## 5 Illustrative Example  
In this section, an illustrative example is introduced to show the closed-loop performance of an LPV state-feedback controller, designed with Theorem 2, under time-varying input saturation limits. Note that the results corresponding to an LPV controller designed using Theorem 1 are omitted because they can be considered a particular case of Section 4 in which the saturation scheduling variables are frozen.  
Figure 1 presents the followed control-loop scheme throughout the example.  
Figure 1: Control scheme.  
Let us consider the LPV plant modelled as in (1) with the following state-space matrices (note that the system is open-loop unstable for every frozen value of \(\theta(t)\))  
\[A(\theta(t))=\begin{bmatrix}10-5\theta(t)&\theta(t)\\ 9\theta(t)-9&\theta(t)+1\end{bmatrix}, \tag{41}\]  
\[B=\begin{bmatrix}1&0\\ 0&0.5\end{bmatrix},\quad C=I_{2\times 2},\quad D=0_{2\times 2}, \tag{42}\]  
where \(B\), \(C\) and \(D\) are constant due to Assumptions 2 and 3, \(\theta(t)\in[0,1]\) and the parameter-varying state matrix \(A(\theta(t))\) can be written in the polytopic form (2) with vertex state matrices  
\[A_{1}=\begin{bmatrix}10&0\\ -9&1\end{bmatrix},\quad A_{2}=\begin{bmatrix}5&1\\ 0&2\end{bmatrix}. \tag{43}\]  
Let us consider a time-varying input saturation, where the saturation limits of \(u_{1}\) and \(u_{2}\) are \(\gamma_{1}(t)\in[10.0,15.0]\) and \(\gamma_{2}(t)\in[5.0,7.5]\) respectively.  
Following the method described in Section 4, the LPV state-feedback controller \(K(\theta,\phi)\) is scheduled by the following parameters  
\[\begin{split}\phi_{1}(t)&=\frac{\overline{\gamma_{1}}^{2}- \gamma_{1}(t)^{2}}{\overline{\gamma_{1}}^{2}-\underline{\gamma_{1}}^{2}},\\ \phi_{2}(t)&=\frac{\overline{\gamma_{2}}^{2}-\gamma_{2}(t)^{2}}{ \overline{\gamma_{2}}^{2}-\underline{\gamma_{2}}^{2}}.\end{split} \tag{44}\]  
The controller's design is obtained solving the LMIs (30)-(33) of Theorem 2, which are particularized as follows  
\[\begin{cases}P\succ 0\\ \begin{bmatrix}P&I;&I^{T}&R\end{bmatrix}\succeq 0\\ A_{1}P+B\Gamma_{1j}+PA_{1}{}^{T}+\Gamma_{1j}{}^{T}B^{T}+2\alpha_{j}P\prec 0 \\ A_{2}P+B\Gamma_{2j}+PA_{2}{}^{T}+\Gamma_{2j}{}^{T}B^{T}+2\alpha_{j}P\prec 0 \\ \begin{bmatrix}Q_{j}{}^{-1}&\Gamma_{1j};&\Gamma_{1j}{}^{T}&P\end{bmatrix} \succeq 0\\ \begin{bmatrix}Q_{j}{}^{-1}&\Gamma_{2j};&\Gamma_{2j}{}^{T}&P\end{bmatrix} \succeq 0\end{cases}, \tag{45}\]  
where \(j=1\ldots 4\) and \(R\) has been chosen as  
\[R=\begin{bmatrix}100&0\\ 0&100\end{bmatrix}, \tag{46}\]  
so that the expected initial condition for the system lies in a circle centered in the origin of the state space, with radius \(0.1\). On the other hand the polytopical expression of (29) for \(M=4\) is  
\[\alpha(\phi)=\phi_{1}\phi_{2}\alpha_{1}+(1-\phi_{1})\phi_{2}\alpha_{2}+\phi_{1 }(1-\phi_{2})\alpha_{3}+(1-\phi_{1})(1-\phi_{2})\alpha_{4}, \tag{47}\]  
and it is chosen to vary within the interval \([1,10]\) obtaining the following coefficients through (40)  
\[\alpha_{1}=1\,\alpha_{2}=\alpha_{3}=5.5,\ \alpha_{4}=10. \tag{48}\]Finally, taking into account the variability of \(\gamma_{1}(t)\) and \(\gamma_{2}(t)\), the matrices \(Q_{j}\) are given by  
\[Q_{1}=\begin{bmatrix}\frac{1}{15^{2}}&0\\ 0&\frac{1}{7.5^{2}}\end{bmatrix},\quad Q_{2} =\begin{bmatrix}\frac{1}{10^{2}}&0\\ 0&\frac{1}{7.5^{2}}\end{bmatrix},\quad Q_{3}=\begin{bmatrix}\frac{1}{15^{2}}&0 \\ 0&\frac{1}{5^{2}}\end{bmatrix},\quad Q_{4}=\begin{bmatrix}\frac{1}{10^{2}}&0\\ 0&\frac{1}{5^{2}}\end{bmatrix}. \tag{49}\]  
By using the SeDuMi solver [32] and the YALMIP [33] toolbox, we find a solution of (45) that, through \(K_{ij}=\Gamma_{ij}P^{-1}\), allows us to calculate the eight controller vertex gains.  
Hereafter, two different scenarios are used to show that the designed LPV state-feedback controller is able to guarantee the closed-loop system stability and its capacity to adapt its performance taking into account the time-varying limits of the input saturation.