# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 2 Switching Structured Linear Systems
}\]and the triplet \((((G,\mathcal{E}_{n}),G_{h}^{in},G_{h}^{out})\), with \(G=\{x_{1},\ldots,x_{n}\},\mathcal{E}_{h}\subseteq G\times G\), \(G_{h}^{in}=\{x_{i_{1}}^{h},\ldots,x_{i_{in}}^{h}\}\subseteq G\) and \(G_{h}^{out}=\{x_{h}^{h},\ldots,x_{i_{p}}^{h}\}\subseteq G\), are associated with each other according to the following rules:  
* \(a_{ij}^{h}\!\
ot\equiv\!0\iff(x_{j},x_{i})\in\mathcal{E}_{h}\);
* \(b_{ij}^{h}\!\
ot\equiv\!0\iff x_{i}=x_{k_{j}}^{h}\), with \(x_{i}\in G\) and \(x_{k_{j}}^{h}\in G_{h}^{in}\);
* \(c_{ij}^{h}\!\
ot\equiv\!0\iff x_{j}=x_{k_{i}}^{h}\), with \(x_{j}\in G\) and \(x_{k_{i}}^{h}\in G_{h}^{out}\). Consistently with the assumption made with reference to the state-space representation, from the correspondences above it ensues that, for any \(h\in\mathcal{H}\), each vertex is directly connected to one input at most, while each input is directly connected to only one vertex. Likewise, it ensues that, for any \(h\in\mathcal{H}\), each vertex is directly connected to one output at most, while each output is directly connected to only one vertex. In the directed graph corresponding to the \(h\)-th triplet, the edge \((x_{j},x_{i})\in\mathcal{E}_{h}\) is represented by an arrow with the tail in \(x_{j}\) and the head in \(x_{i}\); the vertices belonging to \(G_{h}^{in}\) are marked by ingoing arrows; the vertices belonging to \(G_{h}^{out}\) are marked by outgoing arrows. An example of a directed graph is shown in Figure 1.  
From the above discussion, it ensues that a switching structured linear system can be equivalently represented by its state-space representation or by its graph representation. Henceforth, the notation  
\[\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\]  
will be adopted to refer to both representations. In this context, it is worth reviewing the notion of a switching path that will play a role in the formulation of the disturbance decoupling problems and of the unknown-input state reconstruction problem (see Section 4). An example of a switching path is shown in Figure 2. The formal definition follows. Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\) be a switching structured linear system. A _switching path \(P_{s}\) in \((G,\mathcal{E}_{\sigma})\)_ is a finite ordered sequence of edges \((e_{h_{1}},\ldots,e_{h_{q}})\in\mathcal{E}_{h_{1}}\times\ldots\times\mathcal{ E}_{h_{q}}\), such that the head of the edge \(e_{h_{k}}\) is the tail of the edge \(e_{h_{k+1}}\), with \(k=1,\ldots,q-1\). The tail of the first edge of the switching path \(P_{s}\) is called _the tail of the switching path_\(P_{s}\). The head of the last edge of the switching path \(P_{s}\) is called _the head of the switching path_\(P_{s}\). Invariance, Controlled Invariance, and Conditioned Invariance in Switching Structured Linear Systems  
This section presents the definitions and the main properties of the fundamental geometric notions of invariance, controlled invariance, and conditioned invariance extended to the context of switching structured linear systems. These notions were first introduced in [14; 19]. **Definition 1**. : _A switching invariant subset of the switching structured linear system \(\Sigma_{v}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{in},G_{\sigma}^{out})\) is a subset \(I\subseteq G\) such that, for any \(h\in\mathcal{H}\),_  
\[(x_{j},x_{i})\in\mathcal{E}_{h}\text{ with }x_{j}\in I\implies x_{i}\in I.\]  
Figure 3 shows an example of a switching invariant subset. **Definition 2**. : _A switching controlled invariant subset of the switching structured linear system \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{in}_{\sigma},G^{out}_{\sigma})\) is a subset \(V\subseteq G\) such that, for any \(h\in\mathcal{H}\),_  
\[(x_{j},x_{i})\in\mathcal{E}_{h}\text{ with }x_{j}\in V\implies x_{i}\in V\cup G^{in}_ {h}.\]  
Figure 4 shows an example of a switching controlled invariant subset. **Definition 3**.