# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 2 Switching Structured Linear Systems
: _A switching conditioned invariant subset of the switching structured linear system \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) is a subset \(S\subseteq G\) such that, for any \(h\in\mathcal{H}\),_  
\[(x_{j},x_{i})\in\mathcal{E}_{h}\text{ with }x_{j}\in S\cap(G\setminus G^{out}_ {h})\implies x_{i}\in S,\]  
_where the symbol \(G\setminus G^{out}_{h}\) stands for the set difference of \(G\) and \(G^{out}_{h}\)._  
Figure 5 shows an example of a switching conditioned invariant subset. Figure 4: The switching controlled invariant subset \(V=\{x_{1},x_{2},x_{3}\}\). Figure 5: The switching conditioned invariant subset \(S=\{x_{1},x_{2},x_{4}\}\). The following definitions are aimed at establishing the equivalence between switching controlled invariance and switching invariance by essential state feedback. A _switching essential state feedback_\(\mathcal{F}_{\sigma}\) for the switching structured linear system \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) is formed by a family of essential state feedbacks \(\{\mathcal{F}_{h}\}_{h\in\mathcal{H}}\) for the family of triplets \(\{(G,\mathcal{E}_{h}),G^{in}_{h},G^{out}_{h}\}_{h\in\mathcal{H}}\) and by the switching signal \(\sigma\!:\!\mathbb{R}^{+}\to\mathcal{H}\). A _family of essential state feedbacks_ for the family of triplets \(\{((G,\mathcal{E}_{h}),G^{in}_{h},G^{out}_{h})\}_{h\in\mathcal{H}}\) is a family of sets of edges \(\{\mathcal{F}_{h}\}_{h\in\mathcal{H}}\) such that, for any \(h\in\mathcal{H}\),  
\[\mathcal{F}_{h}\subseteq G\times G^{in}_{h}.\]  
The application of \(\{\mathcal{F}_{h}\}_{h\in\mathcal{H}}\) to the family of triplets \(\{((G,\mathcal{E}_{h}),G^{in}_{h},G^{out}_{h})\}_{h\in\mathcal{H}}\) leads to the family of compensated triplets  
\[\{((G,\mathcal{E}_{h}\setminus\mathcal{F}_{h}),G^{in}_{h},G^{out}_{h})\}_{h \in\mathcal{H}},\]  
where the symbol \(\mathcal{E}_{h}\setminus\mathcal{F}_{h}\) stands for the set difference \((\mathcal{E}_{h}\cup\mathcal{F}_{h})\setminus(\mathcal{E}_{h}\cap\mathcal{F} _{h})\). For any \(h\in\mathcal{H}\), the _linear map_  
\[F_{h}=\left[f^{h}_{kj}\right]_{ \begin{subarray}{c}k=1,\ldots,m\\ j=1,\ldots,n\end{subarray} }\]  
and the essential state feedback \(\mathcal{F}_{h}\) are associated with each other according to the following rule:  
\[f^{h}_{kj}\
ot\equiv 0\iff(x_{j},x_{i})\in\mathcal{F}_{h},\text{ with }x_{j}\in G,x_{i}=x^{h}_{i_{h}}\in G^{in}_{h},\]  
and, in particular,  
\[f^{h}_{kj}\!=\!-\frac{a^{h}_{ij}}{b^{h}_{ik}}\text{ if }(x_{j},x_{i})\in \mathcal{F}_{h}\cap\mathcal{E}_{h},\text{ with }x_{j}\in G,x_{i}=x^{h}_{i_{k}}\in G^{in}_{h}.\]  
By applying \(\mathcal{F}_{\sigma}\) to \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\), one obtains the compensated switching structured linear system  
\[\Sigma^{\mathcal{F}}_{\sigma}((G,\mathcal{E}_{\sigma}\setminus\mathcal{F}_{ \sigma}),G^{in}_{\sigma},G^{out}_{\sigma}),\]  
whose state-space representation is  
\[\Sigma^{\mathcal{F}}_{\sigma(t)}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=& (A_{\sigma(t)}+B_{\sigma(t)}\,F_{\sigma(t)})\,x(t)+B_{\sigma(t)}\,u(t),\\ y(t)&=&C_{\sigma(t)}\,x(t).\end{array} \right.\]  
Hence, the anticipated equivalence between switching controlled invariance and switching invariance by essential state feedback can be stated as follows. **Property 1**. : _Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) be a switching structured linear system.