## 3 \(\mathcal{H}_{\infty}\)-Based Control Strategies
#### 3.1.1 \(\mathcal{H}_{\infty}\) Design of Current Control Loop  
The generalised current plant (Figure 5, left) has \(W_{du}\) and \(W_{h}\) as input weights, and \(W_{u}\) and \(W_{err1}\) as outputs weights. The generalised weighted current plant can be expressed as  
\[\begin{pmatrix}\widehat{err_{I}}(z)\\ \widehat{u}(z)\\ err_{I}(z)\end{pmatrix}=\begin{pmatrix}W_{err1}&0&0\\ 0&W_{u}&0\\ 0&0&I\end{pmatrix}\cdot\begin{pmatrix}I&I&G_{Iw}\\ 0&0&I\\ I&I&G_{Iw}\end{pmatrix}\cdot\begin{pmatrix}W_{h}&0&0\\ 0&W_{du}&0\\ 0&0&0&I\end{pmatrix}\cdot\begin{pmatrix}\widehat{h}(z)\\ \widehat{du}(z)\\ u(z)\end{pmatrix} \tag{14}\]Each weight is designed with the following considerations:  
* \(W_{h}\) is the reference weight and includes the added uncertainty to the inductor filter \(L_{\text{IW}}\) and \(R_{\text{W}}\), considering the system in per-unit values. It is fixed to 1 for the output weights to be interpreted in standard position/tracking errors at given frequencies, as customary in the widely used mixed-sensitivity design [18];
* \(W_{du}\) is an input weight at the control action (i.e., at the input of \(G_{Iw}\)). Additionally, the \(\mathcal{H}_{\infty}\) robust control methodologies can reduce the steady-state error at the synchronous frequency, placing poles close to the imaginary axis (theory forbids unobservable output weights in the imaginary axis). Additionally, input disturbances at the control action are included to avoid the cancellation issues produced by the cancellation of complex poles close to the imaginary axis. This weight is also constant and is fixed to 0.1;
* \(W_{u}\) is the control action weight. This weight is constant and fixed to 0.5 to allow the controller to apply two voltage units per each required ampere to fulfil transient saturation constraints. Also, it has an interpretation of robustness to additive uncertainty: limiting the high-frequency gain to 2 tolerates 0.5 units of plant uncertainty;
* \(K_{Low}\) in (15) is a constant value that represents the desired weight gain at low frequencies, which, given that the reference weight \(W_{h}\) is unity, can be interpreted as the low-frequency (DC) position error in conventional control;
- \(W_{err\text{-res}}\) is a weight in the form (16) that has been tuned to accept errors lower than 0.02, setting a low-frequency gain inversely proportional to the accepted error. The gain at high frequencies is adjusted to be 200 times lower. The bandwidth of \(W_{err\text{-res}}\) is selected so the \(H\) inf design shows a comparable settling time to the root locus PR design (7.5 ms). Therefore, the gain \(K\) in (16) is fixed to obtain a gain of 0 dB at 133.3 Hz (1/7.5 ms). Finally, a transformation to a resonant form at 50 Hz is applied to the weight using (7);
- \(W_{High}\) has the same structure as Equation (16), with a gain of 1 at low frequencies, a gain of 1/3 at high frequencies,and a bandwidth of 20 Hz;  
Figure 5: Generalised plant for voltage and current control based on \(\mathcal{H}_{\infty}\) synthesis: Current (**left**); Voltage (**right**).  
* \(\alpha\) is a gain tuned in order to achieve a \(\mathcal{H}_{\infty}\) norm less than 1. The resulting values of \(K_{Low}\), \(W_{err\text{-}res}\) and \(W_{High}\) in \(W_{err,I}\) are listed in Table 2 and shown in Figure 5.  
Using the previously described weighting filters, a current control \(K_{Slave}\) has been obtained by applying Matlab's R2022b hinfsyn function.