## 4 Stability Analysis  
In this section, we will give the main results with the designed fully distributed controller and present the stability analysis. The main results of this article are as follows. **Theorem 1**. : _Consider the closed-loop system consisting of \(N\) uncertain agents as (1) satisfying Assumptions 1-3, the intermediate control signals (21) and the distributed controller (22). Then, we have the following properties:_  
1. All the signals in the closed-loop system are globally bounded
2. _Prespecified tracking performance can be guaranteed, namely,_ \(|e_{i,1}|<1\)_, for_ \(i=1,2,\ldots,N\)_._
3. _The output of each agent ultimately satisfies_ \(|y_{i}-y_{d}|\leq k_{i}(t)\)_._  
**Proof (of Theorem 1). ** From (18), (19) and (21), we have  
\[x_{i,1}=k_{i}e_{i,1}+q_{i,1}+\sigma(t)(x_{i,1}^{0}-q_{i,1}^{0}) \tag{23}\]  
\[x_{i,m}=k_{i}e_{i,m}+\alpha_{i,m-1}(e_{i,m-1})+\sigma(t)x_{i,m}^{0},\ m=2, \ldots,n \tag{24}\]  
It can be observed from (23) that \(x_{i,1}\) is continuous function of \(e_{i,1}\), \(q_{i,1}\) and \(\sigma(t)\), where \(q_{i,1}\) and \(\sigma(t)\) are bounded time-varying functions. Thus, \(x_{i,1}\) can be rewritten as the form of continuous function of \(e_{i,1}\) and \(t\). Similar analysis can be made for \(x_{i,m}\). Therefore, we obtain  
\[ \begin{array}{l}\dot{e}_{i,1}=\frac{1}{k_{i}}(f_{i,1}(x_{i,1})+g_{i,1}(x_{i,1})x_{i,2}-q_{i,2}-\dot{\sigma}(t)(x_{i,1}^{0}-q_{i,1}^{0})-\dot{k}_{i}e_{i,1 }+d_{i,1}(t,x_{i,1})\\ =h_{i,1}(t,e_{i,1},e_{i,2},\dot{\sigma}_{i})\end{array}  \tag{25}\]  
\[ \begin{array}{l}\dot{e}_{i,m}=\frac{1}{k_{i}}(f_{i,m}(\overline{x}_{i,m})+g_ {i,m}(\overline{x}_{i,m})x_{i,m+1}-\frac{\partial\kappa_{i,m-1}}{\partial \overline{x}_{i,m-1}}h_{i,m-1}(t,e_{i,1},\ldots,e_{i,m})\\ \hskip 14.226378 pt-\dot{\sigma}(t)x_{i,m}^{0}-\dot{k}_{i}e_{i,m}+d_{i,m}(t, \overline{x}_{i,m}))\\ =h_{i,m}(t,e_{i,1},\ldots,e_{i,m+1})\end{array}  \tag{26}\]where \(e_{i,n+1}=0\), and \(h_{i,m}(\cdot),\ m=1,2,\ldots,n\) are some continuous functions. Defining \(e_{i}=\left[e_{i,1},\ldots,e_{i,n}\right]^{T}\) and in view of (25) and (26), we obtain  
\[\dot{e}_{i}=h_{i}(t,e_{i})=\left[ \begin{array}{c}h_{i,1}(t,e_{i,1})\\ h_{i,2}(t,e_{i,1},e_{i,2})\\ \vdots\\ h_{i,n}(t,e_{i,1},\ldots,e_{i,n})\end{array} \right] \tag{27}\]  
Let us define the open set:  
\[\Omega_{e}=\underbrace{(-1,1)\times\cdots\times(-1,1)}_{n-times} \tag{28}\]  
It is easily observed that \(e_{i}(0)\in\Omega_{e},i=1,2,\ldots,N\). Additionally, \(h_{i,m}(\cdot)\), \(m=1,2,\ldots,n\) are continuous with respect to all its variables, owing to the fact that \(y_{d},\dot{y}_{d},\sigma(t),q_{i,1},k_{i}(t),f_{i,m},g_{i,m},a_{i,m}\) are all continuous differentiable functions. Therefore, it follows from [32] that the conditions on \(h_{i,m}(\cdot)\) ensure the existence and uniqueness of a maximal solution \(\eta_{i}(t)\) on the time interval \([0,t_{\max})\), namely, \(e_{i}(t)\in\Omega_{e}\) for \(t\in[0,t_{\max})\), which implies  
\[e_{i,m}\in(-1,1),for\ \forall t\in[0,t_{\max}) \tag{29}\]  
for \(i=1,\ldots,N\), and \(m=1,\ldots,n\). In the following, we will prove that \(t_{\max}=+\infty\) by seeking a contradiction. Suppose that \(t_{\max}<+\infty\); then the related analysis is performed as follows, and all of what follows is based on \(t\in[0,t_{\max})\). Step 1: Consider the following positive definite functions  
\[V_{i,1}=\frac{1}{2}\log\frac{1}{1-e_{i,1}^{2}} \tag{30}\]  
Let \(\xi_{i,1}=\frac{1}{1-e_{i,1}^{2}}\). It follows from (21), (24), (25) and (30) that the time derivative of \(V_{i,1}\) is  
\[ \begin{array}{l}\dot{V}_{i,1}=\frac{\xi_{i,1}}{k_{i}}\big{(}f_{i,1}(x_{i,1}) +g_{i,1}(x_{i,1})(a_{i,1}+k_{i}e_{i,2}+\sigma(t)x_{i,2}^{0})+\\ \qquad-q_{i,2}-\dot{\sigma}(t)(x_{i,1}^{0}-q_{i,1}^{0})-\dot{k}e_{i,1}+d_{i,1} (t,x_{i,1})\big{)}\\ \qquad\leq\frac{1}{k_{i}}\xi_{i,1}(x_{i,1})a_{i,1}\xi_{i,1}+F_{i,1}(t)|\xi_{i,1}|\\ \qquad\leq-\lambda_{i,1}E_{i,1}\xi_{i,1}^{2}+F_{i,1}|\xi_{i,1}|\end{array}