# Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems
## 5 Discussion  
In this paper, a novel hysteresis-based current control approach was presented. The basis of the developed control approach is the theory of switching systems, in particular, the system class of switching systems with multiple equilibria. The special feature of this system class is that each stable subsystem converges to its own equilibrium when it is active. This property allows the state trajectory \(\underline{x}(t)\) to converge to a reference trajectory \(\underline{x}_{\text{ref}}(t)\) by an appropriate switching between these subsystems. In the first part of this paper, it was shown that the presented approach minimizes the quality functions \(h_{s}(\underline{x}(t),\underline{x}_{\text{ref}}(t))\) for all \(s\in\mathcal{S}\) by applying the switching law (10). Furthermore, it was shown that reaching the minimum \(\hat{h}\) is equivalent to the convergence of the state trajectory \(\underline{x}(t)\) to the reference trajectory \(\underline{x}_{\text{ref}}(t)\). To prevent high-frequency switching at the reference trajectory, an environment \(\mathcal{R}^{n}_{s}\) around \(\underline{x}_{\text{ref}}(t)\) was introduced where no switching occurs. To illustrate  
Figure 15: Switching frequency \(f_{\text{SW},c,b,a}\) of positive potential on each phase. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**System Constant** & **Symbol** & & **Value** & **Unit** \\ \hline phase number & & \(c\) & \(b\) & \(a\) & – \\ maximum switching frequency & \(\hat{f}_{\text{SW},c,b,a}\) & 38.760 & 32.787 & 32.787 & kHz \\ average switching frequency & \(\hat{f}_{\text{SW},c,b,a}\) & 18.662 & 16.432 & 18.117 & kHz \\ minimum switching frequency & \(\hat{f}_{\text{SW},c,b,a}\) & 1.810 & 1.861 & 2.013 & kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Switching frequency values for all phases. the applicability of the presented approach to a practical system, an example application consisting of a voltage source converter plus load was introduced in the second part of the paper. After adapting the system to the above-mentioned system class, the applicability of the presented approach was shown by verifying the conditions established in Section 2.2. In the simulations subsequently shown, the control approach was extended by the additional hysteresis threshold \(\delta_{\text{hyst}}\). This is necessary to prevent the high-frequency switching along the intersection curves of two subsystems discussed in Section 2.2. However, the simulations shown confirm that the sliding mode, and thus, the use of the second hysteresis, occur only for a very short period. Thus, the state trajectory is within the target environment \(\mathcal{R}_{\delta}^{2}\) after a very short time. Consequently, the switching behavior of the control approach is largely determined by the introduced hysteresis width \(\delta\). The presented approach also shows a reasonable current ripple and current oscillation range while maintaining a low average switching frequency compared with other well-known control strategies. However, a reduction of the hysteresis width, and thus, the current oscillation range are not possible due to the large oscillation range of the switching frequencies, which limits the applicability of the new controller. The variable switching frequency is a challenge for EMI investigations. On the one hand, the switching energy is distributed over a broad spectrum so that peaks do not occur at individual frequencies. On the other hand, an exact switching frequency prediction is necessary to exclude the fact that the expected switching frequencies do not scatter into blocked frequency ranges. A meaningful EMI analysis can therefore only be performed after developing an algorithm to observe or to predict the switching frequency during runtime. As with all hysteresis-based methods, a fast sampling rate of the ADCs and an equally fast clock rate of the controller are necessary to precisely maintain the hysteresis limits in practical implementations.