## 3 Stabilization Problem
### Main Result
The resulting optimization problem is an LMI optimization problem which can be solved by efficient numerical tools._  
To end with this paragraph, note that, if (28) is satisfied, the control proposed in Theorem 1 whose alternate expression is given in the previous remarks, can be simplified. Indeed, to deduce \(\lambda(t)\), quadratic expressions with respect to the current state \(x(t)\) must be evaluated. In fact, if (28) is satisfied, such expressions can be replaced by linear expressions with respect to the state simplifying drastically the control implementation. In that case, the following corollary can be proved. **Corollary 2**. : _Consider \(x_{e}\in X_{\alpha}\) and matrix \(P\) satisfying (28). The control \(\lambda(t)\) defined by_  
\[\lambda(t)=\arg\biggl{\{}\min_{d\in\Lambda_{S}}e(t)^{T}PB_{0}(x_{e})d\biggr{\}} \tag{39}\]  
_solves Problem 1. In addition, the error signal satisfies_  
\[J(e)=\int_{0}^{\infty}e(t)^{T}Q_{1}e(t)dt\leq\frac{1}{2}e(0)^{T}Pe(0) \tag{40}\]  
Proof. : Consider the Lyapunov function \(V(e)=\frac{1}{2}e(t)^{T}Pe(t)\). Then,  
\[ \begin{array}{ll}\frac{dV(e)}{dt}&=e(t)^{T}P\frac{d(t)}{dt}=e(t)^{T}P[A_{0}e (t)+B_{0}(x(t))\lambda(t)-B_{0}(x_{e})\lambda_{e}]\\ &=e(t)^{T}P[A_{0}e(t)+B_{0}(x(t))\lambda(t)-B_{0}(x_{e})\lambda_{e}]\end{array}  \tag{41}\]  
Note that we have  
\[A_{0}e(t)=\left[ \begin{array}{cccc}A_{0}e(t)&A_{0}e(t)&\ldots&A_{0}e(t)\\ \end{array} \right]\left[ \begin{array}{c}d_{1}\\ d_{2}\\ \vdots\\ d_{M}\end{array} \right],\ \sum_{i=1}^{M}d_{i}=1\]  
\[B_{0}(x(t))d=\left[ \begin{array}{cccc}A_{1}x+B_{1}V_{in}&A_{2}x+B_{2}V_{in }&\ldots&A_{M}x+B_{M}V_{in}\\ \end{array} \right]\left[ \begin{array}{c}d_{1}\\ d_{2}\\ \vdots\\ d_{M}\end{array} \right]\]and  
\[e(t)^{T}P[A_{0}e(t)+B_{0}(x(t))d+A_{0}x_{e}]=e(t)^{T}P\Big{[}\ \ (A_{0}+A_{1})e(t)\ \ \ (A_{0}+A_{2})e(t)\ \ \ \ldots\ \ \ (A_{0}+A_{M})e(t)\ \ \ \Big{]}d\] \[+e(t)^{T}P(\Big{[}\ \ A_{1}x_{e}+B_{1}V_{in}\ \ \ A_{2}x_{e}+B_{2}V_{in}\ \ \ \ldots\ \ \ A_{M}x_{e}+B_{M}V_{in}\ \ \Big{]}d-B_{0}(x_{e})\lambda_{e})\]  
Then,  
\[ \begin{array}{rl}\frac{dV(e)}{dt}&=e(t)^{T}P[A_{0}e(t)+B_{0}(x(t))\lambda(t)+ A_{0}x_{e}]\\ &=e(t)^{T}P\Big{[}\ \ (A_{0}+A_{1})e(t)\ \ \ (A_{0}+A_{2})e(t)\ \ \ \ldots\ \ (A_{0}+A_{M})e(t)\ \ \Big{]} \lambda(t)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ +\min_{t\in\Lambda_{S}}e(t)^{T}P[B_{0}(x_{e})d\ -B_{0}(x_{e}),\lambda_{e}]\\ &=e(t)^{T}P\Big{[}\ \ (A_{0}+A_{1})e(t)\ \ \ (A_{0}+A_{2})e(t)\ \ \ \ldots\ \ (A_{0}+A_{M})e(t)\ \ \Big{]} \lambda(t)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\min_{t\in\Lambda_{S}}e(t)^{T}PB_{0}(x_{e})(d- \lambda_{e})\\ &\leq e(t)^{T}P\Big{[}\ \ (A_{0}+A_{1})e(t)\ \ \ (A_{0}+A_{2})e(t)\ \ \ \ldots\ \ (A_{0}+A_{M})e(t)\ \ \Big{]} \lambda(t)\\ &<-e(t)^{T}Qe(t)\end{array}  \tag{42}\]  
by (28) and convexity arguments. The last part of corollary can be proved as for Theorem 1. To summarize the developments of this section, the stabilization problem can be solved following the main steps described below  
1. From the elementary electrical equations describing the converter operation, derive the bilinear model (1). 2. It is then possible to compute matrices \(A_{i},\ B_{i}\) and \(C_{i}\) of the polytopic model (7). 3. Compute a matrix \(P\) solution of (25). This can be done for example using the efficient techniques developed in the LMI-toolbox of MATLAB. A solution always exists. 4. The control expression is then deduced from Theorem 1. To simplify the control expression following the lines of Corollary 2, it is necessary to replace matrix \(P\) of Step 3 by a matrix \(P\) solution of (28). A solution \(P\) can be computed using the LMI toolbox of MATLAB.