## 3 \(\mathcal{H}_{\infty}\)-Based Control Strategies
#### 3.1.2 \(\mathcal{H}_{\infty}\) Design of Outer Voltage Control Loop  
Regarding voltage control, the generalised voltage plant (Figure 5, right) has \(W_{dI}\) and \(W_{p}\) as input weights and \(W_{I}\) and \(W_{errV}\) as outputs weights. The generalised weighted voltage plant in the discrete-time domain is  
\[ \begin{pmatrix}\widehat{err_{V}}\\ \widehat{I}\\ err_{V}\end{pmatrix} = \begin{pmatrix}W_{errV}&0&0\\ 0&W_{I}&0\\ 0&0&I\end{pmatrix} \cdot \begin{pmatrix}I&I&G_{V_{C}}\\ 0&0&I\\ I&I&G_{V_{C}}\end{pmatrix} \cdot \begin{pmatrix}W_{p}&0&0\\ 0&W_{dI}&0\\ 0&0&I\end{pmatrix} \cdot \begin{pmatrix}\widehat{p}\\ \widehat{d}I\\ I^{*}\end{pmatrix}  \tag{17}\]  
Each weight is designed according to the following criteria:  
* \(W_{p}\) is the reference weight, considering that the system is scaled to pu. The weight is a constant, fixed to 1 to ensure reference to output error sensitivity shaping, as in the current loop and standard mixed-sensitivity designs;
* \(W_{dI}\) is the weight of a perturbation at the control action, which, in this cascade structure, is the current command, assumed to be reasonably followed by the inner current loop previously designed. Considering that there is an inner current control loop that is not modelled in this design, this weight is fixed to 0.9;
* \(W_{I}\) is the control action weight. This weight is fixed to 0.4 to allow the controller to apply 2.5 amperes per each required voltage unit;
* \(W_{errV}\) is the voltage error weight. This weight has the same structure as the current error weight \(W_{errI}\) (15). Hence, this weight is tuned the same way as \(W_{errI}\) to achieve a closed-loop \(\mathcal{H}_{\infty}\) norm less than 1. The resulting values of \(K_{Low}\), \(W_{err\text{-}res}\) and \(W_{High}\) from \(W_{err,V}\) for this weight are listed in Table 3.  
The cascaded \(\mathcal{H}_{\infty}\) controller is  
\[K_{cascadedH_{\infty}}=\left[ \begin{array}{cc}1-K_{Master}(z)*K_{Slave}(z)- K_{Slave}(z)\end{array} \right] \tag{18}\]  
where \(K_{Slave}\) and \(K_{Master}\) have been obtained independently by means of \(\mathcal{H}_{\infty}\) optimisation, with the weighting functions described previously (see Figure 6). \begin{table}
\begin{tabular}{c} \hline \hline
**Current \(\mathcal{H}_{\infty}\) Weights** \\ \hline \(W_{h}\): \(K=1\); \(W_{dI}\): \(K=0.1\); \(W_{u}\): \(K=0.5\); \(W_{errI}\): \(K_{Low}=5.25\); \(W_{err\text{-}res}\) (before applying (7)): \(K=0.375\), \(b=6252\), \(a=1.56\); \(W_{High}\): \(K=0.288\), \(b=919.5\), \(a=183.9\); \(\alpha=1.14\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Weight parameters for the \(\mathcal{H}_{\infty}\) current control. \begin{table}
\begin{tabular}{c} \hline \hline
**Voltage \(\mathcal{H}_{\infty}\) Weights** \\ \hline \(W_{p}\): \(K=1\); \(W_{dI}\): \(K=0.9\); \(W_{u}\): \(K=0.4\); \(W_{errV}\): \(K_{Low}=1.2\); \(W_{err\text{-}res}\) (before applying (7)): \(K=0.3\), \(b=2501\), \(a=2.501\); \(W_{High}\): \(K=0.48\), \(b=390.7\), \(a=130.2\); \(\alpha=0.97\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Weight parameters for the \(\mathcal{H}_{\infty}\) Voltage Control. Design Alternative 2: Proportional-Resonant Voltage Control Tuned Using \(\mathcal{H}_{\infty}\) Synthesis  
To obtain a more robust PR controller, the parameters of the cascaded PR controller have been tuned using \(\mathcal{H}_{\infty}\) synthesis for fixed-structure controllers. Note that the fixed-structure \(\mathcal{H}_{\infty}\) synthesis is a non-convex optimisation problem, which may become trapped at spurious local minima solutions in a generic case; nevertheless, it has been pursued here for the sake of comparison with the other alternatives. Hence, the current and voltage generalised plants described in Section 3.1 with the same input and outputs weights (also described in (14) and (17)) have been used. However, \(K_{Master}\) and \(K_{Slave}\) controllers now correspond to the fixed-structure controllers, as described in (12) and (13).