# Impulse Controllability for Singular Hybrid Coupled Systems
## 3 Results
**_Consider the singular hybrid coupled system (1) with the following parameters:_  
\(x_{i}(t)=(x_{i}^{(1)},x_{i}^{(2)},x_{i}^{(3)})^{T},f_{i}(x_{i}(t))=(sin(x_{i}^ {(1)}(t)),sin(x_{i}^{(2)}(t)),\)\(sin(x_{i}^{(3)}(t))^{T}\)  
\(b_{11}=-1,b_{12}=1,b_{13}=0,b_{21}=1,b_{22}=-2,b_{23}=1,b_{31}=0,b_{32}=1,b_{3 3}=-1\)_,_  
\(c=1\) _(i,j=1.2.3),_  
\[E=\left[ \begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&0\end{array} \right],A=\left[ \begin{array}{ccc}1&0&0\\ 0&3&0\\ 0&0&7\end{array} \right],\text{and }\Gamma=\left[ \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right].\]  
_The singular hybrid coupled systems (1) are described as nonlinear singular large-scale systems models (2) with the following parameters:_  
\[\bar{E}=\left[ \begin{array}{ccc}E&0&0\\ 0&E&0\\ 0&0&E\end{array} \right],\tilde{A}=\left[ \begin{array}{ccc}A+cb_{11}\Gamma& cb_{12}\Gamma&cb_{13}\Gamma\\ cb_{21}\Gamma&A+cb_{22}\Gamma&cb_{23}\Gamma\\ cb_{31}\Gamma&cb_{32}\Gamma&A+cb_{33}\Gamma\end{array} \right],f(x(t))= \left( \begin{array}{c}f(x_{1}(t),t)\\ f(x_{2}(t),t)\\ f(x_{3}(t),t)\end{array} \right)\]\[B_{1}=\left[ \begin{array}{c}0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right],B_{2}=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right],B_{3}=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right],B_{3}=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right],C_{1}=\left[ \begin{array}{cccccccc}1&0&0&0&0&0&0&0&0&0 \\ 0\\ 0\\ 0\\ 0\end{array} \right],C_{2}=\left[ \begin{array}{cccccccccccc}0&0&0&1&0&0&0&0&0&0 &0\\ 0\\ 0\\ 0\\ 0\end{array} \right],C_{3}=\left[ \begin{array}{cccccccccccc}0&0&0&0&0&0&1&0&0 &0\\ 0&0&0&0&0&1&0&0&0\end{array} \right].\]  
By Lemma 1, we have  
\[f_{i}(x_{i}(t),t)=\left[ \begin{array}{ccc}\cos\alpha(t)&0&0\\ 0&\cos\beta(t)&0\\ 0&0&\cos\gamma(t)\end{array} \right]x_{i},\quad \begin{array}{c}\alpha(t)\in[0,x_{i}^{1}],\\ \beta(t)\in[0,x_{i}^{2}],\\ \gamma(t)\in[0,x_{i}^{3}].\end{array}  \tag{37}\]  
By Theorem 1 considering the nonlinear effect, we have  
\[\text{rank}\bigg{(}\left[ \begin{array}{cc}0&\bar{E}\\ \bar{E}&\bar{A}(h(t))\end{array} \right]\bigg{)}=11,Nn+\text{rank}(\bar{E})=12,\]  
\[\text{rank}\bigg{(}\left[ \begin{array}{cc}0&\bar{E}\\ \bar{E}&\bar{A}(h(t))\end{array} \right]\bigg{)}\
eq Nn+\text{rank}(\bar{E}),\]  
there is the presence of an impulse within the system (2). By Theorem 2 and (37), we have  
\[\max_{ \begin{subarray}{c}k_{1},\ldots,k_{3}\\ l_{1},\ldots,l_{3}\end{subarray} }\bigg{(}\left[ \begin{array}{cccc}0&\bar{E}+ \sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&0\\ \bar{E}+\sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&\bar{A}(h(t))+\sum\limits_{ i=1}^{3}\bar{B}_{i}K_{i}C_{i}&\bar{B}_{j}\end{array} \right]\bigg{)}\] \[= \text{rank}\bigg{(}\left[ \begin{array}{cccc}0&\bar{E}&0&0&B_{ \varphi_{1}}&0\\ \bar{E}&\bar{A}(h(t))&\bar{B}_{j}&\bar{B}_{\varphi}&0&\bar{B}_{\varphi_{2}}\\ 0&\bar{C}_{\mathcal{N}-\varphi}&0&0&0&0\\ 0&\bar{C}_{\varphi-\varphi_{1}}&0&0&0&0\\ \bar{C}_{(N-\varphi)-\varphi_{2}}&0&0&0&0&0\end{array} \right]\bigg{)}\] \[= 7+\] \[\text{rank}\bigg{(}\left[ \begin{array}{cccc}2+\cos\beta(t)&0&1 &0&0&0\\ 0&5+\cos\gamma(t)&0&1&0&0\\ 1&0&1+\cos\beta(t)&0&1&0\\ 0&1&0&-2+\cos\gamma(t)&0&0\\ 0&0&0&1&0&5+\cos\gamma(t)\end{array} \right]\bigg{)}\] \[= 12,\]  
\[Nn+r=9+3=12,\]  
\[\max_{ \begin{subarray}{c}k_{1},\ldots,k_{3}\\ L_{1},\ldots,L_{3}\end{subarray} }\bigg{(}\left[ \begin{array}{cccc}0&\bar{E}+ \sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&0\\ \bar{E}+\sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&\bar{A}(h(t))+\sum\limits_{ i=1}^{3}\bar{B}_{i}K_{i}C_{i}&\bar{B}_{j}\end{array} \right]\bigg{)}=Nn+r, \tag{38}\]  
Equation (38) satisfies the necessary and sufficient algebraic condition of impulse controllability. Furthermore, if condition (38) is met, it is possible to establish a decentralised derivative output feedback controller that ensures the closed-loop system exhibits impulse controllability. \[u_{i}=-L_{i}y_{i}+K_{i}y_{i}+v_{i},\quad\ (i=1,2,3.),\]when_  
\[L=\left[ \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&0\end{array}