# Robust Stabilization of Linear Switched Systems with Unstable Subsystems
## 2 Preliminaries  
In this paper, \(\mathbb{R}\) represents the real numbers' set, \(\mathbb{I}\) represents the positive integers' set excluding zero, \(\mathbb{I}_{n}\) represents the set of integers \(\{\,1,...,\,n\,\}\) where \(n\) is the dimension of the system, \(Q\) is the identity matrix of adequate dimensions, \(\mathbb{Z}\) represents the positive integers' set including zero, and the set operations are denoted as follows: \(\subset,\subseteq\) for subset and strict subset, respectively, \(\cup,\cap\) for union and intersection, respectively, \(\oplus\) for addition, and \(\mathcal{C}\) for the complement. \(Re(\cdot),Max(\cdot),Min(\cdot)\) stands for the real part, the maximum of real parts, and the minimum of real parts of a real or complex number. \(\lambda(\cdot)\) and \(\delta(\cdot)\) are the eigenvalues and singular values of a matrix. \(\|\cdot\|\) and \(\|\cdot\|_{r}\), denote the spectral norm and the spectral norm with restriction to a subspace \(r\), respectively. Consider a Linear Time-Variant Switched System (LTVSS) as follows:  
\[\dot{x}(t)=A_{\sigma}(t)x(t) \tag{1}\]where \(x(t)\in\mathbb{R}^{n}\) is the vector state, \(A_{\sigma}(t)\in\mathbb{R}^{n\times n}\) is a time-varying matrix that can commute (e.g., from \(A_{1}(t_{1})\) to \(A_{3}(t_{2})\) to \(A_{2}(t_{3})...\)), and \(\sigma(t):t\to I\) is the stabilization switching signal to be designed (\(I=\{1,2,...,s\}\subset\mathbb{I}\), where \(s\) is the number of subsystems). For a switching sequence \(t_{1},t_{2},...\), \(\sigma(t)\) is a piece-wise continuous function and \(\sigma(t)=p\in I\), \(\forall t\in[t_{i},t_{i+1})\) with \(i\in I\) and \(t_{i}<t_{i+1}<t_{i+2},...\). The total number of time-varying entries of \(A_{p}(t)\) is denoted as \(v\); if all of the entries of \(A_{p}(t)\) are time varying, \(v=2^{n\times n}\). With \(k\leq n\) denoting the row and \(l\leq n\) the column of a matrix, an entry of \(A_{p}(t)\) is denoted as \(a_{[p,k,l]}(t)\), and if it is known that \(\underline{a}_{[p,k,l]}\leq a_{[p,k,l]}(t)\leq\overline{a}_{[p,k,l]}\), each \(p^{\text{th}}\) subsystem can be written as a Linear Parameter-Variant Subsystem (LPVS) by a polytopic simple representation [22]:  
\[\dot{x}(t)=A_{p}(t)x(t)=A_{p}(\theta(t))x(t)=\left(\sum_{j=1}^{\overline{x}} \theta_{j}(t)A_{[p,j]}\right)x(t) \tag{2}\]  
where \(\theta\in\Theta=\left\{\theta|\,0\leq\theta_{j}(t)\leq 1,\sum\theta_{j}=1, \forall j\in V=\{1,...,2^{v}\}\right\}\), and \(A_{[p,j]}\) is the \(j^{\text{th}}\) vertex of \(A_{p}(\theta(t))\). For instance, the vertexes can be built with the combinations of \(\underline{a}_{[p,k,l]}\) and \(\overline{a}_{[p,k,l]}\):  
\[A_{p,1}=\left[ \begin{array}{ccc}\underline{a}_{[p,1,1]}&\underline{a}_{[p,1,2]}&...\\ \underline{a}_{[p,2,1]}&\underline{a}_{[p,2,2]}&...\\ \vdots&\vdots&\ddots\end{array} \right],A_{p,2}=\left[ \begin{array}{ccc} \overline{a}_{[p,1,1]}&\underline{a}_{[p,1,2]}&...\\ \underline{a}_{[p,2,1]}&\underline{a}_{[p,2,2]}&...\\ \vdots&\vdots&\ddots\end{array} \right],A_{p,3}=\left[ \begin{array}{ccc} \underline{a}_{[p,1,1]}&\overline{a}_{[p,1,2]}&...\\ \underline{a}_{[p,2,1]}&\underline{a}_{[p,2,2]}&...\\ \vdots&\vdots&\ddots\end{array} \right],... \tag{3}\]  
Below are definitions and previous results to be used. **Definition 1** ([23]). : _Suppose \(A\in\mathbb{R}^{n\times n}\), and \(S\subseteq\mathbb{R}^{n}\) is a subspace. \(S\) is \(A\)-invariant if \(AS\subseteq S\), that is, \(\forall b\in S\Rightarrow Ab\in S\)._  
In the following, \(\sigma(t)\) is considered a strict piecewise continuous (must commute), mode-dependent average dwell time function; that is, each mode has its own average dwell time in order to obtain more flexible and less conservative stability conditions, in comparison with analyzes that use a single average dwell time [24]:  
**Definition 2**.