## 3 Control of the Inverter
### Outer Voltage Controllers  
The tracking of the voltage references is guaranteed by the outer loops with PI voltage compensators defined by:  
\[\left\{\begin{array}{c}s_{x}(t)=sat_{u_{0}}\Big{\{}K_{p}\Big{[}\frac{d(v_{x }(t)-v_{xx}(t))}{dt}+\alpha(v_{x}(t)-v_{xx}(t))\Big{]}\Big{\}}\\ i_{xx}(t)=\int_{-\infty}^{t}s_{x}(\tau)dt\;\;x=1,2\end{array}\right. \tag{46}\]  
where \(sat_{u_{0}}(x)\) is the classical symmetrical saturation function [17] having \(u_{0}=\frac{v_{in}(t)}{L}\) defining its limits. \(K_{p}\) and \(\alpha\) are positive design parameters, \(K_{p}\) being the proportional gain and \(\alpha K_{p}\) the integral gain. When the signal \(s_{x}(t)\) does not saturate, the controller is a classical PI controller whose expression is:  
\[i_{xx}(t)=K_{p}(v_{x}(t)-v_{xx}(t))+\alpha K_{p}\int_{-\infty}^{t}(v_{x}(\tau )-v_{xx}(\tau))dt,\;\;x=1,2\]  
The presence of saturation function ensures that the overall controlled system is asymptotically stable. This can be deduced from the stability result developed in the previous section, simply by noting that functions \(a_{x}(t),x=1,2\), remain positive if condition Equation (15) is satisfied, that is, if:  
\[\frac{di_{x}(t)}{dt}<\frac{v_{in}(t)}{L},\;\;x=1,2\]  
Equation (15) being also a necessary condition for the existence of a sliding regime. In such situation, for all positive \(K_{p}\) and \(\alpha\), we will have:  
\[\underset{t\rightarrow\infty}{\lim}v_{1}(t)=v_{1c}(t)\text{ and }\underset{t \rightarrow\infty}{\lim}v_{2}(t)=v_{2e}(t)\]  
For ease implementation of the proposed controller, we can remark that an expression of the outer control can be formulated with a saturation function whose limit is constant and equal to a positive real number \(s_{0}\) (i.e., independent of time) chosen to facilitate the controller implementation. An alternate expression with such a property could be:  
\[\left\{\begin{array}{c}s_{x}(t)=\frac{v_{in}(t)}{Ls_{0}}sat_{s_{0}}\Big{\{} \frac{Ls_{0}}{v_{in}(t)}K_{p}\Big{[}\frac{d(v_{x}(t)-v_{xx}(t))}{dt}+\alpha(v_ {x}(t)-v_{xx}(t))\Big{]}\Big{\}}\\ i_{xx}(t)=\int_{-\infty}s_{x}(\tau)dt\;\;x=1,2\end{array}\right. \tag{47}\]  
Figure 9 represents the block-diagram of the proposed saturated PI controller.  
To simplify the controller structure and then its implementation, if there exists a value \(v_{in_{min}}>0\) such that for all t, \(v_{in}(t)>v_{in_{min}}\), it is possible to drop the measurement of \(v_{in}(t)\) and replace it by \(v_{in_{min}}\) in the control expression. It is also possible to use a non-saturated PI controller if the converter was adequately designed to track the reference signals of interest (i.e., in a way guaranteeing that the control never saturates).  
To end with the outer loop, the choice of constants \(K_{p}\) and \(\alpha\) have an influence on the transient behavior of the controlled system, but asymptotic stability is always ensured. Even if asymptotic stability is a necessary condition, it is not sufficient in practice. A good performance level is often needed. The parameters \(K_{p}\) and \(\alpha K_{p}\) can be selected as done classically for a nonsaturated PI controller. However, it is important to remark that while the stability will be preserved, the performance associated with such a choice will only be guaranteed in the zone of linearity of the saturated PI controller. When the controller will saturate, except stability, no performance level can be guaranteed.