# Robust LQR Control for PWM Converters with Parameter-Dependent Lyapunov Functions
## 2 Modeling of the DC-DC Converter and LQR State Feedback Control
### Averaged Model of the DC-DC Boost Converter  
Figure 1 shows the block diagram of a DC-DC converter with the control subsystem, where \(v_{0}(t)\) is the output voltage, \(v_{g}(t)\) is the line voltage, \(i_{load}(t)\) is the load disturbance. The output voltage must be kept at a given value \(V_{ref}\). The converter load is modeled as a resistor \(R\).  
In [7], the averaged model of a boost converter is given in the form  
\[\dot{x}(t)=A(\theta)x(t)+B_{u}(\theta)u(t) \tag{1}\]  
The uncertainty in \(A(\theta)\) and \(B_{u}(\theta)\) is included in a convex polytope as follows:  
\[\begin{array}{ll}[A(\theta),B_{u}(\theta)]&\in Co[\varsigma_{1},\,\ldots, \,\varsigma_{N}]\\ &:=\left\{\sum\limits_{i=1}^{N}\lambda_{i}\varsigma_{i}\,\,\lambda_{i}\geq 0,\,\sum\limits_{i=1}^{N}\lambda_{i}=1\ \right\}\end{array} \tag{2}\]  
In general, the admissible values of vector \(\theta\) are constrained in an hyperrectangle in the parameter space \(\Re^{N}\).  
The images of the matrix \([A(\theta),B_{u}(\theta)]\) for each vertex \(v_{i}\) correspond to a set \(\{\varsigma_{1},\,\ldots,\,\varsigma_{N}\}\). The components of the set \(\{\varsigma_{1},\,\ldots,\,\varsigma_{N}\}\) are the extrema of a convex polytope which contains the images for all admissible values of \(\theta\) if the \([A(\theta),B_{u}(\theta)]\) depends linearly on \(\theta\)[7].  
Where  
\[A(\theta)=\left[\begin{array}{ccc}\frac{-R_{L}}{L}&-\frac{D^{\prime}}{L}&0\\ \frac{D^{\prime}}{C}&-\frac{1}{RC}&0\\ 0&-1&0\end{array}\right]\text{; }\ B_{u}(\theta)=\left[\begin{array}{c}\frac{V_{g}}{D^{ \prime}L}\\ -\frac{V_{g}}{(D^{\prime 2}R)C}\\ 0\end{array}\right] \tag{3}\]  
According to [7], for the DC-DC boost converter, the load \(R\) and the duty cycle \(D^{\prime}_{d}\) at the operating point are considered uncertain parameters. Besides, two new uncertain variables, \(\delta=\frac{1}{D^{\prime}_{d}}\) and \(\beta=\frac{1}{D^{\prime 2}_{d}R}\), are defined. Thus, the parameter vector was defined as:  
\[\theta=\left[\begin{array}{ccc}\frac{1}{R}&D^{\prime}_{d}&\delta&\beta\end{array}\right] \tag{4}\]  
where the components of the parameter vector are restricted inside the following intervals:  
\[\begin{array}{c}R\in\left[\begin{array}{cc}\frac{1}{R_{\max}},&\frac{1}{R _{\min}}\end{array}\right]\\ D^{\prime}_{d}\in\left[\begin{array}{c}D^{\prime}_{d\min},&D^{\prime}_{d\max }\\ \delta\in\left[\begin{array}{cc}\frac{1}{D^{\prime}_{d\max}},&\frac{1}{D^{ \prime}_{d\min}}\end{array}\right]\\ \end{array}\right]\\ \beta\in\left[\begin{array}{cc}\frac{1}{(D^{\prime 2}_{d\max}R_{\max})},&\frac{1}{(D^{ \prime 2}_{d\min}R_{\min})}\end{array}\right]\end{array} \tag{5}\]  
This gives an uncertain model, which from now on is noted as \(P_{2009}\), inside a polytopic domain formed by \(N=2^{4}\) vertices. A three-dimensional representation of \(P_{2009}\) is shown in Figure 2. This model was used in [7] to synthesize a robust LQR controller, as is explained in the next subsection.  
Figure 1: Schematic of a DC–DC converter with a state feedback control subsystem.