# Impulse Controllability for Singular Hybrid Coupled Systems
## 3 Results
\right],\]  
\(K_{i}\)\((i=1,2,3)\) _is the arbitrary parameters matrix._  
_We have_  
\[\text{rank}\bigg{(}\left[ \begin{array}{ccc}0&\bar{E}+\sum\limits_{i=1}^{3} \bar{B}_{i}L_{i}\bar{C}_{i}&0\\ \bar{E}+\sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{A}(h(t))+\sum \limits_{i=1}^{3}\bar{B}_{i}K_{i}\bar{C}_{i}&\bar{B}_{j}\end{array} \right]\bigg{)}\] \[= 7+\text{rank}\bigg{(}\left[ \begin{array}{cccccc}2+\cos\beta(t )&0&0&1&0&0\\ 0&5+\cos\gamma(t)&0&0&1&0\\ 1&0&1+\cos\beta(t)&0&1&0\\ 0&1&0&1+\cos\gamma(t)&0&1\\ 0&0&0&0&1&5+\cos\gamma(t)\end{array} \right]\bigg{)}\] \[= 12,\]  
\[Nn+r\] \[= 9+\text{rank}\bigg{(}\left[ \begin{array}{cccccc}E+B_{1}K_{1} C_{1}&0&0\\ 0&E+B_{2}K_{2}C_{2}&0\\ 0&0&E\end{array} \right]\bigg{)}\] \[= 9+\text{rank}\bigg{(}\left[ \begin{array}{cccccc}1&0&0&0&0&0&0&0 &0\\ 1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\end{array} \right]\bigg{)}\] \[= 12,\]  
\[\max_{ \begin{subarray}[]{c}K_{1},\ldots,K_{3}\\ L_{1},\ldots,L_{3}\end{subarray} }\bigg{(}\left[ \begin{array}{cccccc}0&\bar{E }+\sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&0\\ \bar{E}+\sum\limits_{i=1}^{3}\bar{B}_{i}L_{i}C_{i}&\bar{A}(h(t))+\sum\limits_{ i=1}^{3}\bar{B}_{i}K_{i}C_{i}&\bar{B}_{j}\end{array} \right]\bigg{)}=Nn+r.\]  
_based on the derivative feedback controller used, it can be inferred that the closed-loop system is impulse controllable._  
**Example 2**. : _Using the parameters as in [4], consider the singular hybrid coupled systems (1) with the following parameters:_  
\(x_{i}(t)=(x_{i}^{(1)},x_{i}^{(2)})^{T}\)_,_ \(f_{i}(x_{i}(t))=(\frac{1}{\Gamma 5}tanh(x_{i}^{(1)}(t))\)_,_ \(\frac{1}{\Gamma 5}tanh(x_{i}^{(2)}(t)))^{T}\)__  
\(b_{11}=-5,b_{12}=1,b_{13}=1,b_{14}=1,b_{15}=1,b_{16}=1,b_{21}=1,b_{22}=-4,b_{2 3}=1,b_{24}=1,b_{25}=1,b_{26}=0,b_{31}=1,b_{32}=1,b_{33}=-4,b_{34}=1,b_{35}=0,b _{36}=1,b_{41}=1,b_{42}=1,b_{43}=1,b_{44}=-4,b_{45}=1,b_{46}=0,b_{51}=1,b_{52}= 1,b_{53}=0,b_{54}=1,b_{55}=-4,b_{56}=1,b_{61}=1,b_{62}=0,b_{63}=1,b_{64}=0,b_{65 }=1,b_{66}=-3,\)__  
\(c=1\) (\(i,j=1,2,3,4,5,6\))_,_  
\[E=\left[ \begin{array}{cc}8&0\\ 0&0\end{array} \right],A=\left[ \begin{array}{cc}5&1\\ 1&5\end{array} \right],\Gamma=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right],\]the singular hybrid coupled systems (1) are described as nonlinear singular large-scale systems models (2) with the following parameters:_  
\[\bar{E}=\left[ \begin{array}{cccccc}E&0&0&0&0&0\\ 0&E&0&0&0&0\\ 0&0&E&0&0&0\\ 0&0&0&E&0&0\\ 0&0&0&0&E&0\\ 0&0&0&0&0&E\\ \end{array} \right],f(x(t))=\left( \begin{array}{c}f(x_{1}(t),t)\\ f(x_{2}(t),t)\\ f(x_{3}(t),t)\\ f(x_{4}(t),t)\\ f(x_{5}(t),t)\\ f(x_{6}(t),t)\\ \end{array} \right),\]  
\[\bar{A}=\left[ \begin{array}{cccccc}A+cb_{11}\Gamma&cb_{12}\Gamma&cb_{13} \Gamma&cb_{14}\Gamma&cb_{15}\Gamma&cb_{16}\Gamma\\ cb_{21}\Gamma&A+cb_{22}\Gamma&cb_{23}\Gamma&cb_{24}\Gamma&cb_{25}\Gamma&cb_{26 }\Gamma\\ cb_{31}\Gamma&cb_{32}\Gamma&A+cb_{33}\Gamma&cb_{34}\Gamma&cb_{35}\Gamma&cb_{36 }\Gamma\\ cb_{41}\Gamma&cb_{42}\Gamma&cb_{43}\Gamma&A+cb_{44}\Gamma&cb_{45}\Gamma&cb_{46 }\Gamma\\ cb_{51}\Gamma&cb_{52}\Gamma&cb_{53}\Gamma&cb_{54}\Gamma&A+cb_{55}\Gamma&cb_{56 }\Gamma\\ cb_{61}\Gamma&cb_{62}\Gamma&cb_{63}\Gamma&cb_{64}\Gamma&cb_{65}\Gamma&A+cb_{66 }\Gamma\end{array} \right],\]  
\[\begin{array}{c}C_{1}=\left[\begin{array}{cccccccccc}1&0&0&0&0&0&0&0&0&0&0&0\\ C_{2}=\left[\begin{array}{cccccccccccccc}0&0&1&0&0&0&0&0&0&0&0&0\\ C_{3}=\left[\begin{array}{cccccccccccccccc}0&0&0&1&0&0&0&0&0&0&0\\ C_{4}=\left[\begin{array}{cccccccccccccccc}0&0&0&0&0&1&0&0&0&0&0\\ C_{5}=\left[\begin{array}{cccccccccccccccc}0&0&0&0&0&0&0&1  
_By Theorem 2 and (39), we have_  
\[\max_{ \begin{subarray}{c}K_{1},\ldots,K_{6}\\ L_{1},\ldots,L_{6}\end{subarray} }\bigg{(}\left[ \begin{array}{ccccc}0&\bar{E}+ \sum\limits_{i=1}^{6}\bar{B}_{i}L_{i}C_{i}&0\\ \bar{E}+\sum\limits_{i=1}^{6}\bar{B}_{i}L_{i}C_{i}&\tilde{A}(h(t))+\sum\limits _{i=1}^{6}\bar{B}_{i}K_{i}C_{i}&\bar{B}_{j}\end{array} \right]\bigg{)}\] \[= \text{rank}\bigg{(}\left[