## 3 Stabilization Problem
### Integral Action
#### 3.2.1 Rejection of a Measured Perturbation
: _Design a control law \(\lambda:\mathbb{R}\rightarrow\Lambda_{S}\), such that, for any initial condition \(e(0)\in\mathbb{R}^{n}\), the error dynamic is asymptotically stable, ensuring that_  
\[\lim_{t\rightarrow\infty}e(t)=0 \tag{68}\]  
_and, for all \(t>0\), the error \(e(t)\) is bounded implying that the state \(x(t)\) is also bounded with_  
\[\lim_{t\rightarrow\infty}x(t)\in\mathcal{X}_{e} \tag{69}\]  
Assumption A2 becomes  
**Assumption A4**. : _Given \(x_{e}(t)\in\mathcal{X}_{e}\), \(x(t)\in\mathbb{R}^{n}\) and \(w(t)\in\mathbb{R}^{r}\), there exist \(\lambda(t)\in\Lambda_{\mathbb{R}}\) and a positive definite symmetric matrix \(Q_{1}\), \(0<Q_{1}\leq Q\) such that_  
\[e^{T}(t)P[B_{0}(x(t),w(t))\lambda-B_{0}(x_{e}(t),w(t))\lambda_{e}]\leq-e^{T}(t )A_{0}Pe(t)-e^{T}(t)Q_{1}e(t)+e^{T}(t)P\dfrac{dx_{e}(t)}{dt} \tag{70}\]  
As for Problem 1, the existence of \(\lambda(t)\) satisfying Assumption A4 implies that a control exists to reject the perturbations. Theorem 1 can be easily extended and becomes  
**Theorem 3**. : _Consider the system (60), \(x_{e}(t)\in\mathcal{X}_{e}\) and matrix \(P\) solution of (25). Suppose that Assumptions A1 and A4 are satisfied. Then, the control \(\lambda(t)\) defined by_  
\[\lambda(t)=\arg\biggl{\{}\min_{d\in\Lambda_{S}}e^{T}(t)P\biggl{[}A_{0}e(t)+B_{ 0}(x(t),w(t))d+A_{0}x_{e}(t)-\dfrac{dx_{e}(t)}{dt}\biggr{]}\biggr{\}} \tag{71}\]solves Problem 2. In addition, the error signal satisfies_  
\[J(e)=\int_{0}^{\infty}e^{T}(t)Q_{1}e(t)dt\leq\frac{1}{2}e^{T}(0)Pe(0) \tag{72}\]  
The proof is very similar to the one for Theorem 1. **Remark 5**. : _We can note that the control can also be written as_  
\[\lambda(t)=\arg\biggl{\{}\min\limits_{d\in\Lambda_{5}}e^{T}(t)PB_{0}(x(t),w(t) )d\biggr{\}} \tag{73}\]  
_and the knowledge of \(\dfrac{dx_{e}(t)}{dt}\) is no longer needed to deduce the control. Observe that the controls proposed in Theorems 1-3 are structurally similar (see Remarks 2 and 3). In the context of the control of power converters, they can be interpreted as sliding mode controls whose nonlinear sliding surfaces are implicitly defined. The proofs of stability are derived without invoking the equivalent control paradigm._  
**Remark 6**. : _Another important remark concerns the case where \(v_{in}(t)\) is not constant, but it is measured or a priori known. In that case, \(v_{in}(t)\) can be included in the perturbation \(w(t)\) and the model is written as_  
\[\left\{ \begin{array}{l}\dfrac{dx(t)}{dt}=(A_{0}+\sum\limits_{i=1}^{m}u_{i}( t)A_{u_{i}})x(t)+(B_{w0}+\sum\limits_{i=1}^{m}u_{i}(t)B_{wu_{i}})w(t)\\ y(t)=(C_{0}+\sum\limits_{i=1}^{m}u_{i}(t)C_{u_{i}})x(t)+(D_{w0}+\sum\limits_{i= 1}^{m}u_{i}(t)D_{wu_{i}})w(t)\end{array} \right. \tag{74}\]  
_The previous approach can be applied to this particular case._  
The main steps to derive a rejection perturbation control can now be summarized  
1. From the elementary electrical equations describing the converter operation, derive the bilinear model (60). 2. It is then possible to compute the matrices \(A_{i},\;\;B_{wi},\;\;C_{i}\) and \(D_{wi}\) of the polytopic model (63). 3. Compute a matrix \(P\) solution of (25). A solution always exists. 4. The control expression is then deduced from Theorem 3.