# Impulse Controllability for Singular Hybrid Coupled Systems
## 2 Notations and Preliminaries
\right]\] (6) \[= \sum_{i=1}^{N}\sum_{k=1}^{n_{i}}\sum_{j=1}^{n}e_{q}\bigg{(}\sum_{ l=1}^{i-1}n_{l}+k\bigg{)}e_{q}^{T}(j)\frac{\partial f_{ik}}{\partial x_{j}}(z_{ ik}(t)),z_{ik}(t)\in Co(0,x(t)),\]  
and \(f_{ik}\): \(R^{n}\to R\) is the \(k\)-th component of \(f_{i}\). Then,  
\[h_{ik}^{j}(t)=\frac{\partial f_{ik}}{\partial x_{j}}(z_{ik}(t)),\]  
\[h(t)=(h_{11}^{1},\cdots,h_{11}^{n},\cdots,h_{Nn_{N}}^{1},\cdots,h_{Nn_{N}}^{n}),\]  
and  
\[\bar{A}(h(t))=\bar{A}+\Sigma.\]In this context, \(e_{\text{n}}(i)\) and \(e_{\text{n}}(j)\) represent the standard basis vectors of the vector space \(R^{n}\) and can be written as follows:  
\[E_{\text{n}}=\left\{e_{\text{n}}(i)|e_{\text{n}}(i)=(0,...,0,\underbrace{1}_{i },0,...,0)^{T},i=1,...,n\right\},\]  
where the variable \(h^{j}_{ik}(t)\) represents the derivative of the nonlinear function where \(z_{ik}\) denotes a time-varying parameter, and n is equal to \(\text{n}=Nn\). The system (4) has the potential to be rephrased as  
\[\left\{ \begin{aligned} \bar{E}\dot{x}_{i}(t)&=\tilde{A}(h(t ))x(t)+\sum_{j=1}^{N}\bar{B}_{i}u_{i}\\ y_{i}&=\bar{C}_{i}x(i=1.2,\cdots,N).\end{aligned} \right. \tag{7}\]  
It is generally believed that the functions \(h^{j}_{ik}(t)\) are constrained within certain limits. \[\text{max}\Big{|}h^{j}_{ik}(t)\Big{|}<+\infty.\]  
It is important to recognise that this assumption is not limiting in nature. In fact, it holds true for a significant category of nonlinear descriptor systems. Based on this hypothesis, the parameter \(h(t)\) progresses within a constrained domain, denoted as \(H_{\text{n}^{2},\text{n}^{2}}\), which is characterised by \(2^{n^{2}\times n^{2}}\) vertices defined as follows:  
\[\mathcal{V}_{H_{\text{n}^{2},\text{n}^{2}}}=\Big{\{}(\alpha_{11}^{1},\cdots, \alpha_{11}^{n},\cdots,\alpha_{Nn_{N}}^{1},\cdots,\alpha_{Nn_{N}}^{n})\Big{|} \alpha_{ik}^{j}\in\left\{\underline{h}^{j}_{ik},\bar{h}^{j}_{ik}\right\} \Big{\}}, \tag{8}\]  
where  
\[\bar{h}^{j}_{ik}=\max_{t}(h^{j}_{ik}(t)),\underline{h}^{j}_{ik}=\min_{t}(h^{j }_{ik}(t)).\]  
\[\eta=Nn.\]  
If system (4) is impulse controllable, a state feedback can be implemented to ensure the resulting closed-loop system is free of impulses [23]. In cases where a singular system lacks impulse controllability, a decentralised P-D output feedback can be initially introduced to eliminate impulses. \[u_{i}=-L_{i}y_{i}+K_{i}\dot{y}_{i}+v_{i},\ \ \ \ \ i=1,2,...,N. \tag{9}\]  
Applying the feedback (9) to the system (7), we obtain the resultant closed-loop system  
\[\left(\bar{E}+\sum_{i=1}^{N}\bar{B}_{i}K_{i}\bar{C}_{i}\right)\dot{x}(t)= \left(\tilde{A}(h(t))+\sum_{i=1}^{N}\bar{B}_{i}L_{i}\bar{C}_{i}\right)x(t)+ \sum_{i=1}^{N}\bar{B}_{i}v_{i}. \tag{10}\]  
Consider the following singular systems  
\[\left\{ \begin{aligned} \bar{E}\dot{x}(t)&=Ax(t)+Bu\\ y&=Cx\end{aligned} \right., \tag{11}\]  
where \(x\in\mathbb{R}^{n}\) is the state vector, \(u\in\mathbb{R}^{r}\) is the control input, \(y\in\mathbb{R}^{l}\) is the controlled output, and \(E,A\in\mathbb{R}^{n\times n}\), \(B\in\mathbb{R}^{n\times r}\) with limited conditions that the constant matrices and \(rank(E)=n_{0}\leq n\). **Definition 1** ([24]). : _A system (11) is considered regular if there is a constant scalar \(s\in\mathbf{C}\) that satisfies_  
\[\det(sE-A)\
eq 0. \tag{12}\]  
**Lemma 2** ([24]). : _If the system (11) is regular, we have the following results:_  
_(1)_ _The system (_11_) is free from impulses only when the following relation is satisfied:_  
\[\mathrm{rank}\bigg{(}\left[ \begin{array}{cc}0&E\\ E&A\end{array} \right]\bigg{)}=n+\mathrm{rank}(E). \tag{13}\]  
_(2)_ _The system (_11_) is impulse controllable only when the following relation is satisfied:_  
\[\mathrm{rank}\bigg{(}\left[ \begin{array}{cc}0&E&0\\ E&A&B\end{array} \right]\bigg{)}=n+\mathrm{rank}(E). \tag{14}\]  
_(3)_ _The system (_11_) is impulse observable only when the following relation is satisfied:_  
\[\mathrm{rank}\bigg{(}\left[ \begin{array}{cc}0&E\\ E&A\\ 0&C\end{array} \right]\bigg{)}=n+\mathrm{rank}(E). \tag{15}\]  
**Lemma 3** ([15]).