# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 2 Switching Structured Linear Systems
The following statements are equivalent:_  
* _A subset_ \(V\subseteq G\) _is a switching controlled invariant subset of_ \(\Sigma_{\sigma}\)_;_
* _There exists a switching essential state feedback_ \(\mathcal{F}_{\sigma}\) _such that_ \(V\) _is a switching invariant subset for_ \(\Sigma^{\mathcal{F}}_{\sigma}((G,\mathcal{E}_{\sigma}\setminus\mathcal{F}_{ \sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\)_._  
Figure 6 shows an example of how a suitable choice of the switching essential state feedback turns a switching controlled invariant subset for \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) into a switching invariant subset for \(\Sigma^{\mathcal{F}}_{\sigma}((G,\mathcal{E}_{\sigma}\setminus\mathcal{F}_{ \sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\). The arguments developed for switching controlled invariant subsets and switching essential state feedbacks as well as the equivalence between switching controlled invariance and switching invariance by essential state feedback find a counterpart in the context of switching conditioned invariance and switching essential output injections. A _switching essential output injection_\(\mathcal{K}_{\sigma}\) for the switching structured linear system \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\) is formed by a family of essential output injections \(\{\mathcal{K}_{h}\}_{h\in\mathcal{H}}\) for the family of triplets \(\{(G,\mathcal{E}_{h}),G_{h}^{in},G_{h}^{out}\}_{h\in\mathcal{H}}\) and by the switching signal \(\sigma\colon\mathbb{R}^{+}\to\mathcal{H}\). A _family of essential output injections_ for the family of triplets \(\{((G,\mathcal{E}_{h}),G_{h}^{in},G_{h}^{out})\}_{h\in\mathcal{H}}\) is a family of sets of edges \(\{\mathcal{K}_{h}\}_{h\in\mathcal{H}}\) such that, for any \(h\in\mathcal{H}\),  
\[\mathcal{K}_{h}\subseteq G_{h}^{out}\times G.\]  
The application of \(\{\mathcal{K}_{h}\}_{h\in\mathcal{H}}\) to the family of triplets \(\{((G,\mathcal{E}_{h}),G_{h}^{in},G_{h}^{out})\}_{h\in\mathcal{H}}\) leads to the family of compensated triplets  
\[\{((G,\mathcal{E}_{h}\setminus\mathcal{K}_{h}),G_{h}^{in},G_{h}^{out})\}_{h \in\mathcal{H}},\]  
where the symbol \(\mathcal{E}_{h}\setminus\mathcal{K}_{h}\) stands for the set difference \((\mathcal{E}_{h}\cup\mathcal{K}_{h})\setminus(\mathcal{E}_{h}\cap\mathcal{K} _{h})\). For any \(h\in\mathcal{H}\), the _linear map_  
\[K_{h}=\left[k_{i\ell}^{h}\right]_{ \begin{subarray}{c}i=1,\ldots,n\\ \ell=1,\ldots,p\end{subarray} }\]  
and the essential output injection \(\mathcal{K}_{h}\) are associated with each other according to the following rule:  
\[k_{i\ell}^{h}\
ot\equiv 0\iff(x_{j},x_{i})\in\mathcal{K}_{h},\text{ with }x_{i}\in G,x_{j}=x_{i\ell}^{h}\in G_{h}^{out},\]  
and, in particular,  
\[k_{i\ell}^{h}=-\frac{a_{ij}^{h}}{c_{\ell j}^{h}}\text{ if }(x_{j},x_{i})\in \mathcal{K}_{h}\cap\mathcal{E}_{h},\text{ with }x_{i}\in G,x_{j}=x_{i\ell}^{h}\in G_{h}^{out}.\]  
By applying \(\mathcal{K}_{\sigma}\) to \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\), one obtains the compensated switching structured linear system  
\[\Sigma_{\sigma}^{\mathcal{K}}((G,\mathcal{E}_{\sigma}\setminus\mathcal{K}_{ \sigma}),G_{\sigma}^{in},G_{\sigma}^{out}),\]  
whose state-space representation is  
\[\Sigma_{\sigma(t)}^{\mathcal{K}}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=& (A_{\sigma(t)}+K_{\sigma(t)}\,C_{\sigma(t)})\,x(t)+B_{\sigma(t)}\,u(t),\\ y(t)&=&\mathcal{C}_{\sigma(t)}\,x(t).\end{array} \right.\]  
**Property 2**. : _Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) be a switching structured linear system.