# Robust Stabilization of Linear Switched Systems with Unstable Subsystems
## 2 Preliminaries
: _For a switching signal \(\sigma(t)\) with \(T\geq t\geq 0\), let \(N_{[\sigma,p]}(T,t)\) be the quantity of switching events (in [24], this term was originally called switching numbers; in this paper, this is changed for clearness to the quantity of switching events) that the \(p^{\text{th}}\) subsystem is activated along the interval \([t,T]\) and \(\mathcal{T}_{p}(T,t)\) the total running time of the \(p^{\text{th}}\) subsystem over the interval \([t,T]\), \(p\in I\). We say that \(\sigma(t)\) has a mode-dependent average dwell time \(\tau_{p}\) if there exist \(N_{[0,p]}>0\) and \(\tau_{p}>0\) such that:_  
\[N_{[\sigma,p]}(T,t)\leq N_{[0,p]}+\frac{\mathcal{T}_{p}(T,t)}{\tau_{p}},\; \forall T\geq t\geq 0 \tag{4}\]  
In this paper, a mode-dependent average dwell time switching signal is denoted as \(\sigma(t)\in F_{MDADT}[N_{[0,p]},\tau_{p}]\). **Definition 3**. : _The equilibrium of \(x=0\) of (2) is Globally Uniformly Exponentially Stable (GUES) under a certain switching signal \(\sigma(t)\) if for initial conditions \(x(t_{0})\), there exist constants \(\eta_{1}>0\), \(\eta_{2}>0\) such that the solution of the system satisfies \(\|x(t)\|\leq\eta_{1}e^{-\eta_{2}(t-t_{0}}\,\|x(t_{0})\|\), \(\forall t\geq t_{0}\)._  
**3. Main Result**  
The whole state space can be divided into two subspaces \(S_{p}^{S}\) and \(S_{p}^{U}\) defined as follows:  
**Definition 4**. : _The stable subspace \(S_{p^{*}}^{S},p\in I\), is spanned by the eigenvectors corresponding to the eigenvalues:_  
\[\lambda_{k}\left(\sum_{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right),\]  
\[k\in\mathbb{K}_{p}^{s}=\{\,m\in\mathbb{I}_{n}|\,Re\left(\lambda_{m}\left(\sum_{ j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right)\right)\geq 0,p\in I,\theta\in\Theta,T\geq t \geq 0\}.\]  
**Definition 5**. : _The unstable subspace \(S_{p}^{\mathrm{LI}},p\in I\), is spanned by the eigenvectors corresponding to the eigenvalues:_  
\[\lambda_{k}\left(\sum_{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right),\]  
\[k\in\mathbb{K}_{p}^{u}=\{\,m\in\mathbb{I}_{n}|\,Re\left(\lambda_{m}\left(\sum _{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right)\right)<0,p\in I,\theta\in\Theta,T \geq t\geq 0\}.\]  
On the other hand, the following proposition will be used later to demonstrate the main theorem:  
**Lemma 1**. : _Consider the switched linear system (2). If \(S\) is \(A_{p}(\theta(t))\)-invariant \(\forall p\in I\) and \(\forall\theta\in\Theta\), then \(S\) is \(e^{A_{p}(\theta(t))t}\)-invariant \(\forall p\in I\), \(\forall\theta\in\Theta\), and \(\forall t\geq 0\)._  
Proof. : Consider a fixed \(\theta\) and \(p=1\) for a \(\Delta\) period, then \(A_{p}(\theta)=A_{[p,\Delta]}=A_{[1,1]}\). \(S\) is \(A_{[1,1]}\)-invariant \(\forall x\in S\) by assumption; for any sequence of events in \([p,\Delta]\in\mathbb{Z}^{2}\), \(p\in I\), \(\Delta\in\mathbb{Z}\):  
\[A_{[p,\Delta]}^{r}x=A_{[p,\Delta]}^{r-1}A_{[p,\Delta]}x \tag{5}\]  
with \(x_{1}=A_{[p,\Delta]}x\in S\),  
\[A_{[p,\Delta]}^{r}x=A_{[p,\Delta]}^{r-1}x_{1} \tag{6}\]  
with \(x_{2}=A_{[p,\Delta]}x_{1}\in S\),  
\[A_{[p,\Delta]}^{r}x=A_{[p,\Delta]}^{r-2}x_{2} \tag{7}\]  
\[\vdots \tag{8}\]  
with \(x_{\Delta}=A_{[p,\Delta]}x_{r-1}\in S\),  
\[A_{[p,\Delta]}^{r}x=A_{[p,\Delta]}x_{r}\in S \tag{9}\]  
From the matrix exponential definition:  
\[\lambda_{k}\left(\sum_{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right),\]0  
For sufficiently small values of \(\Delta\) in a succession \([p_{1},\Delta_{1}],[p_{2},\Delta_{2}],[p_{3},\Delta_{3}],...\) (without loss of generality), one has the decomposition:  
\[\lambda_{k}\left(\sum_{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right),\]1 \[\lambda_{k}\left(\sum_{j=1}^{2^{n}}\theta_{j}(t)A_{[p,j]}\right),\]2  
where properties for the sum and intersection of subsets are used to complete the proof. This last property is known and demonstrated, in a different way, by some authors as the cocycle property (see for instance [25]). **Lemma 2**.