## 3 Design with Constant Input Saturation  
Let us define the state-feedback control law for (4) as  
\[u=K(\theta)x=\sum_{i=1}^{N}\mu_{i}(\theta)K_{i}x, \tag{7}\]  
where \(K(\theta)\in\mathbb{R}^{m\times n}\) is the parameter-dependent gain matrix and \(K_{i}\in\mathbb{R}^{m\times n}\), \(i=1,\ldots,N\) denotes the gain matrix for each vertex \(i\). Let us also define the region \(\mathcal{E}\) as the one that determines the allowed initial states. It is defined by means of matrix \(R\succ 0\) as follows  
\[\mathcal{E}= \{x\in\mathbb{R}^{n}:\,x^{T}Rx\leq 1\}. \tag{8}\]  
In order to consider exactly (5) within the design, polyhedral Lyapunov functions should be considered, which adds computational complexity since the arising design conditions cannot be expressed as LMIs. For this reason, let us consider an ellipsoidal maximal volume region \(\mathcal{U}\) contained in the hyper-rectangle described by (5), as follows  
\[\mathcal{U}=\{u\in\mathbb{R}^{m}:u^{T}Qu\leq 1\}, \tag{9}\]  
where \(Q=W\Lambda W^{T}\), \(Q\succ 0\), \(\mathrm{W}\) is a rotation matrix that describes the axes orientation of the ellipsoid and  
\[\Lambda= \begin{bmatrix}\frac{1}{\gamma_{1^{2}}}&\ldots&0\\ \vdots&\ddots&\vdots\\ 0&\ldots&\frac{1}{\gamma_{m^{2}}}\end{bmatrix} . \tag{10}\]  
Hereinafter, without loss of generality, we assume that \(W=I\) since in most of the cases the axes of the ellipsoidal region \(\mathcal{U}\) are aligned with the axes of the input space. Note that the ellipsoid \(\mathcal{U}\), although defined in the input space, is mapped onto the state space as a parameter-varying ellipsoid by means of the state-feedback control law, as follows  
\[\mathcal{U}(\theta)=\{x\in\mathbb{R}^{n}:x^{T}K(\theta)^{T}QK(\theta)x\leq 1\}. \tag{11}\]The following theorem provides the conditions to obtain the vertex gains \(K_{i}\) that ensure the closed-loop stability with guaranteed decay rate \(\alpha\in\mathbb{R}^{+}\) of the system obtained as the interconnection of (4) and (7). Consider the continuous time LPV system (4), the control law (7) and the regions \(\mathcal{E}\) and \(\mathcal{U}\) defined in (8) and (11), respectively, with given matrices \(R\succ 0\) and \(Q\succ 0\), and a desired \(\alpha\in\mathbb{R}^{+}\). If \(x(0)\in\mathcal{E}\), there exist a symmetric matrix \(P\succ 0\) and matrices \(\Gamma_{i}\) such that the following set of LMIs is feasible  
\[P\succ 0, \tag{12}\]  
\[A_{i}P+B\Gamma_{i}+PA_{i}{}^{T}+\Gamma_{i}{}^{T}B^{T}+2\alpha P\prec 0, \tag{13}\]  
\[ \begin{bmatrix}P&I\\ I^{T}&R\end{bmatrix} \succeq 0, \tag{14}\]  
\[ \begin{bmatrix}Q^{-1}&\Gamma_{i}\\ \Gamma_{i}{}^{T}&P\end{bmatrix} \succeq 0, \tag{15}\]  
and the vertex gains of the LPV state-feedback controller are calculated as \(K_{i}=\Gamma_{i}P^{-1}\). Then the closed-loop system obtained as the interconnection of (4) and (7) is stable and has a guaranteed decay rate \(\alpha\). Moreover, the control law \(u(t)\) computed as (7) is such that \(u\in\mathcal{U}\). Proof. : By defining the quadratic Lyapunov function, \(V(x)=x^{T}P^{-1}x\), where \(P\succ 0\), the closed-loop stability inequality is obtained for each vertex \(i\) from the condition \(\dot{V}(x)<0\)  
\[A_{i}P+B\Gamma_{i}+PA_{i}{}^{T}+\Gamma_{i}{}^{T}B^{T}\prec 0,\;\forall i=1, \ldots,N, \tag{16}\]  
where \(\Gamma_{i}\) is obtained by means of a change of variable, as follows  
\[\Gamma_{i}=K_{i}P. \tag{17}\]  
The term \(2\alpha P\) can be added to the inequalities (16) to ensure a guaranteed decay rate of the derivative of the Lyapunov function, which can be used to tune the closed-loop transient properties [24], thus obtaining (13).