# Impulse Controllability for Singular Hybrid Coupled Systems
## 3 Results  
In this part, we establish an algebraic condition that is both necessary and sufficient for the presence of a decentralised P-D out feedback controller in singular hybrid coupled systems to achieve impulse controllability. By Lemma 2, it is easy to obtain the following Theorem 1. **Theorem 1**. : _Consider system (7)._  
_The system (7) is regular and impulse-free, if and only if_  
\[\text{rank}\bigg{(}\left[ \begin{array}{cc}0&\bar{E}\\ \bar{E}&\bar{A}(h(t))\end{array} \right]\bigg{)}=Nn+\text{rank}(\bar{E}). \tag{18}\]  
_The primary goal of this paper is to establish an algebraic condition that is essential and complete for determining whether a decentralised P-D output feedback control controller (9) exists, which guarantees that the singular system (7) is impulse controllable. This paper offers an algebraic criterion that is both necessary and sufficient for the presence of a decentralised proportional-derivative (P-D) output feedback controller (9) that guarantees the closed-loop system satisfies certain requirements._  
\[\underset{ \begin{subarray}{c}K_{\mathcal{N}}\\ L_{\mathcal{N}}\end{subarray} }{\max}\bigg{(}\left[ \begin{array}{cc}0&\bar{E}+ \underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&0\\ \bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{A}( h(t))+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}K_{i}\bar{C}_{i}&\bar{B}_{j} \end{array} \right]\bigg{)}=Nn+\underset{ \begin{subarray}{c}L_{\mathcal{N}} \end{subarray} }{\max}(\bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i }\bar{C}_{i}). \tag{19}\]  
**Theorem 2**. : _If a feedback controller (9) exists for the system (7) such that the closed-loop system (10) is impulse controllable, then the closed-loop system (10) will satisfy condition (19), if and only if_  
\[\text{rank}\left(\left[ \begin{array}{cccccc}0&\bar{E}&0&0&\bar{B}_{\varphi 1}&0\\ \bar{E}&\bar{A}(h(t))&\bar{B}_{j}&\bar{B}_{\varphi}&0&\bar{B}_{\varphi_{2}}\\ 0&\bar{C}_{\mathcal{N}-\varphi}&0&0&0&0\\ 0&\bar{C}_{\varphi-\varphi_{1}}&0&0&0&0\\ \bar{C}_{(\mathcal{N}-\varphi)-\varphi_{2}}&0&0&0&0&0\end{array} \right] \right)\geq Nn+r, \tag{20}\]  
_for all \(\varphi\in P(\mathcal{N})\), \(\varphi_{1}\in P(\varphi)\), \(\varphi_{2}\in P(\mathcal{N}-\varphi)\), with_  
\[r\overset{\Delta}{=}\underset{\varphi\in P(\mathcal{N})}{\min}\text{rank} \bigg{(}\left[ \begin{array}{cc}\bar{E}&\bar{B}_{\varphi}\\ \bar{C}_{N-\varphi}&0\end{array} \right]\bigg{)},\]  
**Proof. ** We note that  
\[\left[ \begin{array}{cc}0&\bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{ i}L_{i}\bar{C}_{i}&0\\ \bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{A}( h(t))+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}K_{i}\bar{C}_{i}&\bar{B}_{j} \end{array} \right] \tag{21}\] \[= \left[ \begin{array}{cc}0&\bar{E}+\underset{i\in\mathcal{N}}{ \sum}\bar{B}_{i}L_{i}\bar{C}_{i}&0\\ \bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{A}( h(t))&\bar{B}_{j}\end{array} \right]+\underset{i\in\varphi}{\sum}\left[ \begin{array}{cc}0 \\ \bar{B}_{i}\end{array} \right]L_{i}\big{[} \begin{array}{cc}0&\bar{C}_{i}&0 \end{array} \big{]}.\]  
By Lemma 4 and (21), we have two inequalities that are true:  
\[\underset{ \begin{subarray}{c}K_{1},\ldots,K_{j-1},K_{j+1},\ldots,K_{N-1}\\ L_{\mathcal{N}}\end{subarray} }{\max}\bigg{(}\left[ \begin{array}{cc}0&\bar{E}+ \underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&0&0\\ \bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{A}( h(t))+\underset{i=1}{\sum}\bar{B}_{i}L_{i}\bar{C}_{i}&\bar{B}_{j}&\bar{B}_{N} \end{array} \right]\bigg{)} \tag{22}\] \[\geq Nn+\text{rank}(\bar{E}+\underset{i\in\mathcal{N}}{\sum}\bar{B} _{i}L_{i}\bar{C}_{i}),\]  
The system (10) is impulse controllable if and only if (19) holds. Based on our analysis, we can infer that in order for statement (20) to hold true, it is essential that all \(\varphi\in P(\mathcal{N})\),\(\varphi_{1}\in P(\varphi)\),\(\varphi_{2}\in P(\mathcal{N}-\varphi)\)  
\[\text{rank}\left(\left[