## 2 Problem Statement and Preliminaries  
Consider a class of uncertain MAS as follows  
\[\left\{ \begin{array}{l}\dot{x}_{i,m}=g_{i,m}(\overline{x}_{i,m})x_{i,m+1}+f_{i,m }(\overline{x}_{i,m})+d_{i,m}(t,\overline{x}_{i,m}),\ m=1,2\ldots,n-1\\ \dot{x}_{i,m}=g_{i,m}(\overline{x}_{i,n})u_{i}+f_{i,n}(\overline{x}_{i,n})+d_{ i,n}(t,\overline{x}_{i,n})\\ y_{i}=x_{i,1},\ for\ i=1,2,\ldots,N\end{array} \right. \tag{1}\]  
where \(\overline{x}_{i,m}=[x_{i,1},x_{i,2},\ldots,x_{i,m}]^{T}\in R^{m}\), \(y\in R\), \(u\in R\), are the states, the control input and the output of the \(i\) th subsystem, respectively. The system nonlinearities \(f_{i,m}(\cdot),g_{i,m}(\cdot):R^{m}\times R^{+}\to R\) are unknown continuous functions with respect to \(\overline{x}_{i,m}\). \(d_{i,m}(t,\overline{x}_{i,m})\), \(m=1,2\ldots,n\) represent the system uncertainties and external disturbances. The desired trajectory for the outputs of the subsystems \(y_{d}\) is bounded and only known by part of the \(N\) subsystems, with \(\dot{y}_{d}\) being bounded and unknown to all subsystems. Suppose that the information transmission condition among the group of \(N\) subsystems can be represented by a directed graph \(G\triangleq(V,E)\), where \(V=\{1,\ldots,N\}\) denotes the set of indexes corresponding to each subsystem. The edge \((i,j)\in E\) indicates that subsystem \(j\) could obtain information from subsystem \(i\), but not necessarily vice versa. In this case, subsystem \(j\) is called a neighbor of subsystem \(i\), and vice versa. Denoting the set of neighbors for subsystem \(i\) as \(N_{i}\triangleq\{j\in V:(j,i)\in E\}\). Self-edging \((i,i)\) is not allowed, thus \((i,i)\
otin E\) and \(i\
otin N_{i}\). The connectivity matrix \(A=[a_{ij}]\in R^{N\times N}\) of \(G\) is defined as \(a_{ij}=1\) if \((j,i)\in E\) and \(a_{ij}=0\) if \((j,i)\
otin E\). An in-degree matrix \(\Delta\) is introduced, such that \(\Delta=diag(\Delta_{i})\in R^{N\times N}\) with \(\Delta_{i}=\sum_{j\in N_{i}}a_{ij}\) being the \(i\) th row sum of \(A\). Then, the Laplacian matrix of \(L\) is defined as \(L=\Delta-A\). Defining \(B=diag\{\mu_{1},\mu_{2},\ldots,\mu_{N}\}\), where \(\mu_{i}=1\) means the \(y_{d}\) is accessible directly by subsystem \(i\), and otherwise, we have \(\mu_{i}=0\). Throughout this paper, the following notations are used. \(\|\cdot\|\) is the Euclidean norm of a vector. Letting \(a\in R^{n}\) and \(b\in R^{n}\) be two vectors, then define the vector operator \(. *\) as \(a.*b=[a(1)b(1),\ldots,a(n)b(n)]^{T}\). Letting \(Q\) be a matrix, \(\lambda_{\min}(Q)\) then denotes the minimum eigenvalue of \(Q\). The directed graph \(G\) contains a spanning tree, and the desired trajectory \(y_{d}\) is accessible to at least one subsystem, i.e., \(\sum_{i=1}^{N}\mu_{i}>0\). There exist unknown local Lipschitz functions \(b_{i,m}(\overline{x}_{i,m})\) such that, for \(i=1,2,\ldots,N\)  
\[|d_{i,m}(t,\overline{x}_{i,m})|\leq b_{i,m}(\overline{x}_{i,m}),m=1,2,\ldots,n \tag{2}\]  
The unknown control coefficients \(g_{i,m}(\overline{x}_{i,m})\) is strictly positive or negative. Without a loss of generality, it is assumed to be strictly positive, namely, for \(i=1,2\,,\ldots,N\)  
\[g_{i,m}(\overline{x}_{i,m})>0,m=1,2,\ldots,n \tag{3}\]  
(Ref. [17]) Based on Assumption 1, the matrix \((L+B)\) is nonsingular. Defining  
\[ \begin{split}&\theta=[\theta_{1}\,,\ldots,\theta_{N}]^{T}=(L+B)^{-1}[ 1,\ldots,1]^{T}\\ & P=diag\{P_{1}\,,\ldots,P_{N}\}=diag\Big{\{}\frac{1}{\theta_{1} },\ldots,\frac{1}{\theta_{N}}\Big{\}}\\ & Q=P(L+B)+(L+B)^{T}P\end{split}  \tag{4}\]  
then \(\theta_{i}>0\) for \(i=1,2,\ldots,N\) and \(Q\) is definitely positive.