## 3 Stabilization Problem
### Integral Action  
The control developed in the previous section allows stabilizing the converter around a desired equilibrium point but, in practical applications, as expected, it cannot reject disturbances on the input voltage or the load or cope with uncertainties. To take them into account, at least for perturbations of \"reasonable amplitude\", a cascade control structure can be implemented using an outer voltage proportional-integral (PI) control providing the reference of the inner loop. When the internal loop controller is nonlinear (e.g., a sliding mode control), a linear approximation of the closed-loop system is derived and the linear control methods can be used to design a PI control loop [14]. Here, we show how the stabilization control can be extended to deal with the design of a PI control. An integrator is added at the output of the system leading to an additional state equation written as  
\[\frac{dx_{I}(t)}{dt}=y(t)-y_{ref} \tag{43}\]  
where \(y_{ref}\in Y_{eq}\) is the constant vector of output reference signal and \(Y_{eq}\) is defined by (18). The extended state-space model obtained by taking \(X(t)=\left[ \begin{array}{cc}x(t)&x_{I}(t)\end{array} \right]^{T}\) as state vector is given by  
\[\left\{ \begin{array}{rcl}\frac{dX(t)}{dt}&=&\mathcal{A}\mathcal{X}_{0}(t)+ \mathcal{B}_{0}(X(t))\lambda(t)+\mathcal{H}X_{e}+\mathcal{H}_{e}\\ y(t)&=&\mathcal{C}_{0}X(t)+\mathcal{D}_{0}(X(t))\lambda(t)\end{array} \right. \tag{44}\]where  
\[\mathcal{A}_{0}=\left[ \begin{array}{cc}A_{0}&0\\ C_{0}&0\end{array} \right],\ \mathcal{B}_{0}(X(t))=\left[ \begin{array}{c}B_{0}(x(t))\\ D_{0}(x(t))\end{array} \right],\ \mathcal{H}=\left[ \begin{array}{cc}0&0\\ -C_{0}&0\end{array} \right],\ \mathcal{H}_{e}=\left[ \begin{array}{c}0\\ -D_{0}(x_{e})\lambda_{e}\end{array} \right] \tag{45}\]  
\[\mathcal{C}_{0}=\left[ \begin{array}{cc}C_{0}&0\end{array} \right],\ \mathcal{D}_{0}(X(t))=D_{0}(x(t)),\ X_{e}=\left[ \begin{array}{c}x_{e}\\ x_{L_{e}}\end{array} \right],\ x_{L}\in\mathbb{R}^{p}\]  
Note that, for all \(x_{L_{e}},\in\mathbb{R}^{p}\)  
\[\left[ \begin{array}{cc}A_{0}&0\\ C_{0}&0\end{array} \right]\left[ \begin{array}{c}x_{e}\\ x_{L_{e}}\end{array} \right]+\left[ \begin{array}{c}B_{0}(x_{e})\\ D_{0}(x_{e})\end{array} \right]\lambda_{e}+\left[ \begin{array}{cc}0&0\\ -C_{0}&0\end{array} \right]\left[ \begin{array}{c}x_{e}\\ x_{L_{e}}\end{array} \right]+\left[ \begin{array}{c}0\\ -D_{0}(x_{e})\lambda_{e}\end{array} \right]=0 \tag{46}\]  
Then, the set of equilibrium points for model (44) is defined as  
\[\mathcal{X}_{e}=\left\{X_{e}=\left[ \begin{array}{c}x_{e}\\ x_{L_{e}}\end{array} \right]\in\mathbb{R}^{n+p},\ \lambda_{e}\in\Lambda_{R}:\mathcal{A}_{0}X_{e}+ \mathcal{B}_{0}(X_{e})\lambda_{e}+\mathcal{H}X_{e}+\mathcal{H}_{e}=0\right\} \tag{47}\]  
If \(X_{e}\in\mathcal{X}_{e}\) and the error signal is defined as  
\[e_{I}(t)=X(t)-X_{e}\]  
we have  
\[\frac{de_{I}(t)}{dt}=\frac{dX(t)}{dt}=\mathcal{A}_{0}X(t)+\mathcal{B}_{0}(X(t) )\lambda(t)+\mathcal{H}X_{e}+\mathcal{H}_{e}=\mathcal{A}_{0}e_{I}(t)+\mathcal{ B}_{0}(X(t))\lambda(t)-\mathcal{B}_{0}(X_{e})\lambda_{e} \tag{48}\]  
Define the following LMI optimization problem  
\[\delta_{\max}=Arg\left[\max_{\delta>0}\ \ \delta\right]\]  
\[\text{under}\left\{ \begin{array}{c}A_{0}^{\mathrm{T}}P+PA_{0}-\delta\left(A _{0}^{-1}\right)^{\mathrm{T}}C_{0}^{\mathrm{T}}C_{0}-\delta C_{0}^{\mathrm{T}}C _{0}A_{0}^{-1}+2Q<0\\ \mathcal{P}=\left[\begin{array}{cc}P&-\delta\left(A_{0}^{-1}\right)^{ \mathrm{T}}C_{0}^{\mathrm{T}}\\ -\delta C_{0}A_{0}^{-1}&\delta I\end{array}\right]>0\end{array} \right. \tag{49}\]  
being \(P\) and \(Q\) the matrices involved in (25). Optimization problem (49) always has a solution because, by continuity arguments, there always exists a neighborhood of \(0\), e.g., \(\Delta\), such that, for all \(\delta\in\Delta\), the constraints of problem (49) are satisfied if (25) is satisfied. In fact, optimization problem (49) allows the determination of the maximal set \(\Delta\), denoted as \(\Delta_{\max}\) and defined by \(\Delta_{\max}=[0,\delta_{\max}]\).