# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 2 Switching Structured Linear Systems  
A _switching structured linear system_ has a state-space representation of the form  
\[\Sigma_{\sigma(t)}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=&A_{\sigma(t)} \,x(t)+B_{\sigma(t)}\,u(t),\\ y(t)&=&C_{\sigma(t)}\,x(t),\end{array} \right. \tag{1}\]  
where \(t\in\mathbb{R}^{+}\) is the time variable, \(x\in\mathbb{R}^{n}\) is the state, \(u\in\mathbb{R}^{m}\) is the input, and \(y\in\mathbb{R}^{p}\) is the output. Moreover, \(\sigma\) is a piecewise-constant left-continuous function, defined by \(\sigma\colon\mathbb{R}^{+}\to\mathcal{H}\), where \(\mathcal{H}=\{1,\ldots,N\}\) is a finite index set, and \(\{\Sigma_{h}\}_{h\in\mathcal{H}}\) is a finite indexed family of structured linear systems of the form  
\[\Sigma_{h}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=&A_{h}\,x(t)+B_{h}\,u(t ),\\ y(t)&=&C_{h}\,x(t),\end{array} \right.\quad h\in\mathcal{H}, \tag{2}\]  
where \(A_{h}\), \(B_{h}\), and \(C_{h}\) are matrices of appropriate dimensions whose entries are either fixed zeros or unknown real parameters. It is assumed that each column of \(B_{h}\), and, respectively, each row of \(C_{h}\) has one and only one entry that is different from a fixed zero. This assumption simplifies the treatment with respect to the general case considered, e.g., in [5]. However, this assumption is often encountered in the literature, since it is satisfied by many real systems that can be effectively modeled as structured linear systems: e.g., multiagent networks [20; 25; 26], quantum systems [27], and biological systems [28]. The systems of the family \(\{\Sigma_{h}\}_{h\in\mathcal{H}}\) are called _modes_ of the switching structured linear system. The function \(\sigma\) is called the _switching signal_ and each discontinuity of \(\sigma\) defines a change of the active mode. It is assumed that \(\sigma\) presents only a finite number of discontinuity in any finite time interval. This assumption is aimed at avoiding undesired behaviors, like Zeno behaviors in particular. The state-space representation (1) is associated with the graph representation  
\[((G,\mathcal{E}_{\sigma(t)}),C^{in}_{\sigma(t)},C^{out}_{\sigma(t)}), \tag{3}\]  
where \(\sigma\) is the piecewise-constant left-continuous function considered above, and  
\[\{((G,\mathcal{E}_{h}),G^{in}_{h},C^{out}_{h})\}_{h\in\mathcal{H}} \tag{4}\]  
is a family of triplets whose elements are defined as follows. The pair \((G,\mathcal{E}_{h})\) consists of the finite set of vertices \(G=\{x_{1},\ldots,x_{n}\}\) and of the \(h\)-th set of edges \(\mathcal{E}_{h}\subseteq G\times G\), where \((x_{j},x_{i})\in\mathcal{E}_{h}\) means that the edge with the tail in \(x_{j}\) and the head in \(x_{i}\) belongs to \(\mathcal{E}_{h}\). Note that all of the pairs \((G,\mathcal{E}_{h})\) of the family have the same set of vertices \(G\), while they differ from each other for the set of edges \(\mathcal{E}_{h}\). Moreover, \(G^{in}_{h}=\{x^{h}_{i_{1}},\ldots,x^{h}_{i_{m}}\}\subseteq G\) is the \(h\)-th subset of vertices directly connected to the inputs, while \(G^{out}_{h}=\{x^{h}_{i_{1}},\ldots,x^{h}_{i_{p}}\}\subseteq G\) is the \(h\)-th subset of vertices directly connected to the outputs. In particular, the family of structured linear systems (2) and the family of triplets (4) are associated with each other through the simple rules illustrated below. For any \(h\in\mathcal{H}\), the structured linear system \(\Sigma_{h}\), with  
\[A_{h}=\left[a^{h}_{ij}\right]_{ \begin{subarray}{c}i=1,\ldots,n,\\ j=1,\ldots,n\end{subarray} }\,B_{h}=\left[b^{h}_{ij}\right]_{ \begin{subarray}{c }i=1,\ldots,n,\\ j=1,\ldots,m\end{subarray} }\,,\,C_{h}=\left[c^{h}_{ij}\right]_{ \begin{subarray}{c }i=1,\ldots,p,\\ j=1,\ldots,n\end{subarray}