# Impulse Controllability for Singular Hybrid Coupled Systems
## 2 Notations and Preliminaries  
This section serves to re-examine some of the fundamental symbols and findings presented previously. The set \(\mathbf{R}^{\ n\times m}(C^{n\times m})\) is defined as the collection of \(n\times n\) real (complex) matrices. Let us consider the transpose of \(M\), denoted by \(M^{T}\), as the matrix with elements \([m_{ij}]_{n\times m}\), where \(M\) is an \(nm\) matrix belonging to \(R_{nm}\). Similarly, let us denote the identity matrix as \(I_{m}\). Let \(\max_{K}(M)\) be used to indicate the general rank of matrix \(M\), which represents the highest rank that can be achieved by matrix \(M\) with a specific variable set \(K\). Let \(\mathcal{N}\) be defined as a collection of the natural numbers from \(1\) to \(N\), inclusive. Let \(\varphi\) be a non-empty subset of \(N\) with elements arranged in ascending order, i.e., \(\{1,2,...,j-1,j+1,...,N\}\) and \(\varphi\). In this context, \(B_{\varphi}\) and \(B_{\varphi}\) are defined as follows:  
\[C_{\varphi}=\big{[} \begin{array}{cccc}C_{i_{1}}&C_{i_{2}}&\cdots&C_{i_{s}} \end{array} \big{]}^{T},\]  
and  
\[B_{\varphi}=\big{[} \begin{array}{cccc}B_{i_{1}}&B_{i_{2}}&\cdots&B_{i_{s}} \end{array} \big{]}.\]  
Furthermore, \(P(\mathcal{N})\) denotes the power set of \(\mathcal{N}\), encompasses the complete collection of all subsets of the set \(\mathcal{N}\), and  
\[\mathcal{N}-\varphi=\{x:x\in\mathcal{N},x\
otin\varphi\}.\]  
Let  
\[S_{c}=\{K:K=block\;diag[K_{1},K_{2},...,K_{N}],K_{i}\in\mathbf{R}^{l\times(Nn )},i\in\mathcal{N}\}.\]  
Examine a category of singular hybrid coupled systems of the following form :  
\[ \begin{cases}E\dot{x}_{i}(t)=Ax_{i}(t)+f_{i}(x_{i}(t))+c\sum_{j=1}^{N}b_{ij} \Gamma x_{j}(t)+Bu_{i},\\ y_{i}=\bar{C}_{i}x(i=1.2,\cdots,N)\end{cases}  \tag{1}\]  
where the matrix \(E\) is found to be non-invertible, \(0<\text{rank}(E)=r<n\). \(A\in\mathbf{R}^{n\times n}\), \(B\in\mathbf{R}^{n\times l}\), and \(\bar{C}\in\mathbf{R}^{l_{i}\times Nn}\) are the constant matrices. Let \(N\) represent the quantity of interconnected nodes. \(x_{i}=(x_{i1},x_{i2},...,x_{in})^{T}\in\mathbf{R}^{n}\) are state variables of node  
\[\bar{B}_{i}=\left[ \begin{array}{c}0\\ \vdots\\ B\\ \vdots\\ 0\end{array} \right]\gets i_{i}f(x(t))=\left[ \begin{array}{c}f_{1}(x_{1}(t)) \\ f_{2}(x_{2}(t))\\ \vdots\\ f_{N}(x_{N}(t))\end{array} \right],x=\left[ \begin{array}{c}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{N}(t)\end{array} \right],u=\left[ \begin{array}{c}u_{1}\\ u_{2}\\ \vdots\\ u_{N}\end{array} \right].\]  
System (2) is a nonlinear singular large-scale systems model with n dimensions. **Lemma 1** ([22]). : _Suppose \(f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{q}\), with \(a,b\in\mathbf{R}^{n}\). It is assumed that \(f\) is differentiable on \(Co(a,b)\). Then, there are constant vectors \(c_{1},c_{2},\cdots,c_{q}\in Co(a,b),c_{i}\
eq a,c_{i}\
eq b\) for \(i=1,2,\cdots,q\), \(Co(a,b)=\{\lambda a+(1-\lambda)b,\lambda\in[0,1]\}\) is a convex hull. such that_  
\[f(a)-f(b)=\left(\sum_{i,j=1}^{q,n}e_{q}(i)e_{n}^{T}(j)\frac{\partial f_{i}}{ \partial x_{j}}(c_{i})\right)(a-b). \tag{3}\]  
We set  
\[A_{ii}=A+cb_{ii}\Gamma\quad i=1,...,N,\]  
and  
\[A_{ij}=cb_{ij}\Gamma\quad i\
eq j,\;i,j=1,...,N.\]  
By Lemma 1 and noting \(f(0)=0\), the nonlinear systems (2) can be transformed into linear singular large-scale systems, which can be represented in the following form:  
\[\left\{ \begin{aligned} \dot{E}\dot{x}_{i}(t)&=( \bar{A}+\Sigma)x(t)+\sum_{j=1}^{N}\bar{B}_{i}u_{i}\\ y_{i}&=\bar{C}_{i}x(i=1.2,\cdots,N)\end{aligned} \right. \tag{4}\]  
where  
\[\bar{A} = \left[ \begin{array}{cccc}A_{11}&A_{12}&\cdots&A_{1N}\\ A_{21}&A_{22}&\cdots&A_{23}\\ \vdots&\vdots&\ddots&\vdots\\ A_{N1}&A_{N2}&\cdots&A_{NN}\end{array} \right], \tag{5}\] \[\Sigma = \left[ \begin{array}{cccc}\Sigma_{11}&\Sigma_{12}&\cdots&\Sigma_ {1n}\\ \Sigma_{21}&\Sigma_{22}&\cdots&\Sigma_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ \Sigma_{n1}&\Sigma_{n2}&\cdots&\Sigma_{nn}\end{array}