## 4 Design with Time-Varying Input Saturation  
Following some ideas that appeared in Reference [13], we adapt the controller's design to deal with time-varying input saturation limits. In the proposed method, we add a new scheduling parameter vector to describe changes in time of the saturation function and we use it to schedule both the controller and the achieved performance. More specifically, the vector of varying parameters \(\theta(t)\) in (4), is augmented with another vector \(\phi(t)\in\Phi\subset\mathbb{R}^{m}\) that is linked to \(\gamma(t)\) by the following relation  
\[\phi_{j}(t)=\frac{\overline{\gamma_{j}^{2}}-\gamma_{j}(t)^{2}}{\overline{ \gamma_{j}}^{2}-\underline{\gamma_{j}}^{2}}\quad j=1,\ldots,m. \tag{23}\]  
Note that the values of \(\phi_{j},j=1,\ldots,m\), calculated as in (23) are constrained to belong to the interval \([0,1]\). Also note that (23) can be used to express \(\gamma_{j}(t)\) as a function of \(\phi_{j}(t)\), as follows  
\[\gamma_{j}(t)^{2}=\overline{\gamma_{j}}^{2}+\phi_{j}(t)(\underline{\gamma_{j} }^{2}-\overline{\gamma_{j}}^{2}). \tag{24}\]  
As a consequence, the expression for \(\mathcal{U}\) in the input space becomes  
\[\mathcal{U}(\phi)=\{u\in\mathbb{R}^{m}:u^{T}Q(\phi)u\leq 1\}, \tag{25}\]  
and, taking into account the new scheduling parameters, let us modify (7) as follows  
\[u=K(\theta,\phi)x=\sum_{i=1}^{N}\mu_{i}(\theta)\sum_{j=1}^{M}\eta_{j}(\phi)K_ {ij}x. \tag{26}\]  
Similar to the previous section, the region \(\mathcal{U}(\phi)\) is mapped onto the state domain as a parameter-varying ellipsoid by means of the new state-feedback control law (26), as follows  
\[\mathcal{U}(\theta,\phi)=\{x\in\mathbb{R}^{n}:x^{T}K(\theta,\phi)^{T}Q(\phi)K (\theta,\phi)x\leq 1\}. \tag{27}\]  
The following theorem, akin to Theorem 1, provides the conditions to obtain the vertex gains \(K_{ij}\) of the LPV state-feedback controller (26) that ensure the closed-loop stability of the system (4) and the ability to change the guaranteed decay rate according to changes in the time-varying saturation limits. **Theorem 2**. : _Consider the continuous time LPV system (4), the control law (26) and the regions \(\mathcal{E}\) and \(\mathcal{U}(\phi)\) of the state space described by (8) and (27), respectively, with given matrices \(R\succ 0\) and \(Q(\phi)\succ 0\)_and a desired parameter-varying decay rate \(\alpha(\phi)\in\mathbb{R}^{+}\) that varies within the interval \([\underline{\alpha},\overline{\alpha}]\). Assume that parameter-dependent matrix \(Q(\phi)^{-1}\) and the function \(\alpha(\phi)\) can be expressed in polytopic form as follows_  
\[Q(\phi)^{-1}=\sum_{j=1}^{M}\eta_{j}(\phi)Q_{j}{}^{-1}, \tag{28}\]  
\[\alpha(\phi)=\sum_{j=1}^{M}\eta_{j}(\phi)\alpha_{j}, \tag{29}\]  
_where \(M\) is the number of vertices of \(\Phi\). If \(x(0)\in\mathcal{E}\), there exist a symmetric matrix \(P\succ 0\) and matrices \(\Gamma_{ij}\) such that the following set of LMIs is feasible_  
\[P\succ 0, \tag{30}\]  
\[A_{i}P+B\Gamma_{ij}+PA_{i}{}^{T}+\Gamma_{ij}{}^{T}B^{T}+2\alpha_{j}P\prec 0, \tag{31}\]  
\[ \begin{bmatrix}P&I\\ I^{T}&R\end{bmatrix} \succeq 0, \tag{32}\]  
\[ \begin{bmatrix}Q_{j}{}^{-1}&\Gamma_{ij}\\ \Gamma_{ij}{}^{T}&P\end{bmatrix} \succeq 0, \tag{33}\]  
_and the vertex gains of the LPV state-feedback controller are calculated as \(K_{ij}=\Gamma_{ij}P^{-1}\). Then the closed-loop system obtained as the interconnection of (4) and (26) is stable and has guaranteed decay rate \(\alpha(\phi)\). Moreover, the control law \(u(t)\) computed as (26) is such that \(u\in\mathcal{U}(\phi)\)._  
Theorem 2 ensures the closed-loop system's stability in the same way as Theorem 1, adding the adaptive capacity of the controller to decrease the closed-loop performance when the saturation limits decrease. Hereunder, a sketch of this proof is presented.