# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 2 Switching Structured Linear Systems
The following statements are equivalent:_  
* _A subset_ \(S\subseteq G\) _is a switching conditioned invariant subset of_ \(\Sigma_{\sigma}\)_;_
* _There exists a switching essential output injection_ \(\mathcal{K}_{\sigma}\) _such that_ \(S\) _is a switching invariant subset for_ \(\Sigma^{K}_{\sigma}((G,\mathcal{E}_{\sigma}\setminus\mathcal{K}_{\sigma}),G^ {in}_{\sigma},G^{out}_{\sigma})\)_._  
Figure 7 shows how an appropriate switching essential output injection turns a switching conditioned invariant subset for \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) into a switching invariant subset for \(\Sigma^{K}_{\sigma}((G,\mathcal{E}_{\sigma}\setminus\mathcal{K}_{\sigma}),G^ {in}_{\sigma},G^{out}_{\sigma})\). In addition to switching essential state feedback and switching essential output injection, it is worth introducing a further compensation scheme, which is called switching essential output feedback and that is exploited in the solution of the problem of disturbance decoupling by output feedback (Section 4). A _switching essential output feedback_\(\mathcal{L}_{\sigma}\) 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 output feedbacks \(\{\mathcal{L}_{h}\}_{h\in\mathcal{H}}\) for the family of triplets \(\{(G,\mathcal{E}_{h}),G^{in}_{h},G^{out}_{h}\}_{h\in\mathcal{H}}\) and by the switching signal \(\sigma\colon\mathbb{R}^{+}\to\mathcal{H}\). A _family of essential output 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 \(\{\mathcal{L}_{h}\}_{h\in\mathcal{H}}\) such that, for any \(h\in\mathcal{H}\),  
\[\mathcal{L}_{h}\subseteq G^{out}_{h}\times G^{in}_{h}.\]  
The application of \(\{\mathcal{L}_{h}\}_{h\in\mathcal{H}}\) 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  
\[\{((G,\mathcal{E}_{h}\setminus\mathcal{L}_{h}),G^{in}_{h},G^{out}_{h})\}_{h \in\mathcal{H}},\]  
where the symbol \(\mathcal{E}_{h}\setminus\mathcal{L}_{h}\) stands for the set difference \((\mathcal{E}_{h}\cup\mathcal{L}_{h})\setminus(\mathcal{E}_{h}\cap\mathcal{L} _{h})\). For any \(h\in\mathcal{H}\), the _linear map_  
\[L_{h}= \begin{bmatrix}l^{h}_{k\ell}\end{bmatrix} _{ \begin{subarray}{c}k=1, \ldots,m\\ \ell=1,\ldots,p\end{subarray} }\]  
and the essential output feedback \(\mathcal{L}_{h}\) are associated with each other according to the following rule:  
\[l^{h}_{k\ell}\
ot\equiv 0\iff(x_{j},x_{i})\in\mathcal{L}_{h},\text{ with }x_{j}=x^{h}_{i_{\ell}}\in G^{out}_{h},x_{i}=x^{h}_{i_{k}}\in G^{in}_{h},\]  
and, in particular,  
\[l^{h}_{k\ell}=-\frac{a^{h}_{ij}}{b^{h}_{ik}c^{h}_{ij}}\text{ if }(x_{j},x_{i})\in \mathcal{L}_{h}\cap\mathcal{E}_{h},\text{ with }x_{j}=x^{h}_{i_{\ell}}\in G^{out}_{h},x_{i}=x^{h}_{i_{k}}\in G^{in}_{h}.\]By applying \(\mathcal{L}_{\sigma}\) to \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\), one obtains the compensated switching structured linear system  
\[\Sigma_{\sigma}^{\mathcal{L}}((G,\mathcal{E}_{\sigma}\setminus\bigcup\mathcal{ L}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out}),\]  
whose state-space representation is  
\[\Sigma_{\sigma(t)}^{\mathcal{L}}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=& \left(A_{\sigma(t)}+B_{\sigma(t)}\,L_{\sigma(t)}\,C_{\sigma(t)}\right)x(t)+B_{ \sigma(t)}\,u(t),\\ y(t)&=&C_{\sigma(t)}\,x(t).\end{array} \right.\]  
The following results are relevant to the application of the notion of switching controlled invariance to the solution of the disturbance decoupling problems (Section 4), since they indicate the existence of a switching controlled invariant subset with certain essential properties and they provide a recursive algorithm for its computation. Proposition 1: _Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\) be a switching structured linear system and \(Q\) a subset of \(G\).