## 3 Stabilization Problem
### Integral Action
#### 3.2.1 Rejection of a Measured Perturbation  
In practical problems, a converter is affected by two main disturbances: the changes in the output load (load disturbance) and the changes in the input voltage (line disturbance). The problem of rejection of perturbations is a problem extensively studied in the control design literature. Many approaches exist and, among them, we can cite, for example, the ones based on robust transfer matrix norm minimization [32] or on the use of some kind of adaptation schemes consisting in reconstructing a signal of interest used appropriately by the controller [17; 33]. In addition, it is important to mention the techniques of robust design to synthesize a unique controller able to operate in a wide range of their main variables involving disturbances [14]. Here, we suppose that perturbations are measurable and that these measures can be used by the control. For the previously evoked example, a current sensor can provide information that the control law strategy can use. When a perturbation affects the converter, the model (1) is rewritten as  
\[\left\{ \begin{array}{l}\frac{dx(t)}{dt}=(A_{0}+\sum\limits_{i=1}^{m}u_{i}(t )A_{u_{i}})x(t)+(B_{0}+\sum\limits_{i=1}^{m}u_{i}(t)B_{u_{i}})V_{in}+(B_{u0}+ \sum\limits_{i=1}^{m}u_{i}(t)B_{uu_{i}})w(t)\\ y(t)=(C_{0}+\sum\limits_{i=1}^{m}u_{i}(t)C_{ui})x(t)+(D_{u0}+\sum\limits_{i=1} ^{m}u_{i}(t)D_{uu_{i}})w(t)\end{array} \right. \tag{60}\]  
where \(B_{u0},B_{uui}\in\mathbb{R}^{n\times r},\ D_{u0},\ D_{wu_{i}}\in\mathbb{R}^{p \times r}\) and \(w(t)\in\mathbb{R}^{r}\) is the vector of perturbations supposed measurable, the other matrices being defined as previously. Following the approach developed in Section 3.1, we can introduce the matrices  
\[B_{wu_{1}(t)u_{2}(t)...u_{m}(t)}=\sum\limits_{i=1}^{m}u_{i}(t)B_{wu_{i}}=\sum \limits_{i=1}^{M}\lambda_{i}(t)B_{ui} \tag{61}\]  
\[D_{wu_{1}(t)u_{2}(t)...u_{m}(t)}=\sum\limits_{i=1}^{m}u_{i}(t)D_{wu_{i}}=\sum \limits_{i=1}^{M}\lambda_{i}(t)D_{ui} \tag{62}\]  
where \(\lambda:\mathbb{R}\to\Lambda_{S}\) and the model can be written as \[\left\{ \begin{array}{l}\dfrac{dx(t)}{dt}=A_{0}\;x(t)+B_{0}(x(t),w(t))\lambda(t) \\ y(t)=C_{0}x(t)+D_{0}(x(t),w(t))\lambda(t)\end{array} \right. \tag{63}\]  
with  
\[B_{0}(x,w)=\left[ \begin{array}{cccc}B_{1}(x,w)&\cdots&B_{M}(x,w)\end{array}  \right]\text{ and }D_{0}(x,w)=\left[ \begin{array}{cccc}D_{1}(x,w)&\cdots&D_{M}(x,w)\end{array} \right] \tag{64}\]  
being \(B_{i}(x,w)=A_{i}x+B_{wi}w+B_{i}V_{in}\) and \(D_{i}(x,w)=C_{i}x+D_{wi}w\). The results of the previous sections can be extended. In general, the perturbations are time-varying signals, and, in many practical problems, the objective is to maintain some physical variables around appropriate constant values. Most of them are not always state variables but rather functions of them. For example, if despite the presence of time-varying perturbation at the output, the objective is to maintain constant the voltage and power output, the voltage output being in general a state variable, the output current will have to vary to compensate the perturbations effects. This suggests that, while a subset of state variables is maintained around a specific value, the others may vary in a way allowing to attain the fixed control objectives. More formally, the problem is translated into the one where the state variables are constrained to belong to the following set  
\[\mathcal{X}_{e}=\{x_{e}:\mathbb{R}\rightarrow\mathbb{R}^{n},\;\lambda_{e}: \mathbb{R}\rightarrow\Lambda_{R},;\;A_{0}x_{e}(t)+B_{0}(x_{e}(t),w(t)) \lambda_{e}(t)\;=0\} \tag{65}\]  
If the error signal is \(e(t)=x(t)-x_{e}(t)\), we have  
\[\dfrac{de(t)}{dt} =A_{0}\;x(t)+B_{0}(x(t),w(t))\lambda(t)-\dfrac{dx_{e}(t)}{dt} \tag{66}\] \[=A_{0}\;e(t)+B_{0}(x(t),w(t))\lambda(t)+A_{0}x_{e}(t)-\dfrac{dx_{ e}(t)}{dt} \tag{67}\]  
**Problem 2**.