_{ij}(t_{goi}-t_ {goi})-Z_{1}\\ \vspace{0.2 cm}e_{2}=\sum\limits_{j=1}^{N}a_{ij}(\frac{x_{1i}}{x_{2i}}w_{ri}- \frac{x_{1i}}{x_{2i^{\prime}}}Z_{2i}-\frac{x_{1i}}{x_{2i^{\prime}}}w_{rj}+ \frac{x_{1i}}{x_{2i^{\prime}}}Z_{2j})\\ \vspace{0.2 cm}\zeta_{i}\triangleq\frac{\alpha_{i}(ct)}{e^{2-i}},\;\;i=1,2 \end{array}  \tag{11}\]  
Define the unknown disturbance as follows:  
\[d_{i}=\sum\limits_{j=1}^{N}a_{ij}(\frac{x_{1i}}{x_{2i^{\prime}}}w_{ri}-\frac{ x_{1i}}{x_{2i^{\prime}}}w_{rj}) \tag{12}\]  
By substituting (11) into (24), one obtains:  
\[ \begin{array}{l}\dot{\zeta}_{1}=-h_{1}(sig^{\alpha}(\zeta_{1})+sig^{\beta} (\zeta_{1}))+\zeta_{2}\\ \dot{\zeta}_{2}=-h_{2}(sig^{2\alpha-1}(\zeta_{1})+sig^{2\beta-1}(\zeta_{1}))+ \dot{\epsilon}\dot{d}_{i}\end{array}  \tag{13}\]  
Let \(\zeta\triangleq[\zeta_{1},\zeta_{2}]^{\mathsf{T}},\dot{\xi}\triangleq g(\zeta ),\Delta=\dot{\epsilon}\dot{d}_{i}\), and \(|\Delta|\leq\epsilon d_{1}\). Step 1. Suppose the disturbance of (22) is zero, define:  
\[g_{\gamma}(\zeta)=\left[ \begin{array}{c}-h_{1}(1+\tau)\text{sig}^{\gamma}( \zeta_{1})+\zeta_{2}\\ -h_{2}(1+\tau^{2})sig^{2\gamma-1}(\eta_{1})\end{array} \right] \tag{14}\]  
If (14) is in the dominant scope of the low-order power function, then \(\gamma=\alpha\), \(\tau=|\zeta_{1}|^{\beta-\alpha}\) (i.e., \(|\zeta_{1}|\leq 1\)). If (14) is in the dominant scope of the high-order power function, then \(\gamma=\beta,\tau=|\zeta_{1}|^{\alpha-\beta}\) (i.e., \(|\zeta_{1}|\geq 1\)). Thus, no matter what the value of \(|\zeta_{1}|\) is, \(g_{\alpha}(\zeta)|_{|\zeta_{1}|\to 1^{-}}=g_{\beta}(\zeta)|_{|\zeta_{1}|\to 1^{+}}\) and \(g(\eta)|_{\Delta=0}\) are continuous. According to the Definitions (4-6) in [37], the function \(g(\zeta)\) is homogeneous in both the 0-limit and \(\infty\)-limit. Because \(\alpha\) and \(\beta\) satisfy \(\alpha\in(0.5,1)\) and \(\beta\in(1,1.5)\), based on the work in [38], the origins of \(g_{0}(\zeta)\) and \(g_{\infty}(\zeta)\) are globally asymptotically stable if their polynomials \(s^{2}+h_{1}s+h_{2}\) are Hurwitz. Moreover, the origin of the system \((g(\zeta)|_{\Delta=0})\) is asymptotically stable if the polynomials \(s^{2}+h_{1}(1+\tau)s+h_{2}(1+\tau^{2})\) are Hurwitz. To allow all the roots of these polynomials in the negative real axis, the value of \(h_{1},h_{2}\) should be large enough and should satisfy:  
\[\left\{h_{1}\in\mathbb{R}_{+},h_{2}\in\mathbb{R}_{+}\Big{|}h_{1}\geq 2\sqrt{h_{2 }}\right\} \tag{15}\]  
Let the real numbers \(d_{v_{0}}\) and \(d_{v_{\infty}}\) satisfy:  
\[ \begin{array}{l}d_{v_{0}}=2\alpha\geq\max\{1,\alpha\}\\ d_{v_{\infty}}=2\beta\geq\max\{1,\beta\}\end{array}  \tag{16}\]  
It is known that \(g_{0}(\zeta),g_{\infty}(\zeta)\), and \(g(\zeta)|_{\Delta=0}\) are globally asymptotically stable, according to Definition 5 in [30]. There exists a continuously differentiable, positive definite,and radially unbounded Lyapunov function \((V(\zeta):\mathbb{R}^{2}\to\mathbb{R}_{+}\cup\{0\}\) of \(g(\zeta)|_{\Delta=0})\), which is homogeneous in the bilimit with associated triples: \((\tau_{0},d_{V_{0}},V_{0}(\zeta))\) and \((\tau_{\infty},d_{V_{\infty}},V_{\infty}(\zeta))\). In addition, \(\dot{V}(\zeta)\) is negative definite and homogeneous in the bilimit with associated triples: \(\big{(}\tau_{0},d_{V_{0}},V_{0}(\zeta)\big{)}\) and \((\tau_{\infty},d_{V_{\infty}},V_{\infty}(\zeta))\). Define the following functions:  
\[ \begin{array}{l}\phi(\zeta)\triangleq V(\zeta)^{(dv_{0}+d_{0})/dv_{0}}+V( \zeta)^{(dv_{\infty}+d_{\infty})/dv_{\infty}}\\ \varsigma(\zeta)\triangleq-V(\zeta)=-\left.\frac{\partial V(\zeta)}{\partial \zeta}g(\zeta)\right|_{\Delta=0}\end{array}  \tag{100}\]  
where \(\phi(\zeta)\) is homogeneous in the bilimit with associated triples: \(\big{(}\tau_{0},d_{V_{0}}+d_{0},V(\zeta)^{(dv_{0}+d_{0})/dv_{0}}\big{)}\) and \(\big{(}\tau_{\infty},d_{V_{\infty}}+d_{\infty},V(\zeta)^{(dv_{0}+d_{0})/dv_{0}} \big{)}\).